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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content='DG] 9 Jan 2023 Analysis and spectral theory of neck-stretching problems Thibault Langlais Mathematical Institute, University of Oxford Abstract We study the mapping properties of a large class of elliptic operators PT in gluing problems where two non-compact manifolds with asymptotically cylindrical geometry are glued along a neck of length 2T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' In the limit where T → ∞, we reduce the question of constructing approximate solutions of PT u = f to a finite-dimensional linear system, and provide a geometric interpretation of the obstructions to solving this system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' Under some assumptions on the real roots of the model operator P0 on the cylinder, we are able to construct a Fredholm inverse for PT with good control on the growth of its norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' As applications of our method, we study the decay rate and density of the low eigenvalues of the Laplacian acting on differential forms, and give improved estimates for compact G2-manifolds constructed by twisted connected sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' We relate our results to the swampland distance conjectures in physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' Contents 1 Introduction and motivation 2 2 Setup and discussion of results 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content='1 Model gluing problem .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' .' metadata={'source': 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cylinders by separation of variables .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' .' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' 49 1 References 52 1 Introduction and motivation This paper is concerned with analytical aspects of the study of special geometric structures near degenerate limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' Typically, such structures are defined as solutions of a non-linear PDE and explicit solutions can be quite challenging to find, especially in the absence of symmetries or in the compact setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' A standard way to produce such solutions is to glue simpler building blocks together in order to construct approximate solutions, and then perturb them so as to obtain a genuine solution using a fixed-point argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' This idea has been applied in various contexts for differential-geometric constructions, such as self-dual metrics [13, 15], hyperkähler metrics on K3 surfaces [27], minimal surfaces [21, 22], or compact manifolds with special holonomy [18, 19], to only cite a few of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' The question of whether such construction can be successfully performed can often be reduced to understanding the linearised problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' Therefore we shall be concerned with linear operators during most of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' The broad question that we are interested in is how to invert such operators and with which optimal bound in Sobolev and Hölder norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' For formally self-adjoint operators this corresponds to understanding the low eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' In order to give a precise formulation of these questions we will restrict ourselves to constructions where the gluing region is modelled on a cylinder and study the limit where this neck is infinitely stretched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' Our main focus throughout this paper will be the case where the model operator on the cylinder has real roots, which substantially complicates the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' Our aim is to develop a systematic method to deal with such cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' As a prototypical example of construction that the techniques developed in this paper are aimed at, we shall study compact manifolds with holonomy G2 constructed by twisted connected sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' The exceptional Lie group G2 ⊂ SO(7) is one of the two exceptional groups (along with Spin(7) ⊂ SO(8)) appearing in the Berger’s list of possible Riemannian holonomy groups of non-symmetric spaces [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' At this time, it was not known whether metrics with full holonomy G2 actually existed, and the first non-complete examples on a ball B7 ⊂ R7 were only constructed three decades later by Bryant [7], shortly before the first examples of complete metrics with exceptional holonomy by Bryant–Salamon [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' The first compact G2-manifolds were exhibited by Joyce [18, 19] using a gluing construction to resolve the singularities of a flat orbifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' This construction was more recently extended by Joyce–Karigiannis in [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' The only other known examples of compact manifolds with full holonomy G2 rely on the twisted connected sum construction, first due to Kovalev [25] and later improved by Corti–Haskins–Nordström–Pacini [10, 11] and further extended by Nordström [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' The twisted connected sum fits into our general setup, and we are able to give improved estimates for this construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' Moreover we can derive the decay rate and the density of the low eigenvalues of the Laplacian on twisted connected sums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' The work presented here was partly motivated by our interest in the swampland conjec- tures in physics [37] (see also [6] and [36] for detailed reviews).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' Indeed, compact manifolds with special geometry play an important role for compactifications in quantum gravity theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' For instance, it is relevant to consider Calabi–Yau threefold compactifications in string theory or compactifications on G2-manifolds in M-theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' The geometry of the internal manifold then governs the low-energy physics in the remaining dimensions (four in the previous cases).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' Typically the number of massless fields can be deduced from the Betty numbers of the internal manifold and the masses of the so-called Kaluza–Klein modes are 2 determined by the eigenvalues of certain self-adjoint operators, such as the Laplacian for instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' Thus deforming the geometry of the internal manifold modifies the parameters of the low-energy effective field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' Regardless of a particular choice of internal geometry, the swampland distance con- jectures [34] concern universal features that are expected to hold for any moduli space arising from the low-energy limit of a consistent quantum gravity theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' It is in particu- lar conjectured that as one moves towards an infinite distance limit in such a moduli space, the effective field theory breaks down due to the appearance of an infinite tower of light states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' Therefore it is an interesting problem to study the decay of the low eigenvalues of Laplacian on manifolds with special holonomy near singular limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' Such a problem has been studied in [3] for quintic Calabi–Yau threefolds using numerical methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' In M-theory compactified on G2-manifolds, the eigenvalues of the Laplacian acting on co- closed q-forms for q = 0, 1, 2, 3 correspond the squared masses of Kaluza–Klein states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' As mentioned above, our analysis notably yields precise estimates for the decay rate and the distribution of the low eigenvalues of the Laplacian on twisted connected sums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' We hope that this could be of interest to the study of the distance conjectures and shed some light on the geometric origin of the towers of light states, even if in a very specific example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' Organisation of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' In Section 2, we describe the general gluing problem that we are interested in and the notations and conventions that will be used throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' We also define the notions of adapted operators and their substitute kernel and cokernel that are central to our analysis and state our main results, Theorems 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content='6 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content='8 and Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' Section 3 is concerned with the analysis of translation-invariant PDEs on cylinders and contains most of the technical ingredients underlying our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' Our exposition is self-contained, and for this purpose we include a brief review of the standard results that we need.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' Section 4 is dedicated to the analysis of the mapping properties of adapted operators in the limit where the length of the neck joining the building blocks tends to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' Under some assumptions we prove a general theorem on the invertibility of such operators, but our method is more general and we also comment on how to adapt it in different contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' In the subsequent sections we apply our techniques to the study of the low eigenvalues of the Laplacian (Section 5) and the twisted connected sum construction of compact G2-manifolds (Section 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' I would like to thank Tristan Ozuch for helpful discussions which clarified some points in the analysis presented in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' I am also grateful to my su- pervisor Jason Lotay for his support and advice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' My research is supported by a Clarendon Scholarship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' 2 Setup and discussion of results In this section we explain our setup and formulate the main results of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' The gluing problem under consideration is described in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content='1, in which we introduce the building blocks and the class of adapted operators that we are interested in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' In §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content='2 we motivate and introduce the notions of substitute kernel and cokernel for adapted operators, following ideas present for instance in [21, 23] or [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' Our main results are discussed in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content='3 and related to the swampland distance conjectures in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' Last, we give an overview of our strategy in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content='1 Model gluing problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content='1) Before describing the type of gluing problems we are interested in, we need to recall a few standard definitions, starting with the notion of Exponentially Asymptotically Cylindrical (EAC) manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' Let Z be an oriented non-compact manifold of dimension n and X an oriented compact manifold of dimension n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' We say that Z is asymptotic to the cylinder Y = R × X at infinity if there exist a compact K ⊂ Z and an orientation- preserving diffeomorphism φ : (0, ∞) × X → Z\\K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' The compact manifold X is called the cross-section of Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' It will often be useful to pick a positive function ρ : Z → R such that ρ(φ(t, x)) = t for (t, x) ∈ [1, ∞) × X and ρ < 1 outside of φ([1, ∞) × X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' Following the terminology of [16], we will call such function a cylindrical coordinate function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' We say that a Riemannian metric g on Z is EAC of rate µ > 0 if we have, for all integers l ≥ 0: |∇l Y (φ∗g − gY )|gY = O � e−µt� (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content='1) as t → ∞, where gY = dt2 + gX is a cylindrical metric on Y = R × X, ∇Y the associated Levi-Civita connection and | · |gY the associated norm on tensor bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' Given the above data, we may define a notion of adapted bundle as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' Any vector bundle E0 → X equipped with a metric h0 and a connection ∇0 can be extended to a vector bundle E0 → Y with translation-invariant metric and connection (h0, ∇0) (see Section 3 for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' We call such bundles on Y = R × X translation-invariant bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' Let E → Z be a vector bundle on Z, endowed with a metric h and a connection ∇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' We say that E is an adapted bundle on (Z, g) if there exist a translation-invariant vector bundle (E0, h0, ∇0) and a bundle isomorphism Φ : E0|(0,×X) → E|Z\\K covering φ, such that for all integers l ≥ 0: |∇l 0(Φ∗h − h0)|0 = O(e−µt), and |∇l 0(Φ∗∇ − ∇0)|0 = O � e−µt� (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content='2) as t → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' This definition is valid for both real and complex vector bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' As we will make heavy use of the Fourier transform in Section 3, we usually assume that we are dealing with complex vector bundles and that the bundle metric is hermitian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' Our result will also apply to real vector bundles by considering their complexification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' Last, we may also define the notion of adapted differential operator between adapted bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' Let E, F be adapted bundles on Z and P : C∞(E) → C∞(F) be a differential operator of order k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' If u is a smooth section of E0 defined over the half-cylinder (0, ∞), let: �Pu = Φ−1 F PΦEu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content='3) This defines a differential operator �P : C∞(E0) → C∞(F 0) over the cylinder (0, ∞) × X, modelling the action of P on sections supported in Z\\K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' The operator �P can be written in the form: �P = k � j=0 Ak−j(t)∂j t (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content='4) where ∂t is the covariant derivative in the direction ∂ ∂t for the connection ∇0 on E, and Ak−j(t) : C∞(E0) → C∞(F0) are differential operators depending smoothly on t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' We say that P is adapted (with exponential rate µ > 0) if there exists a translation-invariant differential operator P0 : C∞(E0) → C∞(F0) of the form: P0 = k � j=0 Ak−j∂j t (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content='5) 4 such that for any smooth section u of E0 defined on (0, ∞) × X and for any l ≥ 0 and 0 ≤ j ≤ k we have: |∇l 0(Ak−j(t)u − Ak−ju)|0 = O \uf8eb \uf8ede−µt � i≤l |∇i 0u|0 \uf8f6 \uf8f8 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content='6) as t → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' That is, we essentially want the coefficients of �P − P0 and all their derivatives to have exponential decay when t → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' The operator P0 is called the indicial operator of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' Note that the formal adjoint of an adapted P is also adapted, and its indicial operator is naturally P ∗ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' On an EAC manifold (Z, g) with cross section X, the bundles TZ, TZ⊗l and ΛqT ∗Z are adapted, endowed with the Levi-Civita connection and the metric induced by g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' The operators d + d∗ and ∆ = dd∗ + d∗d are adapted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content='2) We now describe the general gluing problem we are interested in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' Let Z1 and Z2 be two EAC manifolds and assume that the cross-section of Z2 is the same as the cross-section X of Z1, but with opposite orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' By definition there exists compacts Ki ⊂ Zi and diffeomorphisms φi : (0, ∞) × Xi → Zi\\Ki where X1 = X = X2, and we can pick cylindrical coordinate functions ρi : Zi → R>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' For T ≥ 0 we can construct a compact oriented manifold MT by gluing the domains {ρ1 ≤ T + 2} ⊂ Z1 and {ρ2 ≤ T + 2} ⊂ Z2 along the annulus {T ≤ ρi ≤ T + 2} ≃ [−1, 1] × X with the identification: φ1(T + 1 + t, x) ≃ φ2(T + 1 − t, x), ∀(t, x) ∈ [−1, 1] × X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content='7) Define a smooth function ρT on MT by: ρT ≡ � ρ1 − T − 1 in {ρ1 ≤ T + 2} T + 1 − ρ2 in {ρ2 ≤ T + 2} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content='8) This is well defined as ρ1 − T − 1 coincides with T + 1 − ρ2 under the identification (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' Intuitively the function ρT parametrises the neck of MT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' In particular the domain {|ρ| ≤ T} is diffeomorphic to the finite cylinder [−T, T] × X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' Our goal is to study the mapping properties of elliptic operators defined on MT as T becomes very large, and relate it to the corresponding properties of operators on Zi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' To that end, suppose that the manifolds Zi are endowed we EAC metrics gi both asymptotic to a translation-invariant metric gY = dt2 + g0 on Y = R × X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' It will also be useful to fix a cutoff function χ : R → [0, 1] such that χ ≡ 0 on (−∞, − 1 2] and χ ≡ 1 on [1 2, +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' If T ∈ R we denote χT (t) = χ(t − T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' Then for T large enough gi,T = (1 − χT (ρi))gi + χT (ρi)gY (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content='9) is a Riemannian metric on Zi which coincides with gi on {ρi ≤ T − 1 2} and with gY on {ρi ≥ T + 1 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' Moreover, the difference gi − gi,T and all their derivatives are uniformly bounded by O(e−µT ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' Note that here we implicitly identify Zi\\Ki with the half cylinder (0, ∞) × Xi to make notations lighter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' We can patch g1,T and g2,T to form a Riemannian metric gT on MT , defining: gT ≡ � g1,T if ρT ≤ 0 g2,T if ρT ≥ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content='10) Similarly, if we have adapted bundles (Ei, hi, ∇i) on Zi such that their asymptotic models are both isomorphic to the same translation-invariant vector bundle (E0, h0, ∇0) on R×X, we can use the same cutoff procedure to patch them up on MT and form a vector bundle ET equipped with a metric hT and a connection ∇T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' 5 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content='3) Consider matching adapted bundles Ei, Fi on Zi (i = 1, 2) asymptotic to the same translation-invariant bundles E0, F 0, and adapted elliptic operators Pi : C∞(Ei) → C∞(Fi) of order k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' Denote Pi,0(x, ∂x, ∂t) : C∞(E0) → C∞(F 0) the indicial operator of Pi, where we use ∂x as a loose notation for the derivatives along the cross-section X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' In order to patch up these operators we need to assume the following compatibility condition [26]: P2,0(x, ∂x, ∂t) = P1,0(x, ∂x, −∂t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content='11) Assuming that it is satisfied define: Pi,T = (1 − χT (ρi))Pi + χT (ρi)Pi,0 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content='12) which coincides with Pi for ρi ≤ T − 1 2 and with Pi,0 for ρi ≥ T + 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' For large enough T, the operators Pi,T are elliptic, and moreover the coefficients of Pi − Pi,T and all their derivatives are uniformly bounded by O(e−µT ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' Patching P1,T and P2,T as above, we obtain a family of operators PT : C∞(ET ) → C∞(FT ) which are elliptic for large enough T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' Elliptic regularity on compact manifolds implies that the action of PT on Sobolev or Hölder spaces of sections induce Fredholm maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' Our goal is to construct Fredholm inverses for these maps, with a good control on their norm as T → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' In our applications, we will also be interested in a variation of the above gluing construction where the EAC manifolds Z1 and Z2 are glued along a non-trivial isometry γ : X → X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' This corresponds to replacing the identification (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content='7) by: φ1(T + 1 + t, x) ≃ φ2(T + 1 − t, γ(x)), ∀(t, x) ∈ [−1, 1] × X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content='13) All the matching conditions have to be changed accordingly, but otherwise everything we will do applies without modification up to a mere change of notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' This will be important in particular for the construction of compact G2-manifolds by twisted connected sums that we study in §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content='4) Before explaining our results in more details in the next part, let us make our conventions for Sobolev and Hölder norms explicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' For p ≥ 1 and l ≥ 0 an integer, the W l,p-norm of a section u ∈ C∞(ET ) can be defined as: ∥u∥W l,p = � j≤l ∥∇j T u∥Lp (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content='14) where the fibrewise norm of ∇j Tu is computed with the metrics hT , gT and we integrate over the volume form of gT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' The Sobolev space W l,p(ET ) is defined as the completion of C∞(ET ) for the W l,p-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' For the definition of Hölder norms, note that the injectivity radius of (MT , gT ) is uniformly bounded below by some r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE1T4oBgHgl3EQf5gWl/content/2301.03513v1.pdf'} +page_content=' The Cl,α norm of a smooth section u ∈ C∞(ET ) is defined as ∥u∥Cl,α = � j≤l ∥∇j T u∥C0 + sup x∈MT sup dT (x,x′) 0, the p-Selmer group of the Frobenius twist A(p) of A tends to have +larger order than that of A. The aim of this note is to discuss this phenomenon. +1. Introduction +For an elliptic curve A defined over a global function field K of characteristic +p > 0, the p-Selmer group of the Frobenius twist of A tends to have larger order +than that of A. The aim of this note is to discuss this phenomenon. +The Frobenius twist A(p) is the base change A ×K K of A over the absolute +Frobenius Frobp : K −→ K, x �→ xp. +1.1. The main results. We assume that A/K is ordinary, having semi-stable re- +duction everywhere. Let ∆A/K denote the divisor of the global minimal discriminant +of A/K. Comparing the defining equation for both curves yields +∆A(p)/K = p · ∆A/K. +(1) +Let Apν be the the kernel of the multiplication by pν on A and let +Selpν(A/K) ⊂ H1(K, Apν) +denote the pν-Selmer group of A/K. If p = 2, let ð′ be the set of places of K at +which A has non-split multiplicative reduction and has the group of components of +even order; otherwise, put ð′ = ∅. Let Sb denote the set of bad reduction places of +A/K. Write +Sb = ð′ ⊔ ð. +Let k = K(A(p) +p ( ¯Ks)) and let ð0 ⊂ ð be the subset of places splitting completely +over k. Let ℏ denote the p-rank of the subgroup of Hom(Gal(¯ks/k), Z/pZ) consisting +of homomorphisms unramified everywhere and locally trivial at every places of k +sitting over ð. Our main results are as follow. Let q denote the order of the constant +field of K. +Proposition 1. There exists an integer ǫ1, 2ℏ + 1 + |ð0| ≥ ǫ1 ≥ −|ð0|, such that +logp | Selp(A(p)/K)| = (p − 1) deg ∆A/K +12 +· logp q + ǫ1. +Acknowledgement: This research was supported in part by Ministry of Science and Technol- +ogy of Taiwan, MOST 109-2115-M-002-008-MY2. The author thanks F. Trihan for many valuable +suggestions especially for helping him with the proof of Lemma 3.1.1. +1 + +2 +KI-SENG TAN +Proposition 2. There exists an integer ǫ2, 2ℏ + 1 + |ð0| ≥ ǫ2 ≥ −2ℏ − 3|ð0|, such +that for each positive integer ν, +logp | Selpν+1(A(p)/K)| = (p − 1) deg ∆A/K +12 +· logp q + logp | Selpν(A/K)| + ǫ2. +Note that +deg ∆A/K +12 +is a non-negative integer (see [LLTT16, §2.2.1]), it is zero, if +and only if A/K is isotrivial. Let Xp∞(A/K) denote the p-primary part of the +Tate-Shafarevich group of A/K, let Xdiv(A/K) be its p-divisible subgroup, and +denote the p-cotorsion +X(A/K) := Xp∞(A/K)/Xdiv(A/K) = Selp∞(A/K)/ Seldiv(A/K), +where Seldiv(A/K) is the p-divisible subgroup of Selp∞(A/K). +Let r denote the +Zp-co-rank of Seldiv(A/K). +If ν is greater than the exponents of X(A/K) and +X(A(p)/K), then +| Selpν(A/K)| = |X(A/K)| · prν and | Selpν+1(A(p)/K)| = |X(A(p)/K)| · pr(ν+1). +It follows from Proposition 2 that +logp |X(A(p)/K)| + r = (p − 1) deg ∆A/K +12 +· logp q + logp |X(A/K)| + ǫ2. +Such kind of formulae is suggested by the conjectured Birch and Swinnerton-Dyer +formulae (see [Ta66, Tan95]) for both A(p)/K and A/K. +Next, let L/K be a Zd +p-extension ramified only at a finite number of ordinary +places of A/K. +Write Γ := Gal(L/K) and ΛΓ := Zp[[Γ]]. +Let Z be a finitely +generated torsion ΛΓ-module, so that there is an exact sequence +0 +� �m +i=1 ΛΓ/(pαi) ⊕ �n +j=1 ΛΓ/(η +βj +j ) +� Z +� N +� 0, +(2) +where α1, ..., αm, β1, ..., βn are positive integers, η1, ..., ηn ∈ ΛΓ are irreducible, rela- +tively prime to p, and N is pseudo-null. Although the exact sequence is not canon- +ical, the modules �m +i=1 ΛΓ/(pαi) and �n +j=1 ΛΓ/(η +βj +j ) are uniquely determined by Z, +we call them the p part and the non-p part of Z, call pα1, ..., pαm the elementary +µ-invariants and m the µ-rank of Z. If Z is non-torsion, define the µ-rank to be ∞. +Consider the Pontryagin dual X, X(p) of Selp∞(A/L), Selp∞(A(p)/L). They are +finitely generated over ΛΓ (see [Tan14]). Put +ð1 := {v ∈ ð | v splits completely over kL}. +Proposition 3. The µ-rank of X(p) is at least +(p−1) deg ∆A/K +12 +· logp q − |ð1|. +If L contains K(∞) +p +, the constant Zp-extension over K, then X and X(p) are torsion +[OT09, Tan14], in this case |ð1| = 0. +Proposition 4. If L contains K(∞) +p +, then the µ-rank m of X(p) equals +(p−1) deg ∆A/K +12 +· +logp q. Moreover, if pα1, ..., pαm, α1 ≥ · · · ≥ αm, are the elementary µ-invariants +of X(p), then those of X are pα1−1, ..., pαm′−1, where m′ is the greatest i such that +αi > 1. + +THE FROBENIUS TWISTS OF ELLIPTIC CURVES OVER GLOBAL FUNCTION FIELDS +3 +For a finite extension F/K, let wF denote the p-completion of the divisor class +group of of kF and for a Ze +p sub-extension M/K of L/K, put wM := lim +←−K⊂F ⊂M wF, +which is finitely generated torsion over ΛGal(kM/k). Actually, by [Crw87], the char- +acteristic ideal of wM has a generator ΘM := lim +←−F ΘF, where basically for each +F, ΘF ∈ Zp[Gal(kF/k)] is the Stickelberger element defined in [Ta84, §V.1.1], in +particular, we have +pL/M(ΘL) = ΘM · ∗, +(3) +where pL/M : ΛGal(Lk/k) −→ ΛGal(Mk/k) is the continuous Zp-algebra homomorphism +extending the quotient map Gal(Lk/k) −→ Gal(Mk/k) and ∗ ∈ ΛGal(Mk/k) is a +fudge factor not divisible by p. +For simplicity, we shall identify ΛGal(Mk/k) with ΛGal(M/K), and view ΘL, wL as +objects over ΛΓ. In the special case where L = K(∞) +p +, the module wL has trivial +µ-rank, hence ΘL is not divisible by p. To see this, let Y be the complete smooth +curve defined over the constant field of k, having k as its function field. For every +finite sub-extensions F/K of L/K, we have the exact sequence +0 +� w0 +F +� wF +deg � Zp +� 0 +and w0 +F[p] is contained in the subgroup of p-division points of the Jacobian variety +of Y. Therefore, the order of w0 +F[p] is bounded, and hence wL[p] = 0. In general, +(3) says that if L contains K(∞) +p +, then ΘL is not divisible by p. Also, in this case, +ð1 = ∅. Hence Proposition 4 is a special case of the following proposition. +Proposition 5. If ΘL is not divisible by p and ð1 = ∅, then the µ-rank m of +X(p) equals +(p−1) deg ∆A/K +12 +· logp q. Moreover, if pα1, ..., pαm, α1 ≥ · · · ≥ αm, are the +elementary µ-invariants of X(p), then those of X are pα1−1, ..., pαm′−1, where m′ is +the greatest i such that αi > 1. +Since the Frobenius and Verschiebung induce pseudo isomorphisms between the +non-p parts of X and X(p), the proposition implies the characteristic ideal of X(p) +is the q +(p−1) deg ∆A/K +12 +multiple of that of X. If L = K(∞) +p +, this is also a consequence of +the main theorem of [LLTT16]. +1.2. Notation. For a field F, let ¯F and ¯F s denote its algebraic closure and separable +closure, and denote GF = Gal( ¯F s/F). For a place v, let Ov, πv and Fv denote the +ring of integers, an uniformizer and the residue field. Write qv for |Fv|. +Let Sss denote the set of place v of K at which A has supersingular reduction. +For a set S of places of K and an algebraic extension F, let S(F) denote the set of +places of F sitting over S. For an endomorphism ϕ of an abelian group H, let H[ϕ] +denote the kernel. Use ∨ for Pontryagin dual, ∼ for pseudo isomorphism. +In this note we use flat or Galois cohomology. Let +F : A −→ A(p) and V : A(p) −→ A +be the Frobenius and the Verschiebung homomorphisms. We have the exact se- +quences + +4 +KI-SENG TAN +0 +� Cp +� Ap +F +� E(p) +p +� 0, +(4) +as well as +0 +� E(p) +p +� A(p) +p +V +� Cp +� 0, +(5) +where Cp = ker F is connected and E(p) +p += ker V, ´etale (see [LSc10]). For a field F +containing K, we have A(p) +p (F) = E(p) +p (F). In particular, k = K(E(p) +p ( ¯Ks)). Note +that for p = 2, k = K, because the non-trivial point of E(p) +p ( ¯Ks) is the only Galois +conjugate of itself. +1.3. The proofs. The key ingredient in the proof of Proposition 1 is local, concern- +ing the kernel of the natural map H1(Kv, E(p) +p ) −→ H1(Kv, A(p)), especially when v +is a place of supersingular reduction. Lemma 2.2.1, Lemma 2.2.2 and Lemma 2.3.3 +treat different types of reduction and provide us criteria, in terms of the the con- +ductor of the corresponding cyclic extension over kv, for determining if an element +of H1(Kv, E(p) +p ) is in such kernel. +With the local criteria available and with the help of global class field, in Propo- +sition 3.2.5, we determine the order of Sel(E(p) +p /K), the E(p) +p -part of Selp(A(p)/K). +In doing so, we find an interesting phenomena that the discrepancy between the +two discriminants as described in (1) is solely contributed by supersingular places +(see (27)). Next, in Proposition 3.2.6 we determine the order of Sel(Cp/K), the +Cp-part of Selp(A/K), by using the Poitou-Tate duality [Ces15]. Then Proposition +1 is proved at the end of §3.2, as a consequence of the above two propositions. +Proposition 2 is proved in §3.3 by applying Cassels-Tate duality. +In §3.4, using a theorem of Monsky [Mon81], we establish a method of reducing +the proofs of Proposition 3, 5 to the d = 1 case. Since the method can be applied +to more general situations, we loosen the condition in that subsection by allowing +A to be an ordinary abelian variety defined over a global field K. The main result +is summarized in Lemma 3.4.4. As a consequence of this lemma and Proposition 1, +2, the proofs of Proposition 3, 5 are given in the final subsection §3.5. +2. Local fields +At each place v of K, we have the long exact sequence +· · · +� A(p)(Kv) +V∗ � A(Kv) +∂ +� H1(Kv, E(p) +p ) +jv +� H1(Kv, A(p)) +� · · · +(6) +derived from 0 +� E(p) +p +j +� A(p) +V +� A +� 0 . The aim of this section is to +determine ker(jv) = coker(V∗). For a place w of k sitting over v, the abelian group +H1(kw, E(p) +p ) = Hom(Gkw, Z/pZ) = Hom(k∗ +w/(k∗ +w)p, Z/pZ), +(7) +so each ξ ∈ H1(kw, E(p) +p ) determines a degree p cyclic extension kw,ξ/kw. +Let ordv denote the valuation on ¯Kv having ordv(πv) = 1. + +THE FROBENIUS TWISTS OF ELLIPTIC CURVES OVER GLOBAL FUNCTION FIELDS +5 +2.1. Frobenius and Verschiebung. Let F (resp. F (p)) denote the formal group +law associated to A (resp. A(p)) over Kv. Since A has semi-stable reduction, F is +stable under local field extensions. Let ¯A denote the reduction of A at v. For a +place w of an algebraic extension F of K, sitting over v, the pre-image Ao(Fw) of +0 ∈ ¯A(Fw) under the reduction map A(Fw) −→ ¯A(Fw) is identified with F(mw) via +a bijection ι. Let ι(p) be the corresponding bijection associated to A(p) +o (Fw). +Let P ∈ Ov[[t]] be the unique power series fitting into the commutative diagram +F(mw) +Frobp � +ι +� +F (p)(mw) +P +� +ι(p) +� +F(mw) +ι +� +Ao(Fw) +F +� A(p) +o (Fw) +V +� Ao(Fw). +(8) +An element ξ ∈ mw satisfies P(ξ) = 0 if and only if ι(p)(ξ) ∈ E(p) +p (Fw) ∩ A(p) +o (Fw). +Suppose the formal group law of F (resp. F (p)) is given by the Ov-coefficient +power series f(X, Y ) (resp. f (p)(X, Y )). +If ti is a solution to P(t) = 0, then P(f (p)(t, ti)) = P(t), furthermore, since +A/Kv is ordinary, by [Sil86, §IV.7.2], P′(0) ̸= 0, and hence P′(ti) ̸= 0, too. +Lemma 2.1.1. We have P(t) = u(t) · P(t), where u(t) is a unit in Ov[[t]] and +P(t) is the associated distinguished polynomial. The polynomial P(t) is separable +with P(0) = 0. If v is a supersingular place, then deg P = p; if v is ordinary, then +P(t) = t. +Proof. The first assertion follows from the claim that πv does not divides P(t). For +ordinary v, because E(p) +p ( ¯Kv) ∩ A(p) +o ( ¯Kv) = {0}, we know that 0 is the only root of +P(t) in ¯Kv. This implies that the distinguished polynomial P(t) = t. For supersin- +gular v, the group E(p) +p ( ¯Kv) ∩ A(p) +o ( ¯Kv) ≃ Z/pZ, so deg P(t) = p, furthermore, since +P(ti) = 0 and P′(ti) ̸= 0, we have P(ti) = 0 and P ′(ti) ̸= 0. +To prove the claim, we first consider the case where v is a place of good reduction. +The formal group law associated to ¯A is given by ¯f(X, Y ) := f(X, Y ) (mod (πv)), +which has height 1 or 2, so +¯ +P := P (mod (πv)) is non-zero. This proves the claim. +For a multiplicative place v, we prove the claim by showing that the Verschiebung +gives rise to an isomorphism F (p)(mv) −→ F(mv). If v is a split-multiplicative place +and ˜Q is the local Tate period of A, then ˜Qp is the local Tate period of A(p) and +the Verschiebung is given by K∗ +v/ ˜QpZ −→ K∗ +v/ ˜QZ, induced from the identity map +on K∗ +v. This implies F (p)(mv) −→ F(mv) is an isomorphism, and hence the claim +follows. +If v is non-split multiplicative, then A/Kv is the twist of a split multiplicative +elliptic curve B/Kv by the unramified quadratic extension Lw/Kv. Write Zv for the +kernel of the norm map NLw/Kv : O∗ +w −→ O∗ +v. Then F (p)(mv) −→ F(mv) is given +by the identity map Zv −→ Zv, so it is an isomorphism. +□ + +6 +KI-SENG TAN +2.2. Ordinary places. The proof of Lemma 2.1.1 shows that if v is a split multi- +plicative place, then V∗ is given by K∗ +v/ ˜QpZ −→ K∗ +v/ ˜QZ, and hence surjective, so by +(6), ker(jv) = 0. +Lemma 2.2.1. Let v be a multiplicative place. Then ker(jv) = 0, unless v ∈ ð′, in +which case ker(jv) is of order 2 = p, consisting of ξ ∈ H1(Kv, E(p) +p ) with Kv,ξ/Kv +unramified. +Note that if p = 2, then k = K, so Kv,ξ is defined. +Proof. Suppose v is non-split multiplicative and let B/Kv and Lw/Kv be as in the +proof of Lemma 2.1.1. Because A/Lw is split-multiplicative, we have the injection +H1(Lw, E(p) +p ) −→ H1(Lw, A(p)), and hence +ker(jv) = ker(H1(Lw/Kv, E(p) +p (Lw)) −→ H1(Lw/Kv, A(p)(Lw))). +For p ̸= 2, we have H1(Lw/Kv, E(p) +p (Lw)) = 0, so ker(jv) = 0. +Denoting G = +Gal(Lw/Kv), we have the commutative diagram +H1(Lw/Kv, E(p) +p (Lw)) +� +≃ +� +H1(Lw/Kv, A(p)(Lw)) +≃ +� +Hom(G, Z/2Z) +� B(p)(Kv)/ NLw/Kv(B(p)(Lw)). +The non-trivial element of Hom(G, Z/2Z), sending the generator of G to the point of +B(p)(Kv) obtained by the Tate local period Qv of B/Kv, corresponds to an element +of ker(jv) if and only if Qv ∈ NLw/Kv(L∗ +w), or equivalently ordv Qv is even. +□ +Lemma 2.2.2. Suppose v is a good ordinary place and w is a place of k sitting over +v. Then ker(jw) is of order p, consisting of ξ ∈ H1(kw, E(p) +p ) with kw,ξ/kw unramified. +If Kv ̸= kw, then ker(jv) is trivial. +Proof. In view of the diagram (8), Lemma 2.1.1 says A(p) +o (kw) +∼ +V +� Ao(kw) . We +have to determine the cokernel of the induced ¯V : ¯A(p)(Fw) −→ ¯A(Fw). The Frobe- +nius ¯F identifies ¯A(p)(Fw) with ¯A(Fw) and under this, ¯V is identified with the mul- +tiplication by p. The cokernel in question is isomorphic to +¯A(Fw)/p ¯A(Fw) ≃ ¯Ap(Fw) = ¯E(p) +p (Fw). +The snake lemma applied to the diagram +0 +� A(p) +o (kw) +� +V +≃ +� +A(p)(kw) +� +V +� +¯A(p)(Fw) +� +¯V +� +0 +0 +� Ao(kw) +� A(kw) +� ¯A(Fw) +� 0 +implies that the reduction map E(p) +p (kw) −→ ¯E(p) +p (Fw) is an isomorphism, so ker(jw) +is of order p, and by [Mil06, §I.3.8], it is formed by all unramified ξ. If Kv ̸= kw, +then ¯E(p) +p (Fv) = E(p) +p (Kv) = 0, and a similar argument shows ker(jv) is trivial. + +THE FROBENIUS TWISTS OF ELLIPTIC CURVES OVER GLOBAL FUNCTION FIELDS +7 +□ +2.3. Supersingular places. Suppose v is supersingular. Choose a nonzero t1 ∈ mw +(in some Fw) with ι(p)(t1) ∈ E(p) +p (Fw). Let [u] denote the multiplication by u on A(p). +Because tu := ι(p)−1◦[u]◦ι(p)(t1) = ut1+higher terms, if p ∤ u, then ordv tu = ordv t1. +Denote +nv := +� +i∈F∗p +ordv ti = (p − 1) ordv(tu), for (u, p) = 1. +(9) +Write +P(t) = tp + zp−1tp−1 + · + z1t, +with +ordv z1 = nv. +(10) +For s, t ∈ m write s ⊞ t for F (p)(s, t). Then ι(p)(s ⊞ t) = ι(p)(s) + ι(p)(t). Diagram +(8) shows that for a given b ∈ mv, if a0 ∈ mw is a root of P(t) − b = 0, then all +other roots equal au := a0 ⊞ tu = F (p)(a0, tu), u = 1, ..., p − 1. Let Q(t) be the +distinguished polynomial associated to P(t) − b, whose roots are also a0, ..., ap−1. +Since Q(0) = −b · ξ, for some ξ ∈ O∗ +v, +p−1 +� +u=0 +ordv au = ordv b. +(11) +Since F (p)(X, 0) = F (p)(0, X) = X, we can write F (p)(X, Y ) = X + Y + XY · +g(X, Y ). It follows that au = a0 + tu + higher terms. Hence +ordv(au − au′) = +nv +p − 1. +(12) +Lemma 2.3.1. For every b ∈ mv with ordv b > pnv +p−1, there exists an element a ∈ mv, +with ordv a > +nv +p−1, such that P(a) = b. Conversely, for a ∈ mv, with ordv a > +nv +p−1, +the element b = P(a) ∈ mv has ordv b = ordv a + nv > pnv +p−1. +Proof. If b ∈ mv and a0 is a solution to P(t) = b, with ordv(a0) > +nv +p−1, then by (12), +other solutions au have ordv au = +nv +p−1. Therefore, if a ∈ mv, with ordv a > +nv +p−1, and +b = P(a), then by (11), ordv b = ordv a + nv. Conversely, if b ∈ mv, ordv b > pnv +p−1, +by (11), there is a solution a to P(t) = b, such that ordv a > +nv +p−1. Comparing the +valuations, we deduce that a is the only Galois conjugate of itself, whence a ∈ mv. +□ +Lemma 2.3.2. If v is a supersingular place of A/K, then the cokernel of +A(p)(Kv) +V∗ � A(Kv) +is of order pǫv · qnv +v , where if kv = Kv, ǫv = 1; otherwise, ǫ = 0. +Proof. Since ¯A(Fv) has order prime to p, by Diagram (8) we need to show the +cokernel of F (p)(mv) +P +� F(mv) has the desired order. + +8 +KI-SENG TAN +Let β = [ +1 +p−1nv]+1. Lemma 2.3.1 implies that P sends F (p)(mβ) onto F(mβ+nv). +Therefore, it is sufficient to check the co-kernel of the induced homomorphism +¯ +P : F (p)(m)/F (p)(mβ) −→ F(m)/F(mβ+nv). +Since the kernel of +¯ +P is of order pǫv, the proof is completed by counting. +□ +Lemma 2.3.2 says +| ker(jv)| = pǫv · qnv +v . +(13) +Lemma 2.3.3. Let v be a place of K and w a place of k sitting over v. The group +ker(jw) consists of all ξ with kw,ξ/kw having conductor at most pnw +p−1. +Proof. An element ξ ∈ ker(jw) can be written as ∂x for some x ∈ A(kw). Since +¯A(Fw) has order prime to p, we may choose x = ι(b) ∈ Ao(kw), for some b ∈ mw. +Let a0, ..., ap−1 be solutions to P(t) = b. Then all au are integral over Ow and +kw,ξ = kw(a0). It follows from (12) that if Disc is the discriminant of kw,ξ/kw, then +ordw(Disc) ≤ p · nw. +This implies the conductor of kw,ξ/kw is at most pnw +p−1. Classes ξ ∈ H1(kw, E(p) +p ) with +kw,ξ/kw unramified are in ker(jw) (see [Mil06, I.3.8]). They form a subgroup of order +p. By the local class field theory, ramified cyclic extensions of kw of degree p and +conductor at most m are characterized by the group Dm/Dp +m, Dm := O∗ +w/1 + πm +w Ow. +In our case m = pnw +p−1 is an integer divisible by p (by (9), because each tu ∈ kw). Since +Ow = Fw[[πw]], the map +D m +p −→ Dp +m, x �→ xp, +is an bijection. Hence |Dm/Dp +m| = qm−1 +w +· (qw − 1)/q +m +p −1 +w +· (qw − 1) = qnw +w . In view of +(13), the proof is completed by counting. +□ +The lemma actually says that by (7), +ker(jw) = Hom(k∗ +w/(1 + π +pnw +p−1 +w +Ow) · (k∗ +w)p, Z/pZ). +(14) +3. Global fields +Let X /Fq be the complete smooth curve having K as its function field. Let Xg, +Xgo denote the open sets consisting of places where A has good reduction, good +ordinary reduction respectively. +3.1. Poitou-Tate duality. We first recall the following. +Lemma 3.1.1. Let f : B/K −→ B′/K be a given isogeny of elliptic curves having +good reductions at all v ∈ Xg and let f : B −→ B′ be the homomorphism extending f +to N´eron models over Xg. Then N := ker [f] is a finite flat group scheme over Xg. +Furthermore, if ˆ +N denotes the kernel of the homomorphism ˆf : B′ −→ B extending +the dual isogeny ˆf : B′ −→ B, then N and ˆ +N are Cartier dual to each other. +Note that the existence and the uniqueness of f and ˆf are due to the N´eron +mapping property, see [BLR90, §1.2]. + +THE FROBENIUS TWISTS OF ELLIPTIC CURVES OVER GLOBAL FUNCTION FIELDS +9 +Proof. Since B/Xg is an abelian scheme, the morphism f is proper. It follows that +N /Xg is proper and quasi-finite, whence finite [EGA IV, part 4, 8.12.4] and flat +[Mil06, III.C.8]. The exact sequence +0 +� N +� B +f +� B′ +� 0 +together with the isomorphism (see [SGA 7I, VIII.7.1], [BBM82] or [Mil06, III.C.14]) +B/Xg ≃ Ext 1 +Xg(B, Gm) +induce the exact sequence +· · · +� HomXg(B, Gm) +� HomXg(N , Gm) +� B′ +ˆf +� B +� · · ·. +Here we apply the commutative diagram +Ext 1 +Xg(B′, Gm) +f ∗ +� Ext 1 +Xg(B, Gm) +B′ +ˆf +� B +that extends the already known diagram on the generic fibre. Then we check the +equality HomXg(B, Gm) = 0 fibre-wise by using the fact that over a field every +homomorphism from an abelian variety to Gm is trivial. +□ +Let notation be as in Lemma 3.1.1 and let N denote the generic fibre of N . For +v ∈ Xg, we have (see [Mil06, §III.7]) +H1(Ov, N )� � +� H1(Kv, N) . +(15) +Lemma 3.1.2. Let notation be as above. For v ∈ Xg, we have +H1(Ov, N ) = ker(H1(Kv, N) −→ H1(Kv, B)). +Proof. The lemma follows from the fact that H1(Ov, B) = 0 (see [Mil06, §III.2.1]) +and the commutative diagram of exact sequences +B(Ov) +� B′(Ov) +� H1(Ov, N ) +� +� � +� +H1(Ov, B) +� +B(Kv) +� B′(Kv) +� H1(Kv, N) +� H1(Kv, B). +□ +Let U ⊂ X be an open subscheme. Define +S(N/U) := ker(H1(K, N) −→ +� +v∈U +H1(Kv, B)). +Denote Sel(N/K) := S(N/X ), it is the kernel of +S(N/U) +� � +v̸∈U H1(Kv, B) . + +10 +KI-SENG TAN +Let Q(N/U) denote the cokernel of the localization map +S(N/U) +LN/K +� � +v̸∈U H1(Kv, N) . +Lemma 3.1.3. If U ⊂ Xg, then H1(U, N ) = S(N/U). +Proof. Let V ⊂ U be an open affine subscheme. By the localization sequence [Mil06, +III.0.3(c)] and the computation at the beginning of [Mil06, III.7], we have the exact +sequence +0 +� H1(U, N ) +� H1(V, N ) +� � +v∈U\V H1(Kv, N)/ H1(Ov, N ). +[Gon09, Lemma 4.2] says the natural map H1(V, N ) −→ H1(K, N) is injective. The +exact sequence implies H1(U, N ) ⊂ S(N/U). By [Gon09, Lemma 2.3], an element +in S(N/U) can be obtained from H1(V, N ) for some V ⊂ U, and the exact sequence +implies it is in H1(U, N ). +□ +For U ̸= X , apply the local duality [Mil06, §III.6.10] and consider the composition +S(Cp/U)� � +LCp/U +� � +v̸∈U H1(Kv, Cp) +∼ +� � +v̸∈U H1(Kv, E(p) +p )∨, +(16) +where the injectivity of LCp/U is due to [GoT12, Main Theorem]. +Lemma 3.1.4. If U ⊂ Xg and U ̸= X , then under (16), the group S(Cp/U) is the +Pontryagin dual of Q(E(p) +p /U). +Proof. Extend F and V to F : A −→ A(p) and V : A(p) −→ A over Xg. Denote +Cp = ker(F) and E(p) +p += ker(V). +They are Cartier dual to each other. By Lemma 3.1.3, we identify H1(U, Cp) with +S(Cp/U). Then the lemma follows from Poitou-Tate duality [Ces15, (5.1.2)]. +□ +In view of Lemma 3.1.2, the following lemme generalizes the fact that for any place +v ∈ Xg, the local duality identifies H1(Ov, Cp) ⊂ H1(Kv, Cp) with the annihilator of +H1(Ow, E(p) +p ) ⊂ H1(Kv, E(p) +p ) [Mil06, §III, Theorem 7.1]. +Lemma 3.1.5. At each place v of K, under the duality H1(Kv, Cp) = H1(Kv, E(p) +p )∨ +[Mil06, §III. Theorem 6.10], the kernel of j′ +v : H1(Kv, Cp) −→ H1(Kv, A), as a +subgroup of H1(Kv, Cp), is exactly the annihilator of ker(jv) ⊂ H1(Kv, E(p) +p ). +We abbreviate the above as +ker(j′ +v) = ker(jv)⊥. +(17) +Proof. Let ∂ denote the connecting homomorphism in the long exact sequence +· · · +� A(Kv) +F +� A(p)(Kv) +∂ +� H1(Kv, Cp) +j′ +v +� H1(Kv, A) +� · · · , +(18) + +THE FROBENIUS TWISTS OF ELLIPTIC CURVES OVER GLOBAL FUNCTION FIELDS +11 +that, since p = F ◦ V, gives rise to the homomorphism ¯∂ in the diagram +A(p)(Kv)/p · A(p)(Kv) +¯∂ +� +� � +i +� +H1(Kv, Cp) +H1(Kv, A(p) +p ) +V∗ +� H1(Kv, Cp), +(19) +where the bottom left-arrow is due to (5) and the left down-arrow i is induced from +the long exact sequence +A(p) +p (Kv) +� A(p)(Kv) +[p] � A(p)(Kv) +� H1(Kv, A(p) +p ) +� · · · . +We claim that the diagram (19) is commutative, so that the commutative diagram +H1(Kv, A(p) +p ) × H1(Kv, A(p) +p ) +V∗ +� +(−,−)v +� Q/Z +H1(Kv, Cp) × H1(Kv, E(p) +p ) +jv +� +(−,−)v +� Q/Z, +where the left-arrows are local pairings, yields the commutative diagram (see [Mil06, +§III, Theorem 7.8]) +A(p)(Kv) × H1(Kv, A(p)) +∂ +� +(−,−)A(p)/Kv +� Q/Z +H1(Kv, Cp) × H1(Kv, E(p) +p ) +jv +� +(−,−)v +� Q/Z. +This shows that ker(jv)⊥ = Im(∂v) = ker(j′ +v). To prove the claim, we use ˇCech +cocycles (see [Mil80, §III.2.10]). +Let x ∈ A(p)(Kv) and denote ¯x its image in +A(p)(Kv)/p · A(p)(Kv). Let y ∈ A(p)(K′ +w) = Hom(Spec(K′ +w), A(p)) be a p-division +point of x over a finite extension K′ +w. Let prl, l = 1, 2, be the projection +Spec(K′ +w) ×Spec(Kv) Spec(K′ +w) −→ Spec(K′ +w) +to the l’th factor. Then ξ := y ◦ pr1 − y ◦ pr2 is a 1-cocycle representing the class +i(¯x). Let z = V(y) ∈ Hom(Spec(K′ +w), A) so that F(z) = x. Then V ◦ ξ is a 1-cocycle +representing both ¯∂(¯x) and V∗(i(¯x)). +□ +3.2. The conductor. Recall that k = K(E(p) +p ( ¯Ks)) so that E(p) +p ( ¯Ks) = E(p) +p (k), +on which the action of the Galois group Φ := Gal(k/K) is given by an injective +character +c : Φ −→ F∗ +p +such that x +g += c(g) · x, for g ∈ Φ, x ∈ E(p) +p (k). Since the order of Φ is prime to p, +the Hochschild-Serre spectral sequence (see [Mil80, §II.2.21(a)]) yields +H1(K, E(p) +p ) +∼ +� H1(k, E(p) +p )Φ +Hom(Gk, E(p) +p (k))Φ . +(20) + +12 +KI-SENG TAN +For w ∈ Sss(k), let ι(p)(t1) be a non-zero element of E(p) +p (kw) as in §2.3. Put +Mw := + + + + + +(1 + tp +1Ow) · (O∗ +w)p, +if w ∈ Sss(k); +k∗ +w, +if w ∈ ð(k); +O∗ +w, +otherwise. +Let A∗ +k denote the group of ideles of k and W the p-completion of k∗\A∗ +k/ � +w Mw. +Lemma 3.2.1. We have Sel(E(p) +p /k) = Hom(W , E(p) +p (k)). +Proof. By the global class field theory, Hom(W , E(p) +p (k)) ⊂ Hom(Gk, E(p) +p (k)) con- +sists of elements which are locally trivial at w ∈ ð(k), having conductors not greater +than ordw(tp +1) at supersingular places w, unramified at others. The lemma follows +from Lemma 2.2.1, 2.2.2 and 2.3.3. +□ +Every pro-p Φ-module Y can be decomposed as Y = � +χ∈ˆΦ Y χ, where for each χ, +Y χ denote the χ-eigenspace {y ∈ Y | +y +g += χ(g) · y} . By (20) and Lemma 3.2.1, +Sel(E(p) +p /K) = Hom(W , E(p) +p (k))Φ = Hom(W c, Z/pZ). +(21) +For v ∈ Sss, put Wv := � +w|v k∗ +w/((k∗ +w)p · Mw), and Uv := � +w|v O∗ +w/Mw regarded +as a subgroup of Wv. +Lemma 3.2.2. If v ∈ Sss, then |Uc +v| = qnv +v . +Proof. Again, by the Hochschild-Serre spectral sequence +H1(Kv, E(p) +p ) +∼ +� (� +w|v H1(kw, E(p) +p ))Φ +(� +w|v Hom(Gkw, E(p) +p (k)))Φ. +(22) +Therefore, Lemma 2.3.3 implies ker(jv) ≃ Hom(W c +v, Z/pZ). Then the lemma follows +from (13), because if kw ̸= Kv, then W c +v = Uc +v; if kw = Kv, then we have the splitting +exact sequence +0 −→ Uc +v −→ W c +v −→ Z/pZ −→ 0. +□ +3.2.1. An idelic computation. In this subsection only, we consider a general situation +in which for w ∈ Sss(k), the Mw in the previous subsection is replaced by +Mw := (1 + πav +w Ow) · (O∗ +w)p, +where v is a place of K sitting below w and av is a chosen integer depending only on +v, and we keep Mw unchanged for other w. Denote a := (av)v∈Sss. Then let Wa be +the p-completion of k∗\A∗ +k/ � +w Mw so that W = Wo, where o := (p · ordw tv)v∈Sss. +Put Ua := � +w∈Sss O∗ +w/Mw, Wa := � +w∈Sss k∗ +w/((k∗ +w)p · Mw). Assume that +|Uc +a| = +� +v∈Sss +qρv +v . +(23) +Let ¯Uc +a denote the image of the natural map +ςc +a : Uc +a −→ W c +a . + +THE FROBENIUS TWISTS OF ELLIPTIC CURVES OVER GLOBAL FUNCTION FIELDS +13 +Let V be the group of ð(k)-units of k. Then ker(ςc) equals the image of +̺c +a : (V/pV )c −→ Uc +a +induced by the localization map V −→ � +w∈Sss(k) O∗ +w. The torsion part of V is finite +of order prime to p. The map V −→ � +v∈ð Rv, where Rv := � +w|v k∗ +w/O∗ +w, is injective +on the free part of V . The module Rv is fixed by the decomposition subgroup Φv +and is the regular representation of Φ/Φv. If k = K, the Zp-rank of (Zp ⊗Z V )c is +max{|ð0| − 1, 0}; otherwise it is |ð0|, so +| ker(ςc +a)| = | Im ̺c +a| ≤ p|ð0|. +(24) +If ð = ∅, letג denote the p-completion of the divisor class group of k; otherwise, let +ג be the ð(k)-class group. Then ℏ is the p-rank ofג. The exact sequence +0 +� ¯Uc +a +� W c +a +�גc +� 0 +yields +0 +� ¯Uc +a +� W c +a [p] +κc �גc[p] +(25) +Put +τ := +� +logp | Im(κc)| + 1, +if ð = ∅ and k = K; +logp | Im(κc)|, +otherwise. +(26) +Via (20), we identify H1(K, E(p) +p ) with Hom(Gk, E(p) +p (k))Φ, so that the conductor +of its element at each place v is defined. Let Xo ⊂ X be the complement of Sss. +Definition 3.2.3. Define Sela(E(p) +p /K) to be the subgroup of S(E(p) +p /Xo) consisting +of elements having conductors not greater than av at each v ∈ Sss. +Lemma 3.2.4. Assuming (23), we have +logp | Sela(E(p) +p /K)| = +� +v∈Sss +ρv · deg v + ˜ε1 − ˜ε2, +where ˜ε1 = τ ≤ ℏ + 1, ˜ε2 = logp | ker(ςc +a)| ≤ |ð0| ≤ |Sb|. +Proof. Similar to (21), Sela(E(p) +p /K) = Hom(W c +a , Z/pZ). We shall write Wa addi- +tively. If ð ̸= ∅ or k ̸= K, then W c +a is finite with |W c +a /pW c +a | = |W c +a [p]|; otherwise, W c +a +is finitely generated over Zp of rank 1, hence |W c +a /pW c +a | = p · |W c +a [p]|. The lemma +follows from (23), (24), (25) and (26). +□ +Proposition 3.2.5. We have +logp | Sel(E(p) +p /K)| = (p − 1) deg ∆A/K +12 +· logp q + ˜ε1 − ˜ε2, +where ˜ε1 = τ ≤ ℏ + 1, ˜ε2 = logp | ker(ςc +o)| ≤ |ð0| ≤ |Sb|. +Proof. In view of (21), Lemma 3.2.2, and Lemma 3.2.4, we need to show that +� +v∈Sss +nv · deg v = (p − 1) deg ∆A/K +12 +. +(27) + +14 +KI-SENG TAN +Let ω be an invariant differential of A/K and for each place v let ω0,v and ω(p) +0,v be +respectively local N´eron differentials of A and A(p). By [Sil86, §IV. Corollary 4.3], +ordv(V∗ω0,v/ω(p) +0,v) = ordv +dP(t) +dt +|t=0, +which together with Lemma 2.1.1 and (10) yield +� +v∈Sss +nv · deg v = +� +all v +ordv(V∗ω0,v/ω(p) +0,v) · deg v. +(28) +Now the formula (8) in [Tan95] implies +deg ∆A/K +12 += +� +all v +ordv( ω +ω0,v +) · deg v = +� +all v +ordv(V ∗( ω +ω0,v +)) · deg v, +and +p · deg ∆A/K +12 += deg ∆A(p)/K +12 += +� +all v +ordv(V ∗ω +ω(p) +0,v +) · deg v. +These and (28) lead to the desired equality. +□ +3.2.2. Computing S(Cp/U). Next, we investigate S(Cp/U) for U = Xgo, Xg, or X . +Let Sngo be the complement of Xgo in X , S′ +ngo := U ∩ Sngo, S′ +ss := U ∩ Sss, and † the +complement of U in X . Write ‡ for † ∪ ð. We first treat the case in which Xgo ̸= X , +or equivalently, A/K is not isotrivial1. Put +¯W := +� +w∈Sngo(k) +k∗ +w/(k∗ +w)p +and +¯ +W := k∗\A∗ +k/( +� +w∈Xgo(k) +O∗ +w × +� +w∈Sngo(k) +(k∗)p). +If ℓc and ℓc−1 denote the c and c−1 eigenspaces of the regular representation of Φ on +Fp[Φ], then E(p) +p (k) = ℓc. Hence +� +w∈Sngo(k) +H1(kw, E(p) +p ) = Hom( ¯W, ℓc) = Hom( ¯W ⊗Fp ℓc−1, Z/pZ) = ( ¯W ⊗Fp ℓc−1)∨, +and +S(E(p) +p /Xgo × Spec k) = Hom( ¯ +W , ℓc) = (( ¯ +W c/p ¯ +W ) ⊗Fp ℓc−1)∨. +In view of (16) and Lemma 3.1.4, +S(Cp/Xgo × Spec k) = ker( ¯W −→ +¯ +W /p ¯ +W ). +Therefore, by the Hochschild-Serre spectral sequence again, +S(Cp/Xgo) = S(Cp/Xgo × Spec k)Φ = ker( ¯W c −→ +¯ +W c/p ¯ +W c). +For each w, put +Mw := Mw/Mw ∩ (k∗ +w)p = Mw · (k∗ +w)p/(k∗ +w)p. +1Recall that A/K is assumed to be ordinary, having semi-stable reduction everywhere. + +THE FROBENIUS TWISTS OF ELLIPTIC CURVES OVER GLOBAL FUNCTION FIELDS +15 +Lemma 3.1.5 gives rise to the left block of the following commutative diagram +H1(kw, Cp) +∼ +� H1(kw, E(p) +p )∨ +∼ +� Hom(k∗ +w/(k∗ +w)p, Z/pZ)∨ +∼ +� k∗ +w/(k∗ +w)p +ker(j′ +w) +∼ +� +�� +� +ker(jw)⊥ +∼ +� +�� +� +Hom(k∗ +w/Mw · (k∗ +w)p, Z/pZ)⊥ +∼ +� +�� +� +M w. +�� +� +The middle block is obtained by taking the dual of the commutative diagram +H1(kw, E(p) +p ) +∼ +� Hom(k∗ +w/(k∗ +w)p, Zp/pZp) +ker(jw) +�� +� +∼ +� Hom(k∗ +w/(k∗ +w)p · Mw, Zp/pZp), +�� +� +(29) +which is due to Lemma 2.3.3 together with the local class field theory (see (7) and +(14)) while the right block is a direct consequence of duality. +shows that if +¯ +M = � +w∈†(k) k∗ +w/(k∗ +w)p × � +w∈S′ngo(k) M w, then +S(Cp/U) = ker( +¯ +M c −→ +¯ +W c/p ¯ +W c). +(30) +To proceed further, we introduce +˜ +M = +� +w∈†(k) +k∗ +w/(O∗ +p)p × +� +w∈S′ngo(k) +Mw · (O∗ +p)p/(O∗ +p)p +and +˜ +W ; = k∗\A∗ +k/( +� +w∈Xgo(k) +O∗ +w × +� +w∈Sngo(k) +(O∗ +w)p). +Then +¯ +M = +˜ +M /p +˜ +M , and since the kernel of +˜ +W +� � +¯ +W +is inside p ˜ +W , we also have +¯ +W /p ¯ +W = +˜ +W /p ˜ +W . Hence (30) implies +S(Cp/U) = ker( +˜ +M c/p +˜ +M c −→ +˜ +W c/p ˜ +W c). +(31) +Let V and M denote the kernel and image of the natural map ˜ς : +˜ +M −→ +˜ +W . +Let ˜ξ ∈ V and let ξ be a lift of ˜ξ to � +w∈†(k) k∗ +w × � +w∈S′ngo(k) Mw · (O∗ +p)p. Then there +are α ∈ k∗ and θ ∈ � +w∈Xgo(k) O∗ +w × � +w∈Sngo(k)(O∗ +w)p such that +ξ = α · θ. +(32) +Let V‡ denote the group of ‡(k)-units of k. The equality (32) implies that α is in +the subgroup V ′ +‡ ⊂ V‡ consisting of those elements which are inside Mw · (O∗ +w)p, for +all w ∈ S′ +ss(k). Suppose there is another expression ξ = α′ · θ′. Then α′α−1 actually +belongs to the group F∗ +k of global units. Since Sngo = † ⊔ S′ +ngo, the correspondence +˜ξ ↔ α (mod F∗ +k) gives rise to an isomorphism +V ≃ V ′ +‡/F∗ +k. +The exact sequence 0 +� V +� +˜ +M +� M +� 0 induces the exact sequence +˜ +M [p] +� M [p] +∂ +� V /pV +� +˜ +M /p +˜ +M +� M /pM +� 0 . + +16 +KI-SENG TAN +For an h ∈ M [p], let ˜η be one of its preimage in +˜ +M and let η be a lift of ˜η to +� +w∈†(k) k∗ +w × � +w∈S′ngo(k) Mw · (O∗ +p)p. Put ξ = ηp, so that (32) holds for some α and +θ, and ∂(h) is represented by α. In this case, since ξ is a p’th power, α ∈ (k∗ +w)p at +all w ∈ Sngo(k), which is non-empty, so by the local Leopoldt’s conjecture [Kis93], +α = βp, for some β ∈ V‡. Since pV‡ ⊂ V ′ +‡, we conclude that +ker( +˜ +M c/p +˜ +M c −→ M c/pM c) = (V ′ +‡/pV‡)c. +(33) +Denote +ℶ := +˜ +W /M = k∗\A∗ +k/( +� +w∈Xgo(k) +O∗ +w × +� +w∈†(k) +k∗ +w × +� +w∈S′ngo(k) +Mw · (O∗ +p)p. +Then we have the exact sequence +˜ +W [p] +� ℶ[p] +� M /pM +� +˜ +W /p ˜ +W . +(34) +If y ∈ +˜ +W [p] is represented by an idele ζ = (ζw)w, then there are α ∈ k∗ and +θ ∈ � +w∈Xgo(k) O∗ +w · � +w∈Sngo(k)(O∗ +w)p such that +ζp = α · θ. +Then at w ∈ Sngo(k), α ∈ (k∗ +w)p, so by the local Leopoldt’s conjecture again, α = βp, +β ∈ k∗. Since y is also represented by ζ · β−1, we have the isomorphism +� +w∈Sngo(k) O∗ +w/(O∗ +w)p +∼ +� +˜ +W [p]. +Since theג equals the cokernel of � +w∈Sngo(k) O∗ +w/(O∗ +w)p −→ ℶ, the above isomor- +phism and (34) together imply +ker(M c/pM c −→ +˜ +W c/p ˜ +W c) = Im(ℶ[p]c −→ג[p]c) ⊂ג]p]c +(35) +We have ℏ =ג[p] and +logp |(V ′ +‡/p · V‡)c| ≤ logp |Zp ⊗Z V c +‡ /p · Zp ⊗Z V c +‡ | = + + + + + +0, +if ‡ = ∅; +|‡0 − 1|, +if ‡ ̸= ∅ and k = K; +|‡0|, +otherwise, +so by (33) and (35), +logp |S(Cp/U)| ≤ logp |(V ′ +‡/p · V‡)c| + logp |גc[p]| ≤ |‡0| + ℏ. +(36) +Suppose A/K is isotrivial. Then Sb = Sss = ∅. If k ̸= K, choose a place v not +spitting completely over k/K; otherwise, choose any v. Set ♮ := {v}. Let U be the +complement of ♮. Put +¯W := +� +w∈♮(k) +k∗ +w/(k∗)p +and denote +¯ +W := k∗\A∗ +k/( +� +w∈U(k) +O∗ +w · +� +w∈♮(k) +(k∗)p), + +THE FROBENIUS TWISTS OF ELLIPTIC CURVES OVER GLOBAL FUNCTION FIELDS +17 +so that H1(Kv, E(p) +p ) = Hom( ¯W c, Z/pZ) and S(E(p) +p /U) = Hom( ¯ +W c/p ¯ +W c, Z/pZ). +Thus, in view of (16) and Lemma 3.1.4, if +¯ +M := � +w∈♮(k) O∗ +w/(O∗ +w)p, then +S(Cp/X ) = ker( +¯ +M c −→ +¯ +W c/p ¯ +W c). +The kernel of +¯ +M −→ +¯ +W is in the image of V♮, the ♮(k)-units of k. Because of our +choice, (Zp ⊗Z V♮)c = 0, and hence +¯ +M c −→ +¯ +W c is injective. +Similar to the previous case, the local Leopoldt’s conjecture at w ∈ ♮(k) implies +� +w∈♮(k) k∗ +w/(k∗ +w)p +� � +¯ +W [p]. +Since (� +w∈♮(k) k∗ +w/(k∗ +w)p)c = 0, we have +¯ +W [p] = 0. Now, ( ¯ +W / +¯ +M )[p] =ג[p], whereג +is the divisor class group of k. It follows from the exact sequence +¯ +W c[p] +�גc[p] +� +¯ +M c +� +¯ +W c/p ¯ +W c +that +|S(Cp/X )| = |גc[p]|. +Thus, the following proposition is proved. +Proposition 3.2.6. For U = Xgo, Xg, or X , +logp |S(Cp/U)| ≤ |‡0| + ℏ. +In particular, +logp | Sel(Cp/K)| ≤ |ð0| + ℏ. +Proof of Proposition 1. The proposition is a consequence of Proposition 3.2.5 and +Proposition 3.2.6, because we have the commutative diagram of exact sequences +0 +� Sel(E(p) +p /K) +� +� � +� +Selp(A(p)/K) +� +� � +� +Sel(Cp/K) +� � +� +Cp(K) +� H1(K, E(p) +p ) +� +� +H1(K, A(p) +p ) +V∗ +� +� +H1(K, Cp) +� +� +v H1(Kv, A(p)) +� +v H1(Kv, A(p)) +V∗ � � +v H1(Kv, A), +where the middle long exact sequence is induced from (5). +□ +3.3. The Cassels-Tate duality. The Cassels-Tate pairing induces the perfect pair- +ing (see [Mil06, III.9.5]) +⟨−, −⟩A/K : X(A/K) × X(A/K) −→ Qp/Zp. +If ϕ : A −→ B is an isogeny with dual isogeny ϕt, then the commutative diagram +X(A/K) × X(A/K) +ϕ♮ +� +⟨−,−⟩A/K +� Qp/Zp +X(B/K) × X(B/K) +ϕt +♮ +� +⟨−,−⟩B/K +� Qp/Zp + +18 +KI-SENG TAN +yields the duality between X(A/K)/ ker(ϕ♮) and X(B/K)/ ker(ϕt +♮). In particular +|(X(A/K)/ ker(ϕ♮))[pν]| = |(X(B/K)/ ker(ϕt +♮))[pν]|, +(37) +since |G/pνG| = |G[pν]| for a finite abelian group G. +Proof of Proposition 2. Consider the exact sequence +0 +� ker(F♮) +� X(A/K)[pν] +� (X(A/K)/ ker(F♮))[pν] +·pν +�✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐ +ker(F♮) ∩ pνX(A/K) +� 0, +where the morphism ·pν is induced from the multiplication by pν. This implies +logp |X(A/K)[pν]| = logp |(X(A/K)/ ker(F♮))[pν]| + δ1, +(38) +with logp | Sel(Cp/K)| ≥ logp | ker(F♮)| ≥ δ1 ≥ 0. Also, consider the exact sequence +0 +� ker(V♮) +� (X(A(p)/K)/ ker(V♮))[pν] +�❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣ +(X(A(p)/K)/X(A(p)/K)[p])[pν] +·pν +� T +� 0, +where T = (X(A(p)/K)[p]/ ker(V♮)) ∩ pν(X(A(p)/K)/ ker(V♮)). This together with +(37) and (38) lead to +logp |X(A/K)[pν]| = logp |(X(A(p)/K)/X(A(p)/K)[p])[pν]| + δ1 + δ2, +(39) +with +logp | Sel(Cp/K)| ≥ logp | ker(F♮)| ≥ logp |V♮(X(A(p)/K)[p])| ≥ δ2 ≥ 0. +Now +(X(A(p)/K)/X(A(p)/K)[p])[pν] = X(A(p)/K)[pν+1]/X(A(p)/K)[p]. +Recall that r denote the co-rank of Seldiv(A/K) ≃ Seldiv(A(p)/K), so +logp |X(A/K)[pν]| = logp | Selpν(A/K)| − rν, +and a similar formula for A(p). Therefore, +logp | Selpν(A/K)| = logp | Selpν+1(A(p)/K)| − logp | Selp(A(p)/K)| + δ1 + δ2. +Then we apply Proposition 1 and Proposition 3.2.6. +□ + +THE FROBENIUS TWISTS OF ELLIPTIC CURVES OVER GLOBAL FUNCTION FIELDS +19 +3.4. The method of specialization. In this section only, we assume that A/K +an ordinary abelian variety defined over a global field. As before, let L/K be a Zd +p- +extension unramified outside a finite set of places of K. Let Γ be the Galois group, +ΛΓ the Iwasawa algebra. +Endow lµ.. p∞ with the discrete topology and write ˆΓ for the group of all continuous +characters Γ −→ lµ. +. p∞. Let O be the ring of integers of Qp(lµ. +. p∞). Every character +χ ∈ ˆΓ extends to a unique continuous Zp-algebra homomorphism χ : ΛΓ −→ O. +For g ∈ ΛΓ, not divisible by p, by Monsky [Mon81, Theorem 2.3], the set +∇g := {χ ∈ ˆΓ | χ(g) ∈ p · O} +is either ∅ or is contained in T1∪· · ·∪Tn, where for each i, there are ζi,1, ..., ζi,νi ∈ lµ. +. p∞, +νi > 0, and σi,1, ..., σi,νi ∈ Γ, extendable to a Zp-basis of Γ, such that +Ti := {χ ∈ ˆΓ | χ(σi,j) = ζi,j, for j = 1, ..., νi}. +For an element ψ ∈ Γ, extendable to a Zp-basis of Γ, set +Tψ := {χ ∈ ˆΓ | χ(ψ) = 1}. +Then Tψ ⊂ Ti can not hold, unless νi = 1, ζi,1 = 1, and σi,1, ψ topologically generate +the same closed subgroup of Γ, in such case, we actually have Tψ = Ti. Therefore, +there are finitely many rank one Zp-submodules of Γ such that if ψ is chosen away +from them, then +Tψ ⊊ ∇g. +(40) +For a Ze +p-subextension L′/K of L/K. Denote Γ′ = Gal(L′/K). The quotient map +Γ −→ Γ′ extends uniquely to a continuous Zp-algebra homomorphism +pL/L′ : ΛΓ −→ ΛΓ′. +Write Ψ := Gal(L/L′) so that Γ′ = Γ/Ψ. Put IΨ := ker(pL/L′). +Let b be a finite set of places of K. For a subextension M of L/K, let bM ⊂ b +denote the subset consisting of places splitting completely over M. +Lemma 3.4.1. Suppose d ≥ 2. Let Zℓ, ℓ = 1, .., s, be finitely generated torsion +ΛΓ-modules, θ an element in ΛΓ, not divisible by p, and b a finite set of places of K. +There is an element ψ ∈ Γ extendable to a Zp-basis such that if L′ is the fixed field of +ψ, then bL′ = bL, pL/L′(θ) is not divisible by p, and each ΛΓ′-module Z′ +ℓ := Zℓ/IΨZℓ +is torsion, having the same elementary µ-invariants as those of Zℓ over ΛΓ. +Proof. For each v ∈ b with non-trivial decomposition subgroup Γv, if ψ ̸∈ Γv or Γv +is of Zp-rank greater than one, then v does not split completely over L′. To have +bL′ = bL, we only need to choose ψ away from all rank one Γv, v ∈ b. +Similar to (2), we have +0 +� �mℓ +i=1 ΛΓ/(pαℓ,i) ⊕ �nℓ +j=1 ΛΓ/(η +βℓ,j +ℓ,j ) +� Zℓ +� Nℓ +� 0, +(41) + +20 +KI-SENG TAN +where Nℓ is pseudo-null and every ηℓ,j is not divisible by p. Choose for each ℓ, an +annihilator hℓ ∈ ΛΓ of Nℓ, not divisible by p, and put +g := θ · +s� +ℓ=1 +hℓ · ηℓ,1 · · · · · ηℓ,nℓ. +We also choose ψ satisfying (40). Since ˆΓ′ = Tψ, there exists χ ∈ ˆΓ′ such that +χ(pL/L′(g)) ̸∈ p · O, so pL/L′(g) ̸∈ p · ΛΓ′. Because pα · gβ · Zℓ = 0, for some α, β ∈ Z, +we have pα · pL/L′(gβ) · Z′ +ℓ = 0. Hence Z′ +ℓ is torsion over ΛΓ′. +To compare the elementary µ-invariants of Zℓ and Z′ +ℓ, we apply the maps of +multiplication by ψ−1 to (41) and use the snake lemma to obtain the exact sequence +Nℓ[ψ − 1] −→ +mℓ +� +i=1 +ΛΓ′/(pαℓ,i) ⊕ +nℓ +� +j=1 +ΛΓ′/(pL/L′(ηℓ,j)βℓ,j) −→ Z′ +ℓ −→ Nℓ/IΨNℓ. (42) +Because Nℓ[ψ − 1] is annihilated by pL/L′(g) which is relatively prime to p, the +second arrow in the exact sequence is injective on �mℓ +i=1 ΛΓ′/(pαℓ,i). We complete +the proof by comparing the pth power factors of the characteristic ideals of items in +the sequence. +□ +In Lemma 3.4.1, the field L′ is a Zd−1 +p +-extension of K. By repeatedly applying +the lemma, we obtain sequences +L ⊃ L′ ⊃ · · · ⊃ L(d−1), +(43) +and, for each ℓ, +Zℓ −→ Z′ +ℓ −→ · · · −→ Z(d−1) +ℓ +. +(44) +Put Ψ0 = Ψ, Ψi = Gal(L/L(i+1)), and Γ(i+1) = Gal(L(i+1)/K) = Γ/Ψi. Then L(d−1) +is a Zp-extension of K with bL(d−1) = bL, pL/L(d−1)(θ) not divisible by p, and for each +ℓ, the ΛΓ(d−1)-module Z(d−1) +ℓ += Zℓ/IΨd−2Zℓ has the same elementary µ-invariants as +those of Zℓ over ΛΓ. These elementary µ-invariants pαℓ,1, ..., pαℓ,mℓ can be recovered +by using the counting formula below. For each ν, define +αℓ,i,ν = min{ν, αℓ,i}. +Let σ ∈ Γ(d−1) be a topological generator and put x = σ − 1. Let Jν,n denote the +ideal of ΛΓ(d−1) generated by pν and (x + 1)pn − 1. +Lemma 3.4.2. Let the notation be as above. Then +logp |Z(d−1) +ℓ +/Jν,nZ(d−1) +ℓ +| = pn · +mℓ +� +i=1 +αℓ,i,ν + O(1). +Proof. Taking Z = Z(d−1) +ℓ +/pνZ(d−1) +ℓ +in (2), we deduce the following exact sequence, +0 +� �mℓ +i=1 ΛΓ(d−1)/(pαℓ,i,ν) +� Z(d−1) +ℓ +/pνZ(d−1) +ℓ +� Nℓ,ν +� 0, +(45) + +THE FROBENIUS TWISTS OF ELLIPTIC CURVES OVER GLOBAL FUNCTION FIELDS +21 +where Nℓ,ν is finite, since the other two items are pseudo isomorphic. Write Rα for +Zp/pαZp. The lemma is a consequence of the exact sequence induced from (45): +N0[σpn − 1] +� �mℓ +i=1 Rαℓ,i,ν[x]/((x + 1)pn − 1) +� Z(d−1) +ℓ +/Jν,nZ(d−1) +ℓ +�❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤ +N0/(σpn − 1)N0. +□ +To apply the above to dual Selmer groups, we need the following simplified control +lemma. For K ⊂ F ⊂ L, consider the restriction maps +res(ν) +L/F : Selpν(A/F) −→ Selpν(A/L)Gal(L/F ). +Let K(n) denote the nth layer of the Zp-extension L(d−1)/K. Let r denote the +ramification locus of L/K, which is assumed to be finite. +Lemma 3.4.3. Let L(d−1)/K be an intermediate Zp-extension of L/K. Suppose r +contains only places where A has good ordinary reduction or multiplicative reduction, +bL(d−1) = bL and b contains r as well as all places where A has bad reduction. For a +given ν, the orders of ker(res(ν) +L/K(n)) and coker(res(ν) +L/K(n)) are bounded, as n varies. +Proof. Let M = Apν(L) = Apν(K′) for some finite sub-extension K′/K of L/K. +Write K′(n) for K(n)K′. [Tan10, Lemma 3.2.1] says that for i = 0, 1, 2, +| Hi(L/K′(n), M)| ≤ |M|di. +Since [K′(n) : K(n)] ≤ [K′ : K], by counting the number of co-chains, we deduce +| Hi(K′(n)/K(n), M)| ≤ |M|[K′:K]i. +This bounds | ker(res(ν) +L/K(n))|. To bound | coker(res(ν) +L/K(n))|, by the Hochschild-Serre +spectral sequence, we need to bound (see the proof of [Tan10, Theorem 4]) +| +� +all v +� +w|v +H1(Lw/K(n) +w , A(Lw))[pν]|. +Write H (ν) +w +for H1(Lw/K(n) +w , A(Lw))[pν]. If v ̸∈ b, then H (ν) +w += 0 [Mil06, I.3.8], +for all w | v. Also, H (ν) +w += 0, for all w sitting over bL, because K(n) +w = Lw. +Suppose v ∈ b but v ̸∈ bL = bL(d−1). The number of places of K(n) sitting over v +is bounded as n varies. We need to bound the order of H (ν) +w , for all w | v. If A has +good ordinary reduction at v, by [Tan10, (3) and Theorem 2], the order of H (ν) +w +is +bounded by p2ν(d+1) dim A. It is well-known (for instance, see the last two paragraphs +of [Tan10]) that if A has split multiplicative reduction at v, the order of H (ν) +w +is +bounded by pνd dim A. In general, if A has multiplicative reduction at v, then over +some unramified extension K′ +v/Kv, the reduction of A becomes split multiplicative. + +22 +KI-SENG TAN +Since H (ν) +w +⊂ H1(LwK′ +v/Kv, A(LwK′ +v)), writing K′ +v, L′ +w for KvK′ +v, LwK′ +v, we end the +proof by using the exact sequence +H1(K′ +v/Kv, A(K′ +v))� � +� H1(L′ +w/Kv, A(L′ +w)) +� H1(L′ +w/K′ +v, A(L′ +w)) +and the fact that the component group of A/K′ +v has p-rank bounded by dim A +(see [BX96, Proposition 5.2]) so that by [Mil06, Proposition I.3.8] the order of +H1(K′ +v/Kv, A(K′ +v)) is bounded. +□ +Lemma 3.4.4. Let θ ∈ ΛΓ be an element not divisible by p. Let Aℓ, ℓ = 1, ..., s, +be ordinary abelian varieties defined over K such that all Zℓ := Selp∞(Aℓ/L)∨ are +torsion over ΛΓ and the ramification locus r contains only places where each Aℓ has +either good ordinary reduction or multiplicative reduction. Let b be a finite set of +places of K containing r and all places where some Aℓ has bad reduction. Assume +that pαℓ,1, ..., pαℓ,mℓ are elementary µ-invariants of Zℓ. There exists an intermediate +Zp-extension L(d−1)/K of L/K such that the following holds: +(a) pL/L(d−1)(θ) ̸∈ pΛΓ(d−1), where Γ(d−1) = Gal(L(d−1)/K). +(b) bL(d−1) = bL. +(c) For each ℓ, the elementary µ-invariants of Selp∞(Aℓ/L(d−1))∨ over ΛΓ(d−1) are +the same as those of Zℓ over ΛΓ. +(d) In particular, if L/K is a Zp-extension and K(n) denote the nth layer, then +logp | Selpν(Aℓ/K(n))| = pn · +mℓ +� +i=1 +αℓ,i,ν + O(1). +(46) +Proof. (a) and (b) are from Lemma 3.4.1. Observe that Z(d−1) +ℓ +/Jν,nZ(d−1) is nothing +but the Pontryagin dual of Selpν(Aℓ/L)Gal(L/K(n)), so by Lemma 3.4.2, 3.4.3, we have +logp | Selpν(Aℓ/K(n))| = pn · +mℓ +� +i=1 +αℓ,i,ν + O(1). +(47) +In the situation of (d), L = L(d−1), and hence, (46) holds. To show (c), we assume +that the elementary µ-invariants of Selp∞(Aℓ/L(d−1))∨ over ΛΓ(d−1) are pα′ +ℓ,1, ..., pα′ +ℓ,wℓ. +Apply (d) to L(d−1)/K and obtain +logp | Selpν(Aℓ/K(n))| = pn · +wℓ +� +i=1 +α′ +ℓ,i,ν + O(1), +where, as before, α′ +ℓ,i,ν := min{α′ +ℓ,i, ν}. This and (47) leads to +wℓ +� +i=1 +α′ +ℓ,i,ν = +mℓ +� +i=1 +αℓ,i,ν, +for all ν. We conclude that wℓ = mℓ and pα′ +ℓ,1, ..., pα′ +ℓ,wℓ are the same as pαℓ,1, ..., pαℓ,mℓ. +□ + +THE FROBENIUS TWISTS OF ELLIPTIC CURVES OVER GLOBAL FUNCTION FIELDS +23 +3.5. The elementary µ-invariants. In this section, we complete the proof of +Proposition 3 and Proposition 5. For each Zp-subextension F/K of L/K, write +Sb(F) = ð′ +F ⊔ ðF, +where if p = 2, ð′ +F is the set of places at which A/K has non-split multiplicative +reduction such that the group of components is of even order; if p ̸= 2, ð′ +F = ∅. For +w ∈ Sb(F) sitting on v ∈ Sb, if w ∈ ð′ +F and v ∈ ð, or w ∈ ðF and v ∈ ð′, then +Fw ̸= Kv, and hence there is only finitely many places of F sitting over v. +Proof of Proposition 3. We apply Lemma 3.4.4, taking s = 1, Z1 = X(p), b = r ∪Sb. +Let K(n) be the nth layer of L(d−1)/K. Since the degrees of k/K and K(n)/K are +relatively prime, one see that a place v of K splits completely in k, if and only if +all places of K(n) sitting over v split completely in kK(n), because both assertions +are equivalent to that the decomposition subgroup Gal(kK(n)/K)v contains no non- +trivial element in Gal(kK(n)/K(n)), and hence contained in Gal(kK(n)/k). Put +ð0,n := {w ∈ ðK(n) | w splits completely over kK(n)}. +Then, by the discussion at the beginning of this section, +|ð0,n| = |ð0(K(n))| + O(1) = pn · |ð1| + O(1). +If Fq(n) denote the constant field of K(n), then +deg ∆A/K(n) · logp q(n) = pn · deg ∆A/K · logp q. +Therefore, Lemma 3.4.4(d) and Proposition 1 say if m is the µ-rank of A(p)/L, then +pn · m = logp | Selp(A(p)/K(n))| + O(1) ≥ pn · ((p − 1) deg ∆A/K +12 +· logp q − |ð1|) + O(1). +This proves the proposition. +□ +Proof of Proposition 5. Take s = 2, Z1 = X(p), Z2 = X, b = Sb ∪ r, θ = ΘL, , so +that m = m1, and αi = α1,i, for i = 1, ..., m. By Lemma 3.4.4, bL(d−1) = bL = ∅ and +ΘL(d−1) is not divisible by p. Thus, we may assume that d = 1. +Let K(n) denote the nth layer of L/K. +If L = K(∞) +p +, we have shown in §1.1 +that |wK(n)[p]| = O(1); otherwise, for n sufficiently large, L/K(n) is totally ramified +at certain place, so that Hom(Gal(L/K(n)), Qp/Zp) ∩ Hom(wK(n), Qp/Zp) = {0}. +Hence, Hom(wK(n), Qp/Zp) −→ Hom(wL, Qp/Zp) is injective, or equivalently, the +map wL −→ wK(n) is surjective, for sufficiently large n. Since p ∤ ΘL, wL has trivial +p-part, the order of wK(n)[p] must be bounded. The assumption says +|Sb(K(n))| = O(1). +Therefore, by Lemma 3.4.4(d) and Proposition 1, we obtain +pn · m += +pn · �m +i=1 α1,i,1 += +logp | Selp(A(p)/K(n))| + O(1) += +pn · +(p−1) deg ∆A/K +12 +· logp q + O(1), + +24 +KI-SENG TAN +that proves the first assertion. Then Lemma 3.4.4(d) and Proposition 2 lead to +pn · �m2 +i=1 α2,i,ν += +logp | Selpν(A/K(n))| + O(1) += +pn · (logp | Selpν+1(A(p)/K(n))| − +(p−1) deg ∆A/K +12 +· logp q) + O(1) += +pn · �m +i=1(α1,i,ν+1 − 1) + O(1), +which holds for every ν, so the proposition is proved. +□ +References +[BBM82] P. Berthelot, L. Breen, and W. Messing, Th´eorie de Dieudonn´e cristalline. II, Lecture +Notes in Mathematics 930. Springer-Verlag, 1982. +[BLR90] S. Bosch, W. L¨utkebohmert, and M. Raynaud, N´eron Models, Springer-Verlag Berlin +Heidelberg, 1990. +[BX96] S. Bosch, X. Xarles, Component group of N´eron Models via rigid uniformization, Math. +Ann. 306 (1996), 459-486. +[Ces15] K. ˇCesnaviˇcius, Poitou-Tate without the restriction on the order, Math. Res. Lett. 22 +(2015), no. 6, 1621-1666. +[Crw87] R. Crew, L-functions of p-adic characters and geometric Iwasawa theory, Invent. Math. +88 (1987), 395-403. +[Gon09] C.D. Gonz´alez-Avil´es, Arithmetic duality theorems for 1-motives over function fields. J. +Reine Angew. Math. 632 (2009), 203-231. +[GoT12] C.D. Gonz´alez-Avil´es, K.-S. Tan, On the Hasse Principle for finite group schemes over +global function fields, Math. Res. Lett. 19 (2012), no. 02, 453-460. +[EGA IV, part 4] A. Grothendieck and J. Dieudonn´e, El´ements de G´eom´etrie Alg´ebrique IV, Pul. +Math. IHES 32 (1967). +[SGA 7I] A. Grothendieck, Groupes de monodromie en G´eom´etrie Alg´ebrique. I. S´eminaire de +G´eom´etrie Alg´ebrique du Bois-Marie 1967-1969 (SGA 7 I). Lecture Notes in Math. 288. +Springer, Heidelberg 1972. +[Kis93] H. Kisilevsky, Multiplicative independence in function fields, J. Number Theory 44 (1993) +352–355. +[LLTT16] K.F. Lai, I. Longhi, K.-S. Tan and F. Trihan, On the Iwasawa main conjecture for +semistable abelian varieties over function fields, Mathematische Zeitschrift 282 (2016), issue +1, 485-510. +[LSc10] C. Liedtke, S. Schr¨oer, The N´eron model over the Igusa curves, J. Number Theory 130 +(2010), 2157-2197. +[Mil80] J.S. Milne, ´Etale Cohomology, Princeton University Press, Princeton, 1980. +[Mil06] J.S. Milne, Arithmetic duality theorems, Second Ed. (electronic version), 2006. +[Mon81] P. Monsky, On p-adic power series. Math. Ann. 255 (1981), no. 2, 217–227. +[OT09] T. Ochiai and F. Trihan, On the Iwasawa main conjecture of abelian varieties over function +fields of characteristic p > 0, Math. Proc. Camb. Philos. Soc. 146 (2009), 23-43. +[Sil86] J. H. Silverman, The Arithmetic of Elliptic Curves, Springer Verlag, Berlin, 1986. +[Tan95] K.-S. Tan, Refined theorems of the Birch and Swinnerton-Dyer type, Ann. Inst. Fourier, +Grenoble 45 (1995), 317-374. +[Tan10] K.-S. Tan, A generalized Mazur’s theorem and its applications, Trans. Amer. Math. Soc. +362 (2010), no. 8, 4433–4450. +[Tan14] K.-S. Tan, Selmer groups over Zd +p-extensions, Math. Ann. 359 (2014), 1025-1075. +[Ta66] J. Tate, On the conjecture of Birch and swinnerton-Dyer and a geometric analogue, +S´eminare Bourbaki no. 9 (1966), 415-440. +[Ta84] J. Tate, Les Conjectures de Stark sur les Fonctions L d’Artin en s = 0, (Birkhauser, Boston, +1984). + +THE FROBENIUS TWISTS OF ELLIPTIC CURVES OVER GLOBAL FUNCTION FIELDS +25 +Department of Mathematics, National Taiwan University, Taipei 10764, Taiwan +Email address: tan@math.ntu.edu.tw + diff --git a/39AyT4oBgHgl3EQfo_il/content/tmp_files/load_file.txt b/39AyT4oBgHgl3EQfo_il/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2b5bb70f7e7551b42107ec4712919bfb6e9ba2d2 --- /dev/null +++ b/39AyT4oBgHgl3EQfo_il/content/tmp_files/load_file.txt @@ -0,0 +1,897 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf,len=896 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='00518v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='NT] 2 Jan 2023 THE FROBENIUS TWISTS OF ELLIPTIC CURVES OVER GLOBAL FUNCTION FIELDS KI-SENG TAN Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' For an elliptic curve A defined over a global function field K of char- acteristic p > 0, the p-Selmer group of the Frobenius twist A(p) of A tends to have larger order than that of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' The aim of this note is to discuss this phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Introduction For an elliptic curve A defined over a global function field K of characteristic p > 0, the p-Selmer group of the Frobenius twist of A tends to have larger order than that of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' The aim of this note is to discuss this phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' The Frobenius twist A(p) is the base change A ×K K of A over the absolute Frobenius Frobp : K −→ K, x �→ xp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' The main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' We assume that A/K is ordinary, having semi-stable re- duction everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Let ∆A/K denote the divisor of the global minimal discriminant of A/K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Comparing the defining equation for both curves yields ∆A(p)/K = p · ∆A/K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' (1) Let Apν be the the kernel of the multiplication by pν on A and let Selpν(A/K) ⊂ H1(K, Apν) denote the pν-Selmer group of A/K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' If p = 2, let ð′ be the set of places of K at which A has non-split multiplicative reduction and has the group of components of even order;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' otherwise, put ð′ = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Let Sb denote the set of bad reduction places of A/K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Write Sb = ð′ ⊔ ð.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Let k = K(A(p) p ( ¯Ks)) and let ð0 ⊂ ð be the subset of places splitting completely over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Let ℏ denote the p-rank of the subgroup of Hom(Gal(¯ks/k), Z/pZ) consisting of homomorphisms unramified everywhere and locally trivial at every places of k sitting over ð.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Our main results are as follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Let q denote the order of the constant field of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' There exists an integer ǫ1, 2ℏ + 1 + |ð0| ≥ ǫ1 ≥ −|ð0|, such that logp | Selp(A(p)/K)| = (p − 1) deg ∆A/K 12 logp q + ǫ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Acknowledgement: This research was supported in part by Ministry of Science and Technol- ogy of Taiwan, MOST 109-2115-M-002-008-MY2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' The author thanks F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Trihan for many valuable suggestions especially for helping him with the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' 1 2 KI-SENG TAN Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' There exists an integer ǫ2, 2ℏ + 1 + |ð0| ≥ ǫ2 ≥ −2ℏ − 3|ð0|, such that for each positive integer ν, logp | Selpν+1(A(p)/K)| = (p − 1) deg ∆A/K 12 logp q + logp | Selpν(A/K)| + ǫ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Note that deg ∆A/K 12 is a non-negative integer (see [LLTT16, §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='1]), it is zero, if and only if A/K is isotrivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Let Xp∞(A/K) denote the p-primary part of the Tate-Shafarevich group of A/K, let Xdiv(A/K) be its p-divisible subgroup, and denote the p-cotorsion X(A/K) := Xp∞(A/K)/Xdiv(A/K) = Selp∞(A/K)/ Seldiv(A/K), where Seldiv(A/K) is the p-divisible subgroup of Selp∞(A/K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Let r denote the Zp-co-rank of Seldiv(A/K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' If ν is greater than the exponents of X(A/K) and X(A(p)/K), then | Selpν(A/K)| = |X(A/K)| · prν and | Selpν+1(A(p)/K)| = |X(A(p)/K)| · pr(ν+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' It follows from Proposition 2 that logp |X(A(p)/K)| + r = (p − 1) deg ∆A/K 12 logp q + logp |X(A/K)| + ǫ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Such kind of formulae is suggested by the conjectured Birch and Swinnerton-Dyer formulae (see [Ta66, Tan95]) for both A(p)/K and A/K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Next, let L/K be a Zd p-extension ramified only at a finite number of ordinary places of A/K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Write Γ := Gal(L/K) and ΛΓ := Zp[[Γ]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Let Z be a finitely generated torsion ΛΓ-module, so that there is an exact sequence 0 � �m i=1 ΛΓ/(pαi) ⊕ �n j=1 ΛΓ/(η βj j ) � Z � N � 0, (2) where α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=', αm, β1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=', βn are positive integers, η1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=', ηn ∈ ΛΓ are irreducible, rela- tively prime to p, and N is pseudo-null.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Although the exact sequence is not canon- ical, the modules �m i=1 ΛΓ/(pαi) and �n j=1 ΛΓ/(η βj j ) are uniquely determined by Z, we call them the p part and the non-p part of Z, call pα1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=', pαm the elementary µ-invariants and m the µ-rank of Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' If Z is non-torsion, define the µ-rank to be ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Consider the Pontryagin dual X, X(p) of Selp∞(A/L), Selp∞(A(p)/L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' They are finitely generated over ΛΓ (see [Tan14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Put ð1 := {v ∈ ð | v splits completely over kL}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' The µ-rank of X(p) is at least (p−1) deg ∆A/K 12 logp q − |ð1|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' If L contains K(∞) p , the constant Zp-extension over K, then X and X(p) are torsion [OT09, Tan14], in this case |ð1| = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' If L contains K(∞) p , then the µ-rank m of X(p) equals (p−1) deg ∆A/K 12 logp q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Moreover, if pα1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=', pαm, α1 ≥ · · · ≥ αm, are the elementary µ-invariants of X(p), then those of X are pα1−1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=', pαm′−1, where m′ is the greatest i such that αi > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' THE FROBENIUS TWISTS OF ELLIPTIC CURVES OVER GLOBAL FUNCTION FIELDS 3 For a finite extension F/K, let wF denote the p-completion of the divisor class group of of kF and for a Ze p sub-extension M/K of L/K, put wM := lim ←−K⊂F ⊂M wF, which is finitely generated torsion over ΛGal(kM/k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Actually, by [Crw87], the char- acteristic ideal of wM has a generator ΘM := lim ←−F ΘF, where basically for each F, ΘF ∈ Zp[Gal(kF/k)] is the Stickelberger element defined in [Ta84, §V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='1], in particular, we have pL/M(ΘL) = ΘM · ∗, (3) where pL/M : ΛGal(Lk/k) −→ ΛGal(Mk/k) is the continuous Zp-algebra homomorphism extending the quotient map Gal(Lk/k) −→ Gal(Mk/k) and ∗ ∈ ΛGal(Mk/k) is a fudge factor not divisible by p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' For simplicity, we shall identify ΛGal(Mk/k) with ΛGal(M/K), and view ΘL, wL as objects over ΛΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' In the special case where L = K(∞) p , the module wL has trivial µ-rank, hence ΘL is not divisible by p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' To see this, let Y be the complete smooth curve defined over the constant field of k, having k as its function field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' For every finite sub-extensions F/K of L/K, we have the exact sequence 0 � w0 F � wF deg � Zp � 0 and w0 F[p] is contained in the subgroup of p-division points of the Jacobian variety of Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Therefore, the order of w0 F[p] is bounded, and hence wL[p] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' In general, (3) says that if L contains K(∞) p , then ΘL is not divisible by p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Also, in this case, ð1 = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Hence Proposition 4 is a special case of the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' If ΘL is not divisible by p and ð1 = ∅, then the µ-rank m of X(p) equals (p−1) deg ∆A/K 12 logp q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Moreover, if pα1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=', pαm, α1 ≥ · · · ≥ αm, are the elementary µ-invariants of X(p), then those of X are pα1−1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=', pαm′−1, where m′ is the greatest i such that αi > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Since the Frobenius and Verschiebung induce pseudo isomorphisms between the non-p parts of X and X(p), the proposition implies the characteristic ideal of X(p) is the q (p−1) deg ∆A/K 12 multiple of that of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' If L = K(∞) p , this is also a consequence of the main theorem of [LLTT16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' For a field F, let ¯F and ¯F s denote its algebraic closure and separable closure, and denote GF = Gal( ¯F s/F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' For a place v, let Ov, πv and Fv denote the ring of integers, an uniformizer and the residue field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Write qv for |Fv|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Let Sss denote the set of place v of K at which A has supersingular reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' For a set S of places of K and an algebraic extension F, let S(F) denote the set of places of F sitting over S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' For an endomorphism ϕ of an abelian group H, let H[ϕ] denote the kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Use ∨ for Pontryagin dual, ∼ for pseudo isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' In this note we use flat or Galois cohomology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Let F : A −→ A(p) and V : A(p) −→ A be the Frobenius and the Verschiebung homomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' We have the exact se- quences 4 KI-SENG TAN 0 � Cp � Ap F � E(p) p � 0, (4) as well as 0 � E(p) p � A(p) p V � Cp � 0, (5) where Cp = ker F is connected and E(p) p = ker V, ´etale (see [LSc10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' For a field F containing K, we have A(p) p (F) = E(p) p (F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' In particular, k = K(E(p) p ( ¯Ks)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Note that for p = 2, k = K, because the non-trivial point of E(p) p ( ¯Ks) is the only Galois conjugate of itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' The proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' The key ingredient in the proof of Proposition 1 is local, concern- ing the kernel of the natural map H1(Kv, E(p) p ) −→ H1(Kv, A(p)), especially when v is a place of supersingular reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='1, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='2 and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='3 treat different types of reduction and provide us criteria, in terms of the the con- ductor of the corresponding cyclic extension over kv, for determining if an element of H1(Kv, E(p) p ) is in such kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' With the local criteria available and with the help of global class field, in Propo- sition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='5, we determine the order of Sel(E(p) p /K), the E(p) p -part of Selp(A(p)/K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' In doing so, we find an interesting phenomena that the discrepancy between the two discriminants as described in (1) is solely contributed by supersingular places (see (27)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Next, in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='6 we determine the order of Sel(Cp/K), the Cp-part of Selp(A/K), by using the Poitou-Tate duality [Ces15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Then Proposition 1 is proved at the end of §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='2, as a consequence of the above two propositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Proposition 2 is proved in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='3 by applying Cassels-Tate duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' In §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='4, using a theorem of Monsky [Mon81], we establish a method of reducing the proofs of Proposition 3, 5 to the d = 1 case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Since the method can be applied to more general situations, we loosen the condition in that subsection by allowing A to be an ordinary abelian variety defined over a global field K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' The main result is summarized in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' As a consequence of this lemma and Proposition 1, 2, the proofs of Proposition 3, 5 are given in the final subsection §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Local fields At each place v of K, we have the long exact sequence · · � A(p)(Kv) V∗ � A(Kv) ∂ � H1(Kv, E(p) p ) jv � H1(Kv, A(p)) � · · · (6) derived from 0 � E(p) p j � A(p) V � A � 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' The aim of this section is to determine ker(jv) = coker(V∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' For a place w of k sitting over v, the abelian group H1(kw, E(p) p ) = Hom(Gkw, Z/pZ) = Hom(k∗ w/(k∗ w)p, Z/pZ), (7) so each ξ ∈ H1(kw, E(p) p ) determines a degree p cyclic extension kw,ξ/kw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Let ordv denote the valuation on ¯Kv having ordv(πv) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' THE FROBENIUS TWISTS OF ELLIPTIC CURVES OVER GLOBAL FUNCTION FIELDS 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Frobenius and Verschiebung.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Let F (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' F (p)) denote the formal group law associated to A (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' A(p)) over Kv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Since A has semi-stable reduction, F is stable under local field extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Let ¯A denote the reduction of A at v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' For a place w of an algebraic extension F of K, sitting over v, the pre-image Ao(Fw) of 0 ∈ ¯A(Fw) under the reduction map A(Fw) −→ ¯A(Fw) is identified with F(mw) via a bijection ι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Let ι(p) be the corresponding bijection associated to A(p) o (Fw).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Let P ∈ Ov[[t]] be the unique power series fitting into the commutative diagram F(mw) Frobp � ι � F (p)(mw) P � ι(p) � F(mw) ι � Ao(Fw) F � A(p) o (Fw) V � Ao(Fw).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' (8) An element ξ ∈ mw satisfies P(ξ) = 0 if and only if ι(p)(ξ) ∈ E(p) p (Fw) ∩ A(p) o (Fw).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Suppose the formal group law of F (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' F (p)) is given by the Ov-coefficient power series f(X, Y ) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' f (p)(X, Y )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' If ti is a solution to P(t) = 0, then P(f (p)(t, ti)) = P(t), furthermore, since A/Kv is ordinary, by [Sil86, §IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='2], P′(0) ̸= 0, and hence P′(ti) ̸= 0, too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' We have P(t) = u(t) · P(t), where u(t) is a unit in Ov[[t]] and P(t) is the associated distinguished polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' The polynomial P(t) is separable with P(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' If v is a supersingular place, then deg P = p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' if v is ordinary, then P(t) = t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' The first assertion follows from the claim that πv does not divides P(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' For ordinary v, because E(p) p ( ¯Kv) ∩ A(p) o ( ¯Kv) = {0}, we know that 0 is the only root of P(t) in ¯Kv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' This implies that the distinguished polynomial P(t) = t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' For supersin- gular v, the group E(p) p ( ¯Kv) ∩ A(p) o ( ¯Kv) ≃ Z/pZ, so deg P(t) = p, furthermore, since P(ti) = 0 and P′(ti) ̸= 0, we have P(ti) = 0 and P ′(ti) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' To prove the claim, we first consider the case where v is a place of good reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' The formal group law associated to ¯A is given by ¯f(X, Y ) := f(X, Y ) (mod (πv)), which has height 1 or 2, so ¯ P := P (mod (πv)) is non-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' This proves the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' For a multiplicative place v, we prove the claim by showing that the Verschiebung gives rise to an isomorphism F (p)(mv) −→ F(mv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' If v is a split-multiplicative place and ˜Q is the local Tate period of A, then ˜Qp is the local Tate period of A(p) and the Verschiebung is given by K∗ v/ ˜QpZ −→ K∗ v/ ˜QZ, induced from the identity map on K∗ v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' This implies F (p)(mv) −→ F(mv) is an isomorphism, and hence the claim follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' If v is non-split multiplicative, then A/Kv is the twist of a split multiplicative elliptic curve B/Kv by the unramified quadratic extension Lw/Kv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Write Zv for the kernel of the norm map NLw/Kv : O∗ w −→ O∗ v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Then F (p)(mv) −→ F(mv) is given by the identity map Zv −→ Zv, so it is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' □ 6 KI-SENG TAN 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Ordinary places.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' The proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='1 shows that if v is a split multi- plicative place, then V∗ is given by K∗ v/ ˜QpZ −→ K∗ v/ ˜QZ, and hence surjective, so by (6), ker(jv) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Let v be a multiplicative place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Then ker(jv) = 0, unless v ∈ ð′, in which case ker(jv) is of order 2 = p, consisting of ξ ∈ H1(Kv, E(p) p ) with Kv,ξ/Kv unramified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Note that if p = 2, then k = K, so Kv,ξ is defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Suppose v is non-split multiplicative and let B/Kv and Lw/Kv be as in the proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Because A/Lw is split-multiplicative, we have the injection H1(Lw, E(p) p ) −→ H1(Lw, A(p)), and hence ker(jv) = ker(H1(Lw/Kv, E(p) p (Lw)) −→ H1(Lw/Kv, A(p)(Lw))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' For p ̸= 2, we have H1(Lw/Kv, E(p) p (Lw)) = 0, so ker(jv) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Denoting G = Gal(Lw/Kv), we have the commutative diagram H1(Lw/Kv, E(p) p (Lw)) � ≃ � H1(Lw/Kv, A(p)(Lw)) ≃ � Hom(G, Z/2Z) � B(p)(Kv)/ NLw/Kv(B(p)(Lw)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' The non-trivial element of Hom(G, Z/2Z), sending the generator of G to the point of B(p)(Kv) obtained by the Tate local period Qv of B/Kv, corresponds to an element of ker(jv) if and only if Qv ∈ NLw/Kv(L∗ w), or equivalently ordv Qv is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' □ Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Suppose v is a good ordinary place and w is a place of k sitting over v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Then ker(jw) is of order p, consisting of ξ ∈ H1(kw, E(p) p ) with kw,ξ/kw unramified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' If Kv ̸= kw, then ker(jv) is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' In view of the diagram (8), Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='1 says A(p) o (kw) ∼ V � Ao(kw) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' We have to determine the cokernel of the induced ¯V : ¯A(p)(Fw) −→ ¯A(Fw).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' The Frobe- nius ¯F identifies ¯A(p)(Fw) with ¯A(Fw) and under this, ¯V is identified with the mul- tiplication by p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' The cokernel in question is isomorphic to ¯A(Fw)/p ¯A(Fw) ≃ ¯Ap(Fw) = ¯E(p) p (Fw).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' The snake lemma applied to the diagram 0 � A(p) o (kw) � V ≃ � A(p)(kw) � V � ¯A(p)(Fw) � ¯V � 0 0 � Ao(kw) � A(kw) � ¯A(Fw) � 0 implies that the reduction map E(p) p (kw) −→ ¯E(p) p (Fw) is an isomorphism, so ker(jw) is of order p, and by [Mil06, §I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='8], it is formed by all unramified ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' If Kv ̸= kw, then ¯E(p) p (Fv) = E(p) p (Kv) = 0, and a similar argument shows ker(jv) is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' THE FROBENIUS TWISTS OF ELLIPTIC CURVES OVER GLOBAL FUNCTION FIELDS 7 □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Supersingular places.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Suppose v is supersingular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Choose a nonzero t1 ∈ mw (in some Fw) with ι(p)(t1) ∈ E(p) p (Fw).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Let [u] denote the multiplication by u on A(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Because tu := ι(p)−1◦[u]◦ι(p)(t1) = ut1+higher terms, if p ∤ u, then ordv tu = ordv t1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Denote nv := � i∈F∗p ordv ti = (p − 1) ordv(tu), for (u, p) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' (9) Write P(t) = tp + zp−1tp−1 + · + z1t, with ordv z1 = nv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' (10) For s, t ∈ m write s ⊞ t for F (p)(s, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Then ι(p)(s ⊞ t) = ι(p)(s) + ι(p)(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Diagram (8) shows that for a given b ∈ mv, if a0 ∈ mw is a root of P(t) − b = 0, then all other roots equal au := a0 ⊞ tu = F (p)(a0, tu), u = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=', p − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Let Q(t) be the distinguished polynomial associated to P(t) − b, whose roots are also a0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=', ap−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Since Q(0) = −b · ξ, for some ξ ∈ O∗ v, p−1 � u=0 ordv au = ordv b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' (11) Since F (p)(X, 0) = F (p)(0, X) = X, we can write F (p)(X, Y ) = X + Y + XY · g(X, Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' It follows that au = a0 + tu + higher terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Hence ordv(au − au′) = nv p − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' (12) Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' For every b ∈ mv with ordv b > pnv p−1, there exists an element a ∈ mv, with ordv a > nv p−1, such that P(a) = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Conversely, for a ∈ mv, with ordv a > nv p−1, the element b = P(a) ∈ mv has ordv b = ordv a + nv > pnv p−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' If b ∈ mv and a0 is a solution to P(t) = b, with ordv(a0) > nv p−1, then by (12), other solutions au have ordv au = nv p−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Therefore, if a ∈ mv, with ordv a > nv p−1, and b = P(a), then by (11), ordv b = ordv a + nv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Conversely, if b ∈ mv, ordv b > pnv p−1, by (11), there is a solution a to P(t) = b, such that ordv a > nv p−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Comparing the valuations, we deduce that a is the only Galois conjugate of itself, whence a ∈ mv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' □ Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' If v is a supersingular place of A/K, then the cokernel of A(p)(Kv) V∗ � A(Kv) is of order pǫv · qnv v , where if kv = Kv, ǫv = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' otherwise, ǫ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Since ¯A(Fv) has order prime to p, by Diagram (8) we need to show the cokernel of F (p)(mv) P � F(mv) has the desired order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' 8 KI-SENG TAN Let β = [ 1 p−1nv]+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='1 implies that P sends F (p)(mβ) onto F(mβ+nv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Therefore, it is sufficient to check the co-kernel of the induced homomorphism ¯ P : F (p)(m)/F (p)(mβ) −→ F(m)/F(mβ+nv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Since the kernel of ¯ P is of order pǫv, the proof is completed by counting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' □ Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='2 says | ker(jv)| = pǫv · qnv v .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' (13) Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Let v be a place of K and w a place of k sitting over v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' The group ker(jw) consists of all ξ with kw,ξ/kw having conductor at most pnw p−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' An element ξ ∈ ker(jw) can be written as ∂x for some x ∈ A(kw).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Since ¯A(Fw) has order prime to p, we may choose x = ι(b) ∈ Ao(kw), for some b ∈ mw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Let a0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=', ap−1 be solutions to P(t) = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Then all au are integral over Ow and kw,ξ = kw(a0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' It follows from (12) that if Disc is the discriminant of kw,ξ/kw, then ordw(Disc) ≤ p · nw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' This implies the conductor of kw,ξ/kw is at most pnw p−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Classes ξ ∈ H1(kw, E(p) p ) with kw,ξ/kw unramified are in ker(jw) (see [Mil06, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' They form a subgroup of order p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' By the local class field theory, ramified cyclic extensions of kw of degree p and conductor at most m are characterized by the group Dm/Dp m, Dm := O∗ w/1 + πm w Ow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' In our case m = pnw p−1 is an integer divisible by p (by (9), because each tu ∈ kw).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Since Ow = Fw[[πw]], the map D m p −→ Dp m, x �→ xp, is an bijection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Hence |Dm/Dp m| = qm−1 w (qw − 1)/q m p −1 w (qw − 1) = qnw w .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' In view of (13), the proof is completed by counting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' □ The lemma actually says that by (7), ker(jw) = Hom(k∗ w/(1 + π pnw p−1 w Ow) · (k∗ w)p, Z/pZ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' (14) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Global fields Let X /Fq be the complete smooth curve having K as its function field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Let Xg, Xgo denote the open sets consisting of places where A has good reduction, good ordinary reduction respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Poitou-Tate duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' We first recall the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Let f : B/K −→ B′/K be a given isogeny of elliptic curves having good reductions at all v ∈ Xg and let f : B −→ B′ be the homomorphism extending f to N´eron models over Xg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Then N := ker [f] is a finite flat group scheme over Xg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Furthermore, if ˆ N denotes the kernel of the homomorphism ˆf : B′ −→ B extending the dual isogeny ˆf : B′ −→ B, then N and ˆ N are Cartier dual to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Note that the existence and the uniqueness of f and ˆf are due to the N´eron mapping property, see [BLR90, §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' THE FROBENIUS TWISTS OF ELLIPTIC CURVES OVER GLOBAL FUNCTION FIELDS 9 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Since B/Xg is an abelian scheme, the morphism f is proper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' It follows that N /Xg is proper and quasi-finite, whence finite [EGA IV, part 4, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='4] and flat [Mil06, III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' The exact sequence 0 � N � B f � B′ � 0 together with the isomorphism (see [SGA 7I, VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='1], [BBM82] or [Mil06, III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='14]) B/Xg ≃ Ext 1 Xg(B, Gm) induce the exact sequence · · � HomXg(B, Gm) � HomXg(N , Gm) � B′ ˆf � B � · · ·.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Here we apply the commutative diagram Ext 1 Xg(B′, Gm) f ∗ � Ext 1 Xg(B, Gm) B′ ˆf � B that extends the already known diagram on the generic fibre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Then we check the equality HomXg(B, Gm) = 0 fibre-wise by using the fact that over a field every homomorphism from an abelian variety to Gm is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' □ Let notation be as in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='1 and let N denote the generic fibre of N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' For v ∈ Xg, we have (see [Mil06, §III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='7]) H1(Ov, N )� � � H1(Kv, N) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' (15) Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Let notation be as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' For v ∈ Xg, we have H1(Ov, N ) = ker(H1(Kv, N) −→ H1(Kv, B)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' The lemma follows from the fact that H1(Ov, B) = 0 (see [Mil06, §III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='1]) and the commutative diagram of exact sequences B(Ov) � B′(Ov) � H1(Ov, N ) � � � � H1(Ov, B) � B(Kv) � B′(Kv) � H1(Kv, N) � H1(Kv, B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' □ Let U ⊂ X be an open subscheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Define S(N/U) := ker(H1(K, N) −→ � v∈U H1(Kv, B)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Denote Sel(N/K) := S(N/X ), it is the kernel of S(N/U) � � v̸∈U H1(Kv, B) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' 10 KI-SENG TAN Let Q(N/U) denote the cokernel of the localization map S(N/U) LN/K � � v̸∈U H1(Kv, N) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' If U ⊂ Xg, then H1(U, N ) = S(N/U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Let V ⊂ U be an open affine subscheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' By the localization sequence [Mil06, III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='3(c)] and the computation at the beginning of [Mil06, III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='7], we have the exact sequence 0 � H1(U, N ) � H1(V, N ) � � v∈U\\V H1(Kv, N)/ H1(Ov, N ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' [Gon09, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='2] says the natural map H1(V, N ) −→ H1(K, N) is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' The exact sequence implies H1(U, N ) ⊂ S(N/U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' By [Gon09, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='3], an element in S(N/U) can be obtained from H1(V, N ) for some V ⊂ U, and the exact sequence implies it is in H1(U, N ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' □ For U ̸= X , apply the local duality [Mil06, §III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='10] and consider the composition S(Cp/U)� � LCp/U � � v̸∈U H1(Kv, Cp) ∼ � � v̸∈U H1(Kv, E(p) p )∨, (16) where the injectivity of LCp/U is due to [GoT12, Main Theorem].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' If U ⊂ Xg and U ̸= X , then under (16), the group S(Cp/U) is the Pontryagin dual of Q(E(p) p /U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Extend F and V to F : A −→ A(p) and V : A(p) −→ A over Xg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Denote Cp = ker(F) and E(p) p = ker(V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' They are Cartier dual to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='3, we identify H1(U, Cp) with S(Cp/U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Then the lemma follows from Poitou-Tate duality [Ces15, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='2)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' □ In view of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='2, the following lemme generalizes the fact that for any place v ∈ Xg, the local duality identifies H1(Ov, Cp) ⊂ H1(Kv, Cp) with the annihilator of H1(Ow, E(p) p ) ⊂ H1(Kv, E(p) p ) [Mil06, §III, Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' At each place v of K, under the duality H1(Kv, Cp) = H1(Kv, E(p) p )∨ [Mil06, §III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='10], the kernel of j′ v : H1(Kv, Cp) −→ H1(Kv, A), as a subgroup of H1(Kv, Cp), is exactly the annihilator of ker(jv) ⊂ H1(Kv, E(p) p ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' We abbreviate the above as ker(j′ v) = ker(jv)⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' (17) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Let ∂ denote the connecting homomorphism in the long exact sequence · · � A(Kv) F � A(p)(Kv) ∂ � H1(Kv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Cp) j′ v � H1(Kv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' A) � · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' (18) THE FROBENIUS TWISTS OF ELLIPTIC CURVES OVER GLOBAL FUNCTION FIELDS 11 that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' since p = F ◦ V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' gives rise to the homomorphism ¯∂ in the diagram A(p)(Kv)/p · A(p)(Kv) ¯∂ � � � i � H1(Kv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Cp) H1(Kv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' A(p) p ) V∗ � H1(Kv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Cp),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' (19) where the bottom left-arrow is due to (5) and the left down-arrow i is induced from the long exact sequence A(p) p (Kv) � A(p)(Kv) [p] � A(p)(Kv) � H1(Kv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' A(p) p ) � · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' We claim that the diagram (19) is commutative, so that the commutative diagram H1(Kv, A(p) p ) × H1(Kv, A(p) p ) V∗ � (−,−)v � Q/Z H1(Kv, Cp) × H1(Kv, E(p) p ) jv � (−,−)v � Q/Z, where the left-arrows are local pairings, yields the commutative diagram (see [Mil06, §III, Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='8]) A(p)(Kv) × H1(Kv, A(p)) ∂ � (−,−)A(p)/Kv � Q/Z H1(Kv, Cp) × H1(Kv, E(p) p ) jv � (−,−)v � Q/Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' This shows that ker(jv)⊥ = Im(∂v) = ker(j′ v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' To prove the claim, we use ˇCech cocycles (see [Mil80, §III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Let x ∈ A(p)(Kv) and denote ¯x its image in A(p)(Kv)/p · A(p)(Kv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Let y ∈ A(p)(K′ w) = Hom(Spec(K′ w), A(p)) be a p-division point of x over a finite extension K′ w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Let prl, l = 1, 2, be the projection Spec(K′ w) ×Spec(Kv) Spec(K′ w) −→ Spec(K′ w) to the l’th factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Then ξ := y ◦ pr1 − y ◦ pr2 is a 1-cocycle representing the class i(¯x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Let z = V(y) ∈ Hom(Spec(K′ w), A) so that F(z) = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Then V ◦ ξ is a 1-cocycle representing both ¯∂(¯x) and V∗(i(¯x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' The conductor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Recall that k = K(E(p) p ( ¯Ks)) so that E(p) p ( ¯Ks) = E(p) p (k), on which the action of the Galois group Φ := Gal(k/K) is given by an injective character c : Φ −→ F∗ p such that x g = c(g) · x, for g ∈ Φ, x ∈ E(p) p (k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Since the order of Φ is prime to p, the Hochschild-Serre spectral sequence (see [Mil80, §II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='21(a)]) yields H1(K, E(p) p ) ∼ � H1(k, E(p) p )Φ Hom(Gk, E(p) p (k))Φ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' (20) 12 KI-SENG TAN For w ∈ Sss(k), let ι(p)(t1) be a non-zero element of E(p) p (kw) as in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Put Mw := \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 (1 + tp 1Ow) · (O∗ w)p, if w ∈ Sss(k);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' k∗ w, if w ∈ ð(k);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' O∗ w, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Let A∗ k denote the group of ideles of k and W the p-completion of k∗\\A∗ k/ � w Mw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' We have Sel(E(p) p /k) = Hom(W , E(p) p (k)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' By the global class field theory, Hom(W , E(p) p (k)) ⊂ Hom(Gk, E(p) p (k)) con- sists of elements which are locally trivial at w ∈ ð(k), having conductors not greater than ordw(tp 1) at supersingular places w, unramified at others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' The lemma follows from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='2 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' □ Every pro-p Φ-module Y can be decomposed as Y = � χ∈ˆΦ Y χ, where for each χ, Y χ denote the χ-eigenspace {y ∈ Y | y g = χ(g) · y} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' By (20) and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='1, Sel(E(p) p /K) = Hom(W , E(p) p (k))Φ = Hom(W c, Z/pZ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' (21) For v ∈ Sss, put Wv := � w|v k∗ w/((k∗ w)p · Mw), and Uv := � w|v O∗ w/Mw regarded as a subgroup of Wv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' If v ∈ Sss, then |Uc v| = qnv v .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Again, by the Hochschild-Serre spectral sequence H1(Kv, E(p) p ) ∼ � (� w|v H1(kw, E(p) p ))Φ (� w|v Hom(Gkw, E(p) p (k)))Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' (22) Therefore, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='3 implies ker(jv) ≃ Hom(W c v, Z/pZ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Then the lemma follows from (13), because if kw ̸= Kv, then W c v = Uc v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' if kw = Kv, then we have the splitting exact sequence 0 −→ Uc v −→ W c v −→ Z/pZ −→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' An idelic computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' In this subsection only, we consider a general situation in which for w ∈ Sss(k), the Mw in the previous subsection is replaced by Mw := (1 + πav w Ow) · (O∗ w)p, where v is a place of K sitting below w and av is a chosen integer depending only on v, and we keep Mw unchanged for other w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Denote a := (av)v∈Sss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Then let Wa be the p-completion of k∗\\A∗ k/ � w Mw so that W = Wo, where o := (p · ordw tv)v∈Sss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Put Ua := � w∈Sss O∗ w/Mw, Wa := � w∈Sss k∗ w/((k∗ w)p · Mw).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Assume that |Uc a| = � v∈Sss qρv v .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' (23) Let ¯Uc a denote the image of the natural map ςc a : Uc a −→ W c a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' THE FROBENIUS TWISTS OF ELLIPTIC CURVES OVER GLOBAL FUNCTION FIELDS 13 Let V be the group of ð(k)-units of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Then ker(ςc) equals the image of ̺c a : (V/pV )c −→ Uc a induced by the localization map V −→ � w∈Sss(k) O∗ w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' The torsion part of V is finite of order prime to p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' The map V −→ � v∈ð Rv, where Rv := � w|v k∗ w/O∗ w, is injective on the free part of V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' The module Rv is fixed by the decomposition subgroup Φv and is the regular representation of Φ/Φv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' If k = K, the Zp-rank of (Zp ⊗Z V )c is max{|ð0| − 1, 0};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' otherwise it is |ð0|, so | ker(ςc a)| = | Im ̺c a| ≤ p|ð0|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' (24) If ð = ∅, letג denote the p-completion of the divisor class group of k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' otherwise, let ג be the ð(k)-class group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Then ℏ is the p-rank ofג.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' The exact sequence 0 � ¯Uc a � W c a �גc � 0 yields 0 � ¯Uc a � W c a [p] κc �גc[p] (25) Put τ := � logp | Im(κc)| + 1, if ð = ∅ and k = K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' logp | Im(κc)|, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' (26) Via (20), we identify H1(K, E(p) p ) with Hom(Gk, E(p) p (k))Φ, so that the conductor of its element at each place v is defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Let Xo ⊂ X be the complement of Sss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Define Sela(E(p) p /K) to be the subgroup of S(E(p) p /Xo) consisting of elements having conductors not greater than av at each v ∈ Sss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Assuming (23), we have logp | Sela(E(p) p /K)| = � v∈Sss ρv · deg v + ˜ε1 − ˜ε2, where ˜ε1 = τ ≤ ℏ + 1, ˜ε2 = logp | ker(ςc a)| ≤ |ð0| ≤ |Sb|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Similar to (21), Sela(E(p) p /K) = Hom(W c a , Z/pZ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' We shall write Wa addi- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' If ð ̸= ∅ or k ̸= K, then W c a is finite with |W c a /pW c a | = |W c a [p]|;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' otherwise, W c a is finitely generated over Zp of rank 1, hence |W c a /pW c a | = p · |W c a [p]|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' The lemma follows from (23), (24), (25) and (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' □ Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' We have logp | Sel(E(p) p /K)| = (p − 1) deg ∆A/K 12 logp q + ˜ε1 − ˜ε2, where ˜ε1 = τ ≤ ℏ + 1, ˜ε2 = logp | ker(ςc o)| ≤ |ð0| ≤ |Sb|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' In view of (21), Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='2, and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='4, we need to show that � v∈Sss nv · deg v = (p − 1) deg ∆A/K 12 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' (27) 14 KI-SENG TAN Let ω be an invariant differential of A/K and for each place v let ω0,v and ω(p) 0,v be respectively local N´eron differentials of A and A(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' By [Sil86, §IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='3], ordv(V∗ω0,v/ω(p) 0,v) = ordv dP(t) dt |t=0, which together with Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='1 and (10) yield � v∈Sss nv · deg v = � all v ordv(V∗ω0,v/ω(p) 0,v) · deg v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' (28) Now the formula (8) in [Tan95] implies deg ∆A/K 12 = � all v ordv( ω ω0,v ) · deg v = � all v ordv(V ∗( ω ω0,v )) · deg v, and p · deg ∆A/K 12 = deg ∆A(p)/K 12 = � all v ordv(V ∗ω ω(p) 0,v ) · deg v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' These and (28) lead to the desired equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Computing S(Cp/U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Next, we investigate S(Cp/U) for U = Xgo, Xg, or X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Let Sngo be the complement of Xgo in X , S′ ngo := U ∩ Sngo, S′ ss := U ∩ Sss, and † the complement of U in X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Write ‡ for † ∪ ð.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' We first treat the case in which Xgo ̸= X , or equivalently, A/K is not isotrivial1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Put ¯W := � w∈Sngo(k) k∗ w/(k∗ w)p and ¯ W := k∗\\A∗ k/( � w∈Xgo(k) O∗ w × � w∈Sngo(k) (k∗)p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' If ℓc and ℓc−1 denote the c and c−1 eigenspaces of the regular representation of Φ on Fp[Φ], then E(p) p (k) = ℓc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Hence � w∈Sngo(k) H1(kw, E(p) p ) = Hom( ¯W, ℓc) = Hom( ¯W ⊗Fp ℓc−1, Z/pZ) = ( ¯W ⊗Fp ℓc−1)∨, and S(E(p) p /Xgo × Spec k) = Hom( ¯ W , ℓc) = (( ¯ W c/p ¯ W ) ⊗Fp ℓc−1)∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' In view of (16) and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='4, S(Cp/Xgo × Spec k) = ker( ¯W −→ ¯ W /p ¯ W ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Therefore, by the Hochschild-Serre spectral sequence again, S(Cp/Xgo) = S(Cp/Xgo × Spec k)Φ = ker( ¯W c −→ ¯ W c/p ¯ W c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' For each w, put Mw := Mw/Mw ∩ (k∗ w)p = Mw · (k∗ w)p/(k∗ w)p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' 1Recall that A/K is assumed to be ordinary, having semi-stable reduction everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' THE FROBENIUS TWISTS OF ELLIPTIC CURVES OVER GLOBAL FUNCTION FIELDS 15 Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='5 gives rise to the left block of the following commutative diagram H1(kw, Cp) ∼ � H1(kw, E(p) p )∨ ∼ � Hom(k∗ w/(k∗ w)p, Z/pZ)∨ ∼ � k∗ w/(k∗ w)p ker(j′ w) ∼ � �� � ker(jw)⊥ ∼ � �� � Hom(k∗ w/Mw · (k∗ w)p, Z/pZ)⊥ ∼ � �� � M w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' �� � The middle block is obtained by taking the dual of the commutative diagram H1(kw, E(p) p ) ∼ � Hom(k∗ w/(k∗ w)p, Zp/pZp) ker(jw) �� � ∼ � Hom(k∗ w/(k∗ w)p · Mw, Zp/pZp), �� � (29) which is due to Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='3 together with the local class field theory (see (7) and (14)) while the right block is a direct consequence of duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' shows that if ¯ M = � w∈†(k) k∗ w/(k∗ w)p × � w∈S′ngo(k) M w, then S(Cp/U) = ker( ¯ M c −→ ¯ W c/p ¯ W c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' (30) To proceed further, we introduce ˜ M = � w∈†(k) k∗ w/(O∗ p)p × � w∈S′ngo(k) Mw · (O∗ p)p/(O∗ p)p and ˜ W ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' = k∗\\A∗ k/( � w∈Xgo(k) O∗ w × � w∈Sngo(k) (O∗ w)p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Then ¯ M = ˜ M /p ˜ M , and since the kernel of ˜ W � � ¯ W is inside p ˜ W , we also have ¯ W /p ¯ W = ˜ W /p ˜ W .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Hence (30) implies S(Cp/U) = ker( ˜ M c/p ˜ M c −→ ˜ W c/p ˜ W c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' (31) Let V and M denote the kernel and image of the natural map ˜ς : ˜ M −→ ˜ W .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Let ˜ξ ∈ V and let ξ be a lift of ˜ξ to � w∈†(k) k∗ w × � w∈S′ngo(k) Mw · (O∗ p)p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Then there are α ∈ k∗ and θ ∈ � w∈Xgo(k) O∗ w × � w∈Sngo(k)(O∗ w)p such that ξ = α · θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' (32) Let V‡ denote the group of ‡(k)-units of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' The equality (32) implies that α is in the subgroup V ′ ‡ ⊂ V‡ consisting of those elements which are inside Mw · (O∗ w)p, for all w ∈ S′ ss(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Suppose there is another expression ξ = α′ · θ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Then α′α−1 actually belongs to the group F∗ k of global units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Since Sngo = † ⊔ S′ ngo, the correspondence ˜ξ ↔ α (mod F∗ k) gives rise to an isomorphism V ≃ V ′ ‡/F∗ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' The exact sequence 0 � V � ˜ M � M � 0 induces the exact sequence ˜ M [p] � M [p] ∂ � V /pV � ˜ M /p ˜ M � M /pM � 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' 16 KI-SENG TAN For an h ∈ M [p], let ˜η be one of its preimage in ˜ M and let η be a lift of ˜η to � w∈†(k) k∗ w × � w∈S′ngo(k) Mw · (O∗ p)p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Put ξ = ηp, so that (32) holds for some α and θ, and ∂(h) is represented by α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' In this case, since ξ is a p’th power, α ∈ (k∗ w)p at all w ∈ Sngo(k), which is non-empty, so by the local Leopoldt’s conjecture [Kis93], α = βp, for some β ∈ V‡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Since pV‡ ⊂ V ′ ‡, we conclude that ker( ˜ M c/p ˜ M c −→ M c/pM c) = (V ′ ‡/pV‡)c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' (33) Denote ℶ := ˜ W /M = k∗\\A∗ k/( � w∈Xgo(k) O∗ w × � w∈†(k) k∗ w × � w∈S′ngo(k) Mw · (O∗ p)p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Then we have the exact sequence ˜ W [p] � ℶ[p] � M /pM � ˜ W /p ˜ W .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' (34) If y ∈ ˜ W [p] is represented by an idele ζ = (ζw)w, then there are α ∈ k∗ and θ ∈ � w∈Xgo(k) O∗ w · � w∈Sngo(k)(O∗ w)p such that ζp = α · θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Then at w ∈ Sngo(k), α ∈ (k∗ w)p, so by the local Leopoldt’s conjecture again, α = βp, β ∈ k∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Since y is also represented by ζ · β−1, we have the isomorphism � w∈Sngo(k) O∗ w/(O∗ w)p ∼ � ˜ W [p].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Since theג equals the cokernel of � w∈Sngo(k) O∗ w/(O∗ w)p −→ ℶ, the above isomor- phism and (34) together imply ker(M c/pM c −→ ˜ W c/p ˜ W c) = Im(ℶ[p]c −→ג[p]c) ⊂ג]p]c (35) We have ℏ =ג[p] and logp |(V ′ ‡/p · V‡)c| ≤ logp |Zp ⊗Z V c ‡ /p · Zp ⊗Z V c ‡ | = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 0, if ‡ = ∅;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' |‡0 − 1|, if ‡ ̸= ∅ and k = K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' |‡0|, otherwise, so by (33) and (35), logp |S(Cp/U)| ≤ logp |(V ′ ‡/p · V‡)c| + logp |גc[p]| ≤ |‡0| + ℏ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' (36) Suppose A/K is isotrivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Then Sb = Sss = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' If k ̸= K, choose a place v not spitting completely over k/K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' otherwise, choose any v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Set ♮ := {v}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Let U be the complement of ♮.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Put ¯W := � w∈♮(k) k∗ w/(k∗)p and denote ¯ W := k∗\\A∗ k/( � w∈U(k) O∗ w · � w∈♮(k) (k∗)p), THE FROBENIUS TWISTS OF ELLIPTIC CURVES OVER GLOBAL FUNCTION FIELDS 17 so that H1(Kv, E(p) p ) = Hom( ¯W c, Z/pZ) and S(E(p) p /U) = Hom( ¯ W c/p ¯ W c, Z/pZ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Thus, in view of (16) and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='4, if ¯ M := � w∈♮(k) O∗ w/(O∗ w)p, then S(Cp/X ) = ker( ¯ M c −→ ¯ W c/p ¯ W c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' The kernel of ¯ M −→ ¯ W is in the image of V♮, the ♮(k)-units of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Because of our choice, (Zp ⊗Z V♮)c = 0, and hence ¯ M c −→ ¯ W c is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Similar to the previous case, the local Leopoldt’s conjecture at w ∈ ♮(k) implies � w∈♮(k) k∗ w/(k∗ w)p � � ¯ W [p].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Since (� w∈♮(k) k∗ w/(k∗ w)p)c = 0, we have ¯ W [p] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Now, ( ¯ W / ¯ M )[p] =ג[p], whereג is the divisor class group of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' It follows from the exact sequence ¯ W c[p] �גc[p] � ¯ M c � ¯ W c/p ¯ W c that |S(Cp/X )| = |גc[p]|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Thus, the following proposition is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' For U = Xgo, Xg, or X , logp |S(Cp/U)| ≤ |‡0| + ℏ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' In particular, logp | Sel(Cp/K)| ≤ |ð0| + ℏ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Proof of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' The proposition is a consequence of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='5 and Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='6, because we have the commutative diagram of exact sequences 0 � Sel(E(p) p /K) � � � � Selp(A(p)/K) � � � � Sel(Cp/K) � � � Cp(K) � H1(K, E(p) p ) � � H1(K, A(p) p ) V∗ � � H1(K, Cp) � � v H1(Kv, A(p)) � v H1(Kv, A(p)) V∗ � � v H1(Kv, A), where the middle long exact sequence is induced from (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' The Cassels-Tate duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' The Cassels-Tate pairing induces the perfect pair- ing (see [Mil06, III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='5]) ⟨−, −⟩A/K : X(A/K) × X(A/K) −→ Qp/Zp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' If ϕ : A −→ B is an isogeny with dual isogeny ϕt, then the commutative diagram X(A/K) × X(A/K) ϕ♮ � ⟨−,−⟩A/K � Qp/Zp X(B/K) × X(B/K) ϕt ♮ � ⟨−,−⟩B/K � Qp/Zp 18 KI-SENG TAN yields the duality between X(A/K)/ ker(ϕ♮) and X(B/K)/ ker(ϕt ♮).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' In particular |(X(A/K)/ ker(ϕ♮))[pν]| = |(X(B/K)/ ker(ϕt ♮))[pν]|, (37) since |G/pνG| = |G[pν]| for a finite abelian group G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Consider the exact sequence 0 � ker(F♮) � X(A/K)[pν] � (X(A/K)/ ker(F♮))[pν] pν �✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐ ker(F♮) ∩ pνX(A/K) � 0, where the morphism ·pν is induced from the multiplication by pν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' This implies logp |X(A/K)[pν]| = logp |(X(A/K)/ ker(F♮))[pν]| + δ1, (38) with logp | Sel(Cp/K)| ≥ logp | ker(F♮)| ≥ δ1 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Also, consider the exact sequence 0 � ker(V♮) � (X(A(p)/K)/ ker(V♮))[pν] �❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣ (X(A(p)/K)/X(A(p)/K)[p])[pν] pν � T � 0, where T = (X(A(p)/K)[p]/ ker(V♮)) ∩ pν(X(A(p)/K)/ ker(V♮)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' This together with (37) and (38) lead to logp |X(A/K)[pν]| = logp |(X(A(p)/K)/X(A(p)/K)[p])[pν]| + δ1 + δ2, (39) with logp | Sel(Cp/K)| ≥ logp | ker(F♮)| ≥ logp |V♮(X(A(p)/K)[p])| ≥ δ2 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Now (X(A(p)/K)/X(A(p)/K)[p])[pν] = X(A(p)/K)[pν+1]/X(A(p)/K)[p].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Recall that r denote the co-rank of Seldiv(A/K) ≃ Seldiv(A(p)/K), so logp |X(A/K)[pν]| = logp | Selpν(A/K)| − rν, and a similar formula for A(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Therefore, logp | Selpν(A/K)| = logp | Selpν+1(A(p)/K)| − logp | Selp(A(p)/K)| + δ1 + δ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Then we apply Proposition 1 and Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' □ THE FROBENIUS TWISTS OF ELLIPTIC CURVES OVER GLOBAL FUNCTION FIELDS 19 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' The method of specialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' In this section only, we assume that A/K an ordinary abelian variety defined over a global field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' As before, let L/K be a Zd p- extension unramified outside a finite set of places of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Let Γ be the Galois group, ΛΓ the Iwasawa algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Endow lµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='. p∞ with the discrete topology and write ˆΓ for the group of all continuous characters Γ −→ lµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' p∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Let O be the ring of integers of Qp(lµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' p∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Every character χ ∈ ˆΓ extends to a unique continuous Zp-algebra homomorphism χ : ΛΓ −→ O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' For g ∈ ΛΓ, not divisible by p, by Monsky [Mon81, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='3], the set ∇g := {χ ∈ ˆΓ | χ(g) ∈ p · O} is either ∅ or is contained in T1∪· · ·∪Tn, where for each i, there are ζi,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=', ζi,νi ∈ lµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' p∞, νi > 0, and σi,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=', σi,νi ∈ Γ, extendable to a Zp-basis of Γ, such that Ti := {χ ∈ ˆΓ | χ(σi,j) = ζi,j, for j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=', νi}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' For an element ψ ∈ Γ, extendable to a Zp-basis of Γ, set Tψ := {χ ∈ ˆΓ | χ(ψ) = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Then Tψ ⊂ Ti can not hold, unless νi = 1, ζi,1 = 1, and σi,1, ψ topologically generate the same closed subgroup of Γ, in such case, we actually have Tψ = Ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Therefore, there are finitely many rank one Zp-submodules of Γ such that if ψ is chosen away from them, then Tψ ⊊ ∇g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' (40) For a Ze p-subextension L′/K of L/K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Denote Γ′ = Gal(L′/K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' The quotient map Γ −→ Γ′ extends uniquely to a continuous Zp-algebra homomorphism pL/L′ : ΛΓ −→ ΛΓ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Write Ψ := Gal(L/L′) so that Γ′ = Γ/Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Put IΨ := ker(pL/L′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Let b be a finite set of places of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' For a subextension M of L/K, let bM ⊂ b denote the subset consisting of places splitting completely over M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Suppose d ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Let Zℓ, ℓ = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='., s, be finitely generated torsion ΛΓ-modules, θ an element in ΛΓ, not divisible by p, and b a finite set of places of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' There is an element ψ ∈ Γ extendable to a Zp-basis such that if L′ is the fixed field of ψ, then bL′ = bL, pL/L′(θ) is not divisible by p, and each ΛΓ′-module Z′ ℓ := Zℓ/IΨZℓ is torsion, having the same elementary µ-invariants as those of Zℓ over ΛΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' For each v ∈ b with non-trivial decomposition subgroup Γv, if ψ ̸∈ Γv or Γv is of Zp-rank greater than one, then v does not split completely over L′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' To have bL′ = bL, we only need to choose ψ away from all rank one Γv, v ∈ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Similar to (2), we have 0 � �mℓ i=1 ΛΓ/(pαℓ,i) ⊕ �nℓ j=1 ΛΓ/(η βℓ,j ℓ,j ) � Zℓ � Nℓ � 0, (41) 20 KI-SENG TAN where Nℓ is pseudo-null and every ηℓ,j is not divisible by p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Choose for each ℓ, an annihilator hℓ ∈ ΛΓ of Nℓ, not divisible by p, and put g := θ · s� ℓ=1 hℓ · ηℓ,1 · · · · · ηℓ,nℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' We also choose ψ satisfying (40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Since ˆΓ′ = Tψ, there exists χ ∈ ˆΓ′ such that χ(pL/L′(g)) ̸∈ p · O, so pL/L′(g) ̸∈ p · ΛΓ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Because pα · gβ · Zℓ = 0, for some α, β ∈ Z, we have pα · pL/L′(gβ) · Z′ ℓ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Hence Z′ ℓ is torsion over ΛΓ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' To compare the elementary µ-invariants of Zℓ and Z′ ℓ, we apply the maps of multiplication by ψ−1 to (41) and use the snake lemma to obtain the exact sequence Nℓ[ψ − 1] −→ mℓ � i=1 ΛΓ′/(pαℓ,i) ⊕ nℓ � j=1 ΛΓ′/(pL/L′(ηℓ,j)βℓ,j) −→ Z′ ℓ −→ Nℓ/IΨNℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' (42) Because Nℓ[ψ − 1] is annihilated by pL/L′(g) which is relatively prime to p, the second arrow in the exact sequence is injective on �mℓ i=1 ΛΓ′/(pαℓ,i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' We complete the proof by comparing the pth power factors of the characteristic ideals of items in the sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' □ In Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='1, the field L′ is a Zd−1 p extension of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' By repeatedly applying the lemma, we obtain sequences L ⊃ L′ ⊃ · · · ⊃ L(d−1), (43) and, for each ℓ, Zℓ −→ Z′ ℓ −→ · · · −→ Z(d−1) ℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' (44) Put Ψ0 = Ψ, Ψi = Gal(L/L(i+1)), and Γ(i+1) = Gal(L(i+1)/K) = Γ/Ψi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Then L(d−1) is a Zp-extension of K with bL(d−1) = bL, pL/L(d−1)(θ) not divisible by p, and for each ℓ, the ΛΓ(d−1)-module Z(d−1) ℓ = Zℓ/IΨd−2Zℓ has the same elementary µ-invariants as those of Zℓ over ΛΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' These elementary µ-invariants pαℓ,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=', pαℓ,mℓ can be recovered by using the counting formula below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' For each ν, define αℓ,i,ν = min{ν, αℓ,i}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Let σ ∈ Γ(d−1) be a topological generator and put x = σ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Let Jν,n denote the ideal of ΛΓ(d−1) generated by pν and (x + 1)pn − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Let the notation be as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Then logp |Z(d−1) ℓ /Jν,nZ(d−1) ℓ | = pn · mℓ � i=1 αℓ,i,ν + O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Taking Z = Z(d−1) ℓ /pνZ(d−1) ℓ in (2), we deduce the following exact sequence, 0 � �mℓ i=1 ΛΓ(d−1)/(pαℓ,i,ν) � Z(d−1) ℓ /pνZ(d−1) ℓ � Nℓ,ν � 0, (45) THE FROBENIUS TWISTS OF ELLIPTIC CURVES OVER GLOBAL FUNCTION FIELDS 21 where Nℓ,ν is finite, since the other two items are pseudo isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Write Rα for Zp/pαZp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' The lemma is a consequence of the exact sequence induced from (45): N0[σpn − 1] � �mℓ i=1 Rαℓ,i,ν[x]/((x + 1)pn − 1) � Z(d−1) ℓ /Jν,nZ(d−1) ℓ �❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤ N0/(σpn − 1)N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' □ To apply the above to dual Selmer groups, we need the following simplified control lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' For K ⊂ F ⊂ L, consider the restriction maps res(ν) L/F : Selpν(A/F) −→ Selpν(A/L)Gal(L/F ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Let K(n) denote the nth layer of the Zp-extension L(d−1)/K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Let r denote the ramification locus of L/K, which is assumed to be finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Let L(d−1)/K be an intermediate Zp-extension of L/K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Suppose r contains only places where A has good ordinary reduction or multiplicative reduction, bL(d−1) = bL and b contains r as well as all places where A has bad reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' For a given ν, the orders of ker(res(ν) L/K(n)) and coker(res(ν) L/K(n)) are bounded, as n varies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Let M = Apν(L) = Apν(K′) for some finite sub-extension K′/K of L/K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Write K′(n) for K(n)K′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' [Tan10, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='1] says that for i = 0, 1, 2, | Hi(L/K′(n), M)| ≤ |M|di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Since [K′(n) : K(n)] ≤ [K′ : K], by counting the number of co-chains, we deduce | Hi(K′(n)/K(n), M)| ≤ |M|[K′:K]i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' This bounds | ker(res(ν) L/K(n))|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' To bound | coker(res(ν) L/K(n))|, by the Hochschild-Serre spectral sequence, we need to bound (see the proof of [Tan10, Theorem 4]) | � all v � w|v H1(Lw/K(n) w , A(Lw))[pν]|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Write H (ν) w for H1(Lw/K(n) w , A(Lw))[pν].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' If v ̸∈ b, then H (ν) w = 0 [Mil06, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='8], for all w | v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Also, H (ν) w = 0, for all w sitting over bL, because K(n) w = Lw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Suppose v ∈ b but v ̸∈ bL = bL(d−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' The number of places of K(n) sitting over v is bounded as n varies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' We need to bound the order of H (ν) w , for all w | v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' If A has good ordinary reduction at v, by [Tan10, (3) and Theorem 2], the order of H (ν) w is bounded by p2ν(d+1) dim A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' It is well-known (for instance, see the last two paragraphs of [Tan10]) that if A has split multiplicative reduction at v, the order of H (ν) w is bounded by pνd dim A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' In general, if A has multiplicative reduction at v, then over some unramified extension K′ v/Kv, the reduction of A becomes split multiplicative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' 22 KI-SENG TAN Since H (ν) w ⊂ H1(LwK′ v/Kv, A(LwK′ v)), writing K′ v, L′ w for KvK′ v, LwK′ v, we end the proof by using the exact sequence H1(K′ v/Kv, A(K′ v))� � � H1(L′ w/Kv, A(L′ w)) � H1(L′ w/K′ v, A(L′ w)) and the fact that the component group of A/K′ v has p-rank bounded by dim A (see [BX96, Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='2]) so that by [Mil06, Proposition I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='8] the order of H1(K′ v/Kv, A(K′ v)) is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Let θ ∈ ΛΓ be an element not divisible by p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Let Aℓ, ℓ = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=', s, be ordinary abelian varieties defined over K such that all Zℓ := Selp∞(Aℓ/L)∨ are torsion over ΛΓ and the ramification locus r contains only places where each Aℓ has either good ordinary reduction or multiplicative reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Let b be a finite set of places of K containing r and all places where some Aℓ has bad reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Assume that pαℓ,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=', pαℓ,mℓ are elementary µ-invariants of Zℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' There exists an intermediate Zp-extension L(d−1)/K of L/K such that the following holds: (a) pL/L(d−1)(θ) ̸∈ pΛΓ(d−1), where Γ(d−1) = Gal(L(d−1)/K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' (b) bL(d−1) = bL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' (c) For each ℓ, the elementary µ-invariants of Selp∞(Aℓ/L(d−1))∨ over ΛΓ(d−1) are the same as those of Zℓ over ΛΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' (d) In particular, if L/K is a Zp-extension and K(n) denote the nth layer, then logp | Selpν(Aℓ/K(n))| = pn · mℓ � i=1 αℓ,i,ν + O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' (46) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' (a) and (b) are from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Observe that Z(d−1) ℓ /Jν,nZ(d−1) is nothing but the Pontryagin dual of Selpν(Aℓ/L)Gal(L/K(n)), so by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='3, we have logp | Selpν(Aℓ/K(n))| = pn · mℓ � i=1 αℓ,i,ν + O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' (47) In the situation of (d), L = L(d−1), and hence, (46) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' To show (c), we assume that the elementary µ-invariants of Selp∞(Aℓ/L(d−1))∨ over ΛΓ(d−1) are pα′ ℓ,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=', pα′ ℓ,wℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Apply (d) to L(d−1)/K and obtain logp | Selpν(Aℓ/K(n))| = pn · wℓ � i=1 α′ ℓ,i,ν + O(1), where, as before, α′ ℓ,i,ν := min{α′ ℓ,i, ν}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' This and (47) leads to wℓ � i=1 α′ ℓ,i,ν = mℓ � i=1 αℓ,i,ν, for all ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' We conclude that wℓ = mℓ and pα′ ℓ,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=', pα′ ℓ,wℓ are the same as pαℓ,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=', pαℓ,mℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' □ THE FROBENIUS TWISTS OF ELLIPTIC CURVES OVER GLOBAL FUNCTION FIELDS 23 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' The elementary µ-invariants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' In this section, we complete the proof of Proposition 3 and Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' For each Zp-subextension F/K of L/K, write Sb(F) = ð′ F ⊔ ðF, where if p = 2, ð′ F is the set of places at which A/K has non-split multiplicative reduction such that the group of components is of even order;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' if p ̸= 2, ð′ F = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' For w ∈ Sb(F) sitting on v ∈ Sb, if w ∈ ð′ F and v ∈ ð, or w ∈ ðF and v ∈ ð′, then Fw ̸= Kv, and hence there is only finitely many places of F sitting over v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' We apply Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='4, taking s = 1, Z1 = X(p), b = r ∪Sb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Let K(n) be the nth layer of L(d−1)/K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Since the degrees of k/K and K(n)/K are relatively prime, one see that a place v of K splits completely in k, if and only if all places of K(n) sitting over v split completely in kK(n), because both assertions are equivalent to that the decomposition subgroup Gal(kK(n)/K)v contains no non- trivial element in Gal(kK(n)/K(n)), and hence contained in Gal(kK(n)/k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Put ð0,n := {w ∈ ðK(n) | w splits completely over kK(n)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Then, by the discussion at the beginning of this section, |ð0,n| = |ð0(K(n))| + O(1) = pn · |ð1| + O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' If Fq(n) denote the constant field of K(n), then deg ∆A/K(n) · logp q(n) = pn · deg ∆A/K · logp q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Therefore, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='4(d) and Proposition 1 say if m is the µ-rank of A(p)/L, then pn · m = logp | Selp(A(p)/K(n))| + O(1) ≥ pn · ((p − 1) deg ∆A/K 12 logp q − |ð1|) + O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' This proves the proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' □ Proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Take s = 2, Z1 = X(p), Z2 = X, b = Sb ∪ r, θ = ΘL, , so that m = m1, and αi = α1,i, for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=', m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='4, bL(d−1) = bL = ∅ and ΘL(d−1) is not divisible by p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Thus, we may assume that d = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Let K(n) denote the nth layer of L/K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' If L = K(∞) p , we have shown in §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='1 that |wK(n)[p]| = O(1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' otherwise, for n sufficiently large, L/K(n) is totally ramified at certain place, so that Hom(Gal(L/K(n)), Qp/Zp) ∩ Hom(wK(n), Qp/Zp) = {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Hence, Hom(wK(n), Qp/Zp) −→ Hom(wL, Qp/Zp) is injective, or equivalently, the map wL −→ wK(n) is surjective, for sufficiently large n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Since p ∤ ΘL, wL has trivial p-part, the order of wK(n)[p] must be bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' The assumption says |Sb(K(n))| = O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Therefore, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='4(d) and Proposition 1, we obtain pn · m = pn · �m i=1 α1,i,1 = logp | Selp(A(p)/K(n))| + O(1) = pn · (p−1) deg ∆A/K 12 logp q + O(1), 24 KI-SENG TAN that proves the first assertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Then Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='4(d) and Proposition 2 lead to pn · �m2 i=1 α2,i,ν = logp | Selpν(A/K(n))| + O(1) = pn · (logp | Selpν+1(A(p)/K(n))| − (p−1) deg ∆A/K 12 logp q) + O(1) = pn · �m i=1(α1,i,ν+1 − 1) + O(1), which holds for every ν, so the proposition is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' □ References [BBM82] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Berthelot, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Breen, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Messing, Th´eorie de Dieudonn´e cristalline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' II, Lecture Notes in Mathematics 930.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Springer-Verlag, 1982.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' [BLR90] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Bosch, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' L¨utkebohmert, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Raynaud, N´eron Models, Springer-Verlag Berlin Heidelberg, 1990.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' [BX96] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Bosch, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Xarles, Component group of N´eron Models via rigid uniformization, Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' 306 (1996), 459-486.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' [Ces15] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' ˇCesnaviˇcius, Poitou-Tate without the restriction on the order, Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' 22 (2015), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' 6, 1621-1666.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' [Crw87] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Crew, L-functions of p-adic characters and geometric Iwasawa theory, Invent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' 88 (1987), 395-403.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' [Gon09] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Gonz´alez-Avil´es, Arithmetic duality theorems for 1-motives over function fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Reine Angew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' 632 (2009), 203-231.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' [GoT12] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Gonz´alez-Avil´es, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Tan, On the Hasse Principle for finite group schemes over global function fields, Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' 19 (2012), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' 02, 453-460.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' [EGA IV, part 4] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Grothendieck and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Dieudonn´e, El´ements de G´eom´etrie Alg´ebrique IV, Pul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' IHES 32 (1967).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' [SGA 7I] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Grothendieck, Groupes de monodromie en G´eom´etrie Alg´ebrique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' S´eminaire de G´eom´etrie Alg´ebrique du Bois-Marie 1967-1969 (SGA 7 I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Lecture Notes in Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' 288.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Springer, Heidelberg 1972.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' [Kis93] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Kisilevsky, Multiplicative independence in function fields, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Number Theory 44 (1993) 352–355.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' [LLTT16] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Lai, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Longhi, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Tan and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Trihan, On the Iwasawa main conjecture for semistable abelian varieties over function fields, Mathematische Zeitschrift 282 (2016), issue 1, 485-510.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' [LSc10] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Liedtke, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Schr¨oer, The N´eron model over the Igusa curves, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Number Theory 130 (2010), 2157-2197.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' [Mil80] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Milne, ´Etale Cohomology, Princeton University Press, Princeton, 1980.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' [Mil06] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Milne, Arithmetic duality theorems, Second Ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' (electronic version), 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' [Mon81] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Monsky, On p-adic power series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' 255 (1981), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' 2, 217–227.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' [OT09] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Ochiai and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Trihan, On the Iwasawa main conjecture of abelian varieties over function fields of characteristic p > 0, Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Camb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Philos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' 146 (2009), 23-43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' [Sil86] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Silverman, The Arithmetic of Elliptic Curves, Springer Verlag, Berlin, 1986.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' [Tan95] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Tan, Refined theorems of the Birch and Swinnerton-Dyer type, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Fourier, Grenoble 45 (1995), 317-374.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' [Tan10] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Tan, A generalized Mazur’s theorem and its applications, Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' 362 (2010), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' 8, 4433–4450.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' [Tan14] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Tan, Selmer groups over Zd p-extensions, Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' 359 (2014), 1025-1075.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' [Ta66] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Tate, On the conjecture of Birch and swinnerton-Dyer and a geometric analogue, S´eminare Bourbaki no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' 9 (1966), 415-440.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' [Ta84] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' Tate, Les Conjectures de Stark sur les Fonctions L d’Artin en s = 0, (Birkhauser, Boston, 1984).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content=' THE FROBENIUS TWISTS OF ELLIPTIC CURVES OVER GLOBAL FUNCTION FIELDS 25 Department of Mathematics, National Taiwan University, Taipei 10764, Taiwan Email address: tan@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='ntu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39AyT4oBgHgl3EQfo_il/content/2301.00518v1.pdf'} +page_content='tw' metadata={'source': 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We study spectral properties of perturbed discrete Laplacians on two-dimen- +sional Archimedean tilings. The perturbation manifests itself in the introduction of non- +trivial edge weights. We focus on the two lattices on which the unperturbed Laplacian +exhibits flat bands, namely the (3.6)2 Kagome lattice and the (3.12)2 “Super-Kagome” +lattice. We characterize all possible choices for edge weights which lead to flat bands. +Furthermore, we discuss spectral consequences such as the emergence of new band gaps. +Among our main findings is that flat bands are robust under physically reasonable as- +sumptions on the perturbation and we completely describe the perturbation-spectrum +phase diagram. The two flat bands in the Super-Kagome lattice are shown to even ex- +hibit an “all-or-nothing” phenomenon in the sense that there is no perturbation which +can destroy only one flat band while preserving the other. +1. Introduction +This paper is about discrete Schr¨odinger operators on Archimedean tilings, a class of +periodic two-dimensional lattices that were already investigated by Johannes Kepler in +1619 [Kep19]. They are natural candidates for the geometry of two-dimensional nanoma- +terials and due to advances in this field, most prominently represented by graphene, they +have become increasingly a focus of attention. +Much work has been devoted to understanding physical properties of such (new) materi- +als [SYY22, TFGK22, dLFM19]. Most importantly, it can be expected that the underlying +geometry, that is the particular lattice, is a key feature determining physical properties +of the system. In fact, in particular in the mathematical physics literature, investiga- +tions of the connection between the geometry (or topology) of a system and the spectral +properties of the associated Hamiltonian have become ubiquitous. Classical examples in +this context are so-called quantum waveguides [EK15, Exn20, Exn22] as well as quantum +graphs [BK13, BE22]; see also [KP07] for a relatively recent reference relevant in our +context. +A closely related research direction is superconductivity: the existence of a boundary +leads to boundary states in a superconductor with a higher critical temperature than the +one of the bulk [SB20, SB21, HRS]. In this spirit, it seems very promising to also study +the interplay of geometry and many-particle phenomena on Archimedean tilings. Yet +another related investigation can be found in [JBT21, SYY22] where another important +quantum phenomenon, namely Bose-Einstein condensation, is examined. It turns out +that so-called flat bands, that are infinitely degenerate eigenvalues of the Hamiltonian, +play an important role in understanding such many-particle effects, and for other physical +phenomena [KFSH19]. One of the central motivations for this paper is to study robustness +of flat bands under certain natural perturbations. +Two Archimedean tilings, the (3.6)2 Kagome lattice and the (3.122) tiling 1, which +we shall dub Super-Kagome lattice for reasons that will become clear over the course +Date: January 13, 2023. +1We explain the notation for the lattices in Section 2. +1 + +2 +J. KERNER, M. T¨AUFER, AND J. WINTERMAYR +of the article, stand out: they are the only Archimedean lattices on which the discrete, +unweighted Laplacian has flat bands. In particular the Kagome lattice is a prominent +model in physics that has recently enjoyed increasing interest [BM18, MDY22, Dia21]. +From a mathematical point of view, our paper is motivated by [PT21] where flat bands +for the discrete, unweighted Laplacian on Archimedean tilings have been studied in great +detail, in combination with an explicit calculation of the integrated density of states. +A priory, the flat-band phenomena on the Kagome and Super-Kagome lattice seem +very sensitive to perturbations: if one replaces the adjacency matrix or the Laplacian by +a variant with periodically chosen edge weights, one will generically destroy flat bands. +However, the results of this paper suggest that, if one looks at proper, meaningful variants +of the discrete Laplacian which respect certain, natural symmetries of the tiling (we +call them monomeric Laplacians in Definition 3), then flat bands will persist. +Since +monomericity is a physically justifiable assumption, this makes a strong case that flat +bands are a robust phenomenon, caused by the geometry of the lattice alone and specific +to these two lattices, see Theorems 6, and 10. +Other questions of interest on periodic graphs concern existence, persistence and esti- +mates on the width of spectral bands and the gaps between them [KS19, KS19, MW89]. +We will completely identify the spectra as a function of the perturbation in these cases, +see Theorems 8, and 11 as well as Figures 3, and 5. This provides an exhaustive descrip- +tion of all nanomaterials based on Archimedean tilings on which discrete Laplacians can +exhibit flat bands. +Our paper is organized as follows: Sections 2, and 3 are of introductory nature, intro- +ducing the notion of and arguing for the relevance of Archimedean tilings, and defining +a proper notion of a discrete Laplace operator with non-uniform edge weights. Section 3 +also introduces the notion of flat bands and argues why it suffices to restrict our attention +to the (3.6)2 Kagome and the (3.122) Super-Kagome lattice. Sections 4, and 5 contain our +main results on the Kagome and Super Kagome lattice, respectively. The contributions +of this paper are: +(i) We identify the Kagome and Super-Kagome lattice as the only Archimedean lat- +tices on which a natural class of periodic, weighted Laplacians can have flat bands +(Proposition 5). +(ii) We describe all periodic edge weights which lead to the maximal possible number of +bands on the Kagome and Super-Kagome lattice, and prove that this is equivalent +to so-called monomericity of the edge weights (Theorems 6 and 10). +(iii) We completely describe the spectrum in the monomeric Kagome and Super-Kagome +lattice (Theorems 8 and 11). In particular, the monomeric Super-Kagome lattice +has a surprisingly rich spectrum-perturbation phase diagram (Figure 5) which might +bear relevance for various applications. +(iv) In the Super-Kagome lattice, under a weaker condition than monomericity, namely +constant vertex weight, we explicitely describe all remaining “spurious” edge +weights which have only one flat band. We describe the topology of this set within the +parameter space and show in particular that it is disconnected from the monomeric +two-band set (Theorem 12). +2. Archimedean tilings +Archimedean, Keplerian or regular tilings are edge-to-edge tesselations of the Euclidean +plane by regular convex polygons such that every vertex is surrounded by the same pattern +of adjacent polygons. We will adopt the notation of [GS89] and use the (counterclockwise) +order of polygons arranged around a vertex as a symbol for a tiling (this is unique up to + +ROBUSTNESS OF FLAT BANDS +3 +cyclic permutations), see Figure 1 for the (3.6)2 Kagome lattice and the (3.122) Super- +Kagome lattice which will be investigated in this paper. +(3.6)2 Kagome lattice +(3.122) Super-Kagome lattice +Figure 1. The two Archimedean tilings primarily investigated in this article. +The first systematic investigation from 1619 is due to Kepler who identified all 11 such +tilings [Kep19]2. Most importantly, Archimedean tilings provide natural candidates for +geometries of two-dimensional nanomaterials since they form natural, symmetric arrange- +ments of a single buiding block, positioned at every vertex. And indeed, these lattices +can be observed in many naturally occurring materials [FK58, FK59, KHZ+20]. +From a physical point of view, two-dimensional materials such as graphene are inter- +esting since they feature so-called Dirac points which are related to a specific behaviour +of the electronic band structure of the material [FW12, LWL13, HC15]. +Also note that there are deep connections between Laplacians on these lattices, perco- +lation, and self-avoiding walks which have also been studied extensively [SE64, Kes80, +Nie82, SZ99, Ves04, Par07, Jac14, JSG16]. +An important quantity in this context is +the so-called connective constant, which is known only in few cases, for example on the +hexagonal lattice [DCS12]. +3. Defining a suitable Hamiltonian +Every Archimedean tiling can be regarded as an infinite discrete graph G = (V, E) with +(countable) vertex set V and (countable) edge set E. We write v ∼ w if the vertices v +and w are joined by an edge and denote by +|v| := #{w ∈ V : v ∼ w} +the vertex degree of v (which in the case of Archimedean lattice graphs is v-independent). +Archimedean lattices are Z2-periodic, and there exists a cofinite Z2-action +Z2 ∋ β �→ Tβ : V → V , +that is a group of graph isomorphisms (intuitively understood as a group of shifts) iso- +morphic to the group Z2. Let Q ⊂ V be a minimal (in particular finite) fundamental +domain of this action, i.e. the quotient of V under the equivalence relation generated by +the group of isomorphisms (Tβ)β∈Z2. +2All 11 Archimedean tilings are: the (44) rectangular tiling, the (36) triangular tiling, the (63) hexa- +gonal tiling, the (3.62) Kagome lattice, the (3.122) Super-Kagome lattice, the (33.42) tiling, the (4.82) +tiling, the (32.4.3.4) tiling, the (3.4.6.4) tiling, the (4.6.12) tiling, and the (34.6) tiling. + +4 +J. KERNER, M. T¨AUFER, AND J. WINTERMAYR +In the unweighted case, a natural, normalized choice for the Hamiltonian is the discrete +Laplacian +(∆f)(v) := 1 +|v| +� +w∼v +(f(v) − f(w)) = f(v) − 1 +|v| +� +w∼v +f(w) , +(1) +as used for instance in [PT21]. +It can be written as ∆f = Id − 1 +|v|Π where Π is the +adjacency matrix, that is Π(v, w) = 1 if v ∼ w and 0 else. The following is standard: +Lemma 1. The unweighted, normalized Laplacian (1) with a uniformly bounded vertex +degree boasts the following properties: +(i) All restrictions of ∆ to finitely many vertices are real-symmetric M-matrices, that +is, their off-diagonal elements are non-positive¸ and all their eigenvalues are non- +negative. +(ii) The infimum of the spectrum of ∆ is 0. +(iii) All rows and columns of ∆ sum to zero. +Furthermore, the spectrum is always contained in the interval [0, 2]. +Introducing non-trivial edge weights, we would like to keep a form of the Laplacian that +preserves properties (i) to (iii). A natural candidate, similar to formula (2.11) in [KS], is +(∆γf)(v) := +1 +� +µ(v) +� +w∼v +γvw +� +f(v) +� +µ(v) +− +f(w) +� +µ(w) +� +(2) +where the edge weights γvw = γwv > 0 and vertex weights µ(v) satisfy the relation +� +w∼v +γvw = µv +for every v ∈ V . +(3) +As long as the vertex weights µ(v) (and thus also the γvw) are uniformly bounded, this +will lead to an operator with properties (i) to (iii) and spectrum contained in [0, 2]. +Remark 2. In the literature, one often finds the definition +(∆γf)(v) = +1 +µ(v) +� +w∼v +γvw (f(v) − f(w)) +as a normalized, discrete Laplacian. Note that, whenever µ(v) ̸= µ(w) for some v ∼ w, +then this will not lead to a self-adjoint operator, but it can be made self-adjoint on a +suitably weighted ℓ2(V )-space, cf. [KLW21]. If all µ(v) are the same, then this definition +coincides with (2), and can be simplified to +(∆γf)(v) = f(v) − 1 +µ +� +w∼v +γwvf(w) . +(4) +Now, one can prescribe various degrees of the symmetry of the underlying Archimedean +lattice to be respected by the Laplacian: +Definition 3. Consider an Archimedean tiling (V, E) with periodic edge weights γvw = +γwv > 0, that is γvw = γTβvTβw for all v, w ∈ V and β ∈ Z2, and corresponding vertex +weights µ(v) = � +w∼v γvw. Define the Laplacian ∆γ as in (2). Then, we say that the +Archimedean tiling with Laplacian ∆γ +(1) has constant vertex weight, if there is µ > 0 such that µ(v) = µ for all v ∈ V . +(2) is monomeric if for all vertices v ∈ V the list of edge weights, arranged cyclically +around v, coincides (up to cyclic permutations). + +ROBUSTNESS OF FLAT BANDS +5 +Clearly, (2) is stronger than (1). However, in either case, the Laplacian reduces to (4). +The term “monomeric” is inspired by the fact that the associated operators can be +interpreted as describing properties of nanomaterials formed from one type of monomeric +building block, positioned at every vertex of an Archimedean tiling. Clearly, monomeric +Laplacians on Archimedean lattices have constant vertex weights, but the converse is not +true in general. However, we will see in Theorems 6 and 10 that on the Kagome and +Super-Kagome lattice, the validity of the converse implication is equivalent to existence +(or persistence) of all flat bands. Also, monomericity seems a physically reasonable as- +sumption for nanomaterials, which suggests that the emergence of flat bands, while a +priori very sensitive to perturbations of coefficients in the operator, might nevertheless be +robust within the class of physically relevant operators. +Next, let T2 = R2/Z2 be the flat torus and define for every θ ∈ T2 the |Q|-dimensional +Hilbert space +ℓ2(V )θ := {f : V → C | f(Tβv) = ei⟨θ,β⟩f(v)} +with inner product +⟨f, g⟩θ := +� +v∈Q +f(v)g(v) . +Given the Laplacian (4) on ℓ2(V ) with properties described in Definition 3, we define on +ℓ2(V )θ the operator +(∆θ +γf)(v) := f(v) − 1 +µ +� +w∼v +γwvf(w) . +(5) +Clearly, (5) can be represented as a |Q|-dimensional Hermitian matrix. Due to Floquet +theory, we have +σ(∆γ) = +� +θ∈T2 +σ(∆θ +γ) , +and the following statement holds. +Proposition 4 (See [PT21] and references therein). Let E ∈ R. Then, the following are +equivalent: +(i) E ∈ σ(∆θ +γ) for all θ ∈ T2. +(ii) E ∈ σ(∆θ +γ) for a positive measure subset of θ ∈ T2. +(iii) There is an infinite orthonormal family eigenfunctions of ∆γ to the eigenvalue E. +Each of them can be chosen to be supported on a finite number of vertices. +If any of (i) to (iii) is satisfied, we say that ∆γ has a flat band (at energy E). +Note that, in the ℓ∞(V ) setting instead of the ℓ2(V ) setting, such infinitely degenerate +eigenvalues are also referred to as “black hole eigenvalues” in [BL09]. Also, the existence of +flat bands can be interpreted as a breakdown of the unique continuation principle [PTV17]. +In the Hilbert space ℓ2(V ) setting, is known that for constant edge weights, the discrete +Laplacian has flat bands only on two of the 11 Archimedean lattices, namely the (3.6)2 +Kagome lattice and the (3.122) Super-Kagome lattice [PT21]. Before turning to perturbed +versions of those two lattices, one should verify that there won’t be any surprises on the +other lattices: +Proposition 5. On the Archimedean lattices (44), (36), (63), (33.42), (4.82), (32.4.3.4), +(3.4.6.4), (4.6.12), (34.6), there is no choice of periodic (with respect to the fundamental +cell on the lattice) edge weights γvw = γwv > 0 which will make the weighted adjacency + +6 +J. KERNER, M. T¨AUFER, AND J. WINTERMAYR +matrix +Πγ(v, w) = +� +γvw +if v ∼ w, +0 +else +have a flat band. +Consequently, also the Laplacian with constant or monomeric edge +weights has no flat bands on these lattices. +Proposition 5 is proved by a series of straightforward but somewhat lengthy calculations +in which one calculates the associated characteristic polynomials, and shows that there +are no θ-independent roots, employing Proposition 4 (this should be compared to the +proofs of Theorems 6 and 10 below). We omit them here for the sake of conciseness. In +any case, Proposition 5 justifies to restrict our attention to the (perturbed) Kagome and +Super-Kagome lattices from now on. +4. The perturbed Kagome lattice +In this section we discuss the Kagome lattice with non-uniform (periodic) edge weights. +The elementary cell of the Kagome lattice contains three vertices and six edges (one can +think of the edges as arranged around a hexagon). A priori, periodicity allows for six +γ1 +γ2 +γ3 +γ4 +γ5 +γ6 +γ4 +γ5 +γ6 +γ5 +γ1 +γ6 +v1 +v2 +v3 +v2 + ω1 +v3 + ω1 +v3 + ω2 +v1 − ω2 +v1 − ω1 +v2 − ω2 +Figure 2. Fundamental domain of the Kagome lattice with edge weights. +In the monomeric case, all edge weights around downwards pointing tri- +angles are γ2 = γ4 = γ6 =: α and all edge weights on upwards pointing +triangles are γ1 = γ3 = γ5 =: β, where 2α + 2β = µ. +edge weights γ1, ..., γ6 > 0, and the Floquet Laplacian ∆θ +γ can be written as the Hermitian +matrix +∆θ +γ = Id −1 +µ + + +0 +γ3 + wγ6 +wγ4 + zγ1 +γ3 + wγ6 +0 +γ2 + zγ5 +wγ4 + zγ1 +γ2 + zγ5 +0 + + , +(6) +where w := eiθ1 and z := eiθ2. We denote the three real eigenvalues of ∆θ +γ by λ1(θ, γ) ≤ +λ2(θ, γ) ≤ λ3(θ, γ). +Note that the six degrees of freedom are to be further reduced, depending on the +following symmetry conditions: +• If we merely assume a constant vertex weight µ > 0, then identity (3) will impose +the three additional linearly independent conditions +γ1 + γ4 = γ2 + γ5 , +γ3 + γ6 = γ2 + γ5 , +γ1 + γ3 + γ4 + γ6 = µ , +(7) + +ROBUSTNESS OF FLAT BANDS +7 +and we end up with three degrees of freedom. +• If we also assume monomericity, then it is easy to see that the only choice is +the breathing Kagome lattice, cf. [HKdP+22], with an edge weight α > 0 on all +edges belonging to upwards pointing triangles and edge weight β > 0 on all edges +belonging to downwards pointing triangles, where 2(α + β) = µ. After fixing the +vertex weight µ, this amounts to only one degree of freedom. +4.1. Flat bands in the perturbed Kagome lattice. +Theorem 6. Consider the perturbed Kagome lattice with Laplacian (4), fixed vertex +weight µ > 0 and periodic edge weights γ1, ..., γ6 > 0, satisfying the condition (3) on +vertex and edge weights. Then, the following are equivalent: +(i) There exists a flat band. +(ii) The vertex weights are monomeric. More explicitly, there are α, β > 0 with 2(α + +β) = µ such that +γ2 = γ4 = γ6 := α, +γ1 = γ3 = γ5 := β. +The rest of this subsection is devoted to the proof of Theorem 6. We start with identi- +fying flat bands using the weighted adjacency matrix +Πθ +γ := + + +0 +γ3 + wγ6 +wγ4 + zγ1 +γ3 + wγ6 +0 +γ2 + zγ5 +wγ4 + zγ1 +γ2 + zγ5 +0 + + +(8) +which is spectrally equivalent to ∆θ +γ up to scaling and shifting via the relation +∆θ +γ = Id −1 +µΠθ +γ. +In order to find flat bands, we will identify conditions for θ-independent eigenvalues of Πθ +γ +and therefore calculate +det(λ Id −Πθ +γ) = −λ3 + λ(|A|2 + |B|2 + |C|2) + 2ℜ(ABC) +where A := γ3 + wγ6, B := wγ4 + zγ1 and C := γ2 + zγ5. Rearranging the terms yields +det(λ Id −Πθ +γ) =(w + w)(λγ6γ3 + γ3γ2γ4 + γ6γ5γ1) ++(z + z)(λγ5γ2 + γ6γ5γ4 + γ1γ3γ2) ++(wz + zw)(λγ1γ4 + γ3γ5γ4 + γ6γ2γ1) ++(−λ3 + λ(γ2 +1 + ... + γ2 +6) + 2(γ4γ6γ2 + γ3γ5γ1)) . +The prefactors +w + w = 2 cos θ1 , +z + z = 2 cos θ2 , +and +wz + zw = 2 cos(θ1 − θ2) , +are linearly independent as measurable functions of θ on T2. Consequently, since all γi +are positive, θ-independent eigenvalues exist if and only if the w and z-independent terms +in every line are zero. This is only possible for negative λ, which (possibly after scaling +the γi and µ for the moment) can be assumed to equal −1. Therefore, we obtain the +conditions +γ3γ6 = γ2γ3γ4 + γ1γ5γ6 , +γ2γ5 = γ4γ5γ6 + γ1γ2γ3 , +γ1γ4 = γ3γ4γ5 + γ1γ2γ6 , +(9) + +8 +J. KERNER, M. T¨AUFER, AND J. WINTERMAYR +and +1 − (γ2 +1 + · · · + γ2 +6) + 2 (γ2γ4γ6 + γ1γ3γ5) = 0 . +(10) +Lemma 7. The only positive solutions (meaning all γi are non-zero) of (7), (9), (10) are +γ2 = γ4 = γ6 = x +γ1 = γ3 = γ5 = y +(11) +with x, y ∈ (0, 1) and x + y = 1. +Proof. By a direct calculation (11) solves (7), (9), (10). +Conversely, assume that there are positive solutions γ1, ..., γ6 > 0. From (9) we obtain +γ3 = +γ1γ5γ6 +γ6 − γ2γ4 +, +γ1 = +γ3γ4γ5 +γ4 − γ2γ6 +, +and this implies γ6 > γ2γ4 and γ4 > γ2γ6. +Hence, combining both equations yields +γ2 +2γ6 < γ6 which shows that γ2 < 1. In the same way one proves γi < 1 for every other i. +Next, let γ2 + γ5 := Λ. By (7) one immediately concludes γ1 + γ4 = γ3 + γ6 = Λ. Now, +we add (9) and (10) and rearrange the equations to obtain +1 +2 +� +γ2 +1 + · · · + γ2 +6 − 1 +� ++ γ3γ6 + γ2γ5 + γ1γ4 =γ2γ4γ6 + γ1γ3γ5 ++ γ2γ3γ4 + γ1γ5γ6 + γ4γ5γ6 ++ γ1γ2γ3 + γ3γ4γ5 + γ1γ2γ6 . +By repeated factorization, the right hand side simplifies to +γ1γ3(γ2 + γ5) + γ3γ4(γ2 + γ5) + γ1γ6(γ2 + γ5) + γ4γ6(γ2 + γ5) = Λ3, +(12) +and since for the left hand side one has +1 +2 +� +γ2 +1 + · · · + γ2 +6 − 1 +� ++ γ3γ6 + γ2γ5 + γ1γ4 = 3Λ2 − 1 +2 +, +we arrive at the polynomial Λ3 − 3Λ2 +2 + 1 +2 = 0 the only positive solution of which is Λ = 1. +Finally, adding the first the two equations of (9) yields +γ3γ6 + γ2γ5 = (γ6γ5 + γ2γ3)(γ1 + γ4) = γ6γ5 + γ2γ3 +and this implies γ5 = γ3. Furthermore, adding the last two equations gives +γ2γ5 + γ1γ4 = (γ4γ5 + γ1γ2)(γ3 + γ6) = γ4γ5 + γ1γ2 +giving γ4 = γ2. +Conditions (7) hence give γ1 = γ5 and γ6 = γ2. +This proves the +statement. +□ +We are now in the position to prove Theorem 6. +Proof of Theorem 6. Comparing Πθ +γ with ∆θ +γ we conclude that ∆θ +γ has a flat band with +edge weights γ1, ..., γ6 if and only if there exists δ > 0 such that Πθ +γ has a flat band for edge +weights δγ1, ..., δγ6. From this observation the statement follows directly taking Lemma 7 +into account. +□ + +ROBUSTNESS OF FLAT BANDS +9 +4.2. The spectrum and band gaps in the monomeric Kagome lattice. In the case +where the perturbed Kagome lattice has a flat band, we further study the structure of the +rest of the spectrum. We reiterate that, due to Theorem 6, the existence of a flat band is +equivalent to the weights being monomeric. +As shown for instance in [PT21], in the case where all edge weights are equal, the two +other spectral bands, generated by the two other θ-dependent eigenvalues of ∆θ +γ, touch +at E = 3/4, and the derivative of the integrated density of states at E = 3/4 vanishes +– an indication that the spectral density at 3/4 is sufficiently thin for a gap to form +under perturbation. And indeed, this is the statement of the next theorem, which also +characterises the width of the gap. +Theorem 8 (Band gaps in the perturbed Kagome lattice). Consider the perturbed Kago- +me lattice with fixed vertex weight µ > 0, and monomeric edge weights α, β > 0, satisfying +2(α + β) = µ as characterized in Theorem 6. Then, the spectrum is given by +I1 ∪ I2 := +� +0, 3 +4 − +���� +3α +µ − 3 +4 +���� +� � �3 +4 + +���� +3α +µ − 3 +4 +���� , 3 +2 +� +. +Furthermore, there is always a flat band at 3 +2. +Remark 9. Theorem 8 states that, as soon as α ̸= β, or alternatively, α ̸= µ +4, a spectral +gap of width +���� +6α +µ − 3 +2 +���� = 3 +µ|α − β| +will form around 3 +4, see also Figure 3. The flat band at 3 +2 will always be connected to the +energy band below it which means that the “touching” of the flat band at 3 +2 is protected in +the class of monomeric perturbations. +I1 +I2 +α = µ +2 +α = 0 +α = µ +4 +3 +2 +3 +4 +Flat band +σ(∆γ) +Figure 3. Spectrum of the monomeric (32.62) Kagome lattice with vertex +weight µ > 0 as a function of the parameter α ∈ (0, µ +2), describing the edge +weights on edges adjacent to downwards pointing triangles. +Proof. A calculation shows that the eigenvalues of ∆θ +γ with the choice 2(α + β) = µ as in +Theorem 6 are given by +λ1,2(θ, γ) = 3 +4 ± 1 +4 +� +1 + 8 +� +1 + (F(θ) − 3) +�2α +µ − 4α2 +µ2 +�� +and +λ3(θ, γ) = 3 +2 + +10 +J. KERNER, M. T¨AUFER, AND J. WINTERMAYR +where F(θ) := cos(θ1) + cos(θ2) + cos(θ1 − θ2). The function T2 ∋ θ �→ F(θ) takes all +values in [−3/2, 3], see Lemma 3.1 in [PT21], whence λ1(θ, γ) and λ2(θ, γ) take all values +in the intervals +� +0, 3 +4 − +���� +3α +µ − 3 +4 +���� +� +, +and +�3 +4 + +���� +3α +µ − 3 +4 +���� , 3 +2 +� +, +respectively. +□ +5. The perturbed Super-Kagome lattice +In this section, we investigate the Archimedean tiling (3.122) which we call Super- +Kagome lattice. Its minimal elementary cell contains six vertices and nine edges: three +edges on upwards pointing triangles, three edges on downwards pointing triangles, and +three edges bordering two dodecagons, see Figure 4. +v4 +v3 +v5 +v6 +v2 +v1 +v1 − ω2 +v2 − ω1 +v6 + ω1 +v5 + ω2 +γ7 +γ3 +γ2 +γ1 +γ5 +γ6 +γ4 +γ9 +γ8 +γ8 +γ9 +Figure 4. Fundamental domain of the (3.122) tiling with edge weights. In +the monomeric case, all edge weights around triangles triangles are γ1 = +· · · = γ6 =: α and the remaining weights are γ7 = γ8 = γ9 =: β. +Given a constant vertex weight µ > 0, the Floquet Laplacian (5) is a 6×6-matrix given +by +∆θ +γ = Id −1 +µ + + + + + + + +0 +γ4 +γ6 +0 +zγ9 +0 +γ4 +0 +γ5 +0 +0 +wγ8 +γ6 +γ5 +0 +γ7 +0 +0 +0 +0 +γ7 +0 +γ3 +γ2 +zγ9 +0 +0 +γ3 +0 +γ1 +0 +wγ8 +0 +γ2 +γ1 +0 + + + + + + + +, +(13) +where w := eiθ1, z := eiθ2. +• If we fix a constant vertex weight µ > 0, the condition � +w∼v γvw = µ for all v ∈ V +leads to +µ = γ2 + γ3 + γ7 = γ5 + γ6 + γ7 = γ1 + γ2 + γ8 = γ4 + γ5 + γ8 += γ1 + γ3 + γ9 = γ4 + γ6 + γ9. +(14) +This can be seen to be a linear system of 6 linearly independent equations with 9 +unknowns, so the solution space is 3-dimensional. More precisely, by appropriate +additions, we infer the three identities +2γ1 + γ8 + γ9 = 2γ7 + γ2 + γ3, +2γ4 + γ8 + γ8 = 2γ7 + γ5 + dγ6, +γ2 + γ3 = γ5 + γ6 +(15) + +ROBUSTNESS OF FLAT BANDS +11 +which imply γ1 = γ4. The identities γ2 = γ5, and γ3 = γ6 follow by completely +analogous calculations. This leaves us with 6 independent variables γ1, γ2, γ3, and +γ7, γ8, γ9 which are however still subject to the three conditions +γ2 + γ3 + γ7 = γ1 + γ2 + γ8 = γ1 + γ3 + γ9 = µ +from (14). Therefore, we are left with three degrees of freedom. +• If we additionally prescribe monomericity, it is easy to see that there is only one +degree of freedom: All edges around triangles carry the weight α > 0, and all +remaining edges (separating two dodecagons) carry the weight β > 0 under the +condition 2α + β = µ. +5.1. Flat bands in the perturbed Super-Kagome lattice. +Theorem 10. Consider the perturbed Super-Kagome lattice with Laplacian (4), fixed +vertex weight µ > 0, and periodic edge weights γ1, . . . , γ9 > 0 satisfying the condition (3) +on vertex and edge weights. Then, the following are equivalent: +(i) There exist exactly two flat bands. +(ii) The Super-Kagome lattice is monomeric. More explicitly, there are α, β > 0 such +that 2α + β = µ together with +γ1 = γ2 = γ3 = γ4 = γ5 = γ6 = α , +γ7 = γ8 = γ9 = β . +Proof. Recall that in the constant vertex weight case, we have +γ1 = γ4 , +γ2 = γ5 , +and +γ3 = γ6 , +and consider the weighted adjacency matrix +Πθ +γ := + + + + + + + +0 +γ4 +γ6 +0 +zγ9 +0 +γ4 +0 +γ5 +0 +0 +wγ8 +γ6 +γ5 +0 +γ7 +0 +0 +0 +0 +γ7 +0 +γ3 +γ2 +zγ9 +0 +0 +γ3 +0 +γ1 +0 +wγ8 +0 +γ2 +γ1 +0 + + + + + + + += + + + + + + + +0 +γ1 +γ3 +0 +zγ9 +0 +γ1 +0 +γ2 +0 +0 +wγ8 +γ3 +γ2 +0 +γ7 +0 +0 +0 +0 +γ7 +0 +γ3 +γ2 +zγ9 +0 +0 +γ3 +0 +γ1 +0 +wγ8 +0 +γ2 +γ1 +0 + + + + + + + +(16) +which is a shifted and scaled version of ∆θ +γ. We calculate +det(λ Id −Πθ +γ) = λ6 − λ4 � +2γ2 +1 + 2γ2 +2 + 2γ2 +3 + γ2 +7 + γ2 +8 + γ2 +9 +� +− 4λ3γ1γ2γ3 ++ λ2� +γ4 +1 + γ4 +2 + γ4 +3 + 2γ2 +1γ2 +2 + 2γ2 +2γ2 +3 + 2γ2 +3γ2 +1 + 2γ2 +1γ2 +7 + 2γ2 +2γ2 +9 + 2γ2 +3γ2 +8+ ++ γ2 +7γ2 +8 + γ2 +8γ2 +9 + γ2 +9γ2 +7 +� ++ 4λγ1γ2γ3 +� +γ2 +1 + γ2 +2 + γ2 +3 +� +− γ4 +1γ2 +7 − γ4 +2γ2 +9 − γ4 +3γ2 +8 − γ2 +7γ2 +8γ2 +9 + 4γ2 +1γ2 +2γ2 +3 +− (w + w) +� +λ2γ2 +2γ7γ8 + 2λγ1γ2γ3γ7γ8 + γ2 +1γ2 +3γ7γ8 − γ2 +2γ7γ8γ2 +9 +� +− (z + z) +� +λ2γ2 +3γ7γ9 + 2λγ1γ2γ3γ7γ9 + γ2 +1γ2 +2γ7γ9 − γ2 +3γ7γ2 +8γ9 +� +− (wz + wz) +� +λ2γ2 +1γ8γ9 + 2λγ1γ2γ3γ8γ9 + γ2 +2γ2 +3γ8γ9 − γ2 +1γ2 +7γ8γ9 +� +. +Since w + w = 2 cos(θ1), z + z = 2 cos(θ2), and wz + wz = 2 cos(θ1 − θ2) are linearly on +T2, λ is a θ-independent eigenvalue if and only if the conditions + +12 +J. KERNER, M. T¨AUFER, AND J. WINTERMAYR +λ2γ2 +2 + 2λγ1γ2γ3 + γ2 +1γ2 +3 − γ2 +2γ2 +9 = 0, +λ2γ2 +3 + 2λγ1γ2γ3 + γ2 +1γ2 +2 − γ2 +3γ2 +8 = 0, +λ2γ2 +1 + 2λγ1γ2γ3 + γ2 +2γ2 +3 − γ2 +1γ2 +7 = 0, +(17) +as well as +λ6 − λ4 � +2γ2 +1 + 2γ2 +2 + 2γ2 +3 + γ2 +7 + γ2 +8 + γ2 +9 +� +− 4λ3γ1γ2γ3 ++ λ2� +γ4 +1 + γ4 +2 + γ4 +3 + 2γ2 +1γ2 +2 + 2γ2 +2γ2 +3 + 2γ2 +3γ2 +1 + 2γ2 +1γ2 +7 + 2γ2 +2γ2 +9 + 2γ2 +3γ2 +8+ ++ γ2 +7γ2 +8 + γ2 +8γ2 +9 + γ2 +9γ2 +7 +� ++4λγ1γ2γ3 +� +γ2 +1 + γ2 +2 + γ2 +3 +� +−γ4 +1γ2 +7 − γ4 +2γ2 +9 − γ4 +3γ2 +8 − γ2 +7γ2 +8γ2 +9 + 4γ2 +1γ2 +2γ2 +3 = 0 +(18) +hold.3 Conditions (17) imply that any θ-independent eigenvalue of the matrix Πθ +γ must +satisfy +λ = −γ1γ3 +γ2 +± γ9, +λ = −γ1γ2 +γ3 +± γ8, +and +λ = −γ2γ3 +γ1 +± γ7. +Since all γi are positive, the only way for these three equations to have the same set of +solutions, that is for two flat bands to exist, is therefore +− γ1γ3 +γ2 ++ γ9 = −γ1γ2 +γ3 ++ γ8 = −γ2γ3 +γ1 ++ γ7 +(19) +together with +− γ1γ3 +γ2 +− γ9 = −γ1γ2 +γ3 +− γ8 = −γ2γ3 +γ1 +− γ7. +(20) +This implies that the matrix Πθ +γ can only have two θ-independent eigenvalues if there are +α, β > 0 with +α = γ7 = γ8 = γ9 +and +β = γ1 = γ2 = γ3, +that is the monomeric case, and the only candidates for these eigenvalues are −β ± +α. To see that they are indeed eigenvalues, one verifies by an explicit calculation that +condition (18) is also fulfilled. This shows the stated equivalence. +□ +Next, we further describe the spectrum of the monomeric Super-Kagome lattice. +Theorem 11 (Band gaps in the perturbed Super-Kagome lattice). Consider the perturbed +Super-Kagome lattice with Laplacian (4) with fixed vertex weight µ > 0 and monomeric +edge weights α, β > 0, satisfying 2α + β = µ as characterized in Theorem 10. Then, the +spectrum is given by +I1 ∪ I2 := +� +0, +� +1 − α +2µ +� +− |3α − 2β| +2µ +� � �� +1 − α +2µ +� ++ |3α − 2β| +2µ +, 2 − α +µ +� +with flat bands at 3α +µ and 2 − α +µ. +The spectrum and the position of the flat bands have been plotted in Figure 5. The +spectrum generically consists of two distinct intervals (bands) except for the case 3α = 2β, +that is α = 2µ +7 , in which the two bands touch and the spectrum consists of one interval +with an embedded flat band in the middle as well as a flat band at its maximum. This +case α = 2µ +7 connects two regimes with different spectral pictures: +3As we will see later, despite its complexity, (18) will not impose further restrictions and hold in all +relevant cases. This appears to be a consequence of symmetries of the lattice and the operator. + +ROBUSTNESS OF FLAT BANDS +13 +• If α > 2µ +7 the spectrum consists of two intervals the upper one of which has two +flat bands at its endpoints. In the special case of uniform edge weights (that is +α = µ +3, this has already been observed, for instance in [PT21]. +• If α < 2µ +7 , the spectrum will again consist of two intervals each of which will have +a flat band at its maximum. Somewhat surprisingly, the lower flat band has now +attach itself to the lower interval I2 upon passing the critical parameter α = 2µ +7 . +Another noteworthy observation is that no gap opens within the intervals I1 and I2, +despite them being generated by two distinct Floquet eigenvalues and the density of +states measure vanishing at a point in the interior of the bands, see again [PT21] for plots +of the integrated density of states in the case of constant edge weights. In particular, this +distinguishes the monomeric Super-Kagome lattice from the monomeric Kagome lattice +where such a gap indeed opens within the spectrum at points of zero spectral density. +I1 +I2 +α = µ +2 +µ +3 2µ +7 +α = 0 +2 +Flat bands +σ(∆γ) +Constant edge weights +Figure 5. Spectrum of the monomeric (3.122) “Super-Kagome” lattice +with vertex weight µ > 0 as a function of the parameter α ∈ (0, µ +2), describ- +ing the edge weights on edges adjacent to triangles. +Proof of Theorem 11. In the monomeric case, the characteristic polynomial det(λ Id −Πθ +γ) +of the matrix Πθ +γ simplifies to +((α + λ)2 − β2)· +(λ4 − 2αλ3 − (3α2 + 2β2)λ2 + (4α3 + 2αβ2)λ + 4α4 + α2β2 + β4 − 2α2β2F(θ1, θ2)) , +where F(θ1, θ2) = cos(θ1) + cos(θ2) + cos(θ1 + θ2). Its six roots are +� +−α ± β, 1 +2 +� +α ± +� +9α2 + 4β2 ± 4αβ +� +3 + 2F(θ1, θ2) +�� +, +whence the eigenvalues of ∆θ +γ are given by +λ1(θ, γ) = 1 − 1 +2µ +� +α + +� +9α2 + 4β2 + 4αβ +� +3 + 2F(θ1, θ2) +� +, +λ2(θ, γ) = 1 − 1 +2µ +� +α + +� +9α2 + 4β2 − 4αβ +� +3 + 2F(θ1, θ2) +� +, +λ3(θ, γ) = 1 + α − β +µ += 3α +µ = +� +1 − α−|3α−2β| +2µ +if 3α ≥ 2β , +1 − α−|3α−2β| +2µ +if 3α < 2β , + +14 +J. KERNER, M. T¨AUFER, AND J. WINTERMAYR +λ4(θ, γ) = 1 − 1 +2µ +� +α − +� +9α2 + 4β2 − 4αβ +� +3 + 2F(θ1, θ2) +� +, +λ5(θ, γ) = 1 − 1 +2µ +� +β − +� +9α2 + 4β2 + 4αβ +� +3 + 2F(θ1, θ2) +� +, +λ6(θ, γ) = 1 + α + β +µ += 2 − α +µ . +Using that the map T2 ∋ (θ1, θ2) �→ F(θ1, θ2) takes all values in the interval (−3/2, 3), we +conclude that the bands, generated by λ1(θ, γ) and λ2(θ, γ), as well as the bands generated +by λ4(θ, γ) and λ5(θ, γ) always touch, and the spectrum consists of the two intervals +� +min +θ∈T2 λ1(θ, γ), max +θ∈T2 λ2(θ, γ) +� � � +min +θ∈T2 λ4(θ, γ), max +θ∈T2 λ5(θ, γ) +� += +� +0, 1 − α + |3α − 2β| +2µ +� � � +1 − α − |3α − 2β| +2µ +, 2 − α +2µ +� += +� +0, +� +1 − α +2µ +� +− |3α − 2β| +2µ +� � �� +1 − α +2µ +� ++ |3α − 2β| +2µ +, 2 − α +µ +� +. +□ +One might now wonder under which conditions only one flat band exists. The next +theorem completely identifies all parameters for which one flat band exists: +Theorem 12. Consider the perturbed Super-Kagome lattice with Laplacian (4), fixed +vertex weight µ > 0, and periodic edge weights γ1, . . . , γ9 > 0 satisfying the condition +(3) on vertex and edge weights. The set of (γi) such that exactly one flat band exists +consists of six connected components which have no mutual intersections and have +no intersection with the two-flat-band parameter set, identified in Theorem 10. +The solution space is invariant under those permutations of the γi which correspond +to rotations of the lattice by 2π +3 , and 4π +3 . Modulo these permutations, the two connected +components can be described as follows +• A one-dimensional submanifold, isomorphic to an interval, and explicitely descibed +in equation (26), +• Two one-dimensional submanifolds each isomorphic to an interval, explicitely de- +scribed in (28), and (30), which intersect in a single point. +Proof of Theorem 12. Recall that due to the reductions made at the beginning of the +section, after fixing the constant vertex weight µ > 0, the space of edge weights is a +3-dimensional manifold in the 6-dimensional parameter space {γ1, γ2, γ3, γ7, γ8, γ9 > 0}, +subject to the conditions +γ1 + γ3 + γ9 = γ1 + γ2 + γ8 = γ2 + γ3 + γ7 = µ. +(21) +Furthermore, from the proof of Theorem 10 we infer that ∆γ has a flat band at λ if and +only if the weighted adjacency matrix Πθ +γ has the θ-independent eigenvalue ˜λ := µ(1−λ). +This requires in particular that +˜λ = −γ1γ3 +γ2 +± γ9 = −γ1γ2 +γ3 +± γ8 = −γ2γ3 +γ1 +± γ7 +(22) +holds with a certain combination of plus and minus signs. Now, if equality in (22) holds +with all three signs positive or all three signs negative, respectively, then the argument +in the proof of Theorem 10 shows that this already implies that the edge weights are +monomeric, the identities also hold with the opposite sign, the additional condition (18) is +fulfilled, and there are two flat bands. As a consequence, the only chance for the existence + +ROBUSTNESS OF FLAT BANDS +15 +of exactly one flat band is (22) to hold with different signs in front of γ7, γ8, γ9. Also, it +is immediately clear that (22) with different signs does not allow for a monomeric and +non-zero solution and hence the solution space consists of at most six mutually disjoint +components which have no intersection with the two-flat-band manifold, identified in +Theorem 10. +By symmetry, it suffices to investigate two out of these six cases: +Case(- + +): +− γ1γ3 +γ2 +− γ9 = −γ1γ2 +γ3 ++ γ8 = −γ2γ3 +γ1 ++ γ7 = ˜λ , +(23) +and +Case(+ - -): +− γ1γ3 +γ2 ++ γ9 = −γ1γ2 +γ3 +− γ8 = −γ2γ3 +γ1 +− γ7 = ˜λ . +(24) +To solve Case(- + +), combine the second identities in in (21) and (23), to deduce +γ3 − γ1 = +γ2 +γ1γ3 +(γ2 +1 − γ2 +3) +which, recalling γi > 0, is only possible if γ1 = γ3. But then, by (23), γ7 = γ8. Calling +α′ := γ2, and β′ := γ9, we can use (21), to further express +γ1 = γ3 = µ − β′ +2 +, +and +γ7 = γ8 = µ + β′ +2 +− α′. +(25) +Next, we eliminate β′ by resolving the yet unused first identity in (23), which yields +− (µ − β′)2 +4α′ +− β′ = −α′ + µ + β′ +2 +− α′ +⇔ +β′ = µ − 3α′ ± +� +17α′2 − 8α′µ. +This only has real solutions if α′ > +8 +17µ > 1 +3µ, thus only +β′ = µ − 3α′ + +� +17α′2 − 8α′µ. +can be a positive solution. Furthermore, we need β′ ∈ (0, µ), which is the case if and only +if +γ2 = α′ ∈ +�µ +2 , µ +� +. +We therefore find the one-parameter solution set +Case (- + +) + + + + + + + + + +γ1 = γ3 += µ−β′ +2 , +γ2 = α′ +∈ +� µ +2, µ +� +, +γ7 = γ8 += µ+β′ +2 +− α′, +γ9 = β′ +:= µ − 3α′ + +� +17α′2 − 8αµ +(26) +with energy +˜λ = −γ2 + γ7 = −2α′ + µ + β′ +2 += −2α′ + 2µ − 3α′ + +� +17α′2 − 8α′µ +2 +. +Finally, an explicit calculation shows that with these parameters, (18) is indeed fulfilled. +As for Case(+ - -), we combine the second identity in (21) with the second identity +in (24) to deduce +γ3 − γ1 = +γ2 +γ1γ3 +(γ2 +3 − γ2 +1) . +(27) +Identity (27) has two types of solutions: +Case(+ - -)(a): γ1 = γ3. + +16 +J. KERNER, M. T¨AUFER, AND J. WINTERMAYR +As before we find γ7 = γ8. Let α′ := γ2, β′ := γ9, and combine the remaining first identity +in (24) with (25) to solve for β′, finding +− (µ − β′)2 +4α′ ++ β′ = −µ + β′ +2 +⇔ +β′ = µ + 3α′ ± +� +9α′2 + 8α′µ. +Only the solution +β′ = µ + 3α′ − +� +9α′2 + 8α′µ +has a chance to be in (0, µ), and, indeed, this is the case if and only if +γ2 = α′ ∈ +� +0, µ +2 +� +. +We obtain the one-parameter solution set +Case(+ - -)(a) + + + + + + + + + +γ1 = γ3 += µ−β′ +2 , +γ2 = α′ +∈ +� +0, µ +2 +� +, +γ7 = γ8 += µ+β′ +2 +− α′, +γ9 = β′ +:= µ + 3α′ − +� +9α′2 + 8α′µ +(28) +with energy +˜λ = −γ2 − γ7 = −µ + β′ +2 += −2µ + 3α′ − +� +9α′2 + 8α′µ +2 +. +Again, an explicit calculation shows that (18) is fullfilled. +Case(+ - -)(b): The other solution of (27) is +γ1γ3 = γ2(γ1 + γ3). +We set α′′ := γ1, β′′ := γ3, whence +γ2 = +α′′β′′ +α′′ + β′′, +and use (21) to infer +γ7 = µ − 2α′′β′′ + β′′2 +α′′ + β′′ +, +γ8 = µ − α′′2 + 2α′′β′′ +α′′ + β′′ +, +γ9 = µ − α′′ − β′′. +(29) +Plugging (29) into the yet unused first identity in (24), we arrive at +− (α′′ + β′′) + µ − α′′ − β′′ = − +α′′2 +α′′ + β′′ − µ + α′′2 + 2α′′β′′ +α′′ + β′′ +⇔ +β′′ = µ − 3α′′ ± +� +(µ − 3α′′)2 + 4α′′(µ − α′′) +2 += µ − 3α′′ ± +� +µ2 − 2α′′µ + 5α′′2 +2 +We observe that only the solution with a plus has a chance to be positive and it is easy to +see that this solution takes values in (0, µ) for all α′′ ∈ (0, µ). We obtain the one-parameter +solution set +Case (+ - -) (b) + + + + + + + + + + + + + + + + + + + + + +γ1 = α′′ +∈ (0, µ) , +γ2 += +α′′β′′ +α′′+β′′, +γ3 = β′′ +:= +µ−3α′′+√ +µ2−2α′′µ+5α′′2 +2 +, +γ7 += µ − 2α′′β′′+β′′2 +α′′+β′′ +, +γ8 += µ − α′′2+2α′′β′′ +α′′+β′′ +, +γ9 += µ − α′′ − β′′ +(30) + +ROBUSTNESS OF FLAT BANDS +17 +at energy +˜λ = −γ1γ3 +γ2 ++ γ9 = µ − 2α′′ − 2β′′ = α − +� +µ2 − 2α′′µ + 5α′′2. +Again, an explicit calculation verifies that with these choices, (18) is fullfilled. +Finally, to conclude the claimed topological properties of the manifolds, we need to +verify that the solution space (28) in Case(+ - -)(a) intersects the solution space (30) +in Case(+ - -)(b) if and only if +γ1 = γ3 = γ7 = γ8 = 2µ +5 , +γ2 = γ9 = µ +5. +□ +X2 +X1 +One flat band, Case(- + +) +One flat band, Case(+ - -) (a) +One flat band, Case(+ - -) (b) +Monomeric edge weights, +two flat bands +Extremal cases, not belonging +to the parameter space +Figure 6. Schematic overview of the topology of the six “spurious” one- +flat-band solution sets, and the monomeric two-flat-band manifold within +the constant-vertex weight parameter space. +Case(- + +) solutions +asymptotically meet the limit points of the two-flat-band manifold at one +end of the parameter range, whereas Case(+ - -) (a) solutions asymptot- +ically meet it at both ends of the parameter range. +Remark 13. Theorems 10 and 12 imply that the six one-flat-band components and the +two-flat-band component are mutually disjoint. However, a closer analysis of the extremal +cases in Formulas (26), (28), and (30), as well as of the monomeric case, implies that +when sending the parameters to their extremal values, the three one-dimensional manifolds +corresponding to Case(+ - -) (a), and the two-flat-band-manifold of solutions converge +to the two points +X1 := +� +0, 0, 0, µ +2, µ +2 , µ +2 +� +and +X2 := +�µ +2, µ +2 , µ +2 , 0, 0, 0 +� +, +which themselves do no longer belong to the space of admissible parameters. Likewise, the +limit of solutions of Case(+ - -) in (26) corresponding to α′ = µ +2 corresponds to the the +point X2, see also Figure 6. +Acknowledgement. JK would like to thank the Bergische Universit¨at Wuppertal where +parts of this project were done while being on leave from the FernUniversit¨at in Hagen. +JK and MT also acknowledge support by the Cost action CA18232 through the summer +school “Heat Kernels and Geometry: From Manifolds to Graphs” held in Bregenz. MT +would like to thank the Mittag-Leffler Institute where parts of this work were initiated +during the trimester Program “Spectral Methods in Mathematical Physics”. + +18 +J. KERNER, M. T¨AUFER, AND J. WINTERMAYR +References +[BE22] +M. Baradaran and P. Exner, Kagome network with vertex coupling of a preferred orientation, +J. Math. Phys 63 (2022), no. 8, 083502. +[BK13] +G. Berkolaiko and P. Kuchment, Introduction to quantum graphs, American Mathematical +Society, 2013. +[BL09] +J. von Below and J. A. Lubary, Isospectral infinite graphs and networks and infinite eigen- +value multiplicities, Netw. Heterog. Media 4 (2009), no. 3, 453–468. +[BM18] +T. Bilitewski and R. Moessner, Disordered flat bands on the Kagome lattice, Phys. Rev. B +98 (2018), 235109. +[DCS12] +H. Duminil-Copin and S. Smirnov, The connective constant of the Honeycomb lattice equals +� +2 + +√ +2, Ann. Math. 175 (2012), 1653–1665. +[Dia21] +D. P. Dias, Topological properties of flat bands in generalized Kagome lattice materials, dis- +sertation at KTH, School of Engineering Sciences (SCI), 2021. +[dLFM19] +F. C. de Lima, G. J. Ferreira, and R. H. Miwa, Topological flat band, Dirac fermions and +quantum spin Hall phase in 2d Archimedean lattices, Phys. Chem. Chem. Phys. 21 (2019), +no. 40, 22344–22350. +[EK15] +P. Exner and H. Kovaˇr´ık, Quantum waveguides, Springer International Publishing, 2015. +[Exn20] +P. Exner, Spectral properties of soft quantum waveguides, J. Phys. A 53 (2020), no. 35, +355302. +[Exn22] +, Soft quantum waveguides in three dimensions, J. Math. Phys. 63 (2022), no. 4, +042103. +[FK58] +F. C. Frank and J. S. Kasper, Complex alloy structures regarded as sphere packings. I. +Definitions and basic principles, Acta. Cryst. 11 (1958), 184–190. +[FK59] +, Complex alloy structures regarded as sphere packings. II. Analysis and classification +of representative structures, Acta. Cryst. 12 (1959), 483–499. +[FW12] +C. L. Fefferman and M. I. Weinstein, Honeycomb lattice potentials and dirac points, J. Am. +Math. Soc. 25 (2012), no. 4, 1169–1220. +[GS89] +B. Gr¨unbaum and G. C. Shephard, Tilings and patterns, A Series of Books in the Mathe- +matical Sciences, W. H. Freeman and Company, New York, 1989. +[HC15] +J-M. Hou and W. Chen, Hidden symmetry and protection of Dirac points on the honeycomb +lattice, Scientific Reports 5 (2015), 17571. +[HKdP+22] M. A. J. Herrera, S. N. Kempkes, M. Blanco de Paz, A. Garc´ıa-Etxarri, I. Swart, C. Morais +Smith, and D. Bercioux, Corner modes of the breathing Kagome lattice: origin and robust- +ness, arXiv:2201.07576 [cond-mat.mes-hall], 2022. +[HRS] +C. Hainzl, B. Roos, and R. Seiringer, Boundary superconductivity in the BCS model, +arXiv:2201.08090 [math-ph], 2022. +[Jac14] +J. L. Jacobsen, High-precision percolation thresholds and Potts-model critical manifolds from +graph polynomials, J. Phys. A: Math. Theor. 47 (2014), 135001. +[JBT21] +A. Julku, G. M. Bruun, and P. T¨orm¨a, Quantum geometry and flat band Bose-Einstein +condensation, Phys. Rev. Lett. 127 (2021), no. 17. +[JSG16] +J. L. Jacobsen, C. R. Scullard, and A. J. Guttmann, On the growth constant for square-lattice +self-avoiding walks, J. Phys. A Math. Theor. 49 (2016), no. 49, 494004. +[Kep19] +J. Kepler, Harmonices mundi, Johann Plack, 1619. +[Kes80] +H. Kesten, The critical probability of bond percolation on the square lattice equals 1 +2, Comm. +Math. Phys. 74 (1980), no. 1, 41–59. MR 575895 +[KFSH19] +A. J. Koll´ar, M. Fitzpatrick, P. Sarnak, and A. A. Houck, Line-graph lattices: Euclidean and +non-Euclidean flat bands, and implementations in circuit quantum electrodynamics, Comm. +Math. Phys. 376 (2019), no. 3, 1909–1956. +[KHZ+20] +A. A. Kulkarni, E. Hanson, R. Zhang, K. Thornton, and P. V. Braun, Archimedean lattices +emerge in template-directed eutectic solidification, Nature 577 (2020), no. 7790, 355–358. +[KLW21] +M. Keller, D. Lenz, and R. K. Wojciechowski, Graphs and Discrete Dirichlet Spaces, Springer +Nature Switzerland AG, 2021. +[KP07] +P. Kuchment and O. Post, On the spectra of carbon nano-structures, Comm. Math. Phys. +275 (2007), no. 3, 805–826. +[KS] +E. Korotyaev and N. Saburova, Two-sided estimates of total bandwidth for Schr¨odinger op- +erators on periodic graphs, arXiv:2106.08661 [math.SP], 2021. + +ROBUSTNESS OF FLAT BANDS +19 +[KS19] +E. Korotyaev and N. Saburova, Spectral estimates for Schr¨odinger operators on periodic +discrete graphs, St. Petersbg. Math. J. 30 (2019), no. 4, 667–698. +[LWL13] +Z. Liu, J. Wang, and J. Li, Dirac cones in two-dimensional systems: from hexagonal to +square lattices, Phys. Chem. Chem. Phys. 15 (2013), 18855–18862. +[MDY22] +C. Mielke, D. Das, and JX. et al. Yin, Time-reversal symmetry-breaking charge order in a +Kagome superconductor, Nature 602 (2022), 245–250. +[MW89] +B. Mohar and W. Woess, A survey on spectra of infinite graphs, Bull. London Math. Soc. +21 (1989), no. 3, 209–234. +[Nie82] +B. Nienhuis, Exact critical point and critical exponents of O(n) models in two dimensions, +Phys. Rev. Lett. 49 (1982), 1062–1065. +[Par07] +R. Parviainen, Estimation of bond percolation thresholds on the Archimedean lattices, J. +Phys. A: Math. Theor. 40 (2007), 9253–9258. +[PT21] +N. Peyerimhoff and M. T¨aufer, Eigenfunctions and the integrated density of states on +Archimedean tilings, J. Spectr. Theory 11 (2021), 461–488. +[PTV17] +N. Peyerimhoff, M. T¨aufer, and I. Veseli´c, Unique continuation principles and their absence +for schr¨odinger eigenfunctions on combinatorial and quantum graphs and in continuum space, +Nanosystems: Physics, Chemistry, Mathematics (2017), 216–230. +[SB20] +A. Samoilenka and E. Babaev, Boundary states with elevated critical temperatures in +Bardeen-Cooper-Schrieffer superconductors, Phys. Rev. B 101 (2020), no. 13. +[SB21] +A. Samoilenka and E. Babaev, Microscopic derivation of superconductor-insulator boundary +conditions for Ginzburg-Landau theory revisited: Enhanced superconductivity at boundaries +with and without magnetic field, Phys. Rev. B 103 (2021), 224516. +[SE64] +M. F. Sykes and J. W. Essam, Exact critical percolation probabilities for site and bond +problems in two dimensions, J. Math. Phys. 5 (1964), 1117–1127. +[SYY22] +M. Shi, F. Yu, Y. Yang et al., A new class of bilayer Kagome lattice compounds with Dirac +nodal lines and pressure-induced superconductivity., Nat. Commun. 2773 (2022). +[SZ99] +P. N. Suding and R. M. Ziff, Site percolation thresholds for Archimedean lattices, Phys. Rev. +E 60 (1999), no. 1, 275–283. +[TFGK22] +T. Tarnai, P. W. Fowler, S. D. Guest, and F. Kov´acs, Equiauxetic hinged Archimedean tilings, +Symmetry 14 (2022), 232. +[Ves04] +I. Veseli´c, Spectral analysis of percolation Hamiltonians, Math. Ann. 331 (2004), no. 4, 841– +865. +Joachim Kerner, Lehrgebiet Analysis, Fakult¨at Mathematik und Informatik, Fern- +Universit¨at in Hagen, D-58084 Hagen, Germany +Email address: joachim.kerner@fernuni-hagen.de +Matthias T¨aufer, Lehrgebiet Analysis, Fakult¨at Mathematik und Informatik, Fern- +Universit¨at in Hagen, D-58084 Hagen, Germany +Email address: matthias.taeufer@fernuni-hagen.de +Jens Wintermayr, Bergische Universit¨at Wuppertal, Fakult¨at f¨ur Mathematik und +Naturwissenschaften, 42119 Wuppertal, Germany +Email address: wintermayr@uni-wuppertal.de + diff --git a/5NE4T4oBgHgl3EQfbwwx/content/tmp_files/load_file.txt b/5NE4T4oBgHgl3EQfbwwx/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..400d87e012cc1f9b78cb545be99206a73811ee97 --- /dev/null +++ b/5NE4T4oBgHgl3EQfbwwx/content/tmp_files/load_file.txt @@ -0,0 +1,752 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf,len=751 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='05076v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='SP] 12 Jan 2023 ROBUSTNESS OF FLAT BANDS ON THE PERTURBED KAGOME AND THE PERTURBED SUPER-KAGOME LATTICE JOACHIM KERNER, MATTHIAS T¨AUFER, AND JENS WINTERMAYR Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' We study spectral properties of perturbed discrete Laplacians on two-dimen- sional Archimedean tilings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' The perturbation manifests itself in the introduction of non- trivial edge weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' We focus on the two lattices on which the unperturbed Laplacian exhibits flat bands, namely the (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='6)2 Kagome lattice and the (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='12)2 “Super-Kagome” lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' We characterize all possible choices for edge weights which lead to flat bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Furthermore, we discuss spectral consequences such as the emergence of new band gaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Among our main findings is that flat bands are robust under physically reasonable as- sumptions on the perturbation and we completely describe the perturbation-spectrum phase diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' The two flat bands in the Super-Kagome lattice are shown to even ex- hibit an “all-or-nothing” phenomenon in the sense that there is no perturbation which can destroy only one flat band while preserving the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Introduction This paper is about discrete Schr¨odinger operators on Archimedean tilings, a class of periodic two-dimensional lattices that were already investigated by Johannes Kepler in 1619 [Kep19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' They are natural candidates for the geometry of two-dimensional nanoma- terials and due to advances in this field, most prominently represented by graphene, they have become increasingly a focus of attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Much work has been devoted to understanding physical properties of such (new) materi- als [SYY22, TFGK22, dLFM19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Most importantly, it can be expected that the underlying geometry, that is the particular lattice, is a key feature determining physical properties of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' In fact, in particular in the mathematical physics literature, investiga- tions of the connection between the geometry (or topology) of a system and the spectral properties of the associated Hamiltonian have become ubiquitous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Classical examples in this context are so-called quantum waveguides [EK15, Exn20, Exn22] as well as quantum graphs [BK13, BE22];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' see also [KP07] for a relatively recent reference relevant in our context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' A closely related research direction is superconductivity: the existence of a boundary leads to boundary states in a superconductor with a higher critical temperature than the one of the bulk [SB20, SB21, HRS].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' In this spirit, it seems very promising to also study the interplay of geometry and many-particle phenomena on Archimedean tilings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Yet another related investigation can be found in [JBT21, SYY22] where another important quantum phenomenon, namely Bose-Einstein condensation, is examined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' It turns out that so-called flat bands, that are infinitely degenerate eigenvalues of the Hamiltonian, play an important role in understanding such many-particle effects, and for other physical phenomena [KFSH19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' One of the central motivations for this paper is to study robustness of flat bands under certain natural perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Two Archimedean tilings, the (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='6)2 Kagome lattice and the (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='122) tiling 1, which we shall dub Super-Kagome lattice for reasons that will become clear over the course Date: January 13, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' 1We explain the notation for the lattices in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' 1 2 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' KERNER, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' T¨AUFER, AND J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' WINTERMAYR of the article, stand out: they are the only Archimedean lattices on which the discrete, unweighted Laplacian has flat bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' In particular the Kagome lattice is a prominent model in physics that has recently enjoyed increasing interest [BM18, MDY22, Dia21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' From a mathematical point of view, our paper is motivated by [PT21] where flat bands for the discrete, unweighted Laplacian on Archimedean tilings have been studied in great detail, in combination with an explicit calculation of the integrated density of states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' A priory, the flat-band phenomena on the Kagome and Super-Kagome lattice seem very sensitive to perturbations: if one replaces the adjacency matrix or the Laplacian by a variant with periodically chosen edge weights, one will generically destroy flat bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' However, the results of this paper suggest that, if one looks at proper, meaningful variants of the discrete Laplacian which respect certain, natural symmetries of the tiling (we call them monomeric Laplacians in Definition 3), then flat bands will persist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Since monomericity is a physically justifiable assumption, this makes a strong case that flat bands are a robust phenomenon, caused by the geometry of the lattice alone and specific to these two lattices, see Theorems 6, and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Other questions of interest on periodic graphs concern existence, persistence and esti- mates on the width of spectral bands and the gaps between them [KS19, KS19, MW89].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' We will completely identify the spectra as a function of the perturbation in these cases, see Theorems 8, and 11 as well as Figures 3, and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' This provides an exhaustive descrip- tion of all nanomaterials based on Archimedean tilings on which discrete Laplacians can exhibit flat bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Our paper is organized as follows: Sections 2, and 3 are of introductory nature, intro- ducing the notion of and arguing for the relevance of Archimedean tilings, and defining a proper notion of a discrete Laplace operator with non-uniform edge weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Section 3 also introduces the notion of flat bands and argues why it suffices to restrict our attention to the (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='6)2 Kagome and the (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='122) Super-Kagome lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Sections 4, and 5 contain our main results on the Kagome and Super Kagome lattice, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' The contributions of this paper are: (i) We identify the Kagome and Super-Kagome lattice as the only Archimedean lat- tices on which a natural class of periodic, weighted Laplacians can have flat bands (Proposition 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' (ii) We describe all periodic edge weights which lead to the maximal possible number of bands on the Kagome and Super-Kagome lattice, and prove that this is equivalent to so-called monomericity of the edge weights (Theorems 6 and 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' (iii) We completely describe the spectrum in the monomeric Kagome and Super-Kagome lattice (Theorems 8 and 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' In particular, the monomeric Super-Kagome lattice has a surprisingly rich spectrum-perturbation phase diagram (Figure 5) which might bear relevance for various applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' (iv) In the Super-Kagome lattice, under a weaker condition than monomericity, namely constant vertex weight, we explicitely describe all remaining “spurious” edge weights which have only one flat band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' We describe the topology of this set within the parameter space and show in particular that it is disconnected from the monomeric two-band set (Theorem 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Archimedean tilings Archimedean, Keplerian or regular tilings are edge-to-edge tesselations of the Euclidean plane by regular convex polygons such that every vertex is surrounded by the same pattern of adjacent polygons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' We will adopt the notation of [GS89] and use the (counterclockwise) order of polygons arranged around a vertex as a symbol for a tiling (this is unique up to ROBUSTNESS OF FLAT BANDS 3 cyclic permutations), see Figure 1 for the (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='6)2 Kagome lattice and the (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='122) Super- Kagome lattice which will be investigated in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='6)2 Kagome lattice (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='122) Super-Kagome lattice Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' The two Archimedean tilings primarily investigated in this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' The first systematic investigation from 1619 is due to Kepler who identified all 11 such tilings [Kep19]2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Most importantly, Archimedean tilings provide natural candidates for geometries of two-dimensional nanomaterials since they form natural, symmetric arrange- ments of a single buiding block, positioned at every vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' And indeed, these lattices can be observed in many naturally occurring materials [FK58, FK59, KHZ+20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' From a physical point of view, two-dimensional materials such as graphene are inter- esting since they feature so-called Dirac points which are related to a specific behaviour of the electronic band structure of the material [FW12, LWL13, HC15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Also note that there are deep connections between Laplacians on these lattices, perco- lation, and self-avoiding walks which have also been studied extensively [SE64, Kes80, Nie82, SZ99, Ves04, Par07, Jac14, JSG16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' An important quantity in this context is the so-called connective constant, which is known only in few cases, for example on the hexagonal lattice [DCS12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Defining a suitable Hamiltonian Every Archimedean tiling can be regarded as an infinite discrete graph G = (V, E) with (countable) vertex set V and (countable) edge set E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' We write v ∼ w if the vertices v and w are joined by an edge and denote by |v| := #{w ∈ V : v ∼ w} the vertex degree of v (which in the case of Archimedean lattice graphs is v-independent).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Archimedean lattices are Z2-periodic, and there exists a cofinite Z2-action Z2 ∋ β �→ Tβ : V → V , that is a group of graph isomorphisms (intuitively understood as a group of shifts) iso- morphic to the group Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Let Q ⊂ V be a minimal (in particular finite) fundamental domain of this action, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' the quotient of V under the equivalence relation generated by the group of isomorphisms (Tβ)β∈Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' 2All 11 Archimedean tilings are: the (44) rectangular tiling, the (36) triangular tiling, the (63) hexa- gonal tiling, the (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='62) Kagome lattice, the (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='122) Super-Kagome lattice, the (33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='42) tiling, the (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='82) tiling, the (32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='4) tiling, the (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='4) tiling, the (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='12) tiling, and the (34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='6) tiling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' 4 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' KERNER, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' T¨AUFER, AND J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' WINTERMAYR In the unweighted case, a natural, normalized choice for the Hamiltonian is the discrete Laplacian (∆f)(v) := 1 |v| � w∼v (f(v) − f(w)) = f(v) − 1 |v| � w∼v f(w) , (1) as used for instance in [PT21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' It can be written as ∆f = Id − 1 |v|Π where Π is the adjacency matrix, that is Π(v, w) = 1 if v ∼ w and 0 else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' The following is standard: Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' The unweighted, normalized Laplacian (1) with a uniformly bounded vertex degree boasts the following properties: (i) All restrictions of ∆ to finitely many vertices are real-symmetric M-matrices, that is, their off-diagonal elements are non-positive¸ and all their eigenvalues are non- negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' (ii) The infimum of the spectrum of ∆ is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' (iii) All rows and columns of ∆ sum to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Furthermore, the spectrum is always contained in the interval [0, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Introducing non-trivial edge weights, we would like to keep a form of the Laplacian that preserves properties (i) to (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' A natural candidate, similar to formula (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='11) in [KS], is (∆γf)(v) := 1 � µ(v) � w∼v γvw � f(v) � µ(v) − f(w) � µ(w) � (2) where the edge weights γvw = γwv > 0 and vertex weights µ(v) satisfy the relation � w∼v γvw = µv for every v ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' (3) As long as the vertex weights µ(v) (and thus also the γvw) are uniformly bounded, this will lead to an operator with properties (i) to (iii) and spectrum contained in [0, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' In the literature, one often finds the definition (∆γf)(v) = 1 µ(v) � w∼v γvw (f(v) − f(w)) as a normalized, discrete Laplacian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Note that, whenever µ(v) ̸= µ(w) for some v ∼ w, then this will not lead to a self-adjoint operator, but it can be made self-adjoint on a suitably weighted ℓ2(V )-space, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' [KLW21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' If all µ(v) are the same, then this definition coincides with (2), and can be simplified to (∆γf)(v) = f(v) − 1 µ � w∼v γwvf(w) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' (4) Now, one can prescribe various degrees of the symmetry of the underlying Archimedean lattice to be respected by the Laplacian: Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Consider an Archimedean tiling (V, E) with periodic edge weights γvw = γwv > 0, that is γvw = γTβvTβw for all v, w ∈ V and β ∈ Z2, and corresponding vertex weights µ(v) = � w∼v γvw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Define the Laplacian ∆γ as in (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Then, we say that the Archimedean tiling with Laplacian ∆γ (1) has constant vertex weight, if there is µ > 0 such that µ(v) = µ for all v ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' (2) is monomeric if for all vertices v ∈ V the list of edge weights, arranged cyclically around v, coincides (up to cyclic permutations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' ROBUSTNESS OF FLAT BANDS 5 Clearly, (2) is stronger than (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' However, in either case, the Laplacian reduces to (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' The term “monomeric” is inspired by the fact that the associated operators can be interpreted as describing properties of nanomaterials formed from one type of monomeric building block, positioned at every vertex of an Archimedean tiling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Clearly, monomeric Laplacians on Archimedean lattices have constant vertex weights, but the converse is not true in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' However, we will see in Theorems 6 and 10 that on the Kagome and Super-Kagome lattice, the validity of the converse implication is equivalent to existence (or persistence) of all flat bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Also, monomericity seems a physically reasonable as- sumption for nanomaterials, which suggests that the emergence of flat bands, while a priori very sensitive to perturbations of coefficients in the operator, might nevertheless be robust within the class of physically relevant operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Next, let T2 = R2/Z2 be the flat torus and define for every θ ∈ T2 the |Q|-dimensional Hilbert space ℓ2(V )θ := {f : V → C | f(Tβv) = ei⟨θ,β⟩f(v)} with inner product ⟨f, g⟩θ := � v∈Q f(v)g(v) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Given the Laplacian (4) on ℓ2(V ) with properties described in Definition 3, we define on ℓ2(V )θ the operator (∆θ γf)(v) := f(v) − 1 µ � w∼v γwvf(w) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' (5) Clearly, (5) can be represented as a |Q|-dimensional Hermitian matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Due to Floquet theory, we have σ(∆γ) = � θ∈T2 σ(∆θ γ) , and the following statement holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Proposition 4 (See [PT21] and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Let E ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Then, the following are equivalent: (i) E ∈ σ(∆θ γ) for all θ ∈ T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' (ii) E ∈ σ(∆θ γ) for a positive measure subset of θ ∈ T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' (iii) There is an infinite orthonormal family eigenfunctions of ∆γ to the eigenvalue E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Each of them can be chosen to be supported on a finite number of vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' If any of (i) to (iii) is satisfied, we say that ∆γ has a flat band (at energy E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Note that, in the ℓ∞(V ) setting instead of the ℓ2(V ) setting, such infinitely degenerate eigenvalues are also referred to as “black hole eigenvalues” in [BL09].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Also, the existence of flat bands can be interpreted as a breakdown of the unique continuation principle [PTV17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' In the Hilbert space ℓ2(V ) setting, is known that for constant edge weights, the discrete Laplacian has flat bands only on two of the 11 Archimedean lattices, namely the (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='6)2 Kagome lattice and the (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='122) Super-Kagome lattice [PT21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Before turning to perturbed versions of those two lattices, one should verify that there won’t be any surprises on the other lattices: Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' On the Archimedean lattices (44), (36), (63), (33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='42), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='82), (32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='4), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='4), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='12), (34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='6), there is no choice of periodic (with respect to the fundamental cell on the lattice) edge weights γvw = γwv > 0 which will make the weighted adjacency 6 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' KERNER, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' T¨AUFER, AND J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' WINTERMAYR matrix Πγ(v, w) = � γvw if v ∼ w, 0 else have a flat band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Consequently, also the Laplacian with constant or monomeric edge weights has no flat bands on these lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Proposition 5 is proved by a series of straightforward but somewhat lengthy calculations in which one calculates the associated characteristic polynomials, and shows that there are no θ-independent roots, employing Proposition 4 (this should be compared to the proofs of Theorems 6 and 10 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' We omit them here for the sake of conciseness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' In any case, Proposition 5 justifies to restrict our attention to the (perturbed) Kagome and Super-Kagome lattices from now on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' The perturbed Kagome lattice In this section we discuss the Kagome lattice with non-uniform (periodic) edge weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' The elementary cell of the Kagome lattice contains three vertices and six edges (one can think of the edges as arranged around a hexagon).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' A priori, periodicity allows for six γ1 γ2 γ3 γ4 γ5 γ6 γ4 γ5 γ6 γ5 γ1 γ6 v1 v2 v3 v2 + ω1 v3 + ω1 v3 + ω2 v1 − ω2 v1 − ω1 v2 − ω2 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Fundamental domain of the Kagome lattice with edge weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' In the monomeric case, all edge weights around downwards pointing tri- angles are γ2 = γ4 = γ6 =: α and all edge weights on upwards pointing triangles are γ1 = γ3 = γ5 =: β, where 2α + 2β = µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' edge weights γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=', γ6 > 0, and the Floquet Laplacian ∆θ γ can be written as the Hermitian matrix ∆θ γ = Id −1 µ \uf8eb \uf8ed 0 γ3 + wγ6 wγ4 + zγ1 γ3 + wγ6 0 γ2 + zγ5 wγ4 + zγ1 γ2 + zγ5 0 \uf8f6 \uf8f8 , (6) where w := eiθ1 and z := eiθ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' We denote the three real eigenvalues of ∆θ γ by λ1(θ, γ) ≤ λ2(θ, γ) ≤ λ3(θ, γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Note that the six degrees of freedom are to be further reduced, depending on the following symmetry conditions: If we merely assume a constant vertex weight µ > 0, then identity (3) will impose the three additional linearly independent conditions γ1 + γ4 = γ2 + γ5 , γ3 + γ6 = γ2 + γ5 , γ1 + γ3 + γ4 + γ6 = µ , (7) ROBUSTNESS OF FLAT BANDS 7 and we end up with three degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' If we also assume monomericity, then it is easy to see that the only choice is the breathing Kagome lattice, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' [HKdP+22], with an edge weight α > 0 on all edges belonging to upwards pointing triangles and edge weight β > 0 on all edges belonging to downwards pointing triangles, where 2(α + β) = µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' After fixing the vertex weight µ, this amounts to only one degree of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Flat bands in the perturbed Kagome lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Consider the perturbed Kagome lattice with Laplacian (4), fixed vertex weight µ > 0 and periodic edge weights γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=', γ6 > 0, satisfying the condition (3) on vertex and edge weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Then, the following are equivalent: (i) There exists a flat band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' (ii) The vertex weights are monomeric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' More explicitly, there are α, β > 0 with 2(α + β) = µ such that γ2 = γ4 = γ6 := α, γ1 = γ3 = γ5 := β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' The rest of this subsection is devoted to the proof of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' We start with identi- fying flat bands using the weighted adjacency matrix Πθ γ := \uf8eb \uf8ed 0 γ3 + wγ6 wγ4 + zγ1 γ3 + wγ6 0 γ2 + zγ5 wγ4 + zγ1 γ2 + zγ5 0 \uf8f6 \uf8f8 (8) which is spectrally equivalent to ∆θ γ up to scaling and shifting via the relation ∆θ γ = Id −1 µΠθ γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' In order to find flat bands, we will identify conditions for θ-independent eigenvalues of Πθ γ and therefore calculate det(λ Id −Πθ γ) = −λ3 + λ(|A|2 + |B|2 + |C|2) + 2ℜ(ABC) where A := γ3 + wγ6, B := wγ4 + zγ1 and C := γ2 + zγ5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Rearranging the terms yields det(λ Id −Πθ γ) =(w + w)(λγ6γ3 + γ3γ2γ4 + γ6γ5γ1) +(z + z)(λγ5γ2 + γ6γ5γ4 + γ1γ3γ2) +(wz + zw)(λγ1γ4 + γ3γ5γ4 + γ6γ2γ1) +(−λ3 + λ(γ2 1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' + γ2 6) + 2(γ4γ6γ2 + γ3γ5γ1)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' The prefactors w + w = 2 cos θ1 , z + z = 2 cos θ2 , and wz + zw = 2 cos(θ1 − θ2) , are linearly independent as measurable functions of θ on T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Consequently, since all γi are positive, θ-independent eigenvalues exist if and only if the w and z-independent terms in every line are zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' This is only possible for negative λ, which (possibly after scaling the γi and µ for the moment) can be assumed to equal −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Therefore, we obtain the conditions γ3γ6 = γ2γ3γ4 + γ1γ5γ6 , γ2γ5 = γ4γ5γ6 + γ1γ2γ3 , γ1γ4 = γ3γ4γ5 + γ1γ2γ6 , (9) 8 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' KERNER, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' T¨AUFER, AND J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' WINTERMAYR and 1 − (γ2 1 + · · · + γ2 6) + 2 (γ2γ4γ6 + γ1γ3γ5) = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' (10) Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' The only positive solutions (meaning all γi are non-zero) of (7), (9), (10) are γ2 = γ4 = γ6 = x γ1 = γ3 = γ5 = y (11) with x, y ∈ (0, 1) and x + y = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' By a direct calculation (11) solves (7), (9), (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Conversely, assume that there are positive solutions γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=', γ6 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' From (9) we obtain γ3 = γ1γ5γ6 γ6 − γ2γ4 , γ1 = γ3γ4γ5 γ4 − γ2γ6 , and this implies γ6 > γ2γ4 and γ4 > γ2γ6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Hence, combining both equations yields γ2 2γ6 < γ6 which shows that γ2 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' In the same way one proves γi < 1 for every other i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Next, let γ2 + γ5 := Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' By (7) one immediately concludes γ1 + γ4 = γ3 + γ6 = Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Now, we add (9) and (10) and rearrange the equations to obtain 1 2 � γ2 1 + · · · + γ2 6 − 1 � + γ3γ6 + γ2γ5 + γ1γ4 =γ2γ4γ6 + γ1γ3γ5 + γ2γ3γ4 + γ1γ5γ6 + γ4γ5γ6 + γ1γ2γ3 + γ3γ4γ5 + γ1γ2γ6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' By repeated factorization, the right hand side simplifies to γ1γ3(γ2 + γ5) + γ3γ4(γ2 + γ5) + γ1γ6(γ2 + γ5) + γ4γ6(γ2 + γ5) = Λ3, (12) and since for the left hand side one has 1 2 � γ2 1 + · · · + γ2 6 − 1 � + γ3γ6 + γ2γ5 + γ1γ4 = 3Λ2 − 1 2 , we arrive at the polynomial Λ3 − 3Λ2 2 + 1 2 = 0 the only positive solution of which is Λ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Finally, adding the first the two equations of (9) yields γ3γ6 + γ2γ5 = (γ6γ5 + γ2γ3)(γ1 + γ4) = γ6γ5 + γ2γ3 and this implies γ5 = γ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Furthermore, adding the last two equations gives γ2γ5 + γ1γ4 = (γ4γ5 + γ1γ2)(γ3 + γ6) = γ4γ5 + γ1γ2 giving γ4 = γ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Conditions (7) hence give γ1 = γ5 and γ6 = γ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' This proves the statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' □ We are now in the position to prove Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Proof of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Comparing Πθ γ with ∆θ γ we conclude that ∆θ γ has a flat band with edge weights γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=', γ6 if and only if there exists δ > 0 such that Πθ γ has a flat band for edge weights δγ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=', δγ6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' From this observation the statement follows directly taking Lemma 7 into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' □ ROBUSTNESS OF FLAT BANDS 9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' The spectrum and band gaps in the monomeric Kagome lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' In the case where the perturbed Kagome lattice has a flat band, we further study the structure of the rest of the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' We reiterate that, due to Theorem 6, the existence of a flat band is equivalent to the weights being monomeric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' As shown for instance in [PT21], in the case where all edge weights are equal, the two other spectral bands, generated by the two other θ-dependent eigenvalues of ∆θ γ, touch at E = 3/4, and the derivative of the integrated density of states at E = 3/4 vanishes – an indication that the spectral density at 3/4 is sufficiently thin for a gap to form under perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' And indeed, this is the statement of the next theorem, which also characterises the width of the gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Theorem 8 (Band gaps in the perturbed Kagome lattice).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Consider the perturbed Kago- me lattice with fixed vertex weight µ > 0, and monomeric edge weights α, β > 0, satisfying 2(α + β) = µ as characterized in Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Then, the spectrum is given by I1 ∪ I2 := � 0, 3 4 − ���� 3α µ − 3 4 ���� � � �3 4 + ���� 3α µ − 3 4 ���� , 3 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Furthermore, there is always a flat band at 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Remark 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Theorem 8 states that, as soon as α ̸= β, or alternatively, α ̸= µ 4, a spectral gap of width ���� 6α µ − 3 2 ���� = 3 µ|α − β| will form around 3 4, see also Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' The flat band at 3 2 will always be connected to the energy band below it which means that the “touching” of the flat band at 3 2 is protected in the class of monomeric perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' I1 I2 α = µ 2 α = 0 α = µ 4 3 2 3 4 Flat band σ(∆γ) Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Spectrum of the monomeric (32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='62) Kagome lattice with vertex weight µ > 0 as a function of the parameter α ∈ (0, µ 2), describing the edge weights on edges adjacent to downwards pointing triangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' A calculation shows that the eigenvalues of ∆θ γ with the choice 2(α + β) = µ as in Theorem 6 are given by λ1,2(θ, γ) = 3 4 ± 1 4 � 1 + 8 � 1 + (F(θ) − 3) �2α µ − 4α2 µ2 �� and λ3(θ, γ) = 3 2 10 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' KERNER, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' T¨AUFER, AND J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' WINTERMAYR where F(θ) := cos(θ1) + cos(θ2) + cos(θ1 − θ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' The function T2 ∋ θ �→ F(θ) takes all values in [−3/2, 3], see Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='1 in [PT21], whence λ1(θ, γ) and λ2(θ, γ) take all values in the intervals � 0, 3 4 − ���� 3α µ − 3 4 ���� � , and �3 4 + ���� 3α µ − 3 4 ���� , 3 2 � , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' The perturbed Super-Kagome lattice In this section, we investigate the Archimedean tiling (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='122) which we call Super- Kagome lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Its minimal elementary cell contains six vertices and nine edges: three edges on upwards pointing triangles, three edges on downwards pointing triangles, and three edges bordering two dodecagons, see Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' v4 v3 v5 v6 v2 v1 v1 − ω2 v2 − ω1 v6 + ω1 v5 + ω2 γ7 γ3 γ2 γ1 γ5 γ6 γ4 γ9 γ8 γ8 γ9 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Fundamental domain of the (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='122) tiling with edge weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' In the monomeric case, all edge weights around triangles triangles are γ1 = · · = γ6 =: α and the remaining weights are γ7 = γ8 = γ9 =: β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Given a constant vertex weight µ > 0, the Floquet Laplacian (5) is a 6×6-matrix given by ∆θ γ = Id −1 µ \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed 0 γ4 γ6 0 zγ9 0 γ4 0 γ5 0 0 wγ8 γ6 γ5 0 γ7 0 0 0 0 γ7 0 γ3 γ2 zγ9 0 0 γ3 0 γ1 0 wγ8 0 γ2 γ1 0 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 , (13) where w := eiθ1, z := eiθ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' If we fix a constant vertex weight µ > 0, the condition � w∼v γvw = µ for all v ∈ V leads to µ = γ2 + γ3 + γ7 = γ5 + γ6 + γ7 = γ1 + γ2 + γ8 = γ4 + γ5 + γ8 = γ1 + γ3 + γ9 = γ4 + γ6 + γ9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' (14) This can be seen to be a linear system of 6 linearly independent equations with 9 unknowns, so the solution space is 3-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' More precisely, by appropriate additions, we infer the three identities 2γ1 + γ8 + γ9 = 2γ7 + γ2 + γ3, 2γ4 + γ8 + γ8 = 2γ7 + γ5 + dγ6, γ2 + γ3 = γ5 + γ6 (15) ROBUSTNESS OF FLAT BANDS 11 which imply γ1 = γ4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' The identities γ2 = γ5, and γ3 = γ6 follow by completely analogous calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' This leaves us with 6 independent variables γ1, γ2, γ3, and γ7, γ8, γ9 which are however still subject to the three conditions γ2 + γ3 + γ7 = γ1 + γ2 + γ8 = γ1 + γ3 + γ9 = µ from (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Therefore, we are left with three degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' If we additionally prescribe monomericity, it is easy to see that there is only one degree of freedom: All edges around triangles carry the weight α > 0, and all remaining edges (separating two dodecagons) carry the weight β > 0 under the condition 2α + β = µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Flat bands in the perturbed Super-Kagome lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Consider the perturbed Super-Kagome lattice with Laplacian (4), fixed vertex weight µ > 0, and periodic edge weights γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' , γ9 > 0 satisfying the condition (3) on vertex and edge weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Then, the following are equivalent: (i) There exist exactly two flat bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' (ii) The Super-Kagome lattice is monomeric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' More explicitly, there are α, β > 0 such that 2α + β = µ together with γ1 = γ2 = γ3 = γ4 = γ5 = γ6 = α , γ7 = γ8 = γ9 = β .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Recall that in the constant vertex weight case,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' we have γ1 = γ4 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' γ2 = γ5 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' and γ3 = γ6 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' and consider the weighted adjacency matrix Πθ γ := \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed 0 γ4 γ6 0 zγ9 0 γ4 0 γ5 0 0 wγ8 γ6 γ5 0 γ7 0 0 0 0 γ7 0 γ3 γ2 zγ9 0 0 γ3 0 γ1 0 wγ8 0 γ2 γ1 0 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed 0 γ1 γ3 0 zγ9 0 γ1 0 γ2 0 0 wγ8 γ3 γ2 0 γ7 0 0 0 0 γ7 0 γ3 γ2 zγ9 0 0 γ3 0 γ1 0 wγ8 0 γ2 γ1 0 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 (16) which is a shifted and scaled version of ∆θ γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' We calculate ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='det(λ Id −Πθ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='γ) = λ6 − λ4 � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='2γ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='1 + 2γ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='2 + 2γ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='3 + γ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='7 + γ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='8 + γ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='− 4λ3γ1γ2γ3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='+ λ2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='γ4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='1 + γ4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='2 + γ4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='3 + 2γ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='1γ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='2 + 2γ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='2γ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='3 + 2γ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='3γ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='1 + 2γ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='1γ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='7 + 2γ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='2γ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='9 + 2γ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='3γ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='8+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='+ γ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='7γ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='8 + γ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='8γ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='9 + γ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='9γ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='+ 4λγ1γ2γ3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='γ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='1 + γ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='2 + γ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='− γ4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='1γ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='7 − γ4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='2γ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='9 − γ4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='3γ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='8 − γ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='7γ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='8γ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='9 + 4γ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='1γ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='2γ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='− (w + w) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='λ2γ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='2γ7γ8 + 2λγ1γ2γ3γ7γ8 + γ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='1γ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='3γ7γ8 − γ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='2γ7γ8γ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='− (z + z) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='λ2γ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='3γ7γ9 + 2λγ1γ2γ3γ7γ9 + γ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='1γ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='2γ7γ9 − γ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='3γ7γ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='8γ9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='− (wz + wz) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='λ2γ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='1γ8γ9 + 2λγ1γ2γ3γ8γ9 + γ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='2γ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='3γ8γ9 − γ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='1γ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='7γ8γ9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Since w + w = 2 cos(θ1), z + z = 2 cos(θ2), and wz + wz = 2 cos(θ1 − θ2) are linearly on T2, λ is a θ-independent eigenvalue if and only if the conditions 12 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' KERNER, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' T¨AUFER, AND J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' WINTERMAYR λ2γ2 2 + 2λγ1γ2γ3 + γ2 1γ2 3 − γ2 2γ2 9 = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' λ2γ2 3 + 2λγ1γ2γ3 + γ2 1γ2 2 − γ2 3γ2 8 = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' λ2γ2 1 + 2λγ1γ2γ3 + γ2 2γ2 3 − γ2 1γ2 7 = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' (17) as well as λ6 − λ4 � 2γ2 1 + 2γ2 2 + 2γ2 3 + γ2 7 + γ2 8 + γ2 9 � − 4λ3γ1γ2γ3 + λ2� γ4 1 + γ4 2 + γ4 3 + 2γ2 1γ2 2 + 2γ2 2γ2 3 + 2γ2 3γ2 1 + 2γ2 1γ2 7 + 2γ2 2γ2 9 + 2γ2 3γ2 8+ + γ2 7γ2 8 + γ2 8γ2 9 + γ2 9γ2 7 � +4λγ1γ2γ3 � γ2 1 + γ2 2 + γ2 3 � −γ4 1γ2 7 − γ4 2γ2 9 − γ4 3γ2 8 − γ2 7γ2 8γ2 9 + 4γ2 1γ2 2γ2 3 = 0 (18) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='3 Conditions (17) imply that any θ-independent eigenvalue of the matrix Πθ γ must satisfy λ = −γ1γ3 γ2 ± γ9, λ = −γ1γ2 γ3 ± γ8, and λ = −γ2γ3 γ1 ± γ7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Since all γi are positive, the only way for these three equations to have the same set of solutions, that is for two flat bands to exist, is therefore − γ1γ3 γ2 + γ9 = −γ1γ2 γ3 + γ8 = −γ2γ3 γ1 + γ7 (19) together with − γ1γ3 γ2 − γ9 = −γ1γ2 γ3 − γ8 = −γ2γ3 γ1 − γ7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' (20) This implies that the matrix Πθ γ can only have two θ-independent eigenvalues if there are α, β > 0 with α = γ7 = γ8 = γ9 and β = γ1 = γ2 = γ3, that is the monomeric case, and the only candidates for these eigenvalues are −β ± α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' To see that they are indeed eigenvalues, one verifies by an explicit calculation that condition (18) is also fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' This shows the stated equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' □ Next, we further describe the spectrum of the monomeric Super-Kagome lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Theorem 11 (Band gaps in the perturbed Super-Kagome lattice).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Consider the perturbed Super-Kagome lattice with Laplacian (4) with fixed vertex weight µ > 0 and monomeric edge weights α, β > 0, satisfying 2α + β = µ as characterized in Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Then, the spectrum is given by I1 ∪ I2 := � 0, � 1 − α 2µ � − |3α − 2β| 2µ � � �� 1 − α 2µ � + |3α − 2β| 2µ , 2 − α µ � with flat bands at 3α µ and 2 − α µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' The spectrum and the position of the flat bands have been plotted in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' The spectrum generically consists of two distinct intervals (bands) except for the case 3α = 2β, that is α = 2µ 7 , in which the two bands touch and the spectrum consists of one interval with an embedded flat band in the middle as well as a flat band at its maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' This case α = 2µ 7 connects two regimes with different spectral pictures: 3As we will see later, despite its complexity, (18) will not impose further restrictions and hold in all relevant cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' This appears to be a consequence of symmetries of the lattice and the operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' ROBUSTNESS OF FLAT BANDS 13 If α > 2µ 7 the spectrum consists of two intervals the upper one of which has two flat bands at its endpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' In the special case of uniform edge weights (that is α = µ 3, this has already been observed, for instance in [PT21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' If α < 2µ 7 , the spectrum will again consist of two intervals each of which will have a flat band at its maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Somewhat surprisingly, the lower flat band has now attach itself to the lower interval I2 upon passing the critical parameter α = 2µ 7 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Another noteworthy observation is that no gap opens within the intervals I1 and I2, despite them being generated by two distinct Floquet eigenvalues and the density of states measure vanishing at a point in the interior of the bands, see again [PT21] for plots of the integrated density of states in the case of constant edge weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' In particular, this distinguishes the monomeric Super-Kagome lattice from the monomeric Kagome lattice where such a gap indeed opens within the spectrum at points of zero spectral density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' I1 I2 α = µ 2 µ 3 2µ 7 α = 0 2 Flat bands σ(∆γ) Constant edge weights Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Spectrum of the monomeric (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='122) “Super-Kagome” lattice with vertex weight µ > 0 as a function of the parameter α ∈ (0, µ 2), describ- ing the edge weights on edges adjacent to triangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Proof of Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' In the monomeric case, the characteristic polynomial det(λ Id −Πθ γ) of the matrix Πθ γ simplifies to ((α + λ)2 − β2)· (λ4 − 2αλ3 − (3α2 + 2β2)λ2 + (4α3 + 2αβ2)λ + 4α4 + α2β2 + β4 − 2α2β2F(θ1, θ2)) , where F(θ1, θ2) = cos(θ1) + cos(θ2) + cos(θ1 + θ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Its six roots are � −α ± β, 1 2 � α ± � 9α2 + 4β2 ± 4αβ � 3 + 2F(θ1, θ2) �� , whence the eigenvalues of ∆θ γ are given by λ1(θ, γ) = 1 − 1 2µ � α + � 9α2 + 4β2 + 4αβ � 3 + 2F(θ1, θ2) � , λ2(θ, γ) = 1 − 1 2µ � α + � 9α2 + 4β2 − 4αβ � 3 + 2F(θ1, θ2) � , λ3(θ, γ) = 1 + α − β µ = 3α µ = � 1 − α−|3α−2β| 2µ if 3α ≥ 2β , 1 − α−|3α−2β| 2µ if 3α < 2β , 14 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' KERNER, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' T¨AUFER, AND J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' WINTERMAYR λ4(θ, γ) = 1 − 1 2µ � α − � 9α2 + 4β2 − 4αβ � 3 + 2F(θ1, θ2) � , λ5(θ, γ) = 1 − 1 2µ � β − � 9α2 + 4β2 + 4αβ � 3 + 2F(θ1, θ2) � , λ6(θ, γ) = 1 + α + β µ = 2 − α µ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Using that the map T2 ∋ (θ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' θ2) �→ F(θ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' θ2) takes all values in the interval (−3/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' 3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' we conclude that the bands,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' generated by λ1(θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' γ) and λ2(θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' γ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' as well as the bands generated by λ4(θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' γ) and λ5(θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' γ) always touch,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' and the spectrum consists of the two intervals � min θ∈T2 λ1(θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' γ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' max θ∈T2 λ2(θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' γ) � � � min θ∈T2 λ4(θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' γ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' max θ∈T2 λ5(θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' γ) � = � 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' 1 − α + |3α − 2β| 2µ � � � 1 − α − |3α − 2β| 2µ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' 2 − α 2µ � = � 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' � 1 − α 2µ � − |3α − 2β| 2µ � � �� 1 − α 2µ � + |3α − 2β| 2µ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' 2 − α µ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' □ One might now wonder under which conditions only one flat band exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' The next theorem completely identifies all parameters for which one flat band exists: Theorem 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Consider the perturbed Super-Kagome lattice with Laplacian (4), fixed vertex weight µ > 0, and periodic edge weights γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' , γ9 > 0 satisfying the condition (3) on vertex and edge weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' The set of (γi) such that exactly one flat band exists consists of six connected components which have no mutual intersections and have no intersection with the two-flat-band parameter set, identified in Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' The solution space is invariant under those permutations of the γi which correspond to rotations of the lattice by 2π 3 , and 4π 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Modulo these permutations, the two connected components can be described as follows A one-dimensional submanifold, isomorphic to an interval, and explicitely descibed in equation (26), Two one-dimensional submanifolds each isomorphic to an interval, explicitely de- scribed in (28), and (30), which intersect in a single point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Proof of Theorem 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Recall that due to the reductions made at the beginning of the section, after fixing the constant vertex weight µ > 0, the space of edge weights is a 3-dimensional manifold in the 6-dimensional parameter space {γ1, γ2, γ3, γ7, γ8, γ9 > 0}, subject to the conditions γ1 + γ3 + γ9 = γ1 + γ2 + γ8 = γ2 + γ3 + γ7 = µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' (21) Furthermore, from the proof of Theorem 10 we infer that ∆γ has a flat band at λ if and only if the weighted adjacency matrix Πθ γ has the θ-independent eigenvalue ˜λ := µ(1−λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' This requires in particular that ˜λ = −γ1γ3 γ2 ± γ9 = −γ1γ2 γ3 ± γ8 = −γ2γ3 γ1 ± γ7 (22) holds with a certain combination of plus and minus signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Now, if equality in (22) holds with all three signs positive or all three signs negative, respectively, then the argument in the proof of Theorem 10 shows that this already implies that the edge weights are monomeric, the identities also hold with the opposite sign, the additional condition (18) is fulfilled, and there are two flat bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' As a consequence, the only chance for the existence ROBUSTNESS OF FLAT BANDS 15 of exactly one flat band is (22) to hold with different signs in front of γ7, γ8, γ9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Also, it is immediately clear that (22) with different signs does not allow for a monomeric and non-zero solution and hence the solution space consists of at most six mutually disjoint components which have no intersection with the two-flat-band manifold, identified in Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' By symmetry, it suffices to investigate two out of these six cases: Case(- + +): − γ1γ3 γ2 − γ9 = −γ1γ2 γ3 + γ8 = −γ2γ3 γ1 + γ7 = ˜λ , (23) and Case(+ - -): − γ1γ3 γ2 + γ9 = −γ1γ2 γ3 − γ8 = −γ2γ3 γ1 − γ7 = ˜λ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' (24) To solve Case(- + +), combine the second identities in in (21) and (23), to deduce γ3 − γ1 = γ2 γ1γ3 (γ2 1 − γ2 3) which, recalling γi > 0, is only possible if γ1 = γ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' But then, by (23), γ7 = γ8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Calling α′ := γ2, and β′ := γ9, we can use (21), to further express γ1 = γ3 = µ − β′ 2 , and γ7 = γ8 = µ + β′ 2 − α′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' (25) Next, we eliminate β′ by resolving the yet unused first identity in (23), which yields − (µ − β′)2 4α′ − β′ = −α′ + µ + β′ 2 − α′ ⇔ β′ = µ − 3α′ ± � 17α′2 − 8α′µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' This only has real solutions if α′ > 8 17µ > 1 3µ, thus only β′ = µ − 3α′ + � 17α′2 − 8α′µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' can be a positive solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Furthermore, we need β′ ∈ (0, µ), which is the case if and only if γ2 = α′ ∈ �µ 2 , µ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' We therefore find the one-parameter solution set Case (- + +) \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f3 γ1 = γ3 = µ−β′ 2 , γ2 = α′ ∈ � µ 2, µ � , γ7 = γ8 = µ+β′ 2 − α′, γ9 = β′ := µ − 3α′ + � 17α′2 − 8αµ (26) with energy ˜λ = −γ2 + γ7 = −2α′ + µ + β′ 2 = −2α′ + 2µ − 3α′ + � 17α′2 − 8α′µ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Finally, an explicit calculation shows that with these parameters, (18) is indeed fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' As for Case(+ - -), we combine the second identity in (21) with the second identity in (24) to deduce γ3 − γ1 = γ2 γ1γ3 (γ2 3 − γ2 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' (27) Identity (27) has two types of solutions: Case(+ - -)(a): γ1 = γ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' 16 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' KERNER, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' T¨AUFER, AND J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' WINTERMAYR As before we find γ7 = γ8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Let α′ := γ2, β′ := γ9, and combine the remaining first identity in (24) with (25) to solve for β′, finding − (µ − β′)2 4α′ + β′ = −µ + β′ 2 ⇔ β′ = µ + 3α′ ± � 9α′2 + 8α′µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Only the solution β′ = µ + 3α′ − � 9α′2 + 8α′µ has a chance to be in (0, µ), and, indeed, this is the case if and only if γ2 = α′ ∈ � 0, µ 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' We obtain the one-parameter solution set Case(+ - -)(a) \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f3 γ1 = γ3 = µ−β′ 2 , γ2 = α′ ∈ � 0, µ 2 � , γ7 = γ8 = µ+β′ 2 − α′, γ9 = β′ := µ + 3α′ − � 9α′2 + 8α′µ (28) with energy ˜λ = −γ2 − γ7 = −µ + β′ 2 = −2µ + 3α′ − � 9α′2 + 8α′µ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Again, an explicit calculation shows that (18) is fullfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Case(+ - -)(b): The other solution of (27) is γ1γ3 = γ2(γ1 + γ3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' We set α′′ := γ1, β′′ := γ3, whence γ2 = α′′β′′ α′′ + β′′, and use (21) to infer γ7 = µ − 2α′′β′′ + β′′2 α′′ + β′′ , γ8 = µ − α′′2 + 2α′′β′′ α′′ + β′′ , γ9 = µ − α′′ − β′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' (29) Plugging (29) into the yet unused first identity in (24), we arrive at − (α′′ + β′′) + µ − α′′ − β′′ = − α′′2 α′′ + β′′ − µ + α′′2 + 2α′′β′′ α′′ + β′′ ⇔ β′′ = µ − 3α′′ ± � (µ − 3α′′)2 + 4α′′(µ − α′′) 2 = µ − 3α′′ ± � µ2 − 2α′′µ + 5α′′2 2 We observe that only the solution with a plus has a chance to be positive and it is easy to see that this solution takes values in (0, µ) for all α′′ ∈ (0, µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' We obtain the one-parameter solution set Case (+ - -) (b) \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 γ1 = α′′ ∈ (0, µ) , γ2 = α′′β′′ α′′+β′′, γ3 = β′′ := µ−3α′′+√ µ2−2α′′µ+5α′′2 2 , γ7 = µ − 2α′′β′′+β′′2 α′′+β′′ , γ8 = µ − α′′2+2α′′β′′ α′′+β′′ , γ9 = µ − α′′ − β′′ (30) ROBUSTNESS OF FLAT BANDS 17 at energy ˜λ = −γ1γ3 γ2 + γ9 = µ − 2α′′ − 2β′′ = α − � µ2 − 2α′′µ + 5α′′2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Again, an explicit calculation verifies that with these choices, (18) is fullfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Finally, to conclude the claimed topological properties of the manifolds, we need to verify that the solution space (28) in Case(+ - -)(a) intersects the solution space (30) in Case(+ - -)(b) if and only if γ1 = γ3 = γ7 = γ8 = 2µ 5 , γ2 = γ9 = µ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' □ X2 X1 One flat band, Case(- + +) One flat band, Case(+ - -) (a) One flat band, Case(+ - -) (b) Monomeric edge weights, two flat bands Extremal cases, not belonging to the parameter space Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Schematic overview of the topology of the six “spurious” one- flat-band solution sets, and the monomeric two-flat-band manifold within the constant-vertex weight parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Case(- + +) solutions asymptotically meet the limit points of the two-flat-band manifold at one end of the parameter range, whereas Case(+ - -) (a) solutions asymptot- ically meet it at both ends of the parameter range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Remark 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Theorems 10 and 12 imply that the six one-flat-band components and the two-flat-band component are mutually disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' However, a closer analysis of the extremal cases in Formulas (26), (28), and (30), as well as of the monomeric case, implies that when sending the parameters to their extremal values, the three one-dimensional manifolds corresponding to Case(+ - -) (a), and the two-flat-band-manifold of solutions converge to the two points X1 := � 0, 0, 0, µ 2, µ 2 , µ 2 � and X2 := �µ 2, µ 2 , µ 2 , 0, 0, 0 � , which themselves do no longer belong to the space of admissible parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Likewise, the limit of solutions of Case(+ - -) in (26) corresponding to α′ = µ 2 corresponds to the the point X2, see also Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Acknowledgement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' JK would like to thank the Bergische Universit¨at Wuppertal where parts of this project were done while being on leave from the FernUniversit¨at in Hagen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' JK and MT also acknowledge support by the Cost action CA18232 through the summer school “Heat Kernels and Geometry: From Manifolds to Graphs” held in Bregenz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' MT would like to thank the Mittag-Leffler Institute where parts of this work were initiated during the trimester Program “Spectral Methods in Mathematical Physics”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' 18 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' KERNER, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' T¨AUFER, AND J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' WINTERMAYR References [BE22] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Baradaran and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Exner, Kagome network with vertex coupling of a preferred orientation, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Phys 63 (2022), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' 8, 083502.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' [BK13] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Berkolaiko and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Kuchment, Introduction to quantum graphs, American Mathematical Society, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' [BL09] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' von Below and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Lubary, Isospectral infinite graphs and networks and infinite eigen- value multiplicities, Netw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Heterog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Media 4 (2009), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' 3, 453–468.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' [BM18] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Bilitewski and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Moessner, Disordered flat bands on the Kagome lattice, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' B 98 (2018), 235109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' [DCS12] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Duminil-Copin and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Smirnov, The connective constant of the Honeycomb lattice equals � 2 + √ 2, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' 175 (2012), 1653–1665.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' [Dia21] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Dias, Topological properties of flat bands in generalized Kagome lattice materials, dis- sertation at KTH, School of Engineering Sciences (SCI), 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' [dLFM19] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' de Lima, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Ferreira, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Miwa, Topological flat band, Dirac fermions and quantum spin Hall phase in 2d Archimedean lattices, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' 21 (2019), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' 40, 22344–22350.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' [EK15] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Exner and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Kovaˇr´ık, Quantum waveguides, Springer International Publishing, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' [Exn20] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Exner, Spectral properties of soft quantum waveguides, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' A 53 (2020), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' 35, 355302.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' [Exn22] , Soft quantum waveguides in three dimensions, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' 63 (2022), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' 4, 042103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' [FK58] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Frank and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Kasper, Complex alloy structures regarded as sphere packings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Definitions and basic principles, Acta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Cryst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' 11 (1958), 184–190.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' [FK59] , Complex alloy structures regarded as sphere packings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Analysis and classification of representative structures, Acta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Cryst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' 12 (1959), 483–499.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' [FW12] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Fefferman and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Weinstein, Honeycomb lattice potentials and dirac points, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' 25 (2012), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' 4, 1169–1220.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' [GS89] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Gr¨unbaum and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Shephard, Tilings and patterns, A Series of Books in the Mathe- matical Sciences, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Freeman and Company, New York, 1989.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' [HC15] J-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Hou and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Chen, Hidden symmetry and protection of Dirac points on the honeycomb lattice, Scientific Reports 5 (2015), 17571.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' [HKdP+22] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Herrera, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Kempkes, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Blanco de Paz, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Garc´ıa-Etxarri, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Swart, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Morais Smith, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Bercioux, Corner modes of the breathing Kagome lattice: origin and robust- ness, arXiv:2201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='07576 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='mes-hall], 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' [HRS] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Hainzl, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Roos, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Seiringer, Boundary superconductivity in the BCS model, arXiv:2201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='08090 [math-ph], 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' [Jac14] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Jacobsen, High-precision percolation thresholds and Potts-model critical manifolds from graph polynomials, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' A: Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' 47 (2014), 135001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' [JBT21] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Julku, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Bruun, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' T¨orm¨a, Quantum geometry and flat band Bose-Einstein condensation, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' 127 (2021), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' [JSG16] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Jacobsen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Scullard, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Guttmann, On the growth constant for square-lattice self-avoiding walks, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' A Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' 49 (2016), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' 49, 494004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' [Kep19] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Kepler, Harmonices mundi, Johann Plack, 1619.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' [Kes80] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Kesten, The critical probability of bond percolation on the square lattice equals 1 2, Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' 74 (1980), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' 1, 41–59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' MR 575895 [KFSH19] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Koll´ar, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Fitzpatrick, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Sarnak, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Houck, Line-graph lattices: Euclidean and non-Euclidean flat bands, and implementations in circuit quantum electrodynamics, Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' 376 (2019), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' 3, 1909–1956.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' [KHZ+20] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Kulkarni, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Hanson, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Zhang, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Thornton, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Braun, Archimedean lattices emerge in template-directed eutectic solidification, Nature 577 (2020), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' 7790, 355–358.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' [KLW21] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Keller, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Lenz, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Wojciechowski, Graphs and Discrete Dirichlet Spaces, Springer Nature Switzerland AG, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' [KP07] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Kuchment and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Post, On the spectra of carbon nano-structures, Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' 275 (2007), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' 3, 805–826.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' [KS] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Korotyaev and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Saburova, Two-sided estimates of total bandwidth for Schr¨odinger op- erators on periodic graphs, arXiv:2106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='08661 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='SP], 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' ROBUSTNESS OF FLAT BANDS 19 [KS19] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Korotyaev and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Saburova, Spectral estimates for Schr¨odinger operators on periodic discrete graphs, St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Petersbg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' 30 (2019), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' 4, 667–698.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' [LWL13] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Liu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Wang, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Li, Dirac cones in two-dimensional systems: from hexagonal to square lattices, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' 15 (2013), 18855–18862.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' [MDY22] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Mielke, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Das, and JX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Yin, Time-reversal symmetry-breaking charge order in a Kagome superconductor, Nature 602 (2022), 245–250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' [MW89] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Mohar and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Woess, A survey on spectra of infinite graphs, Bull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' London Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' 21 (1989), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' 3, 209–234.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' [Nie82] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Nienhuis, Exact critical point and critical exponents of O(n) models in two dimensions, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' 49 (1982), 1062–1065.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' [Par07] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Parviainen, Estimation of bond percolation thresholds on the Archimedean lattices, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' A: Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' 40 (2007), 9253–9258.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' [PT21] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Peyerimhoff and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' T¨aufer, Eigenfunctions and the integrated density of states on Archimedean tilings, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Spectr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Theory 11 (2021), 461–488.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' [PTV17] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Peyerimhoff, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' T¨aufer, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Veseli´c, Unique continuation principles and their absence for schr¨odinger eigenfunctions on combinatorial and quantum graphs and in continuum space, Nanosystems: Physics, Chemistry, Mathematics (2017), 216–230.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' [SB20] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Samoilenka and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Babaev, Boundary states with elevated critical temperatures in Bardeen-Cooper-Schrieffer superconductors, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' B 101 (2020), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' [SB21] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Samoilenka and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Babaev, Microscopic derivation of superconductor-insulator boundary conditions for Ginzburg-Landau theory revisited: Enhanced superconductivity at boundaries with and without magnetic field, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' B 103 (2021), 224516.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' [SE64] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Sykes and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Essam, Exact critical percolation probabilities for site and bond problems in two dimensions, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' 5 (1964), 1117–1127.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' [SYY22] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Shi, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Yu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=', A new class of bilayer Kagome lattice compounds with Dirac nodal lines and pressure-induced superconductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=', Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' 2773 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' [SZ99] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Suding and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Ziff, Site percolation thresholds for Archimedean lattices, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' E 60 (1999), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' 1, 275–283.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' [TFGK22] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Tarnai, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Fowler, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Guest, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Kov´acs, Equiauxetic hinged Archimedean tilings, Symmetry 14 (2022), 232.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' [Ves04] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Veseli´c, Spectral analysis of percolation Hamiltonians, Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' 331 (2004), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' 4, 841– 865.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content=' Joachim Kerner, Lehrgebiet Analysis, Fakult¨at Mathematik und Informatik, Fern- Universit¨at in Hagen, D-58084 Hagen, Germany Email address: joachim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='kerner@fernuni-hagen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='de Matthias T¨aufer, Lehrgebiet Analysis, Fakult¨at Mathematik und Informatik, Fern- Universit¨at in Hagen, D-58084 Hagen, Germany Email address: matthias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='taeufer@fernuni-hagen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='de Jens Wintermayr, Bergische Universit¨at Wuppertal, Fakult¨at f¨ur Mathematik und Naturwissenschaften, 42119 Wuppertal, Germany Email address: wintermayr@uni-wuppertal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} +page_content='de' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE4T4oBgHgl3EQfbwwx/content/2301.05076v1.pdf'} diff --git a/5dE2T4oBgHgl3EQf6wiO/vector_store/index.faiss b/5dE2T4oBgHgl3EQf6wiO/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..762b82897201149af439b3915e25de520cba742b --- /dev/null +++ b/5dE2T4oBgHgl3EQf6wiO/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:f7c5ac07019534c0a15ea03a394858ca6ca1ee17325344182e69f1cbfafdd3df +size 4194349 diff --git a/6NAyT4oBgHgl3EQf2fnD/content/tmp_files/2301.00753v1.pdf.txt b/6NAyT4oBgHgl3EQf2fnD/content/tmp_files/2301.00753v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..716750d5f3eb38e43ad7903b35dab5b9eed781f2 --- /dev/null +++ b/6NAyT4oBgHgl3EQf2fnD/content/tmp_files/2301.00753v1.pdf.txt @@ -0,0 +1,1236 @@ +arXiv:2301.00753v1 [cs.IT] 2 Jan 2023 +POLYNOMIAL REPRESENTATION OF ADDITIVE CYCLIC CODES AND +NEW QUANTUM CODES +REZA DASTBASTEH AND KHALIL SHIVJI +Abstract. We give a polynomial representation for additive cyclic codes over Fp2. This repre- +sentation will be applied to uniquely present each additive cyclic code by at most two generator +polynomials. We determine the generator polynomials of all different additive cyclic codes. A +minimum distance lower bound for additive cyclic codes will also be provided using linear cyclic +codes over Fp. We classify all the symplectic self-dual, self-orthogonal, and nearly self-orthogonal +additive cyclic codes over Fp2. Finally, we present ten record-breaking binary quantum codes +after applying a quantum construction to self-orthogonal and nearly self-orthogonal additive +cyclic codes over F4. +Keywords: additive cyclic codes, quantum code, self-orthogonal codes, self-dual codes +1. Introduction +Quantum error-correcting codes, or simply quantum codes, are used in quantum computation +to protect quantum information from corruption by noise (decoherence). A general framework +of quantum codes is provided in [9, 13]. Throughout this paper, Fp2 is the finite field of p2 +elements, where p is a prime number. The parameters of a quantum code over Fp that encodes +k logical qubits to n physical qubits and has minimum distance d is denoted by [[n, k, d]]p. An +important family of quantum codes with many similar properties as classical block codes is +the family of quantum stabilizer codes. In particular, quantum stabilizer codes are constructed +using additive codes which are self-orthogonal with respect to a certain symplectic inner product. +Several constructions of quantum stabilizer codes from various classical codes are given in [18]. +An interesting modification of the original definition of quantum stabilizer codes is by relaxing +its self-orthogonality constraint [5, 19]. This method enables us to construct good quantum +codes using not necessarily self-orthogonal additive codes over F4. Previously, this modification +was applied for the construction of new quantum codes from different families of linear codes +[6, 10, 20]. +Additive cyclic codes are of interest due to their rich algebraic properties and application +in the construction of quantum codes. There have been several works in the literature toward +the classification of additive cyclic codes for different applications [1, 4, 7, 16, 17, 21], and also +due to their connection to other families of block codes such as quasi-cyclic codes [15]. +In +[16], a canonical decomposition of additive cyclic code over F4 was introduced using certain +finite field extensions of F4. This decomposition was applied to determine self-orthogonal and +self-dual additive cyclic codes over F4 with respect to the trace inner product. In [3], it was +shown that each additive cyclic code over F4 of length n can be generated by F2-span of at +most two polynomials in F4[x]/⟨xn − 1⟩ and their cyclic shifts. Moreover, a criterion for the +self-orthogonality of such codes with respect to the trace inner product was provided. Another +interesting construction for a subclass of additive cyclic code, namely twisted codes, was provided +in [1]. This construction is analogous to the way linear cyclic codes are constructed. In spite of +many useful properties of twisted codes, all additive cyclic codes cannot be described using the +theory of additive twisted codes. +1 + +POLYNOMIAL REPRESENTATION OF ADDITIVE CYCLIC CODES AND NEW QUANTUM CODES +2 +In this work, we first give a canonical representation of all Fp-additive cyclic codes over Fp2 +using at most two generator polynomials. Our representation is more computationally friendly +than the canonical representation of [16]. +This representation allows us to give a minimum +distance lower bound for additive cyclic codes over Fp2 using the minimum distance of linear +cyclic codes over Fp. +Moreover, we provide a unique set of generator polynomials for each +additive cyclic code over Fp2. +This representation of generator polynomials will be used to +characterize all self-orthogonal and self-dual additive cyclic codes with respect to the symplectic +inner product. We also determine the generator polynomials of the symplectic dual of a given +additive cyclic code over Fp2, and compute nearly the self-orthogonality of each additive cyclic +code using only its generator polynomials. This allows us to apply the nearly self-orthogonal +construction of quantum codes developed in [5, 19]. In particular, we provide a list of eleven +record-breaking binary quantum codes after applying the mentioned quantum construction to +nearly self-orthogonal additive cyclic codes. Furthermore, applying secondary constructions to +our new quantum codes produce many more record-breaking binary codes. Note that such new +quantum codes cannot be constructed using self-orthogonal additive cyclic codes of the same +length. +This paper is organized as follows. Section 2 briefly recalls the essential terminologies used in +this work. Section 3 gives a canonical representation of additive cyclic codes over Fp2. In fact, +we follow a module theory approach to decompose each additive cyclic code using its polynomial +representation in Fp2[x]/⟨xn −1⟩. In Section 4, we compute the symplectic dual of each additive +cyclic code. We provide the necessary and sufficient conditions for an additive cyclic code to +be self-orthogonal, self-dual, or nearly self-orthogonality with respect to the symplectic inner +product. Finally, in Section 5, we present the parameters of our record-breaking quantum codes. +2. Preliminaries +Let ω be a primitive element of Fp2. Then the set {1, ω} forms a basis for Fp2 over Fp. Let +a + bω and a′ + b′ω ∈ Fn +p2, where a, a′, b, b′ ∈ Fn +p. The symplectic inner product of a + bω and +a′ + b′ω is defined by +⟨a + bω, a′ + b′ω⟩s = a′ · b − a · b′. +(2.1) +An Fp-linear subspace C ⊆ Fn +p2 is called a length n additive code over Fp2. +We denote the +Fp-dimension of an additive code C over Fp2 with dimFp(C). Let C ⊆ Fn +p2 be an additive code +over Fp2 such that dimFp(C) = k. Then we call C an (n, pk) code. The set +C⊥s = {x ∈ Fn +p2 : ⟨x, y⟩s = 0 for all y ∈ C}. +is called the symplectic dual of C. One can easily see that C⊥s is an (n, p2n−k) additive code +over Fp2. The code C is called self-orthogonal (respectively self-dual) if C ⊆ C⊥s (respectively +if C = C⊥s). For each x ∈ Fn +p2, we denote the number of non-zero coordinates of x by wt(x). +Moreover, the minimum weight among non-zero vectors of an additive code C is denoted by +d(C). The connection between quantum stabilizer codes and classical additive codes was initially +formulated by the independent works of Calderbank, Rains, Shor, and Sloane [3] and Gottesman +[11]. A non-binary version of this connection is provided below. +Theorem 2.1. [18, Corollary 16] Let C be an (n, pn−k) additive code over Fp2. Then there exists +an [[n, k, d]]p quantum stabilizer code if C is symplectic self-orthogonal, where d = min{wt(x) : +x ∈ C⊥ +s \ C} if k > 0 and d = min{wt(x) : x ∈ C} if k = 0. +The quantum code of Theorem 2.1 is called pure if d = d(C⊥s). There are several secondary +constructions of quantum code. A short list of such constructions is provided below. + +POLYNOMIAL REPRESENTATION OF ADDITIVE CYCLIC CODES AND NEW QUANTUM CODES +3 +Theorem 2.2. [18, Section XV] Let C be an [[n, k, d]]p quantum code. +(1) If k > 0, then an [[n + 1, k, d]]p quantum code exists. +(2) If C is pure and n, d ≥ 2, then an [[n − 1, k + 1, d − 1]]p pure quantum code exists. +(3) If k > 1, then there exists an [[n, k − 1, d]]p quantum code. +3. Additive cyclic codes over Fp2 +Throughout this section, we assume that n is a positive integer such that (n, p) = 1 and +Fp2 = {α + βω : α, β ∈ Fp}, where ω is a root of a degree two irreducible polynomial over +Fp. In this section, we provide a canonical representation of additive cyclic codes over the field +Fp2. In particular, we give a unique representation of each additive cyclic code over Fp2 using +at most two generator polynomials. Moreover, we determine the generator polynomials of all +different additive cyclic codes over Fp2. In particular, each additive cyclic code over F2 +p is a linear +combination of cyclic shifts of its generator polynomials. Such representation is also suitable +for practical computations of additive cyclic codes, especially using Magma computer algebra +system [2]. More particularly, there exists a built-in function in Magma which forms additive +cyclic codes generated by two given generator polynomials. At the end of this section, we give +a minimum distance lower bound for the minimum distance of additive cyclic codes over Fp2 +using the minimum distance of linear cyclic codes over Fp. +Definition 3.1. An Fp-subspace C ⊆ Fn +p2 is called an additive cyclic code of length n over Fp2, +if for every (a0, a1, . . . , an−1) ∈ C, the vector (an−1, a0, . . . , an−2) is also a codeword of C. +We will use the following concepts of module theory frequently in this section, and for more +details one, for example, can see [8, Chapter 12]. Let R be a principal ideal domain and M be an +R-module. The annihilator of M is an ideal of R defined by {r ∈ R : rm = 0 for any m ∈ M}. +An element m ∈ M is called a torsion element, if there exists 0 ̸= r ∈ R such that rm = 0. +The module M is called a torsion module if all of its elements are torsion. +The following +theorem, known as the primary decomposition theorem of modules, plays an important role in +our representation of additive cyclic codes. +Theorem 3.2. [8, Chapter 12, Theorem 7] Let R be a principal ideal domain and M be a torsion +R-module with the annihilator ⟨a⟩ ̸= 0. Let a = u +n +� +i=1 +pai +i , where u is a unit and pi is a prime +element for each 1 ≤ i ≤ n. Then we can decompose M as a direct sum of its submodules in the +form +M = +n +� +i=1 +Ni, +(3.1) +where Ni = {x ∈ M : xpai +i = 0} for each 1 ≤ i ≤ n. +Each element (a0, a1, . . . , an−1) ∈ Fn +p2 can be represented uniquely as a polynomial in Fp2[x]/⟨xn− +1⟩ in the form +n−1 +� +i=0 +aixi. One can easily verify that, under this correspondence, a length n additive +cyclic codes over Fp2 is an Fp[x]-submodule of Fp2[x]/⟨xn − 1⟩. +Notation 3.3. Let f and g ∈ Fp2[x]/⟨xn −1⟩. We fix the following notations for the rest of this +paper. +(1) The ideal generated by f in Fp2[x]/⟨xn − 1⟩ is denoted by ⟨f⟩Fp2[x]. Equivalently it is +the Fp2[x]-submodule of Fp2[x]/⟨xn − 1⟩ generated by the polynomial f. + +POLYNOMIAL REPRESENTATION OF ADDITIVE CYCLIC CODES AND NEW QUANTUM CODES +4 +(2) The Fp[x]-submodule of Fp2[x]/⟨xn − 1⟩ generated by the polynomial g is denoted by +⟨g⟩Fp[x]. +A straightforward computation shows that the annihilator of Fp2[x]/⟨xn − 1⟩ as an Fp[x]- +module is the ideal ⟨xn−1⟩. Moreover, we can decompose xn−1 over Fp[x] as xn−1 = +s +� +i=1 +fi(x), +where each fi(x) is an irreducible polynomial corresponding to a p-cyclotomic coset modulo n. +Next, we apply Theorem 3.2 to Fp2[x]/⟨xn − 1⟩. It is straightforward to see that +Fp2[x]/⟨xn − 1⟩ = +s +� +i=1 +Ni, +(3.2) +where Ni = ⟨(xn − 1)/fi(x)⟩Fp2[x] for each 1 ≤ i ≤ s. We call a non-zero length n additive cyclic +code C over Fp2 irreducible if for any additive cyclic code D ⊆ C, then D = {0} or D = C. The +next lemma shows that each Ni can be decomposed as a direct sum of two irreducible additive +cyclic codes. We determine the generator polynomial of all irreducible additive cyclic codes +inside Ni and provide other useful information about additive cyclic codes inside each Ni. +Lemma 3.4. Let f(x) be an irreducible divisor of xn − 1 over Fp[x] with deg(f) = k and +N = ⟨(xn − 1)/f(x)⟩Fp2[x]. +(1) Let 0 ̸= r(x) ∈ N, then the set L = {r(x), xr(x), . . . , xk−1r(x)} forms a basis for +⟨r(x)⟩Fp[x] as an Fp vector space. +(2) Let 0 ̸= C ⊊ N be an additive cyclic code. The code C has Fp-dimension k and C = +⟨r(x)⟩Fp[x] for any 0 ̸= r(x) ∈ C. +(3) The additive cyclic code N can be decomposed as +N = ⟨(xn − 1)/f(x)⟩Fp[x] ⊕ ⟨ω((xn − 1)/f(x))⟩Fp[x]. +Moreover, dimFp(N) = 2k and N is linear over Fp2. +(4) The number of irreducible additive cyclic codes inside N is 2k + 1. In particular, the +following set gives all the different generator polynomials of such additive cyclic codes. +A = { +� +(xn − 1)/f(x) +�� +ω + g(x) +� +: g(x) ∈ Fp[x], deg(g(x)) < k} ∪ {(xn − 1)/f(x)}. +(3.3) +Proof. (1) Obviously L ⊆ ⟨r(x)⟩Fp[x]. Suppose, on the contrary, that L is linearly dependent +over Fp. Hence we can find a polynomial 0 ̸= s(x) ∈ Fp[x] of degree less than k such that +r(x)s(x) ≡ 0 (mod xn − 1). Since (xn − 1)/f(x) | r(x) and f(x) is irreducible, we conclude that +f(x) | s(x). However, it is a contradiction with the fact that deg(s(x)) < k. This shows that L +is linearly independent over Fp. Note that the set L ∪ {xkr(x)} is linearly dependent over Fp +as this new set generates f(x)r(x) ≡ 0 (mod xn − 1). In a similar fashion, one can show that +{xir(x)} for k < i < n − 1 can be written as a linear combination of elements of L over Fp. +Therefore, L forms a basis for ⟨r(x)⟩Fp[x]. +(2) Let 0 ̸= r(x) ∈ C. +Suppose in contrary that ⟨r(x)⟩Fp[x] ⊊ C. +Then there exists a +polynomial s(x) ∈ C such that s(x) ̸∈ ⟨r(x)⟩Fp[x]. Note that ⟨r(x)⟩Fp[x] ∩ ⟨s(x)⟩Fp[x] = {0} as +otherwise, by part (1), for any polynomial a(x) in the intersection, we have +⟨r(x)⟩Fp[x] = ⟨a(x)⟩Fp[x] = ⟨s(x)⟩Fp[x], +which is a contradiction. Thus C = ⟨r(x)⟩Fp[x] and has dimension k over Fp. +(3) It is easy to see that ⟨(xn − 1)/f(x)⟩Fp[x] ∩ ⟨ω((xn − 1)/f(x)⟩Fp[x] = {0} and +N = ⟨(xn − 1)/f(x)⟩Fp[x] ⊕ ⟨ω((xn − 1)/f(x))⟩Fp[x]. + +POLYNOMIAL REPRESENTATION OF ADDITIVE CYCLIC CODES AND NEW QUANTUM CODES +5 +Hence N has dimension 2k over Fp. The linearity part follows immediately from the structure +of its generator polynomials. +(4) In order to find an additive cyclic code with Fp-dimension k, we need to choose a nonzero +polynomial r(x) ∈ N to be its generator. Also, any non-zero elements of ⟨r(x)⟩Fp[x] generates +the same code. Hence the number of additive cyclic codes with one non-zero generator inside +N is 22k−1 +2k−1 = 2k + 1. +Let C1 and C2 be two k-dimensional additive cyclic codes inside N. If C1 ∩ C2 ̸= {0}, then +C1 = C2 by part (1). Equivalently, if C1 + C2 = N, then C1 ∩ C2 = {0}. Now we show that +different elements of the set A generate different codes. Let g(x) ∈ Fp[x] such that deg(g(x)) < k. +Clearly the additive cyclic code C1 = ⟨(xn − 1)/f(x), ((xn − 1)/f(x))(g(x) + ω)⟩Fp[x] contains +(xn − 1)/f(x) and ω(xn − 1)/f(x). Therefore C1 = N. So ⟨(xn − 1)/f(x)⟩Fp[x] and ⟨((xn − +1)/f(x))(g(x) + ω)⟩Fp[x] are different additive cyclic codes. +Let g1(x) and g2(x) ∈ Fp[x] be two different polynomials of degree less than k. The code +C = ⟨((xn − 1)/f(x))(ω + g1(x)), ((xn − 1)/f(x))(ω + g2(x))⟩Fp[x] contains (xn − 1)/f(x) and +ω(xn − 1)/f(x). It is mainly because +⟨ +� +(xn − 1)/f(x) +� +(g1(x) − g2(x))⟩Fp[x] = ⟨(xn − 1)/f(x)⟩Fp[x]. +Thus C = N. This implies that the additive cyclic codes ⟨((xn − 1)/f(x))(ω + g1(x))⟩Fp[x] and +⟨((xn−1)/f(x))(ω+g2(x))⟩Fp[x] are different. This proves that the set A contains all the different +generators of irreducible additive cyclic codes inside N. +□ +As we mentioned in part (1) of Lemma 3.4, each additive cyclic code inside ⟨(xn−1)/f(x)⟩Fp2[x] +can have many different generator polynomials. Through the next remark, we fix a canonical +representation for each additive cyclic code inside N. +Remark 3.5. For each additive code 0 ̸= C ⊊ ⟨(xn −1)/f(x)⟩Fp2[x], we fix its generator polyno- +mial inside the set A, introduced in (3.3), to be “the” generator polynomial of C. Similarly, the +additive cyclic code C′ = ⟨(xn−1)/f(x)⟩Fp2[x] can be generated by the polynomials (xn−1)/f(x) +and ω((xn − 1)/f(x)). We call them “the” generator polynomials of C′. +This representation helps to uniquely identify each additive cyclic code inside N and avoid +considering the same code more than once. Next, we use the result of Lemma 3.4 and characterize +all the additive cyclic codes of length n over Fp2. Recall that xn − 1 = +s +� +i=1 +fi(x), where fi(x) is +an irreducible polynomial over Fp[x] for each 1 ≤ i ≤ s and Ni = ⟨(xn − 1)/fi(x)⟩Fp2[x]. +Theorem 3.6. Let C be a length n additive cyclic code over Fp2. Then +(i) we can decompose the code C as C = +s +� +i=1 +Ci, where each Ci is an additive cyclic code +inside Ni. +(ii) we have C = ⟨g(x) + ωk(x), ωh(x)⟩Fp[x], where +(a) g(x) + ωk(x) = +s +� +i=1 +gi(x) + ωki(x), +(b) h(x) = +s +� +i=1 +hi(x), +(c) and Ci has the generator polynomial(s) gi(x) + ωki(x) and ωhi(x) selected as dis- +cussed in Remark 3.5. + +POLYNOMIAL REPRESENTATION OF ADDITIVE CYCLIC CODES AND NEW QUANTUM CODES +6 +(iii) dimFp(C) = +s +� +i=1 +(deg(fi) × # of non-zero generators of Ci). +Proof. (i) As we mentioned in (3.2), the following decomposition holds +Fp2[x]/⟨xn − 1⟩ = +s +� +i=1 +Ni. +So we can express C as C = �s +i=1 Ci, where each Ci is an additive cyclic codes inside Ni. +(ii) We show that the additive cyclic codes C = +s +� +i=1 +Ci and ⟨g(x) + ωk(x), ωh(x)⟩Fp[x] are the +same. First note that g(x) + ωk(x), ωh(x) ∈ C and thus ⟨g(x) + ωk(x), ωh(x)⟩Fp[x] ⊆ C. Let +1 ≤ i ≤ s be a fixed integer. Since +� +(xn − 1)/fi(x) +� +| gi(x), ki(x), hi(x) and +� +(xn − 1)/fi(x) +� +gj(x) ≡ +� +(xn − 1)/fi(x) +� +kj(x) ≡ +� +(xn − 1)/fi(x) +� +hj(x) ≡ 0 +(mod xn − 1) +for any j ̸= i, we have +� +(xn − 1)/fi(x) +�� +g(x) + ωk(x) +� +≡ +� +(xn − 1)/fi(x) +�� +gi(x) + ωki(x) +� +(mod xn − 1) +and +� +(xn − 1)/fi(x) +� +ωh(x) ≡ +� +(xn − 1)/fi(x) +� +ωhi(x) +(mod xn − 1). +Moreover, we have +Ci = ⟨gi(x)+ωki(x), ωhi(x)⟩Fp[x] = ⟨ +� +(xn −1)/fi(x) +�� +g(x)+ωk(x) +� +, +� +(xn −1)/fi(x) +� +ωh(x)⟩Fp[x]. +Thus +Ci ⊆ ⟨g(x) + ωk(x), ωh(x)⟩Fp[x]. +This show that +s +� +i=1 +Ci ⊆ ⟨g(x) + ωk(x), ωh(x)⟩Fp[x] and completes the proof. +(iii) Note that dimFp(C) = +s +� +i=1 +dimFp(Ci). Moreover, by Lemmas 3.4, dimFp(Ci) = 0, ki, or +2ki if Ci = 0, Ci is generated by one generator polynomial, or Ci has two generator polynomials, +respectively. Combining these facts with the result of part (i) completes this proof. +□ +Through the next corollary, we characterize all the length n irreducible additive cyclic codes +over Fp2. +Proposition 3.7. Let C be an additive cyclic code of length n over Fp2. Then C is irreducible +if and only if C = ⟨r(x)⟩Fp[x] for some 0 ̸= r(x) ∈ Ni and 1 ≤ i ≤ s. Moreover, there are +s +� +i=1 +(2deg(fi) + 1) many different irreducible additive cyclic codes. +Proof. Let C = ⟨r(x)⟩Fp[x] for some 0 ̸= r(x) ∈ Ni and 1 ≤ i ≤ s. The result of part (1) in +Lemma 3.4 shows that C is irreducible. Conversely, let C be an irreducible additive cyclic code. +Then by part (i) of Theorem 3.6 we have C = +s +� +i=1 +Ci. Since C is irreducible, we have C = Cj +for some 1 ≤ j ≤ s. Moreover, since Nj is not irreducible by Lemma 3.4 part (3), we conclude +that C = ⟨r(x)⟩Fp[x] for some 0 ̸= r(x) ∈ Nj. + +POLYNOMIAL REPRESENTATION OF ADDITIVE CYCLIC CODES AND NEW QUANTUM CODES +7 +Inside each Ni, there are 2deg(fi)+1 many different one generator additive cyclic codes. Hence +the total number of irreducible codes is +s +� +i=1 +(2deg(fi) + 1). +□ +Remark 3.8. Henceforth, we always represent each additive cyclic code with its generator +polynomials g(x) + ωk(x) and ωh(x) introduced in part (ii) of Theorem 3.6. Moreover, the way +we generate these polynomials is unique, and therefore each additive cyclic code has a unique +set of generators. +From now on, we call Fp2-linear cyclic codes simply linear cyclic codes. Let C = ⟨g(x) + +ωk(x), ωh(x)⟩Fp[x] be a length n additive cyclic code over Fp2. Note that Theorem 3.6 and part +(3) of Lemma 3.4 imply that C is linear if and only if g(x) = h(x) and k(x) = 0. Hence linear +cyclic codes can be easily distinguished from non-linear cyclic codes. +Next, we provide a minimum distance bound for additive cyclic codes using linear cyclic +codes over Fp. In general, the minimum distance computation for linear codes is faster than the +additive codes. Hence the following result can speed up the minimum distance computation for +additive cyclic codes. We denote the minimum distance of a code C with d(C). +Theorem 3.9. Let C = ⟨g(x) + ωk(x), ωh(x)⟩Fp[x] be a length n additive cyclic code over Fp2. +Let G(x) = +xn−1 +gcd(xn−1,g(x)), and let S(x) be the generator polynomial of the intersection of the +length n linear cyclic code generated by k(x) and the linear cyclic code generated by h(x) over +Fp. Suppose that D1, D2, D3, and D3 are the length n linear cyclic codes over Fp generated by +g(x), gcd(k(x), h(x)), gcd(G(x)k(x), h(x)), and +g(x)S(x) +gcd(xn−1,k(x)), respectively. Then +min{d(D3), d(D4), max{d(D1), d(D2)}} ≤ d(C). +(3.4) +Proof. Only the following three types of codewords may appear in the code C. +T1 = {a(x) ∈ C : 0 ̸= a(x) ∈ Fp[x]}, +T2 = {ωb(x) ∈ C : 0 ̸= b(x) ∈ Fp[x]}, +T3 = {a(x) + ωb(x) ∈ C : 0 ̸= a(x), 0 ̸= b(x) ∈ Fp[x]}. +We bound the minimum distance of C by considering the minimum distance in each of these +sets. Let f(x) ∈ T1. Then we can write it as f(x) = a1(x)(g(x) + ωk(x)) + b1(x)ωh(x) for some +a1(x), b1(x) ∈ Fp[x]. Hence f(x) = a1(x)g(x) and a1(x)k(x) + b1(x)h(x) ≡ 0 (mod xn − 1). +This implies that a1(x)k(x) is an element of the length n linear cyclic code over Fp generated +by S(x). Hence +S(x) +gcd(xn−1,k(x)) | a1(x). In other words, f(x) = a(x)g(x) ∈ D4. +Next, let ωf1(x) ∈ T2. Then ωf1(x) = a1(x)(g(x)+ωk(x))+b1(x)ωh(x) for some a1(x), b1(x) ∈ +Fp[x]. +Then a1(x)g(x) ≡ 0 (mod xn − 1) or equivalently G(x) | a1(x). +This implies that +f1(x) = a1(x)k(x) + b1(x)h(x). Therefore, f1(x) ∈ D3. +Finally, let a(x) + ωb(x) ∈ T3. Then a(x) + ωb(x) = l(x)(g(x) + ωk(x)) + m(x)ωh(x) for some +l(x), m(x) ∈ Fp[x]. Hence a(x) ∈ D1 and b(x) ∈ D2. This implies that wt(a(x) + ωb(x)) ≥ +max{d(D1), d(D2)}. +□ +Note that if Di = 0 for any value 1 ≤ i ≤ 4, then we simply discard this code in the minimum +distance lower bound of (3.4). For instance if D1 = 0, then the minimum distance lower bound +of (3.4) becomes +min{d(D3), d(D4), d(D2)} ≤ d(C). +The following corollary gives a modification of this result to additive cyclic codes, which are +generated by only one generator. In this result, the cyclic codes Ci are obtained from Di after + +POLYNOMIAL REPRESENTATION OF ADDITIVE CYCLIC CODES AND NEW QUANTUM CODES +8 +substituting h(x) with 0 in Theorem 3.9 for 1 ≤ i ≤ 3. +However, the code C4 is obtained +differently by considering a more direct observation. +Corollary 3.10. Let C = ⟨g(x) + ωk(x)⟩Fp[x] be a length n additive cyclic code over Fp2. Let +C1, C2, C3, and C4 be the length n linear cyclic codes over Fp generated by polynomials g(x), +k(x), +xn−1 +gcd(xn−1,g(x))k(x), and +xn−1 +gcd(xn−1,k(x))g(x), respectively. Then +min{d(C3), d(C4), max{d(C1), d(C2)}} ≤ d(C). +(3.5) +Proof. As we mentioned above, the code Ci all are obtained after applying the condition h(x) = 0 +in the structure of the codes Di for 1 ≤ i ≤ 3 in Theorem 3.9. Since the code D4 in Theorem +3.9 is applied to bound the minimum weight of the set +T1 = {a(x) ∈ C : 0 ̸= a(x) ∈ Fp[x]}, +we compute the minimum weight of T1 directly in this proof. Let f(x) ∈ T1. Then we can write +it as f(x) = a(x)(g(x)+ωk(x)) for some a(x) ∈ Fp[x]. Hence f(x) = a(x)g(x) and a(x)k(x) ≡ 0 +(mod xn − 1). This implies that +xn−1 +gcd(xn−1,k(x)) | a(x). Hence +xn−1 +gcd(xn−1,k(x))g(x) | a(x) and we +have f(x) ∈ C4. +□ +Next, we consider the restriction of the mentioned minimum distance bound to linear cyclic +codes with the generator polynomials g(x) + ωk(x) and h(x), where k(x) = 0. +Corollary 3.11. Let C = ⟨g(x), ωh(x)⟩Fp[x] be a length n additive cyclic code over Fp2. Let +E1 and E2 be the length n linear cyclic codes over Fp generated by polynomials g(x) and h(x), +respectively. Then +min{d(E1), d(E2)} ≤ d(C). +(3.6) +Proof. Applying the condition k(x) = 0 to Theorem 3.9 implies that D1 = D4 = E1 and +D2 = D3 = E2. Now the result follows from the minimum distance bound of Theorem 3.9. +□ +4. Symplectic inner product and dual of additive cyclic codes +In this section, we determine generator polynomials of the symplectic dual of a given additive +cyclic code over Fp2. Moreover, we give the generator polynomials of all self-orthogonal and self- +dual codes. We also measure how close is a given additive cyclic code from being symplectic self- +orthogonal. Recall that p is a prime number and n is a positive integer coprime to p. Moreover, +elements of Fp2 are represented by Fp2 = {α + βω : α, β ∈ Fp}, where ω is a root of a degree 2 +irreducible polynomial over Fp. Recall that in (2.1) we defined the symplectic inner product of +two elements in Fn +p2. We define the symplectic inner product of two polynomials analogously. In +particular, for c(x) = +n−1 +� +i=0 +(ai + ωbi)xi and c′(x) = +n−1 +� +i=0 +(a′ +i + ωb′ +i)xi ∈ Fp2[x]/⟨xn − 1⟩, we define +c(x) ∗ c′(x) = +n−1 +� +i=0 +(aib′ +i − a′ +ibi). +Here we use a different notation for the symplectic inner product to differentiate between the +vectors and polynomials as different objects. +Remark 4.1. Let c(x) = g1(x) + ωg2(x) and c′(x) = g′ +1(x) + ωg′ +2(x) be two polynomials +of Fp2[x]/⟨xn − 1⟩, where g1(x), g2(x), g′ +1(x), g′ +2(x) ∈ Fp[x]/⟨xn − 1⟩. Then c(x) ∗ c′(x) is the +constant term of g1(x)g′ +2(x−1) − g2(x)g′ +1(x−1) (mod xn − 1). A similar argument shows that +c(x) ∗ xic′(x) is the coefficient of xi in g1(x)g′ +2(x−1) − g2(x)g′ +1(x−1) (mod xn − 1). +Thus if + +POLYNOMIAL REPRESENTATION OF ADDITIVE CYCLIC CODES AND NEW QUANTUM CODES +9 +g1(x)g′ +2(x−1) − g2(x)g′ +1(x−1) ≡ 0 (mod xn − 1), then the code generated by c′(x) lies in the +symplectic dual of the code generated by c(x). We use this property very frequently through +this section. +One can easily verify that the symplectic dual of an additive cyclic code C over Fp2 is also +an additive cyclic code over Fp2. Recall that by Theorem 3.6 part (ii), each additive cyclic +code of length n over Fp2 can be represented uniquely as C = ⟨g1(x) + ωg2(x), h(x)⟩Fp[x], where +g1(x), g2(x), h(x) ∈ Fp[x]/⟨xn − 1⟩. Our next theorem gives a criterion for the self-orthogonality +of additive cyclic codes. The proof is very similar to that of [3, Theorem 14 part c]. +Theorem 4.2. Let C = ⟨g1(x) + ωg2(x), h(x)⟩Fp[x] be a length n additive cyclic code over Fp2. +The code C is self-orthogonal if and only if the following conditions are satisfied: +(1) g2(x)h(x−1) ≡ 0 (mod xn − 1), +(2) g1(x)g2(x−1) ≡ g2(x)g1(x−1) (mod xn − 1). +Proof. ⇒: Suppose that C is self-orthogonal. For each 0 ≤ i ≤ n − 1, the inner product of +g1(x) + ωg2(x) and xih(x) is the coefficient of xi in −g2(x)h(x−1) (mod xn − 1). Since C is self- +orthogonal, we have g2(x)h(x−1) ≡ 0 (mod xn − 1). Moreover, +� +xi(g1(x) + ωg2(x)) +� +∗ +� +g1(x) + +ωg2(x) +� +is the coefficient of xi in g1(x)g2(x−1) − g2(x)g1(x−1) (mod xn − 1). Hence, for each +0 ≤ i ≤ n − 1, the coefficient of xi in g1(x)g2(x−1) − g2(x)g1(x−1) (mod xn − 1) is zero. Thus +g1(x)g2(x−1) ≡ g2(x)g1(x−1) (mod xn − 1). +⇐: Conversely, the fact that g1(x)g2(x−1) ≡ g2(x)g1(x−1) (mod xn − 1) implies that all +the vectors inside ⟨g1(x) + ωg2(x)⟩Fp[x] are self-orthogonal. Moreover, since g2(x)h(x−1) ≡ 0 +(mod xn − 1), we conclude that h(x) is orthogonal to all the cyclic shifts of g1(x) + ωg2(x). +Finally h(x) ∗ xih(x) = 0 for each 0 ≤ i ≤ n − 1. So ⟨g1(x) + ωg2(x), h(x)⟩Fp[x] is a symplectic +self-orthogonal code. +□ +Recall that xn − 1 = +s +� +i=1 +fi(x), where each fi(x) is an irreducible polynomial in Fp[x]. +Moreover, as we mentioned earlier in (3.2), we have Fp2[x]/⟨xn − 1⟩ = +s +� +i=1 +Ni, where Ni = +⟨(xn − 1)/fi(x)⟩Fp2[x]. +Let α be a primitive n-th root of unity in a finite filed extension of +Fp. We denote the p-cyclotomic cosets modulo n by Zi for each 1 ≤ i ≤ s in the way that +fi(x) = +� +a∈Zi +(x − αi). This gives a one-to-one correspondence between the sets Ni and all the +p-cyclotomic cosets modulo n. Our first goal in this section is to find the symplectic dual of a +given additive cyclic code. In order to achieve this goal, we need a few preliminary results. In +the next lemma, we find the symplectic dual of each Ni. +Lemma 4.3. Let 1 ≤ i ≤ s and C = Ni. Then C⊥s = +s +� +k=1 +k̸=j +Nk, where Zj = −Zi. +Proof. First note that by Lemma 3.4 part (3) we have C = ⟨(xn−1)/fi(x), ω((xn−1)/fi(x))⟩Fp[x]. +If Zk ̸= −Zi, then fi(x) | (xn − 1)/fk(x−1) and fk(x) | (xn − 1)/fi(x−1). So we have +• +� +(xn − 1)/fi(x) +�� +(xn − 1)/fk(x−1) +� +≡ 0 (mod xn − 1) and +• +� +(xn − 1)/fk(x) +�� +(xn − 1)/fi(x−1) +� +≡ 0 (mod xn − 1). + +POLYNOMIAL REPRESENTATION OF ADDITIVE CYCLIC CODES AND NEW QUANTUM CODES +10 +Hence the symplectic inner product of each element of Ni and each element of Nk is zero by +definition. This proves that +s +� +k=1 +k̸=j +Nk ⊆ C⊥s. Note that both of Ni and Nj have Fp-dimension +2 deg(fi). Now, the facts that dimFp(C) + dimFp(C⊥s) = 2n and dimFp(C) = 2 deg(fi) implies +the other inclusion. +□ +Next, we find the symplectic dual of each irreducible additive cyclic code inside Ni for 1 ≤ +i ≤ s. +Lemma 4.4. Let C ⊊ Ni be a non-zero additive cyclic code for some 1 ≤ i ≤ s. Then +C⊥s = ( +s +� +k=1 +k̸=j +Nk) +� +⟨g1(x) + ωg2(x)⟩Fp[x], +(4.1) +where Zj = −Zi and +g1(x) + ωg2(x) = +� +((xn − 1)/fj(x))(s(x−1) + ω) +if C = ⟨ +� +(xn − 1)/fi(x) +�� +ω + s(x) +� +⟩Fp[x] +(xn − 1)/fj(x) +if C = ⟨(xn − 1)/fi(x)⟩Fp[x] +. +Proof. By Lemma 4.3, one can see that +s +� +k=1 +k̸=j +Nk ⊆ C⊥s. Note that dimFp(⟨g1(x)+ωg2(x)⟩Fp[x]) = +dimFp(C). So it is sufficient to show that C is orthogonal to g1(x) + ωg2(x) and all its cyclic +shifts. We prove the latter statement in two steps. First suppose that C = ⟨ +� +(xn−1)/fi(x) +�� +ω+ +s(x) +� +⟩Fp[x] for some s(x) ∈ Fp[x]. +To show that the codes C and ⟨g1(x) + ωg2(x)⟩Fp[x] are +orthogonal, we apply Remark 4.1. In particular, +� +(xn − 1)/fi(x) +� +s(x)g2(x−1) − +� +(xn − 1)/fi(x) +� +g1(x−1) ≡ +� +(xn − 1)/fi(x) +� +s(x) +� +(xn − 1)/fi(x) +� +− +� +(xn − 1)/fi(x) +�� +(xn − 1)/fi(x) +� +s(x) ≡ 0 +(mod xn − 1). +Next, suppose that C = ⟨(xn − 1)/fi(x)⟩Fp[x]. Then +� +(xn − 1)/fi(x) +� +g2(x−1) − +� +(xn − 1)/fi(x) +� +g1(x−1) ≡ +� +(xn − 1)/fi(x) +� +0 − 0 +� +(xn − 1)/fj(x) +� +s(x) +≡ 0 +(mod xn − 1). +This shows that the code C is orthogonal to the additive cyclic code generated by g1(x)+ωg2(x) +and completes the proof. +□ +Note that as we showed in Lemma 4.4, when C = ⟨ +� +(xn − 1)/fi(x) +�� +ω + s(x) +� +⟩Fp[x], its +symplectic inner product contains the code C′ = ⟨(xn − 1)/fj(x))(s(x−1) + ω)⟩Fp[x]. The code +C′ is not in one of the forms given in Lemma 3.4 part (4). In order to express the code C′ +using the standard notation introduced in 3.4 part (4), we choose its generator to be g(x) = +(xn − 1)/fj(x))(t(x) + ω), where t(x) ≡ s(x−1) (mod fj(x)). Now it is easy to see that g(x) +belongs to the set A introduced in Lemma 3.4 part (4) and C′ = ⟨(xn −1)/fj(x))(t(x)+ω)⟩Fp[x]. +Next, we combine the results of the previous two lemmas and the result of Theorem 3.6 to +determine generator polynomials of the symplectic dual for any additive cyclic code. + +POLYNOMIAL REPRESENTATION OF ADDITIVE CYCLIC CODES AND NEW QUANTUM CODES +11 +Theorem 4.5. Let C be a length n additive cyclic code over Fp2 such that C = +s +� +i=1 +Ci, where +Ci is an additive cyclic codes inside Ni for each 1 ≤ i ≤ s. +Then C⊥s = ⟨ +s +� +i=1 +gi(x) + +ωki(x), +s +� +i=1 +ωhi(x)⟩Fp[x], where for each 1 ≤ i ≤ s we have Zj = −Zi and +• gi(x) = ki(x) = hi(x) = 0 if Cj = Nj, +• gi(x) = hi(x) = (xn − 1)/fi(x) and ki(x) = 0 if Cj = 0, +• gi(x)+ωki(x) = +� +(xn −1)/fi(x) +�� +ω +ti(x) +� +and hi(x) = 0 if Cj = ⟨ +� +(xn −1)/fj(x) +�� +ω + +sj(x) +� +⟩Fp[x] and ti(x) ≡ sj(x−1) (mod fj(x)), +• gi(x) = (xn − 1)/fi(x) and ki(x) = hi(x) = 0 if Cj = ⟨(xn − 1)/fj(x)⟩Fp[x]. +Proof. We apply Lemmas 4.3 and 4.4 to prove the statement. If Cj = Nj, then C⊥s ∩ Ni = {0} +by Lemma 4.3. Moreover, by the same lemma, if Cj = 0, then Ni ⊆ C⊥s. This proves the first +two bullets. Finally, Lemma 4.4 implies that +• ⟨ +� +(xn − 1)/fi(x) +�� +ω + ti(x) +� +⟩Fp[x] ⊆ C⊥s if Cj = ⟨ +� +(xn − 1)/fj(x) +�� +ω + sj(x) +� +⟩Fp[x], and +• ⟨(xn − 1)/fi(x)⟩Fp[x] ⊆ C⊥s if Cj = ⟨(xn − 1)/fi(x)⟩Fp[x]. +This proves the statements of the last two bullets. +□ +To determine self-orthogonal and self-dual additive cyclic codes over Fp2, we need more infor- +mation about irreducible factors of xn − 1 over Fp. Let Z1, Z2, . . . , Zr and Z′ +1, −Z′ +1, . . . , Z′ +t, −Z′ +t +be all the p-cyclotomic cosets modulo n, where Zi = −Zi and r + 2t = s. +Each Zi is in +correspondence to an irreducible polynomial fi(x) and (Z′ +j, −Z′ +j) are in correspondence with an +irreducible pair of polynomials (fj1(x), fj2(x)) over Fp. Therefore, we can rewrite the irreducible +decomposition of xn − 1 as +xn − 1 = +r +� +i=1 +fi(x) +t� +j=1 +fj1(x)fj2(x). +We use the above representation of cyclotomic cosets in the upcoming results. Next, we classify +self-orthogonal and self-dual additive codes over Fp2. +Theorem 4.6. Let C be a length n additive cyclic code over Fp2 such that C = +s +� +i=1 +Ci, where +Ci is an additive cyclic codes inside Ni for each 1 ≤ i ≤ s. Then C is symplectic self-orthogonal +if and only if +(1) for all 1 ≤ k ≤ r only one of the following holds. +(a) Ck = 0. +(b) Ck = ⟨((xn − 1)/fk(x))(s(x) + ω)⟩Fp[x], where fk | s(x−1) − s(x). +(c) Ck = ⟨(xn − 1)/fk(x)⟩Fp[x]. +(2) for all 1 ≤ j ≤ t only one of the following holds. +(a) Cj1 = 0 or Cj2 = 0. +(b) Cj1 = ⟨((xn − 1)/fj1(x))(s(x) + ω)⟩Fp[x] and Cj2 = ⟨((xn − 1)/fj2(x))(s(x−1) + +ω)⟩Fp[x]. +(c) Cj1 = ⟨(xn − 1)/fj1(x)⟩Fp[x] and Cj2 = ⟨(xn − 1)/fj2(x)⟩Fp[x]. + +POLYNOMIAL REPRESENTATION OF ADDITIVE CYCLIC CODES AND NEW QUANTUM CODES +12 +Proof. First, let 1 ≤ k ≤ r. By Lemma 4.3, if Ck = Nk, then C⊥s ∩ Nk = {0}. So Ck cannot +have two generator polynomials. Moreover, by Lemma 4.4, if 0 ̸= Ck = ⟨((xn −1)/fk(x))(s(x)+ +ω)⟩Fp[x], then ⟨((xn − 1)/fk(x))(s(x−1) + ω)⟩Fp[x] ⊆ C⊥s. Thus Ck is self-orthogonal if and only +if +Ck = ⟨((xn − 1)/fk(x))(s(x) + ω)⟩Fp[x] = ⟨((xn − 1)/fk(x))(s(x−1) + ω)⟩Fp[x]. +Note that the above equality holds if and only if fk | s(x−1) − s(x). Thus Ck is self-orthogonal +if and only if one of the conditions of Part (1) follows. +Next, let 1 ≤ j ≤ t. By Lemma 4.3, if Cj1 = Nj1, then C⊥s ∩ Nj2 = {0}. So if one of Cj1 or +Cj2 has two generator polynomials, the other code should be zero. Moreover, the same lemma +shows that if Cj1 = 0 or Cj2 = 0, then Cj1 + Cj2 is self-orthogonal. So we concentrate only +on the case when both Cj1 and Cj2 have exactly one non-zero generator. By Lemma 4.4, if +Cj1 = ⟨((xn − 1)/fj1(x))(s(x) + ω)⟩Fp[x], then +C⊥s ∩ Nj2 = ⟨((xn − 1)/fj2(x))(s(x−1) + ω)⟩Fp[x]. +In this case, the code Cj1 ⊕ Cj2 is self-orthogonal if and only if condition (2)(b) is satisfied. +Condition (2)(c) follows similarly by applying Lemma 4.4. +□ +Next, we use the above conditions to characterize all the symplectic self-dual additive cyclic +codes over Fp2. +Corollary 4.7. Let C be a length n additive cyclic code over Fp2 such that C = +s +� +i=1 +Ci, where +Ci is an additive cyclic codes inside Ni for each 1 ≤ i ≤ s. Then C is symplectic self-dual if and +only if +(1) for all 1 ≤ k ≤ r only one of the following holds. +(a) Ck = ⟨((xn − 1)/fk(x))(s(x) + ω)⟩Fp[x] where fk | s(x−1) − s(x). +(b) Ck = ⟨(xn − 1)/fk(x)⟩Fp[x]. +(2) for all 1 ≤ j ≤ t only one of the following holds. +(a) Cj1 = 0 and Cj2 = Nj2. +(b) Cj2 = 0 and Cj1 = Nj1. +(c) Cj1 = ⟨((xn − 1)/fj1(x))(s(x) + ω)⟩Fp[x] and Cj2 = ⟨((xn − 1)/fj2(x))(s(x−1) + +ω)⟩Fp[x]. +(d) Cj1 = ⟨(xn − 1)/fj1(x)⟩Fp[x] and Cj2 = ⟨(xn − 1)/fj2(x)⟩Fp[x]. +Proof. Note that all the self-dual additive cyclic codes over Fp2 satisfy the conditions of Theorem +4.6 and have maximal dimension. +Thus the result easily follows by implying the maximal +property into the conditions of theorem 4.6. +□ +Our next goal is to compute the parameter e = dimFp(C) − dimFp(C ∩ C⊥s) for all additive +cyclic codes. The parameter e determines how close an additive cyclic code C is from being +self-orthogonal. This parameter plays an important role in the quantum construction that we +are applying in the next section. +Theorem 4.8. Let C be a length n additive cyclic code over Fp2 such that C = +s +� +i=1 +Ci, where +Ci is an additive cyclic codes inside Ni for each 1 ≤ i ≤ s. Let +(1) B1 = {α1, α2, . . . , αt1} ⊆ {1, 2, . . . , r} such that Cαl = Nαl for all 1 ≤ l ≤ t1, +(2) B2 = {β1, β2, . . . , βt2} ⊆ {1, 2, . . . , r} such that Cβl = ⟨((xn − 1)/fβl(x))(sβl(x) + ω)⟩Fp[x] +and fβl ∤ sβl(x−1) − sβl(x) for all 1 ≤ l ≤ t2, + +POLYNOMIAL REPRESENTATION OF ADDITIVE CYCLIC CODES AND NEW QUANTUM CODES +13 +(3) B3 = {γ1, γ2, . . . , γt3} ⊆ {1, 2, . . . , t} such that one of Cγl1 and Cγl2 is generated by two +polynomials and the other one has only one generator polynomial for all 1 ≤ l ≤ t3, +(4) B4 = {κ1, κ2, . . . , κt4} ⊆ {1, 2, . . . , t} such that both of Cκl1 and Cκl2 are generated by +two polynomials for all 1 ≤ l ≤ t4, +(5) B5 = {σ1, σ2, . . . , σt5} ⊆ {1, 2, . . . , t} such that both of Cσl1 and Cσl2 are generated by +one polynomial for all 1 ≤ l ≤ t5 and +(a) if Cσl1 = ⟨((xn−1)/fσl1(x))(sσl(x)+ω)⟩Fp[x], then Cσl2 ̸= ⟨((xn−1)/fσl2(x))(sσl(x−1)+ +ω)⟩Fp[x]. +(b) if Cσl1 = ⟨(xn − 1)/fσl1(x)⟩Fp[x], then Cσl2 ̸= ⟨(xn − 1)/fσl2(x))⟩Fp[x]. +Then +e = dimFp(C)−dimFp(C∩C⊥s) = +t1 +� +l=1 +2|Zαl|+ +t2 +� +l=1 +|Zβl|+ +t3 +� +l=1 +2|Zγl|+ +t4 +� +l=1 +4|Zκl|+ +t5 +� +l=1 +2|Zσl|. (4.2) +Proof. By Theorem 4.6, an additive cyclic code is not symplectic self-orthogonal if and only if at +least one of the sets B1 −B5 is non-empty. Next, we consider all scenarios (1)-(5) independently. +(1) Let j ∈ B1. In this case, C⊥s ∩ Cj = {0} which implies that dimFp(Cj) − dimFp(Cj ∩ +C⊥s) = 2|Zj|. +(2) Let j ∈ B2. In this case, C⊥s ∩ Cj = {0} which implies that dimFp(Cj) − dimFp(Cj ∩ +C⊥s) = |Zj|. +(3) Let j ∈ B3. +Without loss of generality we assume that Cj1 = Nj1 and Cj2 is an +irreducible subcode of Nj2. In this case, the intersection C⊥s∩(Cj1⊕Cj2) is an irreducible +subcode of Nj1 which implies that dimFp(Cj)−dimFp((Cj1⊕Cj2)∩C⊥s) = 3|Zj|−|Zj| = +2|Zj|. +(4) Let j ∈ B4. In this case, C⊥s ∩ (Cj1 ⊕ Cj2) = {0} which implies that dimFp(Cj) − +dimFp((Cj1 ⊕ Cj2) ∩ C⊥s) = 4|Zj|. +(5) Let j ∈ B5. In both parts (a) and (b), C⊥s ∩ (Cj1 ⊕ Cj2) = {0} which implies that +dimFp(Cj) − dimFp((Cj1 ⊕ Cj2) ∩ C⊥s) = 2|Zj|. +Now, the result follows by combining the above observations. +□ +Note that the case (2) of Theorem 4.8 never happens for Ci with deg(fi(x)) = 1. Moreover, +for each 1 ≤ i ≤ r, the cyclotomic coset Zi either is a singleton or it has an even size. This is +mainly because for each 0 ̸= a ∈ Zi, if a ≡ −a (mod n), then n | 2a, which implies that n is +even. Hence in this case p ̸= 2 (we assumed that gcd(n, p) = 1) and Zi = {a}. Therefore, if +Zi satisfies the case (2) of Theorem 4.8 and |Zi| > 1, then for any a ∈ Zi, we have −a ∈ Zi +and −a ̸≡ a (mod n). This implies that |Zi| is an even integer. This fact and the formula in +(4.2) imply that the nearly self-orthogonality parameter e of an additive cyclic code is always +an even integer. Next, we classify additive cyclic codes with small values of e. First, we need +the following preliminary result. +Lemma 4.9. Let p be a prime number and gcd(n, p) = 1 for some positive number n. +(i) If gcd(n, p − 1) = d, then there are d singleton p-cyclotomic cosets modulo n and all of +their coset leaders are {k n +d : 0 ≤ k ≤ d − 1}. +(ii) If gcd(n, p − 1) = d and gcd(n, p2 − 1) = d′. Then there are d′−d +2 +p-cyclotomic cosets +modulo n of size two. + +POLYNOMIAL REPRESENTATION OF ADDITIVE CYCLIC CODES AND NEW QUANTUM CODES +14 +Proof. (i) The proof easily follows from the fact that {a} is a singleton coset if and only if a ≡ pa +(mod n) or equivalently if and only if a(p − 1) ≡ 0 (mod n). By elementary number theory, if +gcd(n, p − 1) = d, then the latter equation has d solutions in the forms {k n +d : 0 ≤ k ≤ d − 1}. +(ii) A p-cyclotomic coset modulo n containing a has size two if and only if a ≡ p2a (mod n) +and a ̸≡ pa (mod n). So we get d′ candidate for the size two cosets by solving a ≡ p2a (mod n). +Moreover, each singleton cyclotomic coset is formed by a solution of the latter equation. Note +also that the p-cyclotomic coset of size two containing a and pa is counted twice in our previous +observation. Hence there are d′−d +2 +many different cosets. +□ +For example, for an odd n, the only singleton p-cyclotomic coset modulo n is {0} when p = 2 +or p = 3. If n is even, then {n +2 } and {0} are the only singleton cyclotomic cosets for p = 3. +The next theorem classifies all the additive cyclic codes with e = 2. Note that the case e = 0 +happens if an additive cyclic code is symplectic self-orthogonal, and this case was characterized +in Theorem 4.6. +Theorem 4.10. Let C = +s +� +i=1 +Ci be an additive cyclic code of length n over Fp2. Then +e = dimFp(C) − dimFp(C ∩ C⊥s) = 2 +if and only if all Ci satisfy the conditions of Theorem 4.6 except one which is in correspondence +to +(1) a singleton coset and satisfies condition (1) of Theorem 4.8, +(2) a size two coset and satisfies condition (2) of Theorem 4.8. +Proof. The result follows from considering the formula (4.2) and considering all conditions of +Theorem 4.8. +□ +Many of our record-breaking quantum codes provided in the next section have e = 2. In +general, the total number of all additive cyclic codes can be a very large number. +So the +classification of e values significantly helps to prune the search algorithm for quantum codes +with good parameters. +5. New binary quantum codes +In this section, we first recall a construction of binary quantum codes from additive codes, +which does not require the symplectic self-orthogonality condition of Theorem 2.1. Then we +apply this construction to several nearly self-orthogonal additive cyclic codes over F4 and con- +struct new binary quantum codes. In the rest of this section, we show the quaternary filed by +F4 = {0, 1, ω, ω + 1}, where ω2 = ω + 1. +Theorem 5.1. [5, Corollary 3.3.7],[19] Let C be an (n, 2k) additive code over F4 and +r = 2n − k − dimFp(C ∩ C⊥s) +2 +Then there exists a binary quantum code with parameters [[n + r, k − n + r, d]]2, where +d ≥ min{d(C), d(C + C⊥s) + 1}. +Note that we take advantage of the result of Theorem 4.8 in the computation of Theorem 5.1. +In particular, the value of r in Theorem 5.1 is +dimFp(C⊥s)−dimFp(C∩C⊥s) +2 +, where the numerator +measures the nearly self-orthogonality of the code C⊥s. Next, we briefly describe two of our new +binary quantum codes. The rest of our new binary quantum codes presented in Table 1 can be +constructed in a similar way. + +POLYNOMIAL REPRESENTATION OF ADDITIVE CYCLIC CODES AND NEW QUANTUM CODES +15 +Example 5.2. Let n = 21 and C = ⟨g(x)+ωk(x)⟩F2[x] be an additive cyclic code over F4, where +g(x) = x20 + x17 + x15 + x13 + x11 + x8 + x7 + x6 + x5 + x4 + x3 + 1 +and +k(x) = x19 + x18 + x17 + x16 + x14 + x10 + x5 + x4 + x3 + x2 + x + 1. +The code C is a (21, 220) additive code. Moreover, our computation using the result of Theorem +4.8 shows that C has nearly self-orthogonality parameter e = 2. Moreover, +7 = min{d(C⊥s), d(C + C⊥s) + 1}. +So, applying the construction of Theorem 5.1 to the code C⊥s gives a new quantum code with +parameters [[22, 2, 7]]2. It has a better minimum distance than the previous best-known quantum +code with the same length and dimension, which had minimum distance 6. +Example 5.3. Let n = 35 and C = ⟨g(x)+ωk(x)⟩F2[x] be an additive cyclic code over F4, where +g(x) = x33 + x29 + x28 + x24 + x19 + x18 + x15 + x13 + x12 + x11 + x6 + x4 + x + 1 +and +k(x) = x34 +x33 +x31 +x30 +x29 +x27 +x25 +x23 +x22 +x20 +x19 +x18 +x15 +x12 +x8 +x3 +x. +The code C has parameters (35, 220) as an additive cyclic code over F4. Also, the result of +Theorem 4.8 shows that C has nearly self-orthogonality parameter e = 4. Moreover, +6 = min{d(C⊥s), d(C + C⊥s) + 1}. +So, applying the construction of Theorem 5.1 to the code C⊥s gives a record-breaking quantum +code with parameters [[37, 17, 6]]2. The previous best-known binary quantum code with the same +parameters had minimum distance 5. +In general, in order to apply the quantum construction given in Theorem 5.1, we target +additive cyclic codes with the nearly self-orthogonality e ≤ 4. +Because it is more likely to +get a new quantum code when e value is small. In Table 1, we present the parameters of our +new binary quantum codes. In the table, we start with an additive cyclic code C over F4 and +compute its nearly self-orthogonality. Then we apply the quantum construction of Theorem 5.1 +to the code C⊥s. The parameters of the corresponding quantum code are given in the fourth +column. Moreover, the minimum distance of the previous quantum code with the same length +and dimension is provided in the last column of the table. The previous minimum distance is +taken from Grassl’s code table [12]. We record the generator polynomials of the additive cyclic +codes of Table 1 in Table 2. +Note also that applying the secondary constructions presented in Theorem 2.2 to the new +codes of Table 1 produces many more record-breaking quantum codes. In particular, the new +[[52, 24, 8]]2 quantum codes alone produces the following new quantum codes: +[[52, 21, 8]]2, [[52, 22, 8]]2, [[52, 23, 8]]2, [[53, 21, 8]]2, [[53, 22, 8]]2, [[53, 23, 8]]2, [[53, 24, 8]]2. +Around the same time as us, authors of [14] independently found several new binary quantum +codes by applying the connection between quasi-cyclic codes and additive cyclic codes. +In +particular, three of our new quantum codes, namely [[45, 6, 10]], [[45, 45, 10, 9]], and [[51, 8, 11]], +are also among the new quantum codes of [14]. +Acknowledgement +The authors would like to thank Petr Lisonˇek and Markus Grassl for many interesting dis- +cussions and comments. + +POLYNOMIAL REPRESENTATION OF ADDITIVE CYCLIC CODES AND NEW QUANTUM CODES +16 +No +Length +e value +Parameters +Previous distance +1 +n = 21 +2 +[[22, 2, 7]]2 +6 +2 +n = 35 +4 +[[37, 17, 6]]2 +5 +3 +n = 45 +0 +[[45, 6, 10]]2 +9 +4 +n = 45 +0 +[[45, 10, 9]]2 +8 +5 +n = 51 +0 +[[51, 8, 11]]2 +10 +6 +n = 51 +2 +[[52, 16, 10]]2 +9 +7 +n = 51 +2 +[[52, 24, 8]]2 +7 +8 +n = 63 +2 +[[64, 33, 8]]2 +7 +9 +n = 63 +2 +[[64, 34, 8]]2 +7 +10 +n = 63 +2 +[[64, 35, 8]]2 +7 +Table 1. Parameters of new binary quantum codes. +References +[1] J. Bierbrauer and Y. Edel. Quantum twisted codes. Journal of Combinatorial Designs, 8(3):174–188, 2000. +[2] W. Bosma, J. Cannon, and C. Playoust. The Magma algebra system I: The user language. Journal of Symbolic +Computation, 24(3-4):235–265, 1997. +[3] A. R. Calderbank, E. M. Rains, P. Shor, and N. J. Sloane. Quantum error correction via codes over GF(4). +IEEE Transactions on Information Theory, 44(4):1369–1387, 1998. +[4] Y. Cao and Y. Gao. Repeated root cyclic Fq-linear codes over Fql. Finite Fields Appl., 31:202–227, 2015. +[5] R. Dastbasteh. Quantum stabilizer codes. Master’s thesis, Sabancı University, 2017. +[6] R. Dastbasteh and P. Lisonek. New quantum codes from self-dual codes over F4. arXiv preprint +arXiv:2211.00891, 2022. +[7] B. K. Dey and B. S. Rajan. F q-linear cyclic codes over : Dft approach. Designs, Codes and Cryptography, +34(1):89–116, 2005. +[8] D. S. Dummit and R. M. Foote. Abstract algebra, volume 3. Wiley Hoboken, 2004. +[9] M. F. Ezerman. Quantum error-control codes. In W. C. Huffman, J.-L. Kim, and P. Sol´e, editors, Concise +encyclopedia of coding theory, chapter 2. Chapman and Hall/CRC, 2021. +[10] M. F. Ezerman, S. Ling, B. ¨Ozkaya, and P. Sol´e. Good stabilizer codes from quasi-cyclic codes over F4 and +F9. In 2019 IEEE International Symposium on Information Theory (ISIT), pages 2898–2902. IEEE, 2019. +[11] D. Gottesman. Class of quantum error-correcting codes saturating the quantum Hamming bound. Physical +Review A, 54(3):1862, 1996. +[12] M. Grassl. Code Tables: Bounds on the parameters of various types of codes. http://www.codetables.de/. +[13] M. Grassl. Algebraic quantum codes: Linking quantum mechanics and discrete mathematics. Int. J. Comput. +Math. Comput. Syst. Theory, 6(4):243–259, 2021. +[14] C. Guan, R. Li, and Z. Ma. Symplectic self-orthogonal quasi-cyclic codes. arXiv preprint arXiv:2212.14225, +2022. +[15] C. G¨uneri, F. ¨Ozdemir, and P. Sole. On the additive cyclic structure of quasi-cyclic codes. Discrete Mathe- +matics, 341(10):2735–2741, 2018. +[16] W. C. Huffman. Additive cyclic codes over F4. Adv. Math. Commun., 1(4):427–459, 2007. +[17] W. C. Huffman. Additive cyclic codes over F4. Adv. Math. Commun., 2(3):309–343, 2008. +[18] A. Ketkar, A. Klappenecker, S. Kumar, and P. K. Sarvepalli. Nonbinary stabilizer codes over finite fields. +IEEE transactions on information theory, 52(11):4892–4914, 2006. +[19] P. +Lisonˇek +and +R. +Dastbasteh. +Constructions +of +quantum +codes. +Presented +at +The +3rd +International +Workshop +on +Boolean +Functions +and +their +Applications, +loen, +norway. +https://people.uib.no/chunlei.li/workshops/BFA2018/Slides/Lisonek.pdf, 2018. +[20] P. Lisonˇek and V. Singh. Quantum codes from nearly self-orthogonal quaternary linear codes. Designs, Codes +and Cryptography, 73(2):417–424, 2014. +[21] K. Samei and S. Mahmoudi. Cyclic R-additive codes. Discrete Mathematics, 340(7):1657–1668, 2017. +Department of Mathematics, Simon Fraser University, Burnaby, BC, Canada +Email address: +rdastbas@sfu.ca, kh411@protonmail.com + +POLYNOMIAL REPRESENTATION OF ADDITIVE CYCLIC CODES AND NEW QUANTUM CODES +17 +No +Generator polynomials as in Theorem 3.6 part (II) +1 +g(x) = x20 + x17 + x15 + x13 + x11 + x8 + x7 + x6 + x5 + x4 + x3 + 1 +k(x) = x19 + x18 + x17 + x16 + x14 + x10 + x5 + x4 + x3 + x2 + x + 1 +h(x) = 0 +2 +g(x) = x33 + x29 + x28 + x24 + x19 + x18 + x15 + x13 + x12 + x11 + x6 + x4 + x + 1 +k(x) = x34+x33+x31+x30+x29+x27+x25+x23+x22+x20+x19+x18+x15+x12+x8+x3+x +h(x)=0 +3 +g(x) = x44 + x43 + x41 + x40 + x39 + x38 + x34 + x33 + x30 + x26 + x24 + x20 + x19 + x18 + +x17 + x16 + x15 + x14 + x11 + x9 + x5 + x3 + 1 +k(x) = x43 + x42 + x41 + x40 + x36 + x33 + x32 + x31 + x30 + x28 + x26 + x25 + x17 + x16 + +x15 + x13 + x11 + x10 + x2 + x +h(x) = 0 +4 +g(x) = x44 + x43 + x40 + x38 + x37 + x34 + x31 + x27 + x22 + x21 + x20 + x19 + x18 + x17 + +x14 + x12 + x7 + x6 + x5 + x3 + x + 1 +k(x) = x44 + x41 + x40 + x37 + x36 + x35 + x33 + x30 + x29 + x27 + x26 + x25 + x22 + x20 + +x15 + x14 + x12 + x11 + x10 + x7 + x5 +h(x) = 0 +5 +g(x) = x50 + x49 + x48 + x46 + x45 + x43 + x42 + x41 + x40 + x37 + x36 + x35 + x30 + x29 + +x28 + x26 + x23 + x19 + x18 + x17 + x16 + x15 + x14 + x13 + x9 + x7 + x6 + x +k(x) = x50 + x47 + x44 + x43 + x42 + x41 + x40 + x38 + x36 + x35 + x33 + x32 + x28 + x26 + +x24 + x21 + x20 + x16 + x14 + x12 + x9 + x8 + x7 + x + 1 +h(x)=0 +6 +g(x) = x48 + x40 + x37 + x36 + x33 + x31 + x30 + x24 + x23 + x21 + x19 + x15 + x11 + x10 + +x9 + x8 + x7 + x4 + x3 + x + 1 +k(x) = x41 + x40 + x36 + x35 + x34 + x33 + x30 + x29 + x27 + x23 + x22 + x21 + x19 + x18 + +x16 + x13 + x12 + x10 + x9 + x8 + x7 + x6 + x5 + x4 + x3 + x +h(x) = x50 + x49 + x48 + x47 + x46 + x45 + x44 + x41 + x40 + x39 + x33 + x31 + x30 + x28 + +x25 + x22 + x20 + x19 + x18 + x17 + x16 + x14 + x13 + x11 + x9 + x5 + x4 +7 +g(x) = x49 + x48 + x46 + x44 + x43 + x41 + x38 + x37 + x36 + x33 + x32 + x31 + x30 + x29 + +x27 + x25 + x21 + x20 + x18 + x17 + x15 + x11 + x10 + x7 + x2 +k(x) = x43 + x42 + x41 + x40 + x38 + x37 + x33 + x32 + x30 + x26 + x24 + x22 + x19 + x18 + +x16 + x15 + x13 + x9 + x5 + x4 + x2 + 1 +h(x) = x50 +x49 +x48 +x47 +x46 +x45 +x44 +x43 +x42 +x41 +x40 +x39 +x38 +x37 +x36 + +x35 +x34 +x33 +x32 +x31 +x30 +x29 +x28 +x27 +x26 +x25 +x24 +x23 +x22 +x21 +x20 +x19 + +x18 +x17 +x16 +x15 +x14 +x13 +x12 +x11 +x10 +x9 +x8 +x7 +x6 +x5 +x4 +x3 +x2 +x+1 +8 +g(x) = x61+x60+x59+x57+x56+x53+x52+x51+x42+x41+x38+x36+x34+x32+x31+x28+ +x27 +x26 +x24+x20 +x19+x16 +x14+x13 +x12+x11 +x9+x8+x7+x6+x5 +x4+x3+x2+x +k(x) = x61 + x59 + x57 + x56 + x55 + x54 + x52 + x51 + x50 + x49 + x47 + x44 + x37 + x36 + +x35 + x33 + x32 + x31 + x29 + x28 + x27 + x26 + x24 + x22 + x21 + x16 + x8 + x5 + x3 + x2 +h(x) = x62 + x61 + x60 + x59 + x58 + x51 + x49 + x47 + x44 + x43 + x40 + x36 + x33 + x31 + +x30 + x28 + x27 + x23 + x22 + x21 + x19 + x14 + x13 + x11 + x9 + x8 + x7 + x4 + x3 + x2 + x +9 +g(x) = x60 + x59 + x58 + x55 + x54 + x53 + x52 + x51 + x48 + x47 + x45 + x44 + x40 + x38 + +x37 + x36 + x35 + x34 + x33 + x32 + x31 + x30 + x29 + x28 + x27 + x24 + x23 + x22 + x21 + x15 + +x13 + x10 + x9 + x7 + x6 + x3 + x + 1 +k(x) = x62 + x59 + x56 + x55 + x54 + x53 + x49 + x47 + x46 + x42 + x41 + x40 + x37 + x35 + +x33 + x31 + x29 + x28 + x27 + x24 + x20 + x19 + x16 + x15 + x14 + x7 + x4 + x2 + x +h(x) = 0 +10 +g(x) = x61+x60+x59+x58+x57+x53+x52+x49+x44+x41+x38+x37+x36+x35+x34+x32+ +x31 +x30+x27+x26+x24+x23+x21+x20 +x19+x13+x12+x11+x8+x6+x5+x4+x3+x+1 +k(x) = x60+x58+x57+x56+x52+x48+x47+x46+x44+x42+x40+x39+x38+x36+x35+x34+ +x32+x31+x30+x26+x25+x24+x22+x19+x18+x17+x13+x12+x9+x7+x6+x5+x4+x3+x2+1 +h(x) = x62 + x61 + x60 + x59 + x58 + x51 + x49 + x47 + x44 + x43 + x40 + x36 + x33 + x31 + +x30 + x28 + x27 + x23 + x22 + x21 + x19 + x14 + x13 + x11 + x9 + x8 + x7 + x4 + x3 + x2 + x +Table 2. Generator polynomials of additive cyclic codes of Table 1 + diff --git a/6NAyT4oBgHgl3EQf2fnD/content/tmp_files/load_file.txt b/6NAyT4oBgHgl3EQf2fnD/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..73b4b063d718ec6a2622dfd806bb681ce1010592 --- /dev/null +++ b/6NAyT4oBgHgl3EQf2fnD/content/tmp_files/load_file.txt @@ -0,0 +1,848 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf,len=847 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='00753v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='IT] 2 Jan 2023 POLYNOMIAL REPRESENTATION OF ADDITIVE CYCLIC CODES AND NEW QUANTUM CODES REZA DASTBASTEH AND KHALIL SHIVJI Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' We give a polynomial representation for additive cyclic codes over Fp2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' This repre- sentation will be applied to uniquely present each additive cyclic code by at most two generator polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' We determine the generator polynomials of all different additive cyclic codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' A minimum distance lower bound for additive cyclic codes will also be provided using linear cyclic codes over Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' We classify all the symplectic self-dual, self-orthogonal, and nearly self-orthogonal additive cyclic codes over Fp2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Finally, we present ten record-breaking binary quantum codes after applying a quantum construction to self-orthogonal and nearly self-orthogonal additive cyclic codes over F4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Keywords: additive cyclic codes, quantum code, self-orthogonal codes, self-dual codes 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Introduction Quantum error-correcting codes, or simply quantum codes, are used in quantum computation to protect quantum information from corruption by noise (decoherence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' A general framework of quantum codes is provided in [9, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Throughout this paper, Fp2 is the finite field of p2 elements, where p is a prime number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' The parameters of a quantum code over Fp that encodes k logical qubits to n physical qubits and has minimum distance d is denoted by [[n, k, d]]p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' An important family of quantum codes with many similar properties as classical block codes is the family of quantum stabilizer codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' In particular, quantum stabilizer codes are constructed using additive codes which are self-orthogonal with respect to a certain symplectic inner product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Several constructions of quantum stabilizer codes from various classical codes are given in [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' An interesting modification of the original definition of quantum stabilizer codes is by relaxing its self-orthogonality constraint [5, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' This method enables us to construct good quantum codes using not necessarily self-orthogonal additive codes over F4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Previously, this modification was applied for the construction of new quantum codes from different families of linear codes [6, 10, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Additive cyclic codes are of interest due to their rich algebraic properties and application in the construction of quantum codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' There have been several works in the literature toward the classification of additive cyclic codes for different applications [1, 4, 7, 16, 17, 21], and also due to their connection to other families of block codes such as quasi-cyclic codes [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' In [16], a canonical decomposition of additive cyclic code over F4 was introduced using certain finite field extensions of F4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' This decomposition was applied to determine self-orthogonal and self-dual additive cyclic codes over F4 with respect to the trace inner product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' In [3], it was shown that each additive cyclic code over F4 of length n can be generated by F2-span of at most two polynomials in F4[x]/⟨xn − 1⟩ and their cyclic shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Moreover, a criterion for the self-orthogonality of such codes with respect to the trace inner product was provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Another interesting construction for a subclass of additive cyclic code, namely twisted codes, was provided in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' This construction is analogous to the way linear cyclic codes are constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' In spite of many useful properties of twisted codes, all additive cyclic codes cannot be described using the theory of additive twisted codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' 1 POLYNOMIAL REPRESENTATION OF ADDITIVE CYCLIC CODES AND NEW QUANTUM CODES 2 In this work, we first give a canonical representation of all Fp-additive cyclic codes over Fp2 using at most two generator polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Our representation is more computationally friendly than the canonical representation of [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' This representation allows us to give a minimum distance lower bound for additive cyclic codes over Fp2 using the minimum distance of linear cyclic codes over Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Moreover, we provide a unique set of generator polynomials for each additive cyclic code over Fp2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' This representation of generator polynomials will be used to characterize all self-orthogonal and self-dual additive cyclic codes with respect to the symplectic inner product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' We also determine the generator polynomials of the symplectic dual of a given additive cyclic code over Fp2, and compute nearly the self-orthogonality of each additive cyclic code using only its generator polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' This allows us to apply the nearly self-orthogonal construction of quantum codes developed in [5, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' In particular, we provide a list of eleven record-breaking binary quantum codes after applying the mentioned quantum construction to nearly self-orthogonal additive cyclic codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Furthermore, applying secondary constructions to our new quantum codes produce many more record-breaking binary codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Note that such new quantum codes cannot be constructed using self-orthogonal additive cyclic codes of the same length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Section 2 briefly recalls the essential terminologies used in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Section 3 gives a canonical representation of additive cyclic codes over Fp2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' In fact, we follow a module theory approach to decompose each additive cyclic code using its polynomial representation in Fp2[x]/⟨xn −1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' In Section 4, we compute the symplectic dual of each additive cyclic code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' We provide the necessary and sufficient conditions for an additive cyclic code to be self-orthogonal, self-dual, or nearly self-orthogonality with respect to the symplectic inner product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Finally, in Section 5, we present the parameters of our record-breaking quantum codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Preliminaries Let ω be a primitive element of Fp2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Then the set {1, ω} forms a basis for Fp2 over Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Let a + bω and a′ + b′ω ∈ Fn p2, where a, a′, b, b′ ∈ Fn p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' The symplectic inner product of a + bω and a′ + b′ω is defined by ⟨a + bω, a′ + b′ω⟩s = a′ · b − a · b′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='1) An Fp-linear subspace C ⊆ Fn p2 is called a length n additive code over Fp2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' We denote the Fp-dimension of an additive code C over Fp2 with dimFp(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Let C ⊆ Fn p2 be an additive code over Fp2 such that dimFp(C) = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Then we call C an (n, pk) code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' The set C⊥s = {x ∈ Fn p2 : ⟨x, y⟩s = 0 for all y ∈ C}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' is called the symplectic dual of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' One can easily see that C⊥s is an (n, p2n−k) additive code over Fp2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' The code C is called self-orthogonal (respectively self-dual) if C ⊆ C⊥s (respectively if C = C⊥s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' For each x ∈ Fn p2, we denote the number of non-zero coordinates of x by wt(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Moreover, the minimum weight among non-zero vectors of an additive code C is denoted by d(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' The connection between quantum stabilizer codes and classical additive codes was initially formulated by the independent works of Calderbank, Rains, Shor, and Sloane [3] and Gottesman [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' A non-binary version of this connection is provided below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' [18, Corollary 16] Let C be an (n, pn−k) additive code over Fp2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Then there exists an [[n, k, d]]p quantum stabilizer code if C is symplectic self-orthogonal, where d = min{wt(x) : x ∈ C⊥ s \\ C} if k > 0 and d = min{wt(x) : x ∈ C} if k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' The quantum code of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='1 is called pure if d = d(C⊥s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' There are several secondary constructions of quantum code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' A short list of such constructions is provided below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' POLYNOMIAL REPRESENTATION OF ADDITIVE CYCLIC CODES AND NEW QUANTUM CODES 3 Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' [18, Section XV] Let C be an [[n, k, d]]p quantum code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' (1) If k > 0, then an [[n + 1, k, d]]p quantum code exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' (2) If C is pure and n, d ≥ 2, then an [[n − 1, k + 1, d − 1]]p pure quantum code exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' (3) If k > 1, then there exists an [[n, k − 1, d]]p quantum code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Additive cyclic codes over Fp2 Throughout this section, we assume that n is a positive integer such that (n, p) = 1 and Fp2 = {α + βω : α, β ∈ Fp}, where ω is a root of a degree two irreducible polynomial over Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' In this section, we provide a canonical representation of additive cyclic codes over the field Fp2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' In particular, we give a unique representation of each additive cyclic code over Fp2 using at most two generator polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Moreover, we determine the generator polynomials of all different additive cyclic codes over Fp2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' In particular, each additive cyclic code over F2 p is a linear combination of cyclic shifts of its generator polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Such representation is also suitable for practical computations of additive cyclic codes, especially using Magma computer algebra system [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' More particularly, there exists a built-in function in Magma which forms additive cyclic codes generated by two given generator polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' At the end of this section, we give a minimum distance lower bound for the minimum distance of additive cyclic codes over Fp2 using the minimum distance of linear cyclic codes over Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' An Fp-subspace C ⊆ Fn p2 is called an additive cyclic code of length n over Fp2, if for every (a0, a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' , an−1) ∈ C, the vector (an−1, a0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' , an−2) is also a codeword of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' We will use the following concepts of module theory frequently in this section, and for more details one, for example, can see [8, Chapter 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Let R be a principal ideal domain and M be an R-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' The annihilator of M is an ideal of R defined by {r ∈ R : rm = 0 for any m ∈ M}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' An element m ∈ M is called a torsion element, if there exists 0 ̸= r ∈ R such that rm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' The module M is called a torsion module if all of its elements are torsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' The following theorem, known as the primary decomposition theorem of modules, plays an important role in our representation of additive cyclic codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' [8, Chapter 12, Theorem 7] Let R be a principal ideal domain and M be a torsion R-module with the annihilator ⟨a⟩ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Let a = u n � i=1 pai i , where u is a unit and pi is a prime element for each 1 ≤ i ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Then we can decompose M as a direct sum of its submodules in the form M = n � i=1 Ni, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='1) where Ni = {x ∈ M : xpai i = 0} for each 1 ≤ i ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Each element (a0, a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' , an−1) ∈ Fn p2 can be represented uniquely as a polynomial in Fp2[x]/⟨xn− 1⟩ in the form n−1 � i=0 aixi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' One can easily verify that, under this correspondence, a length n additive cyclic codes over Fp2 is an Fp[x]-submodule of Fp2[x]/⟨xn − 1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Notation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Let f and g ∈ Fp2[x]/⟨xn −1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' We fix the following notations for the rest of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' (1) The ideal generated by f in Fp2[x]/⟨xn − 1⟩ is denoted by ⟨f⟩Fp2[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Equivalently it is the Fp2[x]-submodule of Fp2[x]/⟨xn − 1⟩ generated by the polynomial f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' POLYNOMIAL REPRESENTATION OF ADDITIVE CYCLIC CODES AND NEW QUANTUM CODES 4 (2) The Fp[x]-submodule of Fp2[x]/⟨xn − 1⟩ generated by the polynomial g is denoted by ⟨g⟩Fp[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' A straightforward computation shows that the annihilator of Fp2[x]/⟨xn − 1⟩ as an Fp[x]- module is the ideal ⟨xn−1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Moreover, we can decompose xn−1 over Fp[x] as xn−1 = s � i=1 fi(x), where each fi(x) is an irreducible polynomial corresponding to a p-cyclotomic coset modulo n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Next, we apply Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='2 to Fp2[x]/⟨xn − 1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' It is straightforward to see that Fp2[x]/⟨xn − 1⟩ = s � i=1 Ni, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='2) where Ni = ⟨(xn − 1)/fi(x)⟩Fp2[x] for each 1 ≤ i ≤ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' We call a non-zero length n additive cyclic code C over Fp2 irreducible if for any additive cyclic code D ⊆ C, then D = {0} or D = C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' The next lemma shows that each Ni can be decomposed as a direct sum of two irreducible additive cyclic codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' We determine the generator polynomial of all irreducible additive cyclic codes inside Ni and provide other useful information about additive cyclic codes inside each Ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Let f(x) be an irreducible divisor of xn − 1 over Fp[x] with deg(f) = k and N = ⟨(xn − 1)/f(x)⟩Fp2[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' (1) Let 0 ̸= r(x) ∈ N, then the set L = {r(x), xr(x), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' , xk−1r(x)} forms a basis for ⟨r(x)⟩Fp[x] as an Fp vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' (2) Let 0 ̸= C ⊊ N be an additive cyclic code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' The code C has Fp-dimension k and C = ⟨r(x)⟩Fp[x] for any 0 ̸= r(x) ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' (3) The additive cyclic code N can be decomposed as N = ⟨(xn − 1)/f(x)⟩Fp[x] ⊕ ⟨ω((xn − 1)/f(x))⟩Fp[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Moreover, dimFp(N) = 2k and N is linear over Fp2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' (4) The number of irreducible additive cyclic codes inside N is 2k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' In particular, the following set gives all the different generator polynomials of such additive cyclic codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' A = { � (xn − 1)/f(x) �� ω + g(x) � : g(x) ∈ Fp[x], deg(g(x)) < k} ∪ {(xn − 1)/f(x)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='3) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' (1) Obviously L ⊆ ⟨r(x)⟩Fp[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Suppose, on the contrary, that L is linearly dependent over Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Hence we can find a polynomial 0 ̸= s(x) ∈ Fp[x] of degree less than k such that r(x)s(x) ≡ 0 (mod xn − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Since (xn − 1)/f(x) | r(x) and f(x) is irreducible, we conclude that f(x) | s(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' However, it is a contradiction with the fact that deg(s(x)) < k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' This shows that L is linearly independent over Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Note that the set L ∪ {xkr(x)} is linearly dependent over Fp as this new set generates f(x)r(x) ≡ 0 (mod xn − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' In a similar fashion, one can show that {xir(x)} for k < i < n − 1 can be written as a linear combination of elements of L over Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Therefore, L forms a basis for ⟨r(x)⟩Fp[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' (2) Let 0 ̸= r(x) ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Suppose in contrary that ⟨r(x)⟩Fp[x] ⊊ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Then there exists a polynomial s(x) ∈ C such that s(x) ̸∈ ⟨r(x)⟩Fp[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Note that ⟨r(x)⟩Fp[x] ∩ ⟨s(x)⟩Fp[x] = {0} as otherwise, by part (1), for any polynomial a(x) in the intersection, we have ⟨r(x)⟩Fp[x] = ⟨a(x)⟩Fp[x] = ⟨s(x)⟩Fp[x], which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Thus C = ⟨r(x)⟩Fp[x] and has dimension k over Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' (3) It is easy to see that ⟨(xn − 1)/f(x)⟩Fp[x] ∩ ⟨ω((xn − 1)/f(x)⟩Fp[x] = {0} and N = ⟨(xn − 1)/f(x)⟩Fp[x] ⊕ ⟨ω((xn − 1)/f(x))⟩Fp[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' POLYNOMIAL REPRESENTATION OF ADDITIVE CYCLIC CODES AND NEW QUANTUM CODES 5 Hence N has dimension 2k over Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' The linearity part follows immediately from the structure of its generator polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' (4) In order to find an additive cyclic code with Fp-dimension k, we need to choose a nonzero polynomial r(x) ∈ N to be its generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Also, any non-zero elements of ⟨r(x)⟩Fp[x] generates the same code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Hence the number of additive cyclic codes with one non-zero generator inside N is 22k−1 2k−1 = 2k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Let C1 and C2 be two k-dimensional additive cyclic codes inside N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' If C1 ∩ C2 ̸= {0}, then C1 = C2 by part (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Equivalently, if C1 + C2 = N, then C1 ∩ C2 = {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Now we show that different elements of the set A generate different codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Let g(x) ∈ Fp[x] such that deg(g(x)) < k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Clearly the additive cyclic code C1 = ⟨(xn − 1)/f(x), ((xn − 1)/f(x))(g(x) + ω)⟩Fp[x] contains (xn − 1)/f(x) and ω(xn − 1)/f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Therefore C1 = N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' So ⟨(xn − 1)/f(x)⟩Fp[x] and ⟨((xn − 1)/f(x))(g(x) + ω)⟩Fp[x] are different additive cyclic codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Let g1(x) and g2(x) ∈ Fp[x] be two different polynomials of degree less than k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' The code C = ⟨((xn − 1)/f(x))(ω + g1(x)), ((xn − 1)/f(x))(ω + g2(x))⟩Fp[x] contains (xn − 1)/f(x) and ω(xn − 1)/f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' It is mainly because ⟨ � (xn − 1)/f(x) � (g1(x) − g2(x))⟩Fp[x] = ⟨(xn − 1)/f(x)⟩Fp[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Thus C = N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' This implies that the additive cyclic codes ⟨((xn − 1)/f(x))(ω + g1(x))⟩Fp[x] and ⟨((xn−1)/f(x))(ω+g2(x))⟩Fp[x] are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' This proves that the set A contains all the different generators of irreducible additive cyclic codes inside N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' □ As we mentioned in part (1) of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='4, each additive cyclic code inside ⟨(xn−1)/f(x)⟩Fp2[x] can have many different generator polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Through the next remark, we fix a canonical representation for each additive cyclic code inside N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' For each additive code 0 ̸= C ⊊ ⟨(xn −1)/f(x)⟩Fp2[x], we fix its generator polyno- mial inside the set A, introduced in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='3), to be “the” generator polynomial of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Similarly, the additive cyclic code C′ = ⟨(xn−1)/f(x)⟩Fp2[x] can be generated by the polynomials (xn−1)/f(x) and ω((xn − 1)/f(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' We call them “the” generator polynomials of C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' This representation helps to uniquely identify each additive cyclic code inside N and avoid considering the same code more than once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Next, we use the result of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='4 and characterize all the additive cyclic codes of length n over Fp2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Recall that xn − 1 = s � i=1 fi(x), where fi(x) is an irreducible polynomial over Fp[x] for each 1 ≤ i ≤ s and Ni = ⟨(xn − 1)/fi(x)⟩Fp2[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Let C be a length n additive cyclic code over Fp2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Then (i) we can decompose the code C as C = s � i=1 Ci, where each Ci is an additive cyclic code inside Ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' (ii) we have C = ⟨g(x) + ωk(x), ωh(x)⟩Fp[x], where (a) g(x) + ωk(x) = s � i=1 gi(x) + ωki(x), (b) h(x) = s � i=1 hi(x), (c) and Ci has the generator polynomial(s) gi(x) + ωki(x) and ωhi(x) selected as dis- cussed in Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' POLYNOMIAL REPRESENTATION OF ADDITIVE CYCLIC CODES AND NEW QUANTUM CODES 6 (iii) dimFp(C) = s � i=1 (deg(fi) × # of non-zero generators of Ci).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' (i) As we mentioned in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='2), the following decomposition holds Fp2[x]/⟨xn − 1⟩ = s � i=1 Ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' So we can express C as C = �s i=1 Ci, where each Ci is an additive cyclic codes inside Ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' (ii) We show that the additive cyclic codes C = s � i=1 Ci and ⟨g(x) + ωk(x), ωh(x)⟩Fp[x] are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' First note that g(x) + ωk(x), ωh(x) ∈ C and thus ⟨g(x) + ωk(x), ωh(x)⟩Fp[x] ⊆ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Let 1 ≤ i ≤ s be a fixed integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Since � (xn − 1)/fi(x) � | gi(x), ki(x), hi(x) and � (xn − 1)/fi(x) � gj(x) ≡ � (xn − 1)/fi(x) � kj(x) ≡ � (xn − 1)/fi(x) � hj(x) ≡ 0 (mod xn − 1) for any j ̸= i, we have � (xn − 1)/fi(x) �� g(x) + ωk(x) � ≡ � (xn − 1)/fi(x) �� gi(x) + ωki(x) � (mod xn − 1) and � (xn − 1)/fi(x) � ωh(x) ≡ � (xn − 1)/fi(x) � ωhi(x) (mod xn − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Moreover, we have Ci = ⟨gi(x)+ωki(x), ωhi(x)⟩Fp[x] = ⟨ � (xn −1)/fi(x) �� g(x)+ωk(x) � , � (xn −1)/fi(x) � ωh(x)⟩Fp[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Thus Ci ⊆ ⟨g(x) + ωk(x), ωh(x)⟩Fp[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' This show that s � i=1 Ci ⊆ ⟨g(x) + ωk(x), ωh(x)⟩Fp[x] and completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' (iii) Note that dimFp(C) = s � i=1 dimFp(Ci).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Moreover, by Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='4, dimFp(Ci) = 0, ki, or 2ki if Ci = 0, Ci is generated by one generator polynomial, or Ci has two generator polynomials, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Combining these facts with the result of part (i) completes this proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' □ Through the next corollary, we characterize all the length n irreducible additive cyclic codes over Fp2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Let C be an additive cyclic code of length n over Fp2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Then C is irreducible if and only if C = ⟨r(x)⟩Fp[x] for some 0 ̸= r(x) ∈ Ni and 1 ≤ i ≤ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Moreover, there are s � i=1 (2deg(fi) + 1) many different irreducible additive cyclic codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Let C = ⟨r(x)⟩Fp[x] for some 0 ̸= r(x) ∈ Ni and 1 ≤ i ≤ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' The result of part (1) in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='4 shows that C is irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Conversely, let C be an irreducible additive cyclic code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Then by part (i) of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='6 we have C = s � i=1 Ci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Since C is irreducible, we have C = Cj for some 1 ≤ j ≤ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Moreover, since Nj is not irreducible by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='4 part (3), we conclude that C = ⟨r(x)⟩Fp[x] for some 0 ̸= r(x) ∈ Nj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' POLYNOMIAL REPRESENTATION OF ADDITIVE CYCLIC CODES AND NEW QUANTUM CODES 7 Inside each Ni, there are 2deg(fi)+1 many different one generator additive cyclic codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Hence the total number of irreducible codes is s � i=1 (2deg(fi) + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' □ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Henceforth, we always represent each additive cyclic code with its generator polynomials g(x) + ωk(x) and ωh(x) introduced in part (ii) of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Moreover, the way we generate these polynomials is unique, and therefore each additive cyclic code has a unique set of generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' From now on, we call Fp2-linear cyclic codes simply linear cyclic codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Let C = ⟨g(x) + ωk(x), ωh(x)⟩Fp[x] be a length n additive cyclic code over Fp2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Note that Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='6 and part (3) of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='4 imply that C is linear if and only if g(x) = h(x) and k(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Hence linear cyclic codes can be easily distinguished from non-linear cyclic codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Next, we provide a minimum distance bound for additive cyclic codes using linear cyclic codes over Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' In general, the minimum distance computation for linear codes is faster than the additive codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Hence the following result can speed up the minimum distance computation for additive cyclic codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' We denote the minimum distance of a code C with d(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Let C = ⟨g(x) + ωk(x), ωh(x)⟩Fp[x] be a length n additive cyclic code over Fp2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Let G(x) = xn−1 gcd(xn−1,g(x)), and let S(x) be the generator polynomial of the intersection of the length n linear cyclic code generated by k(x) and the linear cyclic code generated by h(x) over Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Suppose that D1, D2, D3, and D3 are the length n linear cyclic codes over Fp generated by g(x), gcd(k(x), h(x)), gcd(G(x)k(x), h(x)), and g(x)S(x) gcd(xn−1,k(x)), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Then min{d(D3), d(D4), max{d(D1), d(D2)}} ≤ d(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='4) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Only the following three types of codewords may appear in the code C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' T1 = {a(x) ∈ C : 0 ̸= a(x) ∈ Fp[x]}, T2 = {ωb(x) ∈ C : 0 ̸= b(x) ∈ Fp[x]}, T3 = {a(x) + ωb(x) ∈ C : 0 ̸= a(x), 0 ̸= b(x) ∈ Fp[x]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' We bound the minimum distance of C by considering the minimum distance in each of these sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Let f(x) ∈ T1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Then we can write it as f(x) = a1(x)(g(x) + ωk(x)) + b1(x)ωh(x) for some a1(x), b1(x) ∈ Fp[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Hence f(x) = a1(x)g(x) and a1(x)k(x) + b1(x)h(x) ≡ 0 (mod xn − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' This implies that a1(x)k(x) is an element of the length n linear cyclic code over Fp generated by S(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Hence S(x) gcd(xn−1,k(x)) | a1(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' In other words, f(x) = a(x)g(x) ∈ D4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Next, let ωf1(x) ∈ T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Then ωf1(x) = a1(x)(g(x)+ωk(x))+b1(x)ωh(x) for some a1(x), b1(x) ∈ Fp[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Then a1(x)g(x) ≡ 0 (mod xn − 1) or equivalently G(x) | a1(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' This implies that f1(x) = a1(x)k(x) + b1(x)h(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Therefore, f1(x) ∈ D3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Finally, let a(x) + ωb(x) ∈ T3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Then a(x) + ωb(x) = l(x)(g(x) + ωk(x)) + m(x)ωh(x) for some l(x), m(x) ∈ Fp[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Hence a(x) ∈ D1 and b(x) ∈ D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' This implies that wt(a(x) + ωb(x)) ≥ max{d(D1), d(D2)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' □ Note that if Di = 0 for any value 1 ≤ i ≤ 4, then we simply discard this code in the minimum distance lower bound of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' For instance if D1 = 0, then the minimum distance lower bound of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='4) becomes min{d(D3), d(D4), d(D2)} ≤ d(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' The following corollary gives a modification of this result to additive cyclic codes, which are generated by only one generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' In this result, the cyclic codes Ci are obtained from Di after POLYNOMIAL REPRESENTATION OF ADDITIVE CYCLIC CODES AND NEW QUANTUM CODES 8 substituting h(x) with 0 in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='9 for 1 ≤ i ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' However, the code C4 is obtained differently by considering a more direct observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Let C = ⟨g(x) + ωk(x)⟩Fp[x] be a length n additive cyclic code over Fp2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Let C1, C2, C3, and C4 be the length n linear cyclic codes over Fp generated by polynomials g(x), k(x), xn−1 gcd(xn−1,g(x))k(x), and xn−1 gcd(xn−1,k(x))g(x), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Then min{d(C3), d(C4), max{d(C1), d(C2)}} ≤ d(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='5) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' As we mentioned above, the code Ci all are obtained after applying the condition h(x) = 0 in the structure of the codes Di for 1 ≤ i ≤ 3 in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Since the code D4 in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='9 is applied to bound the minimum weight of the set T1 = {a(x) ∈ C : 0 ̸= a(x) ∈ Fp[x]}, we compute the minimum weight of T1 directly in this proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Let f(x) ∈ T1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Then we can write it as f(x) = a(x)(g(x)+ωk(x)) for some a(x) ∈ Fp[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Hence f(x) = a(x)g(x) and a(x)k(x) ≡ 0 (mod xn − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' This implies that xn−1 gcd(xn−1,k(x)) | a(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Hence xn−1 gcd(xn−1,k(x))g(x) | a(x) and we have f(x) ∈ C4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' □ Next, we consider the restriction of the mentioned minimum distance bound to linear cyclic codes with the generator polynomials g(x) + ωk(x) and h(x), where k(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Let C = ⟨g(x), ωh(x)⟩Fp[x] be a length n additive cyclic code over Fp2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Let E1 and E2 be the length n linear cyclic codes over Fp generated by polynomials g(x) and h(x), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Then min{d(E1), d(E2)} ≤ d(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='6) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Applying the condition k(x) = 0 to Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='9 implies that D1 = D4 = E1 and D2 = D3 = E2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Now the result follows from the minimum distance bound of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Symplectic inner product and dual of additive cyclic codes In this section, we determine generator polynomials of the symplectic dual of a given additive cyclic code over Fp2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Moreover, we give the generator polynomials of all self-orthogonal and self- dual codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' We also measure how close is a given additive cyclic code from being symplectic self- orthogonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Recall that p is a prime number and n is a positive integer coprime to p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Moreover, elements of Fp2 are represented by Fp2 = {α + βω : α, β ∈ Fp}, where ω is a root of a degree 2 irreducible polynomial over Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Recall that in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='1) we defined the symplectic inner product of two elements in Fn p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' We define the symplectic inner product of two polynomials analogously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' In particular, for c(x) = n−1 � i=0 (ai + ωbi)xi and c′(x) = n−1 � i=0 (a′ i + ωb′ i)xi ∈ Fp2[x]/⟨xn − 1⟩, we define c(x) ∗ c′(x) = n−1 � i=0 (aib′ i − a′ ibi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Here we use a different notation for the symplectic inner product to differentiate between the vectors and polynomials as different objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Let c(x) = g1(x) + ωg2(x) and c′(x) = g′ 1(x) + ωg′ 2(x) be two polynomials of Fp2[x]/⟨xn − 1⟩, where g1(x), g2(x), g′ 1(x), g′ 2(x) ∈ Fp[x]/⟨xn − 1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Then c(x) ∗ c′(x) is the constant term of g1(x)g′ 2(x−1) − g2(x)g′ 1(x−1) (mod xn − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' A similar argument shows that c(x) ∗ xic′(x) is the coefficient of xi in g1(x)g′ 2(x−1) − g2(x)g′ 1(x−1) (mod xn − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Thus if POLYNOMIAL REPRESENTATION OF ADDITIVE CYCLIC CODES AND NEW QUANTUM CODES 9 g1(x)g′ 2(x−1) − g2(x)g′ 1(x−1) ≡ 0 (mod xn − 1), then the code generated by c′(x) lies in the symplectic dual of the code generated by c(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' We use this property very frequently through this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' One can easily verify that the symplectic dual of an additive cyclic code C over Fp2 is also an additive cyclic code over Fp2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Recall that by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='6 part (ii), each additive cyclic code of length n over Fp2 can be represented uniquely as C = ⟨g1(x) + ωg2(x), h(x)⟩Fp[x], where g1(x), g2(x), h(x) ∈ Fp[x]/⟨xn − 1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Our next theorem gives a criterion for the self-orthogonality of additive cyclic codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' The proof is very similar to that of [3, Theorem 14 part c].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Let C = ⟨g1(x) + ωg2(x), h(x)⟩Fp[x] be a length n additive cyclic code over Fp2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' The code C is self-orthogonal if and only if the following conditions are satisfied: (1) g2(x)h(x−1) ≡ 0 (mod xn − 1), (2) g1(x)g2(x−1) ≡ g2(x)g1(x−1) (mod xn − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' ⇒: Suppose that C is self-orthogonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' For each 0 ≤ i ≤ n − 1, the inner product of g1(x) + ωg2(x) and xih(x) is the coefficient of xi in −g2(x)h(x−1) (mod xn − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Since C is self- orthogonal, we have g2(x)h(x−1) ≡ 0 (mod xn − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Moreover, � xi(g1(x) + ωg2(x)) � ∗ � g1(x) + ωg2(x) � is the coefficient of xi in g1(x)g2(x−1) − g2(x)g1(x−1) (mod xn − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Hence, for each 0 ≤ i ≤ n − 1, the coefficient of xi in g1(x)g2(x−1) − g2(x)g1(x−1) (mod xn − 1) is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Thus g1(x)g2(x−1) ≡ g2(x)g1(x−1) (mod xn − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' ⇐: Conversely, the fact that g1(x)g2(x−1) ≡ g2(x)g1(x−1) (mod xn − 1) implies that all the vectors inside ⟨g1(x) + ωg2(x)⟩Fp[x] are self-orthogonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Moreover, since g2(x)h(x−1) ≡ 0 (mod xn − 1), we conclude that h(x) is orthogonal to all the cyclic shifts of g1(x) + ωg2(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Finally h(x) ∗ xih(x) = 0 for each 0 ≤ i ≤ n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' So ⟨g1(x) + ωg2(x), h(x)⟩Fp[x] is a symplectic self-orthogonal code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' □ Recall that xn − 1 = s � i=1 fi(x), where each fi(x) is an irreducible polynomial in Fp[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Moreover, as we mentioned earlier in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='2), we have Fp2[x]/⟨xn − 1⟩ = s � i=1 Ni, where Ni = ⟨(xn − 1)/fi(x)⟩Fp2[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Let α be a primitive n-th root of unity in a finite filed extension of Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' We denote the p-cyclotomic cosets modulo n by Zi for each 1 ≤ i ≤ s in the way that fi(x) = � a∈Zi (x − αi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' This gives a one-to-one correspondence between the sets Ni and all the p-cyclotomic cosets modulo n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Our first goal in this section is to find the symplectic dual of a given additive cyclic code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' In order to achieve this goal, we need a few preliminary results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' In the next lemma, we find the symplectic dual of each Ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Let 1 ≤ i ≤ s and C = Ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Then C⊥s = s � k=1 k̸=j Nk, where Zj = −Zi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' First note that by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='4 part (3) we have C = ⟨(xn−1)/fi(x), ω((xn−1)/fi(x))⟩Fp[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' If Zk ̸= −Zi, then fi(x) | (xn − 1)/fk(x−1) and fk(x) | (xn − 1)/fi(x−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' So we have � (xn − 1)/fi(x) �� (xn − 1)/fk(x−1) � ≡ 0 (mod xn − 1) and � (xn − 1)/fk(x) �� (xn − 1)/fi(x−1) � ≡ 0 (mod xn − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' POLYNOMIAL REPRESENTATION OF ADDITIVE CYCLIC CODES AND NEW QUANTUM CODES 10 Hence the symplectic inner product of each element of Ni and each element of Nk is zero by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' This proves that s � k=1 k̸=j Nk ⊆ C⊥s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Note that both of Ni and Nj have Fp-dimension 2 deg(fi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Now, the facts that dimFp(C) + dimFp(C⊥s) = 2n and dimFp(C) = 2 deg(fi) implies the other inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' □ Next, we find the symplectic dual of each irreducible additive cyclic code inside Ni for 1 ≤ i ≤ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Let C ⊊ Ni be a non-zero additive cyclic code for some 1 ≤ i ≤ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Then C⊥s = ( s � k=1 k̸=j Nk) � ⟨g1(x) + ωg2(x)⟩Fp[x], (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='1) where Zj = −Zi and g1(x) + ωg2(x) = � ((xn − 1)/fj(x))(s(x−1) + ω) if C = ⟨ � (xn − 1)/fi(x) �� ω + s(x) � ⟩Fp[x] (xn − 1)/fj(x) if C = ⟨(xn − 1)/fi(x)⟩Fp[x] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='3, one can see that s � k=1 k̸=j Nk ⊆ C⊥s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Note that dimFp(⟨g1(x)+ωg2(x)⟩Fp[x]) = dimFp(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' So it is sufficient to show that C is orthogonal to g1(x) + ωg2(x) and all its cyclic shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' We prove the latter statement in two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' First suppose that C = ⟨ � (xn−1)/fi(x) �� ω+ s(x) � ⟩Fp[x] for some s(x) ∈ Fp[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' To show that the codes C and ⟨g1(x) + ωg2(x)⟩Fp[x] are orthogonal, we apply Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' In particular, � (xn − 1)/fi(x) � s(x)g2(x−1) − � (xn − 1)/fi(x) � g1(x−1) ≡ � (xn − 1)/fi(x) � s(x) � (xn − 1)/fi(x) � − � (xn − 1)/fi(x) �� (xn − 1)/fi(x) � s(x) ≡ 0 (mod xn − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Next, suppose that C = ⟨(xn − 1)/fi(x)⟩Fp[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Then � (xn − 1)/fi(x) � g2(x−1) − � (xn − 1)/fi(x) � g1(x−1) ≡ � (xn − 1)/fi(x) � 0 − 0 � (xn − 1)/fj(x) � s(x) ≡ 0 (mod xn − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' This shows that the code C is orthogonal to the additive cyclic code generated by g1(x)+ωg2(x) and completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' □ Note that as we showed in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='4, when C = ⟨ � (xn − 1)/fi(x) �� ω + s(x) � ⟩Fp[x], its symplectic inner product contains the code C′ = ⟨(xn − 1)/fj(x))(s(x−1) + ω)⟩Fp[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' The code C′ is not in one of the forms given in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='4 part (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' In order to express the code C′ using the standard notation introduced in 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='4 part (4), we choose its generator to be g(x) = (xn − 1)/fj(x))(t(x) + ω), where t(x) ≡ s(x−1) (mod fj(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Now it is easy to see that g(x) belongs to the set A introduced in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='4 part (4) and C′ = ⟨(xn −1)/fj(x))(t(x)+ω)⟩Fp[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Next, we combine the results of the previous two lemmas and the result of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='6 to determine generator polynomials of the symplectic dual for any additive cyclic code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' POLYNOMIAL REPRESENTATION OF ADDITIVE CYCLIC CODES AND NEW QUANTUM CODES 11 Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Let C be a length n additive cyclic code over Fp2 such that C = s � i=1 Ci, where Ci is an additive cyclic codes inside Ni for each 1 ≤ i ≤ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Then C⊥s = ⟨ s � i=1 gi(x) + ωki(x), s � i=1 ωhi(x)⟩Fp[x], where for each 1 ≤ i ≤ s we have Zj = −Zi and gi(x) = ki(x) = hi(x) = 0 if Cj = Nj, gi(x) = hi(x) = (xn − 1)/fi(x) and ki(x) = 0 if Cj = 0, gi(x)+ωki(x) = � (xn −1)/fi(x) �� ω +ti(x) � and hi(x) = 0 if Cj = ⟨ � (xn −1)/fj(x) �� ω + sj(x) � ⟩Fp[x] and ti(x) ≡ sj(x−1) (mod fj(x)), gi(x) = (xn − 1)/fi(x) and ki(x) = hi(x) = 0 if Cj = ⟨(xn − 1)/fj(x)⟩Fp[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' We apply Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='4 to prove the statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' If Cj = Nj, then C⊥s ∩ Ni = {0} by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Moreover, by the same lemma, if Cj = 0, then Ni ⊆ C⊥s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' This proves the first two bullets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Finally, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='4 implies that ⟨ � (xn − 1)/fi(x) �� ω + ti(x) � ⟩Fp[x] ⊆ C⊥s if Cj = ⟨ � (xn − 1)/fj(x) �� ω + sj(x) � ⟩Fp[x], and ⟨(xn − 1)/fi(x)⟩Fp[x] ⊆ C⊥s if Cj = ⟨(xn − 1)/fi(x)⟩Fp[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' This proves the statements of the last two bullets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' □ To determine self-orthogonal and self-dual additive cyclic codes over Fp2, we need more infor- mation about irreducible factors of xn − 1 over Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Let Z1, Z2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' , Zr and Z′ 1, −Z′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' , Z′ t, −Z′ t be all the p-cyclotomic cosets modulo n, where Zi = −Zi and r + 2t = s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Each Zi is in correspondence to an irreducible polynomial fi(x) and (Z′ j, −Z′ j) are in correspondence with an irreducible pair of polynomials (fj1(x), fj2(x)) over Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Therefore, we can rewrite the irreducible decomposition of xn − 1 as xn − 1 = r � i=1 fi(x) t� j=1 fj1(x)fj2(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' We use the above representation of cyclotomic cosets in the upcoming results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Next, we classify self-orthogonal and self-dual additive codes over Fp2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Let C be a length n additive cyclic code over Fp2 such that C = s � i=1 Ci, where Ci is an additive cyclic codes inside Ni for each 1 ≤ i ≤ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Then C is symplectic self-orthogonal if and only if (1) for all 1 ≤ k ≤ r only one of the following holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' (a) Ck = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' (b) Ck = ⟨((xn − 1)/fk(x))(s(x) + ω)⟩Fp[x], where fk | s(x−1) − s(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' (c) Ck = ⟨(xn − 1)/fk(x)⟩Fp[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' (2) for all 1 ≤ j ≤ t only one of the following holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' (a) Cj1 = 0 or Cj2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' (b) Cj1 = ⟨((xn − 1)/fj1(x))(s(x) + ω)⟩Fp[x] and Cj2 = ⟨((xn − 1)/fj2(x))(s(x−1) + ω)⟩Fp[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' (c) Cj1 = ⟨(xn − 1)/fj1(x)⟩Fp[x] and Cj2 = ⟨(xn − 1)/fj2(x)⟩Fp[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' POLYNOMIAL REPRESENTATION OF ADDITIVE CYCLIC CODES AND NEW QUANTUM CODES 12 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' First, let 1 ≤ k ≤ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='3, if Ck = Nk, then C⊥s ∩ Nk = {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' So Ck cannot have two generator polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Moreover, by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='4, if 0 ̸= Ck = ⟨((xn −1)/fk(x))(s(x)+ ω)⟩Fp[x], then ⟨((xn − 1)/fk(x))(s(x−1) + ω)⟩Fp[x] ⊆ C⊥s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Thus Ck is self-orthogonal if and only if Ck = ⟨((xn − 1)/fk(x))(s(x) + ω)⟩Fp[x] = ⟨((xn − 1)/fk(x))(s(x−1) + ω)⟩Fp[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Note that the above equality holds if and only if fk | s(x−1) − s(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Thus Ck is self-orthogonal if and only if one of the conditions of Part (1) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Next, let 1 ≤ j ≤ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='3, if Cj1 = Nj1, then C⊥s ∩ Nj2 = {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' So if one of Cj1 or Cj2 has two generator polynomials, the other code should be zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Moreover, the same lemma shows that if Cj1 = 0 or Cj2 = 0, then Cj1 + Cj2 is self-orthogonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' So we concentrate only on the case when both Cj1 and Cj2 have exactly one non-zero generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='4, if Cj1 = ⟨((xn − 1)/fj1(x))(s(x) + ω)⟩Fp[x], then C⊥s ∩ Nj2 = ⟨((xn − 1)/fj2(x))(s(x−1) + ω)⟩Fp[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' In this case, the code Cj1 ⊕ Cj2 is self-orthogonal if and only if condition (2)(b) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Condition (2)(c) follows similarly by applying Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' □ Next, we use the above conditions to characterize all the symplectic self-dual additive cyclic codes over Fp2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Let C be a length n additive cyclic code over Fp2 such that C = s � i=1 Ci, where Ci is an additive cyclic codes inside Ni for each 1 ≤ i ≤ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Then C is symplectic self-dual if and only if (1) for all 1 ≤ k ≤ r only one of the following holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' (a) Ck = ⟨((xn − 1)/fk(x))(s(x) + ω)⟩Fp[x] where fk | s(x−1) − s(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' (b) Ck = ⟨(xn − 1)/fk(x)⟩Fp[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' (2) for all 1 ≤ j ≤ t only one of the following holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' (a) Cj1 = 0 and Cj2 = Nj2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' (b) Cj2 = 0 and Cj1 = Nj1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' (c) Cj1 = ⟨((xn − 1)/fj1(x))(s(x) + ω)⟩Fp[x] and Cj2 = ⟨((xn − 1)/fj2(x))(s(x−1) + ω)⟩Fp[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' (d) Cj1 = ⟨(xn − 1)/fj1(x)⟩Fp[x] and Cj2 = ⟨(xn − 1)/fj2(x)⟩Fp[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Note that all the self-dual additive cyclic codes over Fp2 satisfy the conditions of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='6 and have maximal dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Thus the result easily follows by implying the maximal property into the conditions of theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' □ Our next goal is to compute the parameter e = dimFp(C) − dimFp(C ∩ C⊥s) for all additive cyclic codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' The parameter e determines how close an additive cyclic code C is from being self-orthogonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' This parameter plays an important role in the quantum construction that we are applying in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Let C be a length n additive cyclic code over Fp2 such that C = s � i=1 Ci, where Ci is an additive cyclic codes inside Ni for each 1 ≤ i ≤ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Let (1) B1 = {α1, α2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' , αt1} ⊆ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' , r} such that Cαl = Nαl for all 1 ≤ l ≤ t1, (2) B2 = {β1, β2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' , βt2} ⊆ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' , r} such that Cβl = ⟨((xn − 1)/fβl(x))(sβl(x) + ω)⟩Fp[x] and fβl ∤ sβl(x−1) − sβl(x) for all 1 ≤ l ≤ t2, POLYNOMIAL REPRESENTATION OF ADDITIVE CYCLIC CODES AND NEW QUANTUM CODES 13 (3) B3 = {γ1, γ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' , γt3} ⊆ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' , t} such that one of Cγl1 and Cγl2 is generated by two polynomials and the other one has only one generator polynomial for all 1 ≤ l ≤ t3, (4) B4 = {κ1, κ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' , κt4} ⊆ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' , t} such that both of Cκl1 and Cκl2 are generated by two polynomials for all 1 ≤ l ≤ t4, (5) B5 = {σ1, σ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' , σt5} ⊆ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' , t} such that both of Cσl1 and Cσl2 are generated by one polynomial for all 1 ≤ l ≤ t5 and (a) if Cσl1 = ⟨((xn−1)/fσl1(x))(sσl(x)+ω)⟩Fp[x], then Cσl2 ̸= ⟨((xn−1)/fσl2(x))(sσl(x−1)+ ω)⟩Fp[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' (b) if Cσl1 = ⟨(xn − 1)/fσl1(x)⟩Fp[x], then Cσl2 ̸= ⟨(xn − 1)/fσl2(x))⟩Fp[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Then e = dimFp(C)−dimFp(C∩C⊥s) = t1 � l=1 2|Zαl|+ t2 � l=1 |Zβl|+ t3 � l=1 2|Zγl|+ t4 � l=1 4|Zκl|+ t5 � l=1 2|Zσl|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='2) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' By Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='6, an additive cyclic code is not symplectic self-orthogonal if and only if at least one of the sets B1 −B5 is non-empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Next, we consider all scenarios (1)-(5) independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' (1) Let j ∈ B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' In this case, C⊥s ∩ Cj = {0} which implies that dimFp(Cj) − dimFp(Cj ∩ C⊥s) = 2|Zj|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' (2) Let j ∈ B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' In this case, C⊥s ∩ Cj = {0} which implies that dimFp(Cj) − dimFp(Cj ∩ C⊥s) = |Zj|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' (3) Let j ∈ B3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Without loss of generality we assume that Cj1 = Nj1 and Cj2 is an irreducible subcode of Nj2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' In this case, the intersection C⊥s∩(Cj1⊕Cj2) is an irreducible subcode of Nj1 which implies that dimFp(Cj)−dimFp((Cj1⊕Cj2)∩C⊥s) = 3|Zj|−|Zj| = 2|Zj|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' (4) Let j ∈ B4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' In this case, C⊥s ∩ (Cj1 ⊕ Cj2) = {0} which implies that dimFp(Cj) − dimFp((Cj1 ⊕ Cj2) ∩ C⊥s) = 4|Zj|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' (5) Let j ∈ B5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' In both parts (a) and (b), C⊥s ∩ (Cj1 ⊕ Cj2) = {0} which implies that dimFp(Cj) − dimFp((Cj1 ⊕ Cj2) ∩ C⊥s) = 2|Zj|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Now, the result follows by combining the above observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' □ Note that the case (2) of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='8 never happens for Ci with deg(fi(x)) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Moreover, for each 1 ≤ i ≤ r, the cyclotomic coset Zi either is a singleton or it has an even size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' This is mainly because for each 0 ̸= a ∈ Zi, if a ≡ −a (mod n), then n | 2a, which implies that n is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Hence in this case p ̸= 2 (we assumed that gcd(n, p) = 1) and Zi = {a}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Therefore, if Zi satisfies the case (2) of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='8 and |Zi| > 1, then for any a ∈ Zi, we have −a ∈ Zi and −a ̸≡ a (mod n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' This implies that |Zi| is an even integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' This fact and the formula in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='2) imply that the nearly self-orthogonality parameter e of an additive cyclic code is always an even integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Next, we classify additive cyclic codes with small values of e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' First, we need the following preliminary result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Let p be a prime number and gcd(n, p) = 1 for some positive number n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' (i) If gcd(n, p − 1) = d, then there are d singleton p-cyclotomic cosets modulo n and all of their coset leaders are {k n d : 0 ≤ k ≤ d − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' (ii) If gcd(n, p − 1) = d and gcd(n, p2 − 1) = d′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Then there are d′−d 2 p-cyclotomic cosets modulo n of size two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' POLYNOMIAL REPRESENTATION OF ADDITIVE CYCLIC CODES AND NEW QUANTUM CODES 14 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' (i) The proof easily follows from the fact that {a} is a singleton coset if and only if a ≡ pa (mod n) or equivalently if and only if a(p − 1) ≡ 0 (mod n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' By elementary number theory, if gcd(n, p − 1) = d, then the latter equation has d solutions in the forms {k n d : 0 ≤ k ≤ d − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' (ii) A p-cyclotomic coset modulo n containing a has size two if and only if a ≡ p2a (mod n) and a ̸≡ pa (mod n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' So we get d′ candidate for the size two cosets by solving a ≡ p2a (mod n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Moreover, each singleton cyclotomic coset is formed by a solution of the latter equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Note also that the p-cyclotomic coset of size two containing a and pa is counted twice in our previous observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Hence there are d′−d 2 many different cosets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' □ For example, for an odd n, the only singleton p-cyclotomic coset modulo n is {0} when p = 2 or p = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' If n is even, then {n 2 } and {0} are the only singleton cyclotomic cosets for p = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' The next theorem classifies all the additive cyclic codes with e = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Note that the case e = 0 happens if an additive cyclic code is symplectic self-orthogonal, and this case was characterized in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Let C = s � i=1 Ci be an additive cyclic code of length n over Fp2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Then e = dimFp(C) − dimFp(C ∩ C⊥s) = 2 if and only if all Ci satisfy the conditions of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='6 except one which is in correspondence to (1) a singleton coset and satisfies condition (1) of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='8, (2) a size two coset and satisfies condition (2) of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' The result follows from considering the formula (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='2) and considering all conditions of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' □ Many of our record-breaking quantum codes provided in the next section have e = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' In general, the total number of all additive cyclic codes can be a very large number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' So the classification of e values significantly helps to prune the search algorithm for quantum codes with good parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' New binary quantum codes In this section, we first recall a construction of binary quantum codes from additive codes, which does not require the symplectic self-orthogonality condition of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Then we apply this construction to several nearly self-orthogonal additive cyclic codes over F4 and con- struct new binary quantum codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' In the rest of this section, we show the quaternary filed by F4 = {0, 1, ω, ω + 1}, where ω2 = ω + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' [5, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='7],[19] Let C be an (n, 2k) additive code over F4 and r = 2n − k − dimFp(C ∩ C⊥s) 2 Then there exists a binary quantum code with parameters [[n + r, k − n + r, d]]2, where d ≥ min{d(C), d(C + C⊥s) + 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Note that we take advantage of the result of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='8 in the computation of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' In particular, the value of r in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='1 is dimFp(C⊥s)−dimFp(C∩C⊥s) 2 , where the numerator measures the nearly self-orthogonality of the code C⊥s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Next, we briefly describe two of our new binary quantum codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' The rest of our new binary quantum codes presented in Table 1 can be constructed in a similar way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' POLYNOMIAL REPRESENTATION OF ADDITIVE CYCLIC CODES AND NEW QUANTUM CODES 15 Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Let n = 21 and C = ⟨g(x)+ωk(x)⟩F2[x] be an additive cyclic code over F4, where g(x) = x20 + x17 + x15 + x13 + x11 + x8 + x7 + x6 + x5 + x4 + x3 + 1 and k(x) = x19 + x18 + x17 + x16 + x14 + x10 + x5 + x4 + x3 + x2 + x + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' The code C is a (21, 220) additive code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Moreover, our computation using the result of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='8 shows that C has nearly self-orthogonality parameter e = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Moreover, 7 = min{d(C⊥s), d(C + C⊥s) + 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' So, applying the construction of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='1 to the code C⊥s gives a new quantum code with parameters [[22, 2, 7]]2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' It has a better minimum distance than the previous best-known quantum code with the same length and dimension, which had minimum distance 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Let n = 35 and C = ⟨g(x)+ωk(x)⟩F2[x] be an additive cyclic code over F4, where g(x) = x33 + x29 + x28 + x24 + x19 + x18 + x15 + x13 + x12 + x11 + x6 + x4 + x + 1 and k(x) = x34 +x33 +x31 +x30 +x29 +x27 +x25 +x23 +x22 +x20 +x19 +x18 +x15 +x12 +x8 +x3 +x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' The code C has parameters (35, 220) as an additive cyclic code over F4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Also, the result of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='8 shows that C has nearly self-orthogonality parameter e = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Moreover, 6 = min{d(C⊥s), d(C + C⊥s) + 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' So, applying the construction of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='1 to the code C⊥s gives a record-breaking quantum code with parameters [[37, 17, 6]]2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' The previous best-known binary quantum code with the same parameters had minimum distance 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' In general, in order to apply the quantum construction given in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='1, we target additive cyclic codes with the nearly self-orthogonality e ≤ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Because it is more likely to get a new quantum code when e value is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' In Table 1, we present the parameters of our new binary quantum codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' In the table, we start with an additive cyclic code C over F4 and compute its nearly self-orthogonality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Then we apply the quantum construction of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='1 to the code C⊥s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' The parameters of the corresponding quantum code are given in the fourth column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Moreover, the minimum distance of the previous quantum code with the same length and dimension is provided in the last column of the table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' The previous minimum distance is taken from Grassl’s code table [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' We record the generator polynomials of the additive cyclic codes of Table 1 in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Note also that applying the secondary constructions presented in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='2 to the new codes of Table 1 produces many more record-breaking quantum codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' In particular, the new [[52, 24, 8]]2 quantum codes alone produces the following new quantum codes: [[52, 21, 8]]2, [[52, 22, 8]]2, [[52, 23, 8]]2, [[53, 21, 8]]2, [[53, 22, 8]]2, [[53, 23, 8]]2, [[53, 24, 8]]2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Around the same time as us, authors of [14] independently found several new binary quantum codes by applying the connection between quasi-cyclic codes and additive cyclic codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' In particular, three of our new quantum codes, namely [[45, 6, 10]], [[45, 45, 10, 9]], and [[51, 8, 11]], are also among the new quantum codes of [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Acknowledgement The authors would like to thank Petr Lisonˇek and Markus Grassl for many interesting dis- cussions and comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' POLYNOMIAL REPRESENTATION OF ADDITIVE CYCLIC CODES AND NEW QUANTUM CODES 16 No Length e value Parameters Previous distance 1 n = 21 2 [[22, 2, 7]]2 6 2 n = 35 4 [[37, 17, 6]]2 5 3 n = 45 0 [[45, 6, 10]]2 9 4 n = 45 0 [[45, 10, 9]]2 8 5 n = 51 0 [[51, 8, 11]]2 10 6 n = 51 2 [[52, 16, 10]]2 9 7 n = 51 2 [[52, 24, 8]]2 7 8 n = 63 2 [[64, 33, 8]]2 7 9 n = 63 2 [[64, 34, 8]]2 7 10 n = 63 2 [[64, 35, 8]]2 7 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Parameters of new binary quantum codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' References [1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Bierbrauer and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Edel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Quantum twisted codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Journal of Combinatorial Designs, 8(3):174–188, 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' [2] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Bosma, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Cannon, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Playoust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' The Magma algebra system I: The user language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Journal of Symbolic Computation, 24(3-4):235–265, 1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' [3] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Calderbank, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Rains, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Shor, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Sloane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Quantum error correction via codes over GF(4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' IEEE Transactions on Information Theory, 44(4):1369–1387, 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' [4] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Cao and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Gao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Repeated root cyclic Fq-linear codes over Fql.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Finite Fields Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=', 31:202–227, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' [5] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Dastbasteh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Quantum stabilizer codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Master’s thesis, Sabancı University, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' [6] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Dastbasteh and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Lisonek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' New quantum codes from self-dual codes over F4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' arXiv preprint arXiv:2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='00891, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' [7] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Dey and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Rajan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' F q-linear cyclic codes over : Dft approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Designs, Codes and Cryptography, 34(1):89–116, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' [8] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Dummit and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Foote.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Abstract algebra, volume 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Wiley Hoboken, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' [9] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Ezerman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Quantum error-control codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' In W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Huffman, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Kim, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Sol´e, editors, Concise encyclopedia of coding theory, chapter 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Chapman and Hall/CRC, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' [10] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Ezerman, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Ling, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' ¨Ozkaya, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Sol´e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Good stabilizer codes from quasi-cyclic codes over F4 and F9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' In 2019 IEEE International Symposium on Information Theory (ISIT), pages 2898–2902.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' IEEE, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' [11] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Gottesman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Class of quantum error-correcting codes saturating the quantum Hamming bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Physical Review A, 54(3):1862, 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' [12] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Grassl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Code Tables: Bounds on the parameters of various types of codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='codetables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='de/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' [13] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Grassl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Algebraic quantum codes: Linking quantum mechanics and discrete mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Theory, 6(4):243–259, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' [14] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Guan, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Li, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Ma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Symplectic self-orthogonal quasi-cyclic codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' arXiv preprint arXiv:2212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='14225, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' [15] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' G¨uneri, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' ¨Ozdemir, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Sole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' On the additive cyclic structure of quasi-cyclic codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Discrete Mathe- matics, 341(10):2735–2741, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' [16] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Huffman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Additive cyclic codes over F4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=', 1(4):427–459, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' [17] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Huffman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Additive cyclic codes over F4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=', 2(3):309–343, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' [18] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Ketkar, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Klappenecker, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Kumar, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Sarvepalli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Nonbinary stabilizer codes over finite fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' IEEE transactions on information theory, 52(11):4892–4914, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' [19] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Lisonˇek and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Dastbasteh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Constructions of quantum codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Presented at The 3rd International Workshop on Boolean Functions and their Applications, loen, norway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' https://people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='uib.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='no/chunlei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='li/workshops/BFA2018/Slides/Lisonek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='pdf, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' [20] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Lisonˇek and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Singh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Quantum codes from nearly self-orthogonal quaternary linear codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Designs, Codes and Cryptography, 73(2):417–424, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' [21] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Samei and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Mahmoudi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Cyclic R-additive codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Discrete Mathematics, 340(7):1657–1668, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Department of Mathematics, Simon Fraser University, Burnaby, BC, Canada Email address: rdastbas@sfu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='ca, kh411@protonmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='com POLYNOMIAL REPRESENTATION OF ADDITIVE CYCLIC CODES AND NEW QUANTUM CODES 17 No Generator polynomials as in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='6 part (II) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='g(x) = x20 + x17 + x15 + x13 + x11 + x8 + x7 + x6 + x5 + x4 + x3 + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='k(x) = x19 + x18 + x17 + x16 + x14 + x10 + x5 + x4 + x3 + x2 + x + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='h(x) = 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='g(x) = x33 + x29 + x28 + x24 + x19 + x18 + x15 + x13 + x12 + x11 + x6 + x4 + x + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='k(x) = x34+x33+x31+x30+x29+x27+x25+x23+x22+x20+x19+x18+x15+x12+x8+x3+x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='h(x)=0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='g(x) = x44 + x43 + x41 + x40 + x39 + x38 + x34 + x33 + x30 + x26 + x24 + x20 + x19 + x18 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='x17 + x16 + x15 + x14 + x11 + x9 + x5 + x3 + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='k(x) = x43 + x42 + x41 + x40 + x36 + x33 + x32 + x31 + x30 + x28 + x26 + x25 + x17 + x16 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='x15 + x13 + x11 + x10 + x2 + x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='h(x) = 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='g(x) = x44 + x43 + x40 + x38 + x37 + x34 + x31 + x27 + x22 + x21 + x20 + x19 + x18 + x17 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='x14 + x12 + x7 + x6 + x5 + x3 + x + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='k(x) = x44 + x41 + x40 + x37 + x36 + x35 + x33 + x30 + x29 + x27 + x26 + x25 + x22 + x20 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='x15 + x14 + x12 + x11 + x10 + x7 + x5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='h(x) = 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='g(x) = x50 + x49 + x48 + x46 + x45 + x43 + x42 + x41 + x40 + x37 + x36 + x35 + x30 + x29 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='x28 + x26 + x23 + x19 + x18 + x17 + x16 + x15 + x14 + x13 + x9 + x7 + x6 + x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='k(x) = x50 + x47 + x44 + x43 + x42 + x41 + x40 + x38 + x36 + x35 + x33 + x32 + x28 + x26 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='x24 + x21 + x20 + x16 + x14 + x12 + x9 + x8 + x7 + x + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='h(x)=0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='g(x) = x48 + x40 + x37 + x36 + x33 + x31 + x30 + x24 + x23 + x21 + x19 + x15 + x11 + x10 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='x9 + x8 + x7 + x4 + x3 + x + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='k(x) = x41 + x40 + x36 + x35 + x34 + x33 + x30 + x29 + x27 + x23 + x22 + x21 + x19 + x18 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='x16 + x13 + x12 + x10 + x9 + x8 + x7 + x6 + x5 + x4 + x3 + x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='h(x) = x50 + x49 + x48 + x47 + x46 + x45 + x44 + x41 + x40 + x39 + x33 + x31 + x30 + x28 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='x25 + x22 + x20 + x19 + x18 + x17 + x16 + x14 + x13 + x11 + x9 + x5 + x4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='g(x) = x49 + x48 + x46 + x44 + x43 + x41 + x38 + x37 + x36 + x33 + x32 + x31 + x30 + x29 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='x27 + x25 + x21 + x20 + x18 + x17 + x15 + x11 + x10 + x7 + x2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='k(x) = x43 + x42 + x41 + x40 + x38 + x37 + x33 + x32 + x30 + x26 + x24 + x22 + x19 + x18 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='x16 + x15 + x13 + x9 + x5 + x4 + x2 + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='h(x) = x50 +x49 +x48 +x47 +x46 +x45 +x44 +x43 +x42 +x41 +x40 +x39 +x38 +x37 +x36 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='x35 +x34 +x33 +x32 +x31 +x30 +x29 +x28 +x27 +x26 +x25 +x24 +x23 +x22 +x21 +x20 +x19 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='x18 +x17 +x16 +x15 +x14 +x13 +x12 +x11 +x10 +x9 +x8 +x7 +x6 +x5 +x4 +x3 +x2 +x+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='g(x) = x61+x60+x59+x57+x56+x53+x52+x51+x42+x41+x38+x36+x34+x32+x31+x28+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='x27 +x26 +x24+x20 +x19+x16 +x14+x13 +x12+x11 +x9+x8+x7+x6+x5 +x4+x3+x2+x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='k(x) = x61 + x59 + x57 + x56 + x55 + x54 + x52 + x51 + x50 + x49 + x47 + x44 + x37 + x36 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='x35 + x33 + x32 + x31 + x29 + x28 + x27 + x26 + x24 + x22 + x21 + x16 + x8 + x5 + x3 + x2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='h(x) = x62 + x61 + x60 + x59 + x58 + x51 + x49 + x47 + x44 + x43 + x40 + x36 + x33 + x31 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='x30 + x28 + x27 + x23 + x22 + x21 + x19 + x14 + x13 + x11 + x9 + x8 + x7 + x4 + x3 + x2 + x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='g(x) = x60 + x59 + x58 + x55 + x54 + x53 + x52 + x51 + x48 + x47 + x45 + x44 + x40 + x38 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='x37 + x36 + x35 + x34 + x33 + x32 + x31 + x30 + x29 + x28 + x27 + x24 + x23 + x22 + x21 + x15 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='x13 + x10 + x9 + x7 + x6 + x3 + x + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='k(x) = x62 + x59 + x56 + x55 + x54 + x53 + x49 + x47 + x46 + x42 + x41 + x40 + x37 + x35 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='x33 + x31 + x29 + x28 + x27 + x24 + x20 + x19 + x16 + x15 + x14 + x7 + x4 + x2 + x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='h(x) = 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='g(x) = x61+x60+x59+x58+x57+x53+x52+x49+x44+x41+x38+x37+x36+x35+x34+x32+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='x31 +x30+x27+x26+x24+x23+x21+x20 +x19+x13+x12+x11+x8+x6+x5+x4+x3+x+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='k(x) = x60+x58+x57+x56+x52+x48+x47+x46+x44+x42+x40+x39+x38+x36+x35+x34+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='x32+x31+x30+x26+x25+x24+x22+x19+x18+x17+x13+x12+x9+x7+x6+x5+x4+x3+x2+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='h(x) = x62 + x61 + x60 + x59 + x58 + x51 + x49 + x47 + x44 + x43 + x40 + x36 + x33 + x31 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='x30 + x28 + x27 + x23 + x22 + x21 + x19 + x14 + x13 + x11 + x9 + x8 + x7 + x4 + x3 + x2 + x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content='Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} +page_content=' Generator polynomials of additive cyclic codes of Table 1' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAyT4oBgHgl3EQf2fnD/content/2301.00753v1.pdf'} diff --git a/6tAzT4oBgHgl3EQfgPwz/content/tmp_files/2301.01464v1.pdf.txt b/6tAzT4oBgHgl3EQfgPwz/content/tmp_files/2301.01464v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..bea8da3d4291b195394cb1c17cdbc1336355cc99 --- /dev/null +++ b/6tAzT4oBgHgl3EQfgPwz/content/tmp_files/2301.01464v1.pdf.txt @@ -0,0 +1,1472 @@ +Low-frequency shear Alfv´en waves at DIII-D: theoretical +interpretation of experimental observations +Ruirui Ma,1, 2, ∗ W.W. Heidbrink,3 Liu Chen,4, 3, 2 Fulvio Zonca,2, 4 and Zhiyong Qiu4, 2 +1Southwestern Institute of Physics, P.O. Box 432, Chengdu, 610041, China +2Center for Nonlinear Plasma Science and C.R. +ENEA Frascati, C.P. 65, 00044 Frascati, Italy +3Department of Physics and Astronomy, +University of California, Irvine, CA 92697-4574, USA +4Institute for Fusion Theory and Simulation and Department of Physics, +Zhejiang University, Hangzhou, 310027, People’s Republic of China +(Dated: January 5, 2023) +Abstract +The linear properties of the low-frequency shear Alfv´en waves such as those associated with +the beta-induced Alfv´en eigenmodes (BAEs) and the low-frequency modes observed in reversed- +magnetic-shear DIII-D discharges (W. Heidbrink, et al 2021 Nucl. Fusion 61 066031) are theoret- +ically investigated and delineated based on the theoretical framework of the general fishbone-like +dispersion relation (GFLDR). By adopting representative experimental equilibrium profiles, it is +found that the low-frequency modes and BAEs are, respectively, the reactive-type and dissipative- +type unstable modes with dominant Alfv´enic polarization, thus the former being more precisely +called low-frequency Alfv´en modes (LFAMs). More specifically, due to different instability mech- +anisms, the maximal drive of BAEs occurs, in comparison to LFAMs, when the minimum of the +safety factor (qmin) deviates from a rational number. Meanwhile, the BAE eigenfunction peaks +at the radial position of the maximum energetic particle pressure gradient, resulting in a large +deviation from the qmin surface. +Moreover, the ascending frequency spectrum patterns of the +experimentally observed BAEs and LFAMs can be theoretically reproduced by varying qmin and +also be well interpreted based on the GFLDR. The present analysis illustrates the solid predictive +capability of the GFLDR and its practical usefulness in enhancing the interpretative capability of +both experimental and numerical simulation results. +∗ corresponding author. Email address: rrma@swip.ac.cn +1 +arXiv:2301.01464v1 [physics.plasm-ph] 4 Jan 2023 + +I. +INTRODUCTION AND MOTIVATION +The low-frequency Alfv´en wave spectrum in the kinetic thermal-ion (KTI) gap frequency +range [1] has been of research interest since the first observations of beta-induced Alfv´en +eigenmodes (BAEs) [2, 3]. These modes are characterized with frequencies comparable to +thermal ion transit and/or bounce frequencies, and can interact with both thermal and +fast particles [4–9], with possible (positive/negative) impact on the corresponding transport +processes resulting from finite fluctuation and zonal field structures levels [1, 9, 10]. The +effects of energetic particles (EPs) on low-frequency shear Alfv´en waves (SAWs) ranging +from kinetic ballooning mode (KBM) [11–13] to BAE are one of areas widely studied in +the magnetic fusion literature. Recent papers on this topic cover the interpretation and +modeling of experimental measurements by currently developed innovative diagnostics [14– +18], as well as latest progress in comparing numerical investigation and/or simulation results +with observed phenomena [19–24]. +A series of dedicated experiments have been recently conducted on DIII-D to investigate +the stability of the low-frequency SAWs [16–18]. The experiments show that the observed +low-frequency mode1, which was previously misidentified as ‘beta-induced Alfv´en acoustic +eigenmode (BAAE)’ [25, 26], is actually a lower-frequency reactive unstable KBM which +favors high thermal electron temperature but almost has no coupling with energetic ions +[16]; while the BAE is resonantly excited by energetic ions with its stability depending +sensitively on the beam power and injection geometry [17], consistent with earlier theoretical +predictions [27] based on the GFLDR theoretical framework [28, 29]. These instabilities are +also found to occur when the minimum of the safety factor (qmin) approaches rational values +and the modes in ascending pattern of higher frequency BAEs and LFAMs are separated by +approximately the toroidal rotation frequency (frot). However, the subtle differences between +them are that, for LFAMs, the maximum frequency appears at rational values of qmin and +the detected modes are radially localized near qmin, while BAEs occur at times near rational +qmin values but the timing of unstable modes is less precise than that for LFAMs. In addition, +compared with the LFAMs, the BAE eigenfunction shows more deviation from the radial +position of qmin spatially. Although dedicated numerical simulations of the linear properties +1We will refer from now on only to the low frequency Alfv´en mode (LFAM) which belongs to low-frequency +SAWs predominantly Alfv´enic polarization, keeping in mind that this terminology is the same as the low- +frequency mode observed in recent DIII-D experiments [16]. +2 + +of the BAEs and LFAMs [24, 30] have been carried out, the above experimental phenomena +have not been fully explained. +Motivated by this, the present work aims to provide an +in-depth theoretical understanding of the linear properties of low-frequency SAWs, with +particular attention to the effects of energetic ions on their stability. The analysis is carried +out based on the theoretical framework of the generalized fishbone-like dispersion relation +(GFLDR) [28, 29, 31–35], and provides qualitative and quantitative interpretation of the +main instability mechanisms underlying the numerical simulation results and experimental +observations. +As a result, our analysis provides yet another evidence of the predictive +strength of the GFLDR theoretical framework and of its enhanced “interpretative capability +for both experimental and numerical simulation results” [28, 29]. +In this work, unlike the previous paper not considering effects due to energetic particles +(EPs) [36], we focus on the BAE excitation via transit resonance with passing fast ions +created by NBI heating [17]. In this case, the dynamics of various species enter the dispersion +relation of low-frequency SAW, and affect its behavior linearly at different pressure gradient +scale lengths. For DIII-D discharge #178631, Fig. 1 shows the radial dependence of different +scale lengths of thermal and energetic particle pressure (LPth and LPE), as well as the +estimated radial mode width (∆m) for weak and/or vanishing magnetic shear range, i.e., +|s| = |(r/q)(dq/dr)| ≲ 0.05. More specifically, the EP pressure profiles are given by the +following two limits. +One is the relaxed EP profile provided with EFIT reconstruction +[37], where the fast-ion pressure is the difference between the equilibrium pressure and the +thermal pressure. The other is the “classical” EP profile obtained by TRANSP/NUBEAM +[38] in the absence of fast-ion transport by instabilities. The pressure scale lengths of EPs +are denoted by LPE;rel and LPE;cl for these two cases (respectively). The true EP profile when +the modes are destabilized likely lies between these two limits. The actual pressure is closest +to the EFIT-based one but this is measured after the unstable modes have (presumably) +caused the gradients to flatten. Meanwhile, for the weak and/or vanishing magnetic shear +region and given toroidal and poloidal mode numbers (n, m), the normalized parallel wave +vector is ΩA,m = k∥n0qminR0 = nqmin − m, and the radial width of the mode can then +be estimated by ∆m ≃ 1/|nq′′|1/2 [39, 40]. Here, k∥n0 represents the parallel wave-vector +at r0, where q has a minimum given by qmin, q′′ denotes the second derivative of q in the +radial direction, and R0 is the torus major radius. It can be found that in this region, +LPth ≫ ∆m, which yields the usual local limit of the mode dispersion relation. This is the +3 + +case for the reactive unstable LFAM in the absence of EPs already studied in Ref. [36]. +However, for the energetic ion-driven BAEs, there are two distinct cases: the moderate +EP pressure gradient case with LPE;rel > ∆m, which also approximately yields the usual +local GFLDR [4, 28, 29, 32, 33, 35, 39, 40]; and the strong EP pressure gradient case with +LPE;rel ≃ ∆m, for which the global dispersion relation of low-frequency SAWs is needed +and will be discussed in Sec. II. Performing detailed numerical investigations of the two +FIG. 1. +The radial dependences of the typical scale lengths of thermal and energetic particle +pressure (LPth and LPE), as well as the estimated radial mode width (∆m). +cases, it is found that the LFAMs and BAEs can both be driven unstable, however, due to +different instability mechanisms, these modes yield different experimental observations. All +these features can be, quantitatively and qualitatively, interpreted theoretically based on the +GFLDR. Moreover, it is also confirmed that the stability of BAAE is not affected by EPs, +even though it becomes weakly damped after coupling with KBM, consistent with theoretical +predictions by Chen and Zonca [27] as well as numerical simulation results reported in Refs. +[20, 23, 24]. +The paper is structured as follows. Local and global dispersion relations for the low- +frequency SAWs near weak and/or vanishing magnetic shear are introduced and discussed +in Sec. II in different parameter regimes, depending on the relative magnitude of LPE and +∆m. Detailed numerical investigations and theoretical analysis of the low-frequency SAWs +in the presence of EPs are discussed in Sec. III, where comparisons between theory and +experiments are also made. Finally, conclusions and further discussions are given in Sec. +IV. +4 + +1.5 +th +E;cl +length (m) +E;rel +m +0.5 +0 +0.2 +0.24 +0.28 +0.32 +r/aII. +THE GENERAL FISHBONE-LIKE DISPERSION RELATION FOR LOW- +FREQUENCY SAWS +In this Section, we will present analytical dispersion relations for low-frequency SAW +excitation in weakly reversed-shear DIII-D discharges. As stated in the previous Section, +two cases determined by the relative magnitude of LPE and ∆m will be used to investigate the +low-frequency SAW stability: case I, the local GFLDR model corresponding to LPE > ∆m; +and case II, the global GFLDR corresponding to LPE ≃ ∆m. +Consider case I first. For LPE;rel > ∆m, the scales of LPE and ∆m can be separated, +and the vorticity equation [4, 9, 28, 29, 32, 33] which governs shear Alfv´en waves (SAWs) +can yield the low-frequency electromagnetic fluctuation dispersion relation in the usual local +limit, as derived and discussed in great details in Refs. [9, 28, 29, 32, 33, 35]. We just note +that, for DIII-D case of interest, the reversed magnetic shear configuration and thermal +plasma compression effects should be accounted for properly [36]. Thus, for s = 0 at r0 but +with finite S ≡ (r/q)[q +′′]1/2, the local GFLDR for low-frequency SAWs can be written as +[27–29, 35, 40] +iS(Λ2 +n − k2 +∥n0q2 +minR2 +0)1/2(1/n)1/2� +k∥n0qminR0 − i(Λ2 +n − k2 +∥n0q2 +minR2 +0)1/2�1/2 = δ ˆWnf + δ ˆWnk(ω), +(1) +where the generalized inertia term Λn(ω) here, including both diamagnetic effects as well as +kinetic effects of circulating and trapped particle dynamics, has been derived explicitly in +Ref. [7] and the main results are summarized in Appendix A. The right hand side of Eq. +(1) contains both “fluid” (δ ˆWnf) and “kinetic” (δ ˆWnk) contributions to the potential energy +in the “regular” ideal region. In the low-frequency limits (|Λ2 +n| ≪ 1), δ ˆWnf is independent +of the frequency and the explicit expression, specialized to the (s, α) model equilibrium [41] +with circular flux surfaces, reads, +δ ˆWnf ≃ π +4 +�S2k∥0qminR0 +n +− 3 +2α2S +��k∥0qminR0 +n +��1/2 + 9 +32α4 +� +(2) +where α = αc + αE, αc = −R0q2 +mindβ/dr and αE = − 1 +2R0q2 +mind(βE∥ + βE⊥)/dr. Note that +Eq. (2) includes the contribution of the energetic particle adiabatic and convective responses +as well [31]. +The term δ ˆWnk is always a function of the mode frequency ω, as it reflects resonant +as well as non-resonant wave-particle interactions. For simplicity but still relevant to the +5 + +DIII-D case, we take F0E to be a single pitch angle (λ = µ/ε) slowing-down beam ion +equilibrium distribution function; i.e., F0E = +B0βE(r) +25√ +2π2mEεb +� +(1 − λ0B0)ε−3/2δ(λ − λ0). Here, +βE(r) ≡ 8πPE(r)/B2 +0 is the ratio of EP kinetic and magnetic pressures and B0 the on- +axis equilibrium magnetic field, δ(x) is the Dirac function, µ is the magnetic moment and +ε = υ2/2 ≤ εb with εb being the EP birth energy per unit mass. Then the explicit expression +of non-adiabatic contribution δ ˆWnku for the passing energetic ions is given by [32, 33] +δ ˆWnku ≃ παE +25/2 (1 − λ0B0/2)¯ω +� +2 − ¯ω ln +� ¯ω + 1 +¯ω − 1 +�� +, +(3) +where ¯ω = ω/ωtEm and ωtEm ≡ √2εb/qR0 is the EP transit frequency at the maximum +particle energy. +It is worthwhile emphasizing that the finite k∥n0qminR0 in Eq. (1) plays an important +stabilizing role since it represents the finite line bending effect at r = r0 [28, 29, 35]. Further- +more, the expression of Λn depends on the mode polarization via Sf ≡ (iδE∥/k∥)a.c. +� +δφd.c., +where a.c. and d.c. refer to the sinusoidal and nearly constant (flute-like) components of the +parallel electric field, wave vector, and scalar potential fluctuation [21, 27]. The detailed +expression of Sf, again, is given in the Appendix A. Here, we just note that |Sf| is much +smaller than unity for shear Alfv´en wave and order of unity for ion acoustic wave [7, 21, 27]. +We remark here that, in the moderate pressure gradient case, the local GFLDR for +the low-frequency SAWs is enough to delineate the underlying physics of the experimental +and simulation results. However, the local GFLDR for the low-frequency SAWs, given by +Eq. (1), will fail in the presence of strong EP pressure gradient, i.e., case II. In this case, +two typical scale lengths LPE,cl and ∆m can not be separated anymore and, thus, a global +dispersion relation is needed which can be derived from the vorticity equation, i.e., Eq. (1) +of Ref. [40]. Noting that the mode structure is dominated by single toroidal and poloidal +mode numbers, (n, m), the governing equation reads +(eθ − erξ) · +� +Λ2 − Ω2 +A,m +� +1 + +x2 +ΩA,m ++ +x4 +4Ω2 +A,m +�� +(eθ − erξ)δφm − (F + K)δφm = 0, +(4) +where k⊥/kθ = −(eθ − erξ) with er and eθ being, respectively, the radial and poloidal unit +vectors, x2 = nq′′ +min(r − r0)2, ξ ≡ (i/n1/2)S(∂/∂x), and δφm is the mth poloidal harmonic +of the scalar field perturbation. It is worth noting that, toroidal coupling among different +poloidal harmonics is typically not important for modes in the reversed magnetic shear +region, consistent with the mode being dominated by single m and n. The terms F and K +6 + +in Eq. (4) represent, respectively, the fluid-like particle and energetic ion contributions with +their explicit form reading +F ≃ D2 +S − 4α2DS + 2αD2 +S − (α + 1)α + 2α3, +K ≃ 2πq2 +Eq2R2 +0ω +mEc2 +�Ω2 +dEQF0E +ω2 +tE − ω2 +� +υ += 2 +πδ ˆWnku, +(5) +where DS = S +� +ΩA,m/n, qE and mE are the electric charge and mass of energetic ions, ΩdE = +(υ2 +E⊥/2+υ2 +E∥)/ωcER0, ωtE = υE∥/qR0, QF0E = (ω∂ε+ˆω∗E)F0E, ˆω∗EF0E = ω−1 +cE (k×b)·∇F0E, +ωcE = qEB/mEc, ⟨(...)⟩υ = +� +d3υ(...), and the subscripts ∥ and ⊥ represent the parallel and +perpendicular components with respect to the equilibrium magnetic field b. +Equation (4) is an ordinary differential equation and, generally, requires a numerical +approach to be solved. However, for DIII-D case, the radial dependence of the normalized +pressure gradient of energetic ions with the classical profile, as is shown by black curve in Fig. +2, can be well fitted by the analytic formula αE(ρ) = c1 (1 − (ρ − c2)2/c2 +3), with c1 = 0.7099, +c2 = 0.3018 and c3 = 0.2944. This allows us to obtain simple analytical dispersion relations +for low-frequency SAWs excitation. We just note that the maximum drive of energetic ions +is located around ρ = c2 = 0.3018, which deviates from the radial position of qmin. Then +αE(r) in Eq. (3) can be rewritten as +αE(r) = δaαE0 +� +1 − (r − r0 + δb)2 +δ2 +cL2 +PE;cl +� +, +(6) +where δa = c1/αE0, δb = r0 − c2a and δc = c3a/LPE;cl, a is the minor radius, αE0 and LPE;cl +are evaluated at r = r0. Introducing the notation x = r − r0 = σz − δb, Eq. (4) is readily +cast into the form +∂2 +∂z2δφm − nσ2 +S2 +� +1 − F + 2δa +π δ ˆWnku0 +ϵA0 +� +δφm − 1 +4z2δφm = 0, +2nσ4δaδ ˆWnku0 +ϵA0πS2δ2 +cL2 +PE;cl += 1 +4, +(7) +where ϵA0 = Λ2 − Ω2 +A,m, δ ˆWnku0 = παE0 +4 +√ +2 +� +2 − ¯ω ln +� ¯ω+1 +¯ω−1 +�� +. Then, Eq. (7) yields the following +global dispersion relation for low-frequency SAWs, +−n1/2π1/2δcLPE;clϵ1/2 +A0 +2 +√ +2Sδ1/2 +a δ ˆW 1/2 +nku0 +� +1 − F + 2δa +π δ ˆWnku0 +ϵA0 +� += 2L + 1, +L = 0, 1, 2, 3 ... +(8) +7 + +FIG. 2. The radial dependence of the normalized pressure gradient of EPs with the classical profile. +Here, the normalized radial position of qmin is ρ0 ≡ r0/a = 0.28. +Here, the integer L is the radial eigenmode number. The corresponding eigenfunction reads +δφm(r) = HL(z)e−z2 ∝ exp +� +−(r − r0 + δb)2 +4σ2 +� +, +(9) +where HL(z) represents Lth order Hermite polynomials and the causality constraints upon +the discrete bound modes requiring Re(σ2) > 0, where σ2 is solved for from the second of +Eqs. (7) consistently with the dispersion relation, Eq. (8). The typical radial width, w, of +δφm(r) is determined by w2 = 4σ2. +Equations (1) and (8) constitute the results of the present section, i.e., the local and +global GFLDR for the low-frequency SAWs excited by energetic ions. With their explicit +form, we can compute the individual terms involved in equations and investigate the linear +properties of the experimentally observed low-frequency SAWs. +III. +THE LOW-FREQUENCY SAW INSTABILITIES NUMERICAL RESULTS +AND ANALYSIS +In this Section, we separately present numerical results for the local and global low- +frequency SAW stability properties in the presence of energetic ions, for which the dispersion +relation is given by Eqs. (1) and (8). The numerical investigations use experimental equilib- +rium and profiles as shown in Fig. 3 for the DIII-D shot #178631 at the time t = 1200 ms +[16], where the q-profile has a reversed shear configuration with qmin = 1.37 at r0/a = 0.28 +8 + +0.72 +0.7 +0.68 +a +0.66 +αE;cl;exp. +vs. p +0.64 +fit +-Prediction bounds -99% +0.62 +0.2 +0.25 +0.3 +0.35 +p=r/aFIG. 3. Radial profiles of (a) temperature and q and (b) density and toroidal rotation frequency +frot of DIII-D shot #178631 used for numerical studies. +and qmin decreases from 1.49 to 1.18 in the time window 1050 ms < t < 1350 ms, as shown +in Fig. 6 (b) in Ref. 16. +A. +The local low-frequency SAW stability properties +We first consider the linear properties of the low-frequency SAW with relaxed energetic ion +profile, i.e., case I. The local equilibrium parameters used in the numerical studies evaluated +at r0/a = 0.28 are S = 0.5895, τ = Te/Ti =3.86 keV/2.37 keV=1.62, ne = 3.80 × 1019 +m−3, ni = 3.19 × 1019 m−3, ϵr = r0/R = 0.10, βi ≃ 0.01, ϵni = Lni/R0 = 0.414, ηi = +Lni/LTi = 0.8324, ω∗ni/ωti = 0.1919, (m, n) = (8, 6), kθρLi = 0.2555 and kθρLe = 0.0054. +Other fixed equilibrium parameters are a = 0.64 m, R0 = 1.74 m, B0 = 1.8 T. Here, kθ +is the poloidal wavenumber, ρLi and ρLe are the Larmor radii of thermal ions and thermal +electrons, respectively. +Dependencies of the (a) mode frequencies, (b) growth rates and (c) mode polarization +predicted by Eq. +(1) are shown in Fig. +4 as a function of the normalized thermal ion +diamagnetic frequency Ω∗pi ≡ ω∗pi/ωti for the cases without and with the consideration of +EP effects. According to the scaling of mode frequencies with physical parameters and the +value of the |Sf| [21], three branches in Fig. 4 can be classified as: (i) the KBM (red curves +marked with circles), with a frequency scaling with ω ∼ ω∗pi; (ii) the BAE (blue curves), +with the frequency being close to the well-known estimate ω/ωti = qmin +� +7/4 + τ ≃ 2.51; +9 + +(a) +T。 (keV) +6 +.T. (keV) +q +TE:cl (keV/10) +4 +2 +0 +0 +0.2 +0.4 +0.6 +0.8 +1 +r/a(b) +5 +n(1019m-3) + - -n, (1019m3) +4 +4 × nE:cl (1019m=3) +-4 × nE:rel (1019m-3) +3 +-.0.5× frot (kHz) +2 +1 +0 +0 +0.2 +0.4 +0.6 +0.8 +1 +r/aFIG. 4. Dependence of the (a) real frequencies, (b) growth rates and (c) polarization of the low- +frequency SAWs on Ω∗pi ≡ ω∗pi/ωti for the cases without (w/o) and with (w/) EP effects. Here, a +dashed vertical line represents the experimental value of Ω∗pi;exp of about 0.35. +and (iii) the BAAE (green curves marked with diamonds), with a frequency of about half +of the BAE and experiencing strong damping. The EP effects on the low-frequency SAW +stabilities are apparent in the region highlighted by the purple curve of Fig. 4 (b), where the +KBM is the only unstable mode in the absence of EPs, while both the KBM and BAE are +unstable in the low-frequency region in the presence of EPs. In particular, the diamagnetic +ion frequency calculated on the basis of experimental parameters is Ω∗pi;exp = 0.3517, as +shown by the dashed vertical line. In this case, both KBM and BAE are unstable with the +frequencies in the plasma frame being 5.6 kHz and 63.7 kHz, respectively, which are in good +agreement with the experimental observations. Meanwhile, the polarization plot of Fig. 4 +(c) shows that KBM and BAE have small values for |Sf| ≲ 0.1, which indicates that the +KBM and BAE are essentially of Alfv´enic polarization. Moreover, in order to exclude the +10 + +(a) +KBMw/oEP +BAAE w/o EP +BAE w/o EP +4 +KBM w/ EP +Re(wlwti +BAAE w/ EP +BAE w/ EP +0 +2 +4 +6 +2 +* +pi:(b) +m(w/wti +KBM w/o EP +BAAE w/o EP +BAEw/o EP +KBMw/EP +BAAE w/ EP +BAEw/EP +0 +*pi;exp +2 +4 +6 +U +pi +*100 +(c) +KBM w/o EP +BAAE w/o EP +BAEw/oEP +S +KBM w/ EP +BAAE w/ EP +BAEw/EP +0 +2 +4 +6 +* pispurious nonzero solutions produced by singularities of the transcendental function of the +local GFLDR (D), the Nyquist diagram in the complex D plane presented in Fig. 5 shows +that in the presence of EPs, the path encircles the origin twice (see Fig. 5 (b)) but only once +without EPs (see Fig. 5 (a)), thus confirming there are two unstable modes with EPs. It +FIG. 5. The Nyquist diagram in the complex D(ω) plane for the cases (a) without and (b) with +EP effects. +should be noted that, compared with the frequency insensitive to the EP effects, the growth +rate of the KBMs changes significantly in the cases with and without EP effects. +This +occurs because in our theoretical model the adiabatic and convective contribution of EPs +modifies the value of δ ˆWf via α, as is shown in Eq. (2). At this point, in order to obtain more +convincing comparison of theoretical prediction and experimental observation, it is necessary +to provide a more precise theoretical model and also a more comprehensive experimental +analysis. We also note here that, in this case, the stability/property of the BAAE is not +affected by energetic ions — as is shown by the green dashed lines with symbols (without +EP effects) and solid lines with symbols (with EP effects) which are apparently overlaying +in all three graphs — even though it becomes weakly damped by coupling with the KBM +due to diamagnetic and trapped particle effects for sufficiently strong Ω∗pi. The numerical +results are consistent with the numerical simulation results reported in Refs. [20, 23, 24] +and the theoretical prediction in Ref. [27], that is, “EPs preferentially excite the BAE over +the BAAE branch due to the stronger wave-EP interaction”. +We now investigate the underlying instability mechanisms of the ascending spectrum of +the higher frequency BAEs and LFAMs observed in DIII-D (see Fig. 8 of Ref. [17]) by using +11 + +×10-3 +(a) +10 +5 +0 +-5 +-5 +0 +5 +Re(D) +×10-3X10-3 +10 +(b) +5 +Im(D) +0 +.5 +-5 +0 +5 +Re(D) +×10-3qmin as the scanning parameter. Figure 6 shows the dependence of the mode frequencies +(solid curves with markers) and growth rates (dashed curves with markers) on qmin of the +KBMs (red curves) and the BAEs (blue, green, purple and orange curves) for different +poloidal and toroidal mode numbers (m, n). It is shown that the modes in ascending pattern +FIG. 6. Dependence of mode frequencies (solid curves with markers) and growth rates (dashed +curves with markers) on qmin of the KBMs (red curves) and the BAEs (blue, green, purple and +orange curves) for different (m, n). The experimentally observed frequencies are also shown. For +the BAE, since the modes span a range of frequencies, the lines indicate the upper and lower limits +of the unstable bands; for the LFAM, the experimental frequency variation is < 0.5 kHz. In the +abscissa, the experimentally measured qmin(t) fit shown in Fig. 8 of [17] is used to convert time to +qmin, with an associated uncertainty of ∆qmin ≃ 0.01. In the ordinate, the theoretical lab-frame +frequency incorporates a Doppler shift to the calculated plasma-frame frequency of nfrot, with an +associated uncertainty of ∼ 0.5 × n kHz. +of higher frequency BAEs and lower frequency KBMs are both separated by approximately +12 + +200 ++7,5 +f: in the lab-frame +(10,8) +KBM +f8,6 +KBM +9,7 +f9,7 +KBM +8.6 +f10,8 +150 +KBM +(7,5) +KBM +KBM +KBM +ZH) +.10,8 +100 +/2元 +“KBM +Expi..data +7,5 +BAE +(range) +8,6 +(lines with ★) +(9,7) +(10,8) +(8,6) +BAE +(7,5) +9,7 +BAE +10,8 +50 +BAE +/27 +BAE +“BAE +BAE +10,8 +BAE +1.45 +1.4 +1.35 +1.3 +1.25 +1.2 +minfrot of about 7.5 kHz. More specifically, for KBMs, the instabilities peak exactly at the +rational values of qmin; while the BAEs occur at times near rational values of qmin but the +timing of unstable modes is less precise than for KBMs. In addition, the low-n BAEs deviate +more from rational qmin crossings than higher n modes. The comparison of the theoretically +predicted frequencies with the experimentally measured values can also be seen clearly from +Fig. 6. As discussed in more detail in the next section, these numerical results are in good +agreement with the experimental observations. +In order to gain insight into the different excitation mechanisms of the instabilities pre- +sented in Fig. 6, let us further analyze the GFLDR in the high-frequency (|ω| ≫ ωti) and +low-frequency |ω| ≪ ωbi limits. +For |ω| ≫ |ωti|, the corresponding inertia term of the BAE can be reduced to the simplified +expression with Λ2 ≃ +ω2−ω2 +BAE +ω2 +A +[4, 35, 42]. Here, ω2 +BAE = (7/4 + τ)υ2 +i /R2 +0 is the fluid limit +expression of the BAE frequency. Taking ω = ωr + iγ and δ ˆWku = Reδ ˆWku + iImδ ˆWku, and +assuming |γ/ωr|, we have |Imδ ˆWku/Reδ ˆWku| ≪ 1. Then, for the gap mode, the existence +condition is δ ˆWnf + Re(δ ˆWnk(ωr)) < 0 and the real mode frequency is given by +ω2 +r = ω2 +BAE +� +��1 + +ω2 +A +ω2 +BAE +� +� +�k2 +∥n0q2 +minR2 +0 − +n +��k∥n0qminR0 +�� +� +δ ˆWnf + Re(δ ˆWnk(ωr)) +�2 +S2 +� +� +� +� +�� , +(10) +while the growth rate is obtained from +γ = −Im(δ ˆWnk(ωr))ω2 +A +ωr +n +� +δ ˆWnf + Re(δ ˆWnk(ωr)) +� +��k∥n0qminR0 +�� S2 +, +(11) +It can be readily obtained from Eq. (10) that the BAE frequency is positively correlated with +��k∥n0qminR0 +��. Therefore, the more deviation from the rational qmin surface is, the larger the +BAE frequency is, as is shown in Fig. 6. Note also that the BAE has a positive frequency. +Equation (11) imposes Im(δ ˆWnk(ωr)) > 0 for BAE excitation by EPs via resonant wave- +particle interaction. It can be concluded that the duration of BAEs is influenced by the +associated resonances with the EPs, as well as by the value of qmin [17]. +Similarly, for KBM with |ω| ≪ |ωbi|, we have Λ2 ≃ c0 +q2 +min +√ϵ +(ω−¯ωdi)(ω−ω∗pi) +ω2 +A +[7, 16, 21, 35, 43]. +Here, ¯ωdi is the average thermal-ion precession frequency, c0 ≃ 1.6 due to trapped and barely +13 + +circulating particles [44, 45]. Thus, the real mode frequency is given by +ω = 1 +2(¯ωdi+ω∗pi)±1 +2 +� +��(ω∗pi − ¯ωdi)2 − 4ω2 +A +√ϵ +q2 +minc0 +� +� +� +n +� +δ ˆWnf + Re(δ ˆWnk(ωr)) +�2 +��k∥n0qminR0 +�� S2 +− k2 +∥n0q2 +minR2 +0 +� +� +� +� +�� +1/2 +, +(12) +and the system is reactively unstable if +|ω∗pi − ¯ωdi|2 +ω2 +A +< +4√ϵ +q2 +minc0 +� +� +� +n +� +δ ˆWnf + Re(δ ˆWnk(ωr)) +�2 +��k∥n0qminR0 +�� S2 +− k2 +∥n0q2 +minR2 +0 +� +� +� . +(13) +Note that δ ˆWf + Reδ ˆWku < 0, due to, again, the causality constraint. Therefore, for the +reactive-type instability, the maximum drive sets in when k∥n0qminR0 → 0, which corre- +sponds to the unstable KBM exactly peaking at the rational values of qmin. +The above numerical results and theoretical analyses have explained the experimental +observations that the BAEs deviate more from the rational qmin values temporally, com- +pared with the KBM. To further delineate this deviation and its impact on the radial mode +structure, numerical investigation of the global model for low-frequency SAWs is needed. +B. +The global low-frequency SAW stability properties +In this part, we consider the case II and apply Eq. (8) to investigate the global low- +frequency SAW stability properties with the classical energetic ion profile. +Figure 7 shows (a) the dependence of the real frequencies (blue markers) and growth rates +(red markers) of the KBM (triangle markers) and BAE (line with markers) on the radial +mode number L; and (b) the radial mode structure δφm(r) for the L = 0 BAE. It can be +found that (i) the ground eigenstate with L = 0 is most unstable for the BAE and KBM; +(ii) for BAE, the frequency and growth rate in the plasma frame is (80.7 + 15.2i) kHz with +the ratio of the growth rate to real frequency γ/ω ≃ 0.19, which is the typical feature of the +marginally unstable gap mode excited by EPs; and (iii) for KBM, the frequency and growth +rate in the plasma frame is (−3.2 + 5.7i) kHz with γ/ω ≃ 1.8, which is the typical feature +of the reactive-type instability, consistent with the results reported in Ref. [24]. +Correspondingly, the radial eigenfunction plot of the BAE for L = 0, as shown in Fig. +7 (b), presents that δφm has a Gaussian form with a shape similar to the experimentally +14 + +FIG. 7. (a) Dependence of the real frequencies (blue markers) and growth rates (red markers) of +the KBM (triangle markers) and BAE (line with markers) on the radial mode number L; (b) the +radial mode structure δφm(r) for the L = 0 BAE. The approximate experimental measurement of +the mode structure of BAE is also shown. +measured radial mode structure. In this case, the radial width of δφm by theory is w = +0.2107, is comparable to the scale length of energetic-ion pressure, i.e., LPE;cl = 0.1773; +consistent with the analysis of Fig. 1. Note that determined by the EP distribution, the +BAE eigenfunction peaks at the radial position of the maximum energetic particle pressure +gradient, resulting in a large deviation from the qmin surface. It can also be expected that +the KBM eigenfunction should peak at the rational values of qmin where the instability drive +is maximum. +Finally, the continuous spectra plots for low-frequency shear Alfv´en and acoustic waves +given by Λ2 +n(ω) = k2 +∥nq2R2 +0 = (nq−m)2 [4, 6, 28, 29, 42, 46, 47] are shown in Fig. 8. Here, the +inertia term includes the diamagnetic effects and thermal ion compressibility as well as drift +Alfv´en wave and drift wave sideband coupling via the wave-thermal-passing-ion interaction +and diamagnetic effect [6]. The figure shows that based on the GFLDR, the nature of various +branches can be clearly classified via their frequencies (a), growth rates (b) and polarizations +(c). Here, the short notation “e-KBM” represents the branch of the KBM propagating in +the thermal-electron diamagnetic drift direction. The unstable continuum spectrum of the +e-KBM is due to the inclusion of the kinetic dynamics of thermal particles in inertia term. In +addition, the frequencies of the (m, n) = (8, 6) BAE and the (m, n) = (8, 6) KBM calculated +by the local and global cases are, respectively, in the gaps of the BAE and KBM continua, +15 + +4 +(a) +3 +△ Re(wlwt) of KBM +A +Im(w/wti) of KBM +2 +-Re(wlwt:) of BAE +-- Im(w/wti) of BAE +0 +2 +3(m,n)=(8,6) +0.75 +analytic +-ECE-measured +m,n +0.5 +0.25 +0 +0 +0.25 +0.5 +0.75 +1 +p=r/awhich is consistent with the numerical simulation results reported in Refs. [16, 24]. +FIG. 8. The continuous spectra of low-frequency shear Alfv´en and acoustic branches for n=6, +m=8-15. The equilibrium profiles of DIII-D #178631 at 1200 ms are adopted. +IV. +SUMMARY AND DISCUSSIONS +The present work has addressed linear properties of the low-frequency shear Alfv´en waves +(SAWs) with the consideration of energetic ions in DIII-D reversed magnetic shear tokamak +experiments. By analyzing the experimental equilibrium profiles, the local and global models +for low-frequency SAWs for weak and/or vanishing magnetic shear are discussed based on the +unified theoretical framework of the generalize fishbone-like dispersion relation (GFLDR). +Resorting to numerical and theoretical analyses, the dependences of mode frequency, growth +rate and polarization on the minimum of the safety factor (qmin), as well as the instability +mechanisms are delineated. +16 + +200 +(a) +n=6; +BAE +pi;exp +BAAE +米 +KBM; rel +150 +m=8~15 +KBM +BAE; rel +KBM1 +米 +KBM; cl +LFM +BAE; cl +(ZH>) +e-KBM +100 +p +50 +0 +米 +米 +0.2 +0.4 +0.6 +0.8 +r/a10 +(b) +0 +(kHz) +-10 +-20 +2元 +BAE +-30 +BAAE +KBM +n=6; +KBM1 +-40 +m=8~15 +LFM +e-KBM +0.2 +0.4 +0.6 +0.8 +r/a102 +(c) +BAE +KBM1 +BAAE +LFM +KBM +e-KBM +Sf100 +0.2 +0.4 +0.6 +0.8 +r/aThe main results of this work are that the LFAMs and BAEs observed in DIII-D ex- +periments are, respectively, the reactive-type and dissipative-type unstable modes with pre- +dominantly Alfv´enic polarization. Due to the different instability mechanisms, BAE peak +occurs further away from the rational qmin than LFAM peak does. The BAE eigenfunction +is localized at the radial position with the strongest energetic-ion-drive spatially, which leads +to deviation from the radial position of qmin. +The theoretical analysis explains many experimental observations. +1. The theory successfully explains the temporal pattern of two bands of instability, the +BAE band and the LFAM band, that both appear near rational values of qmin but +with distinctly different stability properties. +2. The predicted values of KBM frequency are in excellent agreement with the experi- +mental LFAM frequencies. The KBM can be unstable even in the absence of energetic +particles (EPs). +3. The predicted values of BAE frequency span the same range as the experimentally +observed values. +4. The theory also successfully explains the absence of a third branch of instability at +BAAE frequencies, as that branch is predicted to be stable. +5. Experimentally, an individual unstable BAE spans a much larger range of frequencies +than an unstable LFAM, another feature successfully reproduced by theory. +6. Experimentally, unstable LFAMs only persist for a few milliseconds. The short du- +ration of the LFAM is consistent with the very strong qmin dependence of the KBM +growth rate. +7. In experiment, unstable BAEs persist longer than LFAMs, which is consistent with +the weaker dependence of the BAE growth rate on qmin in theory. +8. Temporally, in experiment, LFAMs occur at rational values of qmin; BAEs also occur +near rational values but less precisely. This feature is also reproduced by the theoretical +stability predictions: the KBM growth rate peaks sharply at rational qmin values but +the peak of the BAE growth rate deviates slightly. +17 + +9. In experiment, for both the LFAM and the BAE, unstable modes with higher values of +toroidal mode number n are of shorter duration than lower values of n. The narrower +growth rate curves as n increases successfully explains this feature. +10. Experimentally, the BAE radial eigenfunction has an approximately gaussian shape, +consistent with the theoretical prediction that the L = 0 radial harmonic is most +unstable. +11. Experimentally, the LFAM is more unstable in plasmas with hydrogen than in pure +deuterium plasmas [18], a feature explained by the higher value of ωA in hydrogen +plasmas. As Eq. (13) shows, a larger value of ωA lowers the instability threshold. +On the other hand, there are three discrepancies between theory and experiment. +1. Although the predicted KBM growth rate correctly peaks sharply for rational values +of qmin, it remains positive for a much longer duration than the LFAMs are observed +experimentally. Evidently, an additional damping mechanism is missing in the theory. +2. Although the predicted KBM growth rate has changed significantly for the cases with +and without EPs, there is no apparent dependence of LFAM stability on EPs ex- +perimentally. Therefore, a more precise theoretical model and more comprehensive +experimental analysis are needed for meaningful comparison. +3. Although the predicted BAE frequency spans the observed values, the predicted fre- +quency has a parabolic shape with time, while the experimental frequency has a less +regular shape. A likely explanation for this discrepancy is imprecise modeling of the +fast-ion distribution function. +Finally, there is one theoretical prediction that is inconclusive experimentally: the mode +polarization. Theory predicts predominately Alfv´enic polarization for both the KBM and +the BAE. In experiment, low toroidal mode number (n ≤ 3) BAEs are usually observed on +external magnetic coils; LFAMs are never detected, but the inferred toroidal mode numbers +typically span a larger range than those normally detected for RSAEs or BAEs. DIII-D is +equipped with one diagnostic that can detect internal magnetic fields, a radial interferometer- +polarimeter (RIP) [48] that measures the line integral of the density and radial magnetic +field, +� +neBrdl. This diagnostic clearly detects RSAEs and BAEs, which is consistent with +18 + +their expected shear-wave polarization. Fluctuations are observed by RIP for some LFAMs, +indicating that there is at least some magnetic component, but the signal is weaker than +for RSAEs and BAEs. It is not presently known if this difference is due to a line-integral +effect associated with the mode structure or if the LFAM polarization is less Alfv´enic than +the other modes. +ACKNOWLEDGMENTS +One of authors (R.R. Ma) would like to acknowledge Dr. Lei Yang and Dr. Yunpeng Zou +for their useful discussions and the DIII-D team for providing the experimental data. The +authors thank Dr. Xiaodi Du for helpful comments concerning the mode polarization. R.R +Ma is also grateful to the Center for Nonlinear Plasma Science (CNPS) for its enlightening +academic discussion, which provides a valuable sources of scientific stimuli. +This work has been supported in part by the National key R&D Program of China under +Grant Nos. 2022YFE03040002 and 2018YFE0304103, by the National Science Foundation +of China under Grant Nos. 12261131622 and 12175053 and Natural Science Foundation of +Sichuan under Grant No. 2022NSFSC1814 and Sichuan Science and Technology Program +under Grant No. 2022ZYD0019. This work has also been carried out within the framework +of the EUROfusion Consortium, funded by the European Union via the Euratom Research +and Training Programme (Grant Agreement No. 101052200 – EUROfusion). Views and +opinions expressed are however those of the author(s) only and do not necessarily reflect +those of the European Union or the European Commission. Neither the European Union +nor the European Commission can be held responsible for them. This material is based +upon work supported by the U.S. Department of Energy, Office of Science, Office of Fusion +Energy Sciences, using the DIII-D National Fusion Facility, a DOE Office of Science user +facility, under Awards DE-FC02-04ER54698 and DE-SC0020337. +This report was prepared as an account of work sponsored by an agency of the United States +Government. +Neither the United States Government nor any agency thereof, nor any of their +employees, makes any warranty, express or implied, or assumes any legal liability or responsibility +for the accuracy, completeness, or usefulness of any information, apparatus, product, or process +disclosed, or represents that its use would not infringe privately owned rights. Reference herein to +any specific commercial product, process, or service by trade name, trademark, manufacturer, or +19 + +otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring +by the United States Government or any agency thereof. +The views and opinions of authors +expressed herein do not necessarily state or reflect those of the United States Government or any +agency thereof. +Appendix A: Detailed Expressions of Λ2 +n and Sf +Detailed derivations of the generalized inertia, Λ2 +n and wave polarization, Sf, can be found +in Ref. 7. Here, we only present the results. In low-β (β = 8πP/B2 +0 ≈ ϵ2) axisymmetric +tokamak plasmas, +Λ2 +n = Iφ +� ω2 +ω2 +A +� +1 − ω∗pi +ω +� ++ Λ2 +cir + Λ2 +tra +� +, +(A1) +where Λ2 +cir and Λ2 +tra represent, respectively, the modified circulating and trapped ion re- +sponses, and Iφ describes the non-vanishing ‘flute-like’ component of the parallel elec- +tric field (δE∥) due to the effect of trapped thermal particle precession resonance [7, 21]. +Meanwhile, ωA = υA/qR0 is the Alfv´en frequency with υA being the Alfv´en velocity, and +ω∗ps = (Tsc/esB)(k×b)·(∇ns/ns+∇Ts/Ts) ≡ ω∗ns+ω∗Ts is the thermal particle diamagnetic +drift frequency due to density and temperature gradients. +For Λ2 +n, the various terms involved in Eq. (A1) are given by [7] +Λ2 +cir = q2ωωti +ω2 +A +�� +1 − ω∗ni +ω +�� +F +� ω +ωti +� ++ ∆F +� ω +ωti +�� +− ω∗Ti +ω +� +G +� ω +ωti +� ++ ∆G +� ω +ωti +�� ++ ωωti +4¯ω2 +Di +� +N1 +� ω +ωti +� ++ ∆N1 +� ω +ωti +�� +Sf(ω, ¯ωDi, ωbi, ωti) +� +, +(A2) +Λ2 +tra = ω2ω2 +bi +ω2 +A¯ω2 +Di +q2 +√ +2ϵ +� +P3 + (P2 − P3)Sf(ω, ¯ωDi, ωbi, ωti) +� +, +(A3) +Iφ = 1 + +√ +2ϵ(L(ω/¯ωDi) + τ −1L(ω/¯ωDe)) +1 + τω∗ni/ω + +√ +2ϵτ[1 − ω∗ni/ω − M(ω/¯ωDi) − τ −1M(ω/¯ωDe)], +(A4) +and, as to Sf ≡ (iδE∥/k∥)a.c. +� +δφd.c., it is given by [7] +Sf = − +N1 +� +ω +ωti +� ++ ∆N1 +� +ω +ωti +� ++ +√ +2ϵP2 +1 + 1 +τ + D1 +� +ω +ωti +� ++ ∆D1 +� +ω +ωti +� ++ +√ +2ϵ (P1 − P2) +(A5) +where the functions F(x), ∆F(x), G(x), ∆G(x), N1(x), ∆N1(x), D1(x), ∆D1(x), P1, P2, P3, +L(ω/¯ωDs) and M(ω/¯ωDs) with x = ω/ωti, and using the plasma dispersion function Z(x), +20 + +are defined as +Z(x) = π−1/2 +� ∞ +−∞ +e−y2 +y − xdy, +F(x) = x(x2 + 3/2) + (x4 + x2 + 1/2)Z(x), +∆F(x) = +1 +π1/2 +� ∞ +0 +e−y ln +�x + √2ϵy +x − √2ϵy +�y2 +4 dy, +G(x) = x(x4 + x2 + 2) + (x6 + x4/2 + x2 + 3/4)Z(x), +∆G(x) = +1 +π1/2 +� ∞ +0 +e−y ln +�x + √2ϵy +x − √2ϵy +�y2 +4 +� +y − 3 +2 +� +dy, +N1(x) = 2 ¯ωDi +ωti +�� +1 − ω∗ni +ω +� +[x + (1/2 + x2)Z(x)] − ω∗Ti +ω [x(1/2 + x2) + (1/4 + x4)Z(x)] +� +, +∆N1(x) = ¯ωDi/ωti +π1/2 +� ∞ +0 +ye−y ln +�x + √2ϵy +x − √2ϵy +� � +1 − ω∗ni +ω +− ω∗Ti +ω +� +y − 3 +2 +�� +dy, +D1(x) = x +� +1 − ω∗ni +ω +� +Z(x) − ω∗Ti +ω [x + (x2 − 1/2)Z(x)], +∆D1(x) = ¯ωDi/ωti +π1/2 +� ∞ +0 +e−y ln +�x + √2ϵy +x − √2ϵy +� � +1 − ω∗ni +ω +− ω∗Ti +ω +� +y − 3 +2 +�� +dy, +P1 = −2 ω2 +¯ω2 +Di +�� +1 − ω∗ni +ω ++ 3 +2 +ω∗Ti +ω +� +G2 − ω∗Ti +ω G4 +� +, +P2 = −2 ω +¯ωDi +�� +1 − ω∗ni +ω ++ 3 +2 +ω∗Ti +ω +� +G4 − ω∗Ti +ω G6 +� +, +P3 = −2 +�� +1 − ω∗ni +ω ++ 3 +2 +ω∗Ti +ω +� +G6 − ω∗Ti +ω G8 +� +, +Gn = +1 +π1/2 +� ∞ +−∞ +e−x2xn +(ω/¯ωDi − x2)2 − (ωbi/¯ωDi)2x2dx, +M +� ω +¯ωDs +� += −2 ω +¯ωDs +� � +1 − ω∗ni +ω ++ 3 +2 +ω∗Ti +ω +� � +1 + +� ω +¯ωDs +Z +�� ω +¯ωDs +�� +− ω∗Ti +ω +� +1 +2 + ω +¯ωDs ++ +� ω +¯ωDs +�3/2 +Z +�� ω +¯ωDs +�� � +, +L +� ω +¯ωDs +� += −2 +� � +1 − ω∗ni +ω ++ 3 +2 +ω∗Ti +ω +� � +1 +2 + ω +¯ωDs ++ +� ω +¯ωDs +�3/2 +Z +�� ω +¯ωDs +�� +− ω∗Ti +ω +� +3 +4 + 1 +2 +ω +¯ωDs ++ +� ω +¯ωDs +�2 ++ +� ω +¯ωDs +�5/2 +Z +�� ω +¯ωDs +�� � +. +(A6) +Here the magnetic drift orbit precession frequency ¯ωds = ¯ωDsmsυ2/2Ts for deeply +trapped particles (s = i, e) with ¯ωDs = (nq/r)Ts/msR0ωcs and ωcs = esB/msc; the +bounce frequency of deeply trapped ions ωbi ≡ (r/R0)1/2(Ti/mi)1/2/(qR0) ≈ ϵ1/2ωti with +21 + +ωti = (2Ti/mi)1/2/qR0; and τ ≡ Te/Ti. +REFERENCES +[1] L. Chen and F. Zonca, Nucl. Fusion 47, S727 (2007). +[2] W. W. Heidbrink, E. J. Strait, M. S. Chu, +and A. D. Turnbull, Phys. Rev. Lett. 71, 855 +(1993). +[3] A. D. Turnbull, E. J. Strait, W. W. Heidbrink, M. S. Chu, H. H. Duong, J. W. Greene, L. L. +Lao, T. S. Taylor, and S. J. Thompson, Phys. Fluids B 5, 2546 (1993). +[4] F. Zonca, L. Chen, and R. A. Santoro, Plasma Phys. Control. Fusion 38, 2011 (1996). +[5] F. Zonca, L. Chen, J. Q. Dong, and R. A. Santoro, Phys. Plasmas 6, 1917 (1999). +[6] F. Zonca, A. Biancalani, I. Chavdarovski, L. Chen, C. D. Troia, +and X. Wang, Journal of +Physics: Conference Series 260, 012022 (2010). +[7] I. Chavdarovski and F. Zonca, Plasma Phys. Control. Fusion 51, 115001 (2009). +[8] P. Lauber, M. Brudgam, D. Curran, V. Igochine, K. Sassenberg, S. Gunter, M. Maraschek, +M. Garcia-Munoz, N. Hicks, and the ASDEX Upgrade Team, Plasma Phys. Control. Fusion +51, 124009 (2009). +[9] L. Chen and F. Zonca, Rev. Mod. Phys. 88, 015008 (2016). +[10] F. Zonca, L. Chen, M. V. Falessi, and Z. Qiu, Journal of Physics: Conference Series 1785, +012005 (2021). +[11] C. Cheng, Phys. Fluids 25, 1020 (1982). +[12] W. Tang, J. Connor, and R. Hastie, Nucl. Fusion 20(11), 1439 (1980). +[13] H. Biglari and L. Chen, Phys. Rev. Lett. 67, 3681 (1991). +[14] S. Sharapov, B. Alper, H. Berk, D. Borba, B. Breizman, C. Challis, I. Classen, E. Edlund, +J. Eriksson, A. Fasoli, E. Fredrickson, G. Fu, M. Garcia-Munoz, T. Gassner, K. Ghantous, +V. Goloborodko, N. Gorelenkov, M. Gryaznevich, S. Hacquin, W. Heidbrink, C. Hellesen, +V. Kiptily, G. Kramer, P. Lauber, M. Lilley, M. Lisak, F. Nabais, R. Nazikian, R. Nyqvist, +M. Osakabe, C. P. von Thun, S. Pinches, M. Podesta, M. Porkolab, K. Shinohara, K. Schoepf, +Y. Todo, K. Toi, M. V. Zeeland, I. Voitsekhovich, R. White, V. Yavorskij, I. E. TG, and J.-E. +22 + +Contributorsa, Nucl. Fusion 53, 104022 (2013). +[15] N. Gorelenkov, S. Pinches, and K. Toi, Nucl. Fusion 54, 125001 (2014). +[16] W. Heidbrink, M. V. Zeeland, M. Austin, A. Bierwage, L. Chen, G. Choi, P. Lauber, Z. Lin, +G. McKee, and D. Spong, Nucl. Fusion 61, 016029 (2021). +[17] W. Heidbrink, M. V. Zeeland, M. Austin, N. Crocker, X. Du, G. McKee, and D. Spong, Nucl. +Fusion 61, 066031 (2021). +[18] W. Heidbrink, G. Choi, M. V. Zeeland, M. Austin, G. Degrandchamp, D. Spong, A. Bierwage, +N. Crocker, X. Du, P. Lauber, Z. Lin, and G. McKee, Nucl. Fusion 61, 106021 (2021). +[19] D. Curran, P. Lauber, P. J. M. Carthy, S. da Graca, V. Igochine, +and the ASDEX Up- +grade Team, Plasma Phys. Control. Fusion 54, 055001 (2012). +[20] P. Lauber, Physics Reports 533, 33 (2013). +[21] I. Chavdarovski and F. Zonca, Phys. Plasmas 21, 052506 (2014). +[22] A. Fasoli, S. Brunner, W. Cooper, J. Graves, P.Ricci, O. Sauter, +and L. Villard, Nature +Physics 12, 411 (2016). +[23] A. Bierwage and P. Lauber, Nucl. Fusion 57, 116063 (2017). +[24] G. Choi, P. Liu, X. Wei, J. Nicolau, G. Dong, W. Zhang, Z. Lin, W. Heidbrink, and T. Hahm, +Nucl. Fusion 61, 066007 (2021). +[25] N. Gorelenkov, H. Berk, E. Fredrickson, S. Sharapov, and J. E. Contributors, Phys. Lett. A +370, 70 (2007). +[26] N. N. Gorelenkov, M. A. V. Zeeland, H. L. Berk, N. A. Crocker, D. Darrow, E. Fredrickson, +G.-Y. Fu, W. W. Heidbrink, J. Menard, and R. Nazikian, Phys. Plasmas 16, 056107 (2009). +[27] L. Chen and F. Zonca, Phys. Plasmas 24, 072511 (2017). +[28] F. Zonca and L. Chen, Phys. Plasmas 21, 072120 (2014). +[29] F. Zonca and L. Chen, Phys. Plasmas 21, 072121 (2014). +[30] J. Varela, D. Spong, L. Garcia, J. Huang, M. Murakami, A. Garofalo, J. Qian, C. Holcomb, +A. Hyatt, J. Ferron, C. Collins, Q. Ren, J. McClenaghan, +and W. Guo, Nucl. Fusion 58, +076017 (2018). +[31] L. Chen, R. B. White, and M. Rosenbluth, Phys. Rev. Lett. 52, 1122 (1984). +[32] L. Chen, Phys. Plasmas 1, 1519 (1994). +[33] S. Tsai and L. Chen, Phys. Fluids B 5, 3284 (1993). +[34] F. Zonca and L. Chen, Plasma Phys. Control. Fusion 48, 537 (2006). +23 + +[35] F. Zonca, P. Buratti, A. Cardinali, L. Chen, J.-Q. Dong, Y.-X. Long, A. Milovanov, F. Ro- +manelli, P. Smeulders, L. Wang, Z.-T. Wang, C. Castaldo, R. Cesario, E. Giovannozzi, M. Mar- +inucci, and V. P. Ridolfini, Nucl. Fusion 47, 1588 (2007). +[36] R. Ma, L. Chen, F. Zonca, Y. Li, +and Z. Qiu, Plasma Phys. Control. Fusion 64, 035019 +(2022). +[37] L. L. Lao, H. S. John, and R. D. Stambaugh, Nucl. Fusion 25, 1611 (1985). +[38] A. Pankin, D. McCune, R. Andre, G. Bateman, and A. Kritz, Comput. Phys. Commun. 159, +157 (2004). +[39] F. Zonca and L. Chen, Phys. Plasmas 7, 4600 (2000). +[40] F. Zonca, S. Briguglio, L. Chen, S. Dettrick, G. Fogaccia, D. Testa, +and G. Vlad, Phys. +Plasmas 9(12), 4939 (2002). +[41] J. W. Connor, R. J. Hastie, and J. B. Taylor, Phys. Rev. Lett. 40, 396 (1978). +[42] F. Zonca, L. Chen, A. Botrugno, P. Buratti, A. Cardinali, R. Cesario, V. P. Ridolfini, and +the JET-EFDA contributors, Nucl. Fusion 49, 085009 (2009). +[43] L. Chen, private notes (2020). +[44] M. N. Rosenbluth and F. L. Hinton, Phys. Rev. Lett. 80, 724 (1998). +[45] J. P. Graves, R. J. Hastie, and K. I. Hopcraft, Plasma Phys. Control. Fusion 42, 1049 (2000). +[46] M. V. Falessi, N. Carlevaro, V. Fusco, G. Vlad, +and F. Zonca, Phys. Plasmas 26, 082502 +(2019). +[47] M. V. Falessi, N. Carlevaro, V. Fusco, E. Giovannozzi, P. Lauber, G. Vlad, and F. Zonca, J. +Plasma Phys. 86, 845860501 (2020). +[48] J. Chen, W. X. Ding, D. L. Brower, D. Finkenthal, C. Muscatello, D. Taussig, and R. Boivin, +Review of Scientific Instruments 87, 11E108 (2016). +24 + diff --git a/6tAzT4oBgHgl3EQfgPwz/content/tmp_files/load_file.txt b/6tAzT4oBgHgl3EQfgPwz/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..abccee2c2acc38e3f60612daae7517242047558b --- /dev/null +++ b/6tAzT4oBgHgl3EQfgPwz/content/tmp_files/load_file.txt @@ -0,0 +1,975 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf,len=974 +page_content='Low-frequency shear Alfv´en waves at DIII-D: theoretical interpretation of experimental observations Ruirui Ma,1, 2, ∗ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Heidbrink,3 Liu Chen,4, 3, 2 Fulvio Zonca,2, 4 and Zhiyong Qiu4, 2 1Southwestern Institute of Physics, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Box 432, Chengdu, 610041, China 2Center for Nonlinear Plasma Science and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' ENEA Frascati, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' 65,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' 00044 Frascati,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Italy 3Department of Physics and Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' University of California,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Irvine,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' CA 92697-4574,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' USA 4Institute for Fusion Theory and Simulation and Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Zhejiang University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Hangzhou,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' 310027,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' People’s Republic of China (Dated: January 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' 2023) Abstract The linear properties of the low-frequency shear Alfv´en waves such as those associated with the beta-induced Alfv´en eigenmodes (BAEs) and the low-frequency modes observed in reversed- magnetic-shear DIII-D discharges (W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Heidbrink, et al 2021 Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Fusion 61 066031) are theoret- ically investigated and delineated based on the theoretical framework of the general fishbone-like dispersion relation (GFLDR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' By adopting representative experimental equilibrium profiles, it is found that the low-frequency modes and BAEs are, respectively, the reactive-type and dissipative- type unstable modes with dominant Alfv´enic polarization, thus the former being more precisely called low-frequency Alfv´en modes (LFAMs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' More specifically, due to different instability mech- anisms, the maximal drive of BAEs occurs, in comparison to LFAMs, when the minimum of the safety factor (qmin) deviates from a rational number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Meanwhile, the BAE eigenfunction peaks at the radial position of the maximum energetic particle pressure gradient, resulting in a large deviation from the qmin surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Moreover, the ascending frequency spectrum patterns of the experimentally observed BAEs and LFAMs can be theoretically reproduced by varying qmin and also be well interpreted based on the GFLDR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' The present analysis illustrates the solid predictive capability of the GFLDR and its practical usefulness in enhancing the interpretative capability of both experimental and numerical simulation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' ∗ corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Email address: rrma@swip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='cn 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='01464v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='plasm-ph] 4 Jan 2023 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' INTRODUCTION AND MOTIVATION The low-frequency Alfv´en wave spectrum in the kinetic thermal-ion (KTI) gap frequency range [1] has been of research interest since the first observations of beta-induced Alfv´en eigenmodes (BAEs) [2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' These modes are characterized with frequencies comparable to thermal ion transit and/or bounce frequencies, and can interact with both thermal and fast particles [4–9], with possible (positive/negative) impact on the corresponding transport processes resulting from finite fluctuation and zonal field structures levels [1, 9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' The effects of energetic particles (EPs) on low-frequency shear Alfv´en waves (SAWs) ranging from kinetic ballooning mode (KBM) [11–13] to BAE are one of areas widely studied in the magnetic fusion literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Recent papers on this topic cover the interpretation and modeling of experimental measurements by currently developed innovative diagnostics [14– 18], as well as latest progress in comparing numerical investigation and/or simulation results with observed phenomena [19–24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' A series of dedicated experiments have been recently conducted on DIII-D to investigate the stability of the low-frequency SAWs [16–18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' The experiments show that the observed low-frequency mode1, which was previously misidentified as ‘beta-induced Alfv´en acoustic eigenmode (BAAE)’ [25, 26], is actually a lower-frequency reactive unstable KBM which favors high thermal electron temperature but almost has no coupling with energetic ions [16];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' while the BAE is resonantly excited by energetic ions with its stability depending sensitively on the beam power and injection geometry [17], consistent with earlier theoretical predictions [27] based on the GFLDR theoretical framework [28, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' These instabilities are also found to occur when the minimum of the safety factor (qmin) approaches rational values and the modes in ascending pattern of higher frequency BAEs and LFAMs are separated by approximately the toroidal rotation frequency (frot).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' However, the subtle differences between them are that, for LFAMs, the maximum frequency appears at rational values of qmin and the detected modes are radially localized near qmin, while BAEs occur at times near rational qmin values but the timing of unstable modes is less precise than that for LFAMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' In addition, compared with the LFAMs, the BAE eigenfunction shows more deviation from the radial position of qmin spatially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Although dedicated numerical simulations of the linear properties 1We will refer from now on only to the low frequency Alfv´en mode (LFAM) which belongs to low-frequency SAWs predominantly Alfv´enic polarization, keeping in mind that this terminology is the same as the low- frequency mode observed in recent DIII-D experiments [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' 2 of the BAEs and LFAMs [24, 30] have been carried out, the above experimental phenomena have not been fully explained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Motivated by this, the present work aims to provide an in-depth theoretical understanding of the linear properties of low-frequency SAWs, with particular attention to the effects of energetic ions on their stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' The analysis is carried out based on the theoretical framework of the generalized fishbone-like dispersion relation (GFLDR) [28, 29, 31–35], and provides qualitative and quantitative interpretation of the main instability mechanisms underlying the numerical simulation results and experimental observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' As a result, our analysis provides yet another evidence of the predictive strength of the GFLDR theoretical framework and of its enhanced “interpretative capability for both experimental and numerical simulation results” [28, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' In this work, unlike the previous paper not considering effects due to energetic particles (EPs) [36], we focus on the BAE excitation via transit resonance with passing fast ions created by NBI heating [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' In this case, the dynamics of various species enter the dispersion relation of low-frequency SAW, and affect its behavior linearly at different pressure gradient scale lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' For DIII-D discharge #178631, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' 1 shows the radial dependence of different scale lengths of thermal and energetic particle pressure (LPth and LPE), as well as the estimated radial mode width (∆m) for weak and/or vanishing magnetic shear range, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=', |s| = |(r/q)(dq/dr)| ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' More specifically, the EP pressure profiles are given by the following two limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' One is the relaxed EP profile provided with EFIT reconstruction [37], where the fast-ion pressure is the difference between the equilibrium pressure and the thermal pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' The other is the “classical” EP profile obtained by TRANSP/NUBEAM [38] in the absence of fast-ion transport by instabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' The pressure scale lengths of EPs are denoted by LPE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='rel and LPE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='cl for these two cases (respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' The true EP profile when the modes are destabilized likely lies between these two limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' The actual pressure is closest to the EFIT-based one but this is measured after the unstable modes have (presumably) caused the gradients to flatten.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Meanwhile, for the weak and/or vanishing magnetic shear region and given toroidal and poloidal mode numbers (n, m), the normalized parallel wave vector is ΩA,m = k∥n0qminR0 = nqmin − m, and the radial width of the mode can then be estimated by ∆m ≃ 1/|nq′′|1/2 [39, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Here, k∥n0 represents the parallel wave-vector at r0, where q has a minimum given by qmin, q′′ denotes the second derivative of q in the radial direction, and R0 is the torus major radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' It can be found that in this region, LPth ≫ ∆m, which yields the usual local limit of the mode dispersion relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' This is the 3 case for the reactive unstable LFAM in the absence of EPs already studied in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' However, for the energetic ion-driven BAEs, there are two distinct cases: the moderate EP pressure gradient case with LPE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='rel > ∆m, which also approximately yields the usual local GFLDR [4, 28, 29, 32, 33, 35, 39, 40];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' and the strong EP pressure gradient case with LPE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='rel ≃ ∆m, for which the global dispersion relation of low-frequency SAWs is needed and will be discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Performing detailed numerical investigations of the two FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' The radial dependences of the typical scale lengths of thermal and energetic particle pressure (LPth and LPE), as well as the estimated radial mode width (∆m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' cases, it is found that the LFAMs and BAEs can both be driven unstable, however, due to different instability mechanisms, these modes yield different experimental observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' All these features can be, quantitatively and qualitatively, interpreted theoretically based on the GFLDR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Moreover, it is also confirmed that the stability of BAAE is not affected by EPs, even though it becomes weakly damped after coupling with KBM, consistent with theoretical predictions by Chen and Zonca [27] as well as numerical simulation results reported in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' [20, 23, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' The paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Local and global dispersion relations for the low- frequency SAWs near weak and/or vanishing magnetic shear are introduced and discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' II in different parameter regimes, depending on the relative magnitude of LPE and ∆m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Detailed numerical investigations and theoretical analysis of the low-frequency SAWs in the presence of EPs are discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' III, where comparisons between theory and experiments are also made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Finally, conclusions and further discussions are given in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='5 th E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='cl length (m) E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='rel m 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='32 r/aII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' THE GENERAL FISHBONE-LIKE DISPERSION RELATION FOR LOW- FREQUENCY SAWS In this Section, we will present analytical dispersion relations for low-frequency SAW excitation in weakly reversed-shear DIII-D discharges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' As stated in the previous Section, two cases determined by the relative magnitude of LPE and ∆m will be used to investigate the low-frequency SAW stability: case I, the local GFLDR model corresponding to LPE > ∆m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' and case II, the global GFLDR corresponding to LPE ≃ ∆m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Consider case I first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' For LPE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='rel > ∆m, the scales of LPE and ∆m can be separated, and the vorticity equation [4, 9, 28, 29, 32, 33] which governs shear Alfv´en waves (SAWs) can yield the low-frequency electromagnetic fluctuation dispersion relation in the usual local limit, as derived and discussed in great details in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' [9, 28, 29, 32, 33, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' We just note that, for DIII-D case of interest, the reversed magnetic shear configuration and thermal plasma compression effects should be accounted for properly [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Thus, for s = 0 at r0 but with finite S ≡ (r/q)[q ′′]1/2, the local GFLDR for low-frequency SAWs can be written as [27–29, 35, 40] iS(Λ2 n − k2 ∥n0q2 minR2 0)1/2(1/n)1/2� k∥n0qminR0 − i(Λ2 n − k2 ∥n0q2 minR2 0)1/2�1/2 = δ ˆWnf + δ ˆWnk(ω), (1) where the generalized inertia term Λn(ω) here, including both diamagnetic effects as well as kinetic effects of circulating and trapped particle dynamics, has been derived explicitly in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' [7] and the main results are summarized in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' The right hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' (1) contains both “fluid” (δ ˆWnf) and “kinetic” (δ ˆWnk) contributions to the potential energy in the “regular” ideal region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' In the low-frequency limits (|Λ2 n| ≪ 1), δ ˆWnf is independent of the frequency and the explicit expression, specialized to the (s, α) model equilibrium [41] with circular flux surfaces, reads, δ ˆWnf ≃ π 4 �S2k∥0qminR0 n − 3 2α2S ��k∥0qminR0 n ��1/2 + 9 32α4 � (2) where α = αc + αE, αc = −R0q2 mindβ/dr and αE = − 1 2R0q2 mind(βE∥ + βE⊥)/dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Note that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' (2) includes the contribution of the energetic particle adiabatic and convective responses as well [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' The term δ ˆWnk is always a function of the mode frequency ω, as it reflects resonant as well as non-resonant wave-particle interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' For simplicity but still relevant to the 5 DIII-D case, we take F0E to be a single pitch angle (λ = µ/ε) slowing-down beam ion equilibrium distribution function;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=', F0E = B0βE(r) 25√ 2π2mEεb � (1 − λ0B0)ε−3/2δ(λ − λ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Here, βE(r) ≡ 8πPE(r)/B2 0 is the ratio of EP kinetic and magnetic pressures and B0 the on- axis equilibrium magnetic field, δ(x) is the Dirac function, µ is the magnetic moment and ε = υ2/2 ≤ εb with εb being the EP birth energy per unit mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Then the explicit expression of non-adiabatic contribution δ ˆWnku for the passing energetic ions is given by [32, 33] δ ˆWnku ≃ παE 25/2 (1 − λ0B0/2)¯ω � 2 − ¯ω ln � ¯ω + 1 ¯ω − 1 �� , (3) where ¯ω = ω/ωtEm and ωtEm ≡ √2εb/qR0 is the EP transit frequency at the maximum particle energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' It is worthwhile emphasizing that the finite k∥n0qminR0 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' (1) plays an important stabilizing role since it represents the finite line bending effect at r = r0 [28, 29, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Further- more, the expression of Λn depends on the mode polarization via Sf ≡ (iδE∥/k∥)a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' � δφd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=', where a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' and d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' refer to the sinusoidal and nearly constant (flute-like) components of the parallel electric field, wave vector, and scalar potential fluctuation [21, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' The detailed expression of Sf, again, is given in the Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Here, we just note that |Sf| is much smaller than unity for shear Alfv´en wave and order of unity for ion acoustic wave [7, 21, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' We remark here that, in the moderate pressure gradient case, the local GFLDR for the low-frequency SAWs is enough to delineate the underlying physics of the experimental and simulation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' However, the local GFLDR for the low-frequency SAWs, given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' (1), will fail in the presence of strong EP pressure gradient, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=', case II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' In this case, two typical scale lengths LPE,cl and ∆m can not be separated anymore and, thus, a global dispersion relation is needed which can be derived from the vorticity equation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=', Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' (1) of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Noting that the mode structure is dominated by single toroidal and poloidal mode numbers, (n, m), the governing equation reads (eθ − erξ) · � Λ2 − Ω2 A,m � 1 + x2 ΩA,m + x4 4Ω2 A,m �� (eθ − erξ)δφm − (F + K)δφm = 0, (4) where k⊥/kθ = −(eθ − erξ) with er and eθ being, respectively, the radial and poloidal unit vectors, x2 = nq′′ min(r − r0)2, ξ ≡ (i/n1/2)S(∂/∂x), and δφm is the mth poloidal harmonic of the scalar field perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' It is worth noting that, toroidal coupling among different poloidal harmonics is typically not important for modes in the reversed magnetic shear region, consistent with the mode being dominated by single m and n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' The terms F and K 6 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' (4) represent, respectively, the fluid-like particle and energetic ion contributions with their explicit form reading F ≃ D2 S − 4α2DS + 2αD2 S − (α + 1)α + 2α3, K ≃ 2πq2 Eq2R2 0ω mEc2 �Ω2 dEQF0E ω2 tE − ω2 � υ = 2 πδ ˆWnku, (5) where DS = S � ΩA,m/n, qE and mE are the electric charge and mass of energetic ions, ΩdE = (υ2 E⊥/2+υ2 E∥)/ωcER0, ωtE = υE∥/qR0, QF0E = (ω∂ε+ˆω∗E)F0E, ˆω∗EF0E = ω−1 cE (k×b)·∇F0E, ωcE = qEB/mEc, ⟨(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=')⟩υ = � d3υ(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='), and the subscripts ∥ and ⊥ represent the parallel and perpendicular components with respect to the equilibrium magnetic field b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Equation (4) is an ordinary differential equation and, generally, requires a numerical approach to be solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' However, for DIII-D case, the radial dependence of the normalized pressure gradient of energetic ions with the classical profile, as is shown by black curve in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' 2, can be well fitted by the analytic formula αE(ρ) = c1 (1 − (ρ − c2)2/c2 3), with c1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='7099, c2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='3018 and c3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='2944.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' This allows us to obtain simple analytical dispersion relations for low-frequency SAWs excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' We just note that the maximum drive of energetic ions is located around ρ = c2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='3018, which deviates from the radial position of qmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Then αE(r) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' (3) can be rewritten as αE(r) = δaαE0 � 1 − (r − r0 + δb)2 δ2 cL2 PE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='cl � , (6) where δa = c1/αE0, δb = r0 − c2a and δc = c3a/LPE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='cl, a is the minor radius, αE0 and LPE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='cl are evaluated at r = r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Introducing the notation x = r − r0 = σz − δb, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' (4) is readily cast into the form ∂2 ∂z2δφm − nσ2 S2 � 1 − F + 2δa π δ ˆWnku0 ϵA0 � δφm − 1 4z2δφm = 0, 2nσ4δaδ ˆWnku0 ϵA0πS2δ2 cL2 PE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='cl = 1 4, (7) where ϵA0 = Λ2 − Ω2 A,m, δ ˆWnku0 = παE0 4 √ 2 � 2 − ¯ω ln � ¯ω+1 ¯ω−1 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Then, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' (7) yields the following global dispersion relation for low-frequency SAWs, −n1/2π1/2δcLPE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='clϵ1/2 A0 2 √ 2Sδ1/2 a δ ˆW 1/2 nku0 � 1 − F + 2δa π δ ˆWnku0 ϵA0 � = 2L + 1, L = 0, 1, 2, 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' (8) 7 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' The radial dependence of the normalized pressure gradient of EPs with the classical profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Here, the normalized radial position of qmin is ρ0 ≡ r0/a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Here, the integer L is the radial eigenmode number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' The corresponding eigenfunction reads δφm(r) = HL(z)e−z2 ∝ exp � −(r − r0 + δb)2 4σ2 � , (9) where HL(z) represents Lth order Hermite polynomials and the causality constraints upon the discrete bound modes requiring Re(σ2) > 0, where σ2 is solved for from the second of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' (7) consistently with the dispersion relation, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' The typical radial width, w, of δφm(r) is determined by w2 = 4σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Equations (1) and (8) constitute the results of the present section, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=', the local and global GFLDR for the low-frequency SAWs excited by energetic ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' With their explicit form, we can compute the individual terms involved in equations and investigate the linear properties of the experimentally observed low-frequency SAWs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' THE LOW-FREQUENCY SAW INSTABILITIES NUMERICAL RESULTS AND ANALYSIS In this Section, we separately present numerical results for the local and global low- frequency SAW stability properties in the presence of energetic ions, for which the dispersion relation is given by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' (1) and (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' The numerical investigations use experimental equilib- rium and profiles as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' 3 for the DIII-D shot #178631 at the time t = 1200 ms [16], where the q-profile has a reversed shear configuration with qmin = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='37 at r0/a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='28 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='68 a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='66 αE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='cl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' p 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='64 fit Prediction bounds -99% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='35 p=r/aFIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Radial profiles of (a) temperature and q and (b) density and toroidal rotation frequency frot of DIII-D shot #178631 used for numerical studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' and qmin decreases from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='49 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='18 in the time window 1050 ms < t < 1350 ms, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' 6 (b) in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' The local low-frequency SAW stability properties We first consider the linear properties of the low-frequency SAW with relaxed energetic ion profile, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=', case I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' The local equilibrium parameters used in the numerical studies evaluated at r0/a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='28 are S = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='5895, τ = Te/Ti =3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='86 keV/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='37 keV=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='62, ne = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='80 × 1019 m−3, ni = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='19 × 1019 m−3, ϵr = r0/R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='10, βi ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='01, ϵni = Lni/R0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='414, ηi = Lni/LTi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='8324, ω∗ni/ωti = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='1919, (m, n) = (8, 6), kθρLi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='2555 and kθρLe = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='0054.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Other fixed equilibrium parameters are a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='64 m, R0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='74 m, B0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='8 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Here, kθ is the poloidal wavenumber, ρLi and ρLe are the Larmor radii of thermal ions and thermal electrons, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Dependencies of the (a) mode frequencies, (b) growth rates and (c) mode polarization predicted by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' (1) are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' 4 as a function of the normalized thermal ion diamagnetic frequency Ω∗pi ≡ ω∗pi/ωti for the cases without and with the consideration of EP effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' According to the scaling of mode frequencies with physical parameters and the value of the |Sf| [21], three branches in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' 4 can be classified as: (i) the KBM (red curves marked with circles), with a frequency scaling with ω ∼ ω∗pi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' (ii) the BAE (blue curves), with the frequency being close to the well-known estimate ω/ωti = qmin � 7/4 + τ ≃ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='51;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' 9 (a) T。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' (keV) 6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' (keV) q TE:cl (keV/10) 4 2 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='8 1 r/a(b) 5 n(1019m-3) -n, (1019m3) 4 4 × nE:cl (1019m=3) 4 × nE:rel (1019m-3) 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='5× frot (kHz) 2 1 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='8 1 r/aFIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Dependence of the (a) real frequencies, (b) growth rates and (c) polarization of the low- frequency SAWs on Ω∗pi ≡ ω∗pi/ωti for the cases without (w/o) and with (w/) EP effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Here, a dashed vertical line represents the experimental value of Ω∗pi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='exp of about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' and (iii) the BAAE (green curves marked with diamonds), with a frequency of about half of the BAE and experiencing strong damping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' The EP effects on the low-frequency SAW stabilities are apparent in the region highlighted by the purple curve of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' 4 (b), where the KBM is the only unstable mode in the absence of EPs, while both the KBM and BAE are unstable in the low-frequency region in the presence of EPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' In particular, the diamagnetic ion frequency calculated on the basis of experimental parameters is Ω∗pi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='exp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='3517, as shown by the dashed vertical line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' In this case, both KBM and BAE are unstable with the frequencies in the plasma frame being 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='6 kHz and 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='7 kHz, respectively, which are in good agreement with the experimental observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Meanwhile, the polarization plot of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' 4 (c) shows that KBM and BAE have small values for |Sf| ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='1, which indicates that the KBM and BAE are essentially of Alfv´enic polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Moreover, in order to exclude the 10 (a) KBMw/oEP BAAE w/o EP BAE w/o EP 4 KBM w/ EP Re(wlwti BAAE w/ EP BAE w/ EP 0 2 4 6 2 pi:(b) m(w/wti KBM w/o EP BAAE w/o EP BAEw/o EP KBMw/EP BAAE w/ EP BAEw/EP 0 pi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='exp 2 4 6 U pi 100 (c) KBM w/o EP BAAE w/o EP BAEw/oEP S KBM w/ EP BAAE w/ EP BAEw/EP 0 2 4 6 pispurious nonzero solutions produced by singularities of the transcendental function of the local GFLDR (D), the Nyquist diagram in the complex D plane presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' 5 shows that in the presence of EPs, the path encircles the origin twice (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' 5 (b)) but only once without EPs (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' 5 (a)), thus confirming there are two unstable modes with EPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' It FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' The Nyquist diagram in the complex D(ω) plane for the cases (a) without and (b) with EP effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' should be noted that, compared with the frequency insensitive to the EP effects, the growth rate of the KBMs changes significantly in the cases with and without EP effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' This occurs because in our theoretical model the adiabatic and convective contribution of EPs modifies the value of δ ˆWf via α, as is shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' At this point, in order to obtain more convincing comparison of theoretical prediction and experimental observation, it is necessary to provide a more precise theoretical model and also a more comprehensive experimental analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' We also note here that, in this case, the stability/property of the BAAE is not affected by energetic ions — as is shown by the green dashed lines with symbols (without EP effects) and solid lines with symbols (with EP effects) which are apparently overlaying in all three graphs — even though it becomes weakly damped by coupling with the KBM due to diamagnetic and trapped particle effects for sufficiently strong Ω∗pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' The numerical results are consistent with the numerical simulation results reported in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' [20, 23, 24] and the theoretical prediction in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' [27], that is, “EPs preferentially excite the BAE over the BAAE branch due to the stronger wave-EP interaction”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' We now investigate the underlying instability mechanisms of the ascending spectrum of the higher frequency BAEs and LFAMs observed in DIII-D (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' 8 of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' [17]) by using 11 ×10-3 (a) 10 5 0 5 5 0 5 Re(D) ×10-3X10-3 10 (b) 5 Im(D) 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='5 5 0 5 Re(D) ×10-3qmin as the scanning parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Figure 6 shows the dependence of the mode frequencies (solid curves with markers) and growth rates (dashed curves with markers) on qmin of the KBMs (red curves) and the BAEs (blue, green, purple and orange curves) for different poloidal and toroidal mode numbers (m, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' It is shown that the modes in ascending pattern FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Dependence of mode frequencies (solid curves with markers) and growth rates (dashed curves with markers) on qmin of the KBMs (red curves) and the BAEs (blue, green, purple and orange curves) for different (m, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' The experimentally observed frequencies are also shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' For the BAE, since the modes span a range of frequencies, the lines indicate the upper and lower limits of the unstable bands;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' for the LFAM, the experimental frequency variation is < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='5 kHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' In the abscissa, the experimentally measured qmin(t) fit shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' 8 of [17] is used to convert time to qmin, with an associated uncertainty of ∆qmin ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' In the ordinate, the theoretical lab-frame frequency incorporates a Doppler shift to the calculated plasma-frame frequency of nfrot, with an associated uncertainty of ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='5 × n kHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' of higher frequency BAEs and lower frequency KBMs are both separated by approximately 12 200 +7,5 f: in the lab-frame (10,8) KBM f8,6 KBM 9,7 f9,7 KBM 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='6 f10,8 150 KBM (7,5) KBM KBM KBM ZH) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='10,8 100 /2元 “KBM Expi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='.data 7,5 BAE (range) 8,6 (lines with ★) (9,7) (10,8) (8,6) BAE (7,5) 9,7 BAE 10,8 50 BAE /27 BAE “BAE BAE 10,8 BAE 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='45 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='35 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='2 minfrot of about 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='5 kHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' More specifically, for KBMs, the instabilities peak exactly at the rational values of qmin;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' while the BAEs occur at times near rational values of qmin but the timing of unstable modes is less precise than for KBMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' In addition, the low-n BAEs deviate more from rational qmin crossings than higher n modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' The comparison of the theoretically predicted frequencies with the experimentally measured values can also be seen clearly from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' As discussed in more detail in the next section, these numerical results are in good agreement with the experimental observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' In order to gain insight into the different excitation mechanisms of the instabilities pre- sented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' 6, let us further analyze the GFLDR in the high-frequency (|ω| ≫ ωti) and low-frequency |ω| ≪ ωbi limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' For |ω| ≫ |ωti|, the corresponding inertia term of the BAE can be reduced to the simplified expression with Λ2 ≃ ω2−ω2 BAE ω2 A [4, 35, 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Here, ω2 BAE = (7/4 + τ)υ2 i /R2 0 is the fluid limit expression of the BAE frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Taking ω = ωr + iγ and δ ˆWku = Reδ ˆWku + iImδ ˆWku, and assuming |γ/ωr|, we have |Imδ ˆWku/Reδ ˆWku| ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Then, for the gap mode, the existence condition is δ ˆWnf + Re(δ ˆWnk(ωr)) < 0 and the real mode frequency is given by ω2 r = ω2 BAE � ��1 + ω2 A ω2 BAE � � �k2 ∥n0q2 minR2 0 − n ��k∥n0qminR0 �� � δ ˆWnf + Re(δ ˆWnk(ωr)) �2 S2 � � � � �� , (10) while the growth rate is obtained from γ = −Im(δ ˆWnk(ωr))ω2 A ωr n � δ ˆWnf + Re(δ ˆWnk(ωr)) � ��k∥n0qminR0 �� S2 , (11) It can be readily obtained from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' (10) that the BAE frequency is positively correlated with ��k∥n0qminR0 ��.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Therefore, the more deviation from the rational qmin surface is, the larger the BAE frequency is, as is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Note also that the BAE has a positive frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Equation (11) imposes Im(δ ˆWnk(ωr)) > 0 for BAE excitation by EPs via resonant wave- particle interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' It can be concluded that the duration of BAEs is influenced by the associated resonances with the EPs, as well as by the value of qmin [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Similarly, for KBM with |ω| ≪ |ωbi|, we have Λ2 ≃ c0 q2 min √ϵ (ω−¯ωdi)(ω−ω∗pi) ω2 A [7, 16, 21, 35, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Here, ¯ωdi is the average thermal-ion precession frequency, c0 ≃ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='6 due to trapped and barely 13 circulating particles [44, 45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Thus, the real mode frequency is given by ω = 1 2(¯ωdi+ω∗pi)±1 2 � ��(ω∗pi − ¯ωdi)2 − 4ω2 A √ϵ q2 minc0 � � � n � δ ˆWnf + Re(δ ˆWnk(ωr)) �2 ��k∥n0qminR0 �� S2 − k2 ∥n0q2 minR2 0 � � � � �� 1/2 , (12) and the system is reactively unstable if |ω∗pi − ¯ωdi|2 ω2 A < 4√ϵ q2 minc0 � � � n � δ ˆWnf + Re(δ ˆWnk(ωr)) �2 ��k∥n0qminR0 �� S2 − k2 ∥n0q2 minR2 0 � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' (13) Note that δ ˆWf + Reδ ˆWku < 0, due to, again, the causality constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Therefore, for the reactive-type instability, the maximum drive sets in when k∥n0qminR0 → 0, which corre- sponds to the unstable KBM exactly peaking at the rational values of qmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' The above numerical results and theoretical analyses have explained the experimental observations that the BAEs deviate more from the rational qmin values temporally, com- pared with the KBM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' To further delineate this deviation and its impact on the radial mode structure, numerical investigation of the global model for low-frequency SAWs is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' The global low-frequency SAW stability properties In this part, we consider the case II and apply Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' (8) to investigate the global low- frequency SAW stability properties with the classical energetic ion profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Figure 7 shows (a) the dependence of the real frequencies (blue markers) and growth rates (red markers) of the KBM (triangle markers) and BAE (line with markers) on the radial mode number L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' and (b) the radial mode structure δφm(r) for the L = 0 BAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' It can be found that (i) the ground eigenstate with L = 0 is most unstable for the BAE and KBM;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' (ii) for BAE, the frequency and growth rate in the plasma frame is (80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='7 + 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='2i) kHz with the ratio of the growth rate to real frequency γ/ω ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='19, which is the typical feature of the marginally unstable gap mode excited by EPs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' and (iii) for KBM, the frequency and growth rate in the plasma frame is (−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='2 + 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='7i) kHz with γ/ω ≃ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='8, which is the typical feature of the reactive-type instability, consistent with the results reported in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Correspondingly, the radial eigenfunction plot of the BAE for L = 0, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' 7 (b), presents that δφm has a Gaussian form with a shape similar to the experimentally 14 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' (a) Dependence of the real frequencies (blue markers) and growth rates (red markers) of the KBM (triangle markers) and BAE (line with markers) on the radial mode number L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' (b) the radial mode structure δφm(r) for the L = 0 BAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' The approximate experimental measurement of the mode structure of BAE is also shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' measured radial mode structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' In this case, the radial width of δφm by theory is w = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='2107, is comparable to the scale length of energetic-ion pressure, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=', LPE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='cl = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='1773;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' consistent with the analysis of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Note that determined by the EP distribution, the BAE eigenfunction peaks at the radial position of the maximum energetic particle pressure gradient, resulting in a large deviation from the qmin surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' It can also be expected that the KBM eigenfunction should peak at the rational values of qmin where the instability drive is maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Finally, the continuous spectra plots for low-frequency shear Alfv´en and acoustic waves given by Λ2 n(ω) = k2 ∥nq2R2 0 = (nq−m)2 [4, 6, 28, 29, 42, 46, 47] are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Here, the inertia term includes the diamagnetic effects and thermal ion compressibility as well as drift Alfv´en wave and drift wave sideband coupling via the wave-thermal-passing-ion interaction and diamagnetic effect [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' The figure shows that based on the GFLDR, the nature of various branches can be clearly classified via their frequencies (a), growth rates (b) and polarizations (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Here, the short notation “e-KBM” represents the branch of the KBM propagating in the thermal-electron diamagnetic drift direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' The unstable continuum spectrum of the e-KBM is due to the inclusion of the kinetic dynamics of thermal particles in inertia term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' In addition, the frequencies of the (m, n) = (8, 6) BAE and the (m, n) = (8, 6) KBM calculated by the local and global cases are, respectively, in the gaps of the BAE and KBM continua, 15 4 (a) 3 △ Re(wlwt) of KBM A Im(w/wti) of KBM 2 Re(wlwt:) of BAE -- Im(w/wti) of BAE 0 2 3(m,n)=(8,6) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='75 analytic ECE-measured m,n 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='25 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='75 1 p=r/awhich is consistent with the numerical simulation results reported in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' [16, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' The continuous spectra of low-frequency shear Alfv´en and acoustic branches for n=6, m=8-15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' The equilibrium profiles of DIII-D #178631 at 1200 ms are adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' SUMMARY AND DISCUSSIONS The present work has addressed linear properties of the low-frequency shear Alfv´en waves (SAWs) with the consideration of energetic ions in DIII-D reversed magnetic shear tokamak experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' By analyzing the experimental equilibrium profiles, the local and global models for low-frequency SAWs for weak and/or vanishing magnetic shear are discussed based on the unified theoretical framework of the generalize fishbone-like dispersion relation (GFLDR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Resorting to numerical and theoretical analyses, the dependences of mode frequency, growth rate and polarization on the minimum of the safety factor (qmin), as well as the instability mechanisms are delineated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' 16 200 (a) n=6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' BAE pi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='exp BAAE 米 KBM;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' rel 150 m=8~15 KBM BAE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' rel KBM1 米 KBM;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' cl LFM BAE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' cl (ZH>) e-KBM 100 p 50 0 米 米 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='8 r/a10 (b) 0 (kHz) 10 20 2元 BAE 30 BAAE KBM n=6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' KBM1 40 m=8~15 LFM e-KBM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='8 r/a102 (c) BAE KBM1 BAAE LFM KBM e-KBM Sf100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='8 r/aThe main results of this work are that the LFAMs and BAEs observed in DIII-D ex- periments are, respectively, the reactive-type and dissipative-type unstable modes with pre- dominantly Alfv´enic polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Due to the different instability mechanisms, BAE peak occurs further away from the rational qmin than LFAM peak does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' The BAE eigenfunction is localized at the radial position with the strongest energetic-ion-drive spatially, which leads to deviation from the radial position of qmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' The theoretical analysis explains many experimental observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' The theory successfully explains the temporal pattern of two bands of instability, the BAE band and the LFAM band, that both appear near rational values of qmin but with distinctly different stability properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' The predicted values of KBM frequency are in excellent agreement with the experi- mental LFAM frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' The KBM can be unstable even in the absence of energetic particles (EPs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' The predicted values of BAE frequency span the same range as the experimentally observed values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' The theory also successfully explains the absence of a third branch of instability at BAAE frequencies, as that branch is predicted to be stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Experimentally, an individual unstable BAE spans a much larger range of frequencies than an unstable LFAM, another feature successfully reproduced by theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Experimentally, unstable LFAMs only persist for a few milliseconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' The short du- ration of the LFAM is consistent with the very strong qmin dependence of the KBM growth rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' In experiment, unstable BAEs persist longer than LFAMs, which is consistent with the weaker dependence of the BAE growth rate on qmin in theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Temporally, in experiment, LFAMs occur at rational values of qmin;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' BAEs also occur near rational values but less precisely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' This feature is also reproduced by the theoretical stability predictions: the KBM growth rate peaks sharply at rational qmin values but the peak of the BAE growth rate deviates slightly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' 17 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' In experiment, for both the LFAM and the BAE, unstable modes with higher values of toroidal mode number n are of shorter duration than lower values of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' The narrower growth rate curves as n increases successfully explains this feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Experimentally, the BAE radial eigenfunction has an approximately gaussian shape, consistent with the theoretical prediction that the L = 0 radial harmonic is most unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Experimentally, the LFAM is more unstable in plasmas with hydrogen than in pure deuterium plasmas [18], a feature explained by the higher value of ωA in hydrogen plasmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' As Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' (13) shows, a larger value of ωA lowers the instability threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' On the other hand, there are three discrepancies between theory and experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Although the predicted KBM growth rate correctly peaks sharply for rational values of qmin, it remains positive for a much longer duration than the LFAMs are observed experimentally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Evidently, an additional damping mechanism is missing in the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Although the predicted KBM growth rate has changed significantly for the cases with and without EPs, there is no apparent dependence of LFAM stability on EPs ex- perimentally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Therefore, a more precise theoretical model and more comprehensive experimental analysis are needed for meaningful comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Although the predicted BAE frequency spans the observed values, the predicted fre- quency has a parabolic shape with time, while the experimental frequency has a less regular shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' A likely explanation for this discrepancy is imprecise modeling of the fast-ion distribution function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Finally, there is one theoretical prediction that is inconclusive experimentally: the mode polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Theory predicts predominately Alfv´enic polarization for both the KBM and the BAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' In experiment, low toroidal mode number (n ≤ 3) BAEs are usually observed on external magnetic coils;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' LFAMs are never detected, but the inferred toroidal mode numbers typically span a larger range than those normally detected for RSAEs or BAEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' DIII-D is equipped with one diagnostic that can detect internal magnetic fields, a radial interferometer- polarimeter (RIP) [48] that measures the line integral of the density and radial magnetic field, � neBrdl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' This diagnostic clearly detects RSAEs and BAEs, which is consistent with 18 their expected shear-wave polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Fluctuations are observed by RIP for some LFAMs, indicating that there is at least some magnetic component, but the signal is weaker than for RSAEs and BAEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' It is not presently known if this difference is due to a line-integral effect associated with the mode structure or if the LFAM polarization is less Alfv´enic than the other modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' ACKNOWLEDGMENTS One of authors (R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Ma) would like to acknowledge Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Lei Yang and Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Yunpeng Zou for their useful discussions and the DIII-D team for providing the experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' The authors thank Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Xiaodi Du for helpful comments concerning the mode polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='R Ma is also grateful to the Center for Nonlinear Plasma Science (CNPS) for its enlightening academic discussion, which provides a valuable sources of scientific stimuli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' This work has been supported in part by the National key R&D Program of China under Grant Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' 2022YFE03040002 and 2018YFE0304103, by the National Science Foundation of China under Grant Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' 12261131622 and 12175053 and Natural Science Foundation of Sichuan under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' 2022NSFSC1814 and Sichuan Science and Technology Program under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' 2022ZYD0019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' This work has also been carried out within the framework of the EUROfusion Consortium, funded by the European Union via the Euratom Research and Training Programme (Grant Agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' 101052200 – EUROfusion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Commission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Neither the European Union nor the European Commission can be held responsible for them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' This material is based upon work supported by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Department of Energy, Office of Science, Office of Fusion Energy Sciences, using the DIII-D National Fusion Facility, a DOE Office of Science user facility, under Awards DE-FC02-04ER54698 and DE-SC0020337.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' This report was prepared as an account of work sponsored by an agency of the United States Government.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or 19 otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Appendix A: Detailed Expressions of Λ2 n and Sf Detailed derivations of the generalized inertia, Λ2 n and wave polarization, Sf, can be found in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Here, we only present the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' In low-β (β = 8πP/B2 0 ≈ ϵ2) axisymmetric tokamak plasmas, Λ2 n = Iφ � ω2 ω2 A � 1 − ω∗pi ω � + Λ2 cir + Λ2 tra � , (A1) where Λ2 cir and Λ2 tra represent, respectively, the modified circulating and trapped ion re- sponses, and Iφ describes the non-vanishing ‘flute-like’ component of the parallel elec- tric field (δE∥) due to the effect of trapped thermal particle precession resonance [7, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Meanwhile, ωA = υA/qR0 is the Alfv´en frequency with υA being the Alfv´en velocity, and ω∗ps = (Tsc/esB)(k×b)·(∇ns/ns+∇Ts/Ts) ≡ ω∗ns+ω∗Ts is the thermal particle diamagnetic drift frequency due to density and temperature gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' For Λ2 n, the various terms involved in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' (A1) are given by [7] Λ2 cir = q2ωωti ω2 A �� 1 − ω∗ni ω �� F � ω ωti � + ∆F � ω ωti �� − ω∗Ti ω � G � ω ωti � + ∆G � ω ωti �� + ωωti 4¯ω2 Di � N1 � ω ωti � + ∆N1 � ω ωti �� Sf(ω, ¯ωDi, ωbi, ωti) � , (A2) Λ2 tra = ω2ω2 bi ω2 A¯ω2 Di q2 √ 2ϵ � P3 + (P2 − P3)Sf(ω, ¯ωDi, ωbi, ωti) � , (A3) Iφ = 1 + √ 2ϵ(L(ω/¯ωDi) + τ −1L(ω/¯ωDe)) 1 + τω∗ni/ω + √ 2ϵτ[1 − ω∗ni/ω − M(ω/¯ωDi) − τ −1M(ω/¯ωDe)], (A4) and, as to Sf ≡ (iδE∥/k∥)a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' � δφd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' it is given by [7] Sf = − N1 � ω ωti � + ∆N1 � ω ωti � + √ 2ϵP2 1 + 1 τ + D1 � ω ωti � + ∆D1 � ω ωti � + √ 2ϵ (P1 − P2) (A5) where the functions F(x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' ∆F(x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' G(x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' ∆G(x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' N1(x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' ∆N1(x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' D1(x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' ∆D1(x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' P1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' P2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' P3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' L(ω/¯ωDs) and M(ω/¯ωDs) with x = ω/ωti,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' and using the plasma dispersion function Z(x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' 20 are defined as Z(x) = π−1/2 � ∞ −∞ e−y2 y − xdy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' F(x) = x(x2 + 3/2) + (x4 + x2 + 1/2)Z(x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' ∆F(x) = 1 π1/2 � ∞ 0 e−y ln �x + √2ϵy x − √2ϵy �y2 4 dy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' G(x) = x(x4 + x2 + 2) + (x6 + x4/2 + x2 + 3/4)Z(x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' ∆G(x) = 1 π1/2 � ∞ 0 e−y ln �x + √2ϵy x − √2ϵy �y2 4 � y − 3 2 � dy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' N1(x) = 2 ¯ωDi ωti �� 1 − ω∗ni ω � [x + (1/2 + x2)Z(x)] − ω∗Ti ω [x(1/2 + x2) + (1/4 + x4)Z(x)] � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' ∆N1(x) = ¯ωDi/ωti π1/2 � ∞ 0 ye−y ln �x + √2ϵy x − √2ϵy � � 1 − ω∗ni ω − ω∗Ti ω � y − 3 2 �� dy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' D1(x) = x � 1 − ω∗ni ω � Z(x) − ω∗Ti ω [x + (x2 − 1/2)Z(x)],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' ∆D1(x) = ¯ωDi/ωti π1/2 � ∞ 0 e−y ln �x + √2ϵy x − √2ϵy � � 1 − ω∗ni ω − ω∗Ti ω � y − 3 2 �� dy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' P1 = −2 ω2 ¯ω2 Di �� 1 − ω∗ni ω + 3 2 ω∗Ti ω � G2 − ω∗Ti ω G4 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' P2 = −2 ω ¯ωDi �� 1 − ω∗ni ω + 3 2 ω∗Ti ω � G4 − ω∗Ti ω G6 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' P3 = −2 �� 1 − ω∗ni ω + 3 2 ω∗Ti ω � G6 − ω∗Ti ω G8 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Gn = 1 π1/2 � ∞ −∞ e−x2xn (ω/¯ωDi − x2)2 − (ωbi/¯ωDi)2x2dx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' M � ω ¯ωDs � = −2 ω ¯ωDs � � 1 − ω∗ni ω + 3 2 ω∗Ti ω � � 1 + � ω ¯ωDs Z �� ω ¯ωDs �� − ω∗Ti ω � 1 2 + ω ¯ωDs + � ω ¯ωDs �3/2 Z �� ω ¯ωDs �� � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' L � ω ¯ωDs � = −2 � � 1 − ω∗ni ω + 3 2 ω∗Ti ω � � 1 2 + ω ¯ωDs + � ω ¯ωDs �3/2 Z �� ω ¯ωDs �� − ω∗Ti ω � 3 4 + 1 2 ω ¯ωDs + � ω ¯ωDs �2 + � ω ¯ωDs �5/2 Z �� ω ¯ωDs �� � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' (A6) Here the magnetic drift orbit precession frequency ¯ωds = ¯ωDsmsυ2/2Ts for deeply trapped particles (s = i, e) with ¯ωDs = (nq/r)Ts/msR0ωcs and ωcs = esB/msc;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' the bounce frequency of deeply trapped ions ωbi ≡ (r/R0)1/2(Ti/mi)1/2/(qR0) ≈ ϵ1/2ωti with 21 ωti = (2Ti/mi)1/2/qR0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' and τ ≡ Te/Ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' REFERENCES [1] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Chen and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Zonca, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Fusion 47, S727 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' [2] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Heidbrink, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Strait, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Chu, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Turnbull, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' 71, 855 (1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' [3] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Turnbull, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Strait, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Heidbrink, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Chu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Duong, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Greene, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Lao, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Taylor, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Thompson, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Fluids B 5, 2546 (1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' [4] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Zonca, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Chen, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Santoro, Plasma Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Fusion 38, 2011 (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' [5] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Zonca, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Chen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Dong, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Santoro, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Plasmas 6, 1917 (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' [6] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Zonca, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Biancalani, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Chavdarovski, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Chen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Troia, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Wang, Journal of Physics: Conference Series 260, 012022 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' [7] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Chavdarovski and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Zonca, Plasma Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Fusion 51, 115001 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' [8] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Lauber, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Brudgam, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Curran, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Igochine, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Sassenberg, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Gunter, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Maraschek, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Garcia-Munoz, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Hicks, and the ASDEX Upgrade Team, Plasma Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Fusion 51, 124009 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' [9] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Chen and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Zonca, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' 88, 015008 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' [10] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Zonca, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Chen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Falessi, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Qiu, Journal of Physics: Conference Series 1785, 012005 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' [11] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Cheng, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Fluids 25, 1020 (1982).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' [12] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Tang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Connor, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Hastie, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Fusion 20(11), 1439 (1980).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' [13] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Biglari and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Chen, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' 67, 3681 (1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' [14] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Sharapov, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Alper, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Berk, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Borba, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Breizman, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Challis, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Classen, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Edlund, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Eriksson, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Fasoli, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Fredrickson, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Fu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Garcia-Munoz, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Gassner, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Ghantous, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Goloborodko, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Gorelenkov, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Gryaznevich, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Hacquin, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Heidbrink, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Hellesen, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Kiptily, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Kramer, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Lauber, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Lilley, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Lisak, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Nabais, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Nazikian, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Nyqvist, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Osakabe, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' von Thun, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Pinches, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Podesta, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Porkolab, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Shinohara, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Schoepf, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Todo, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Toi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Zeeland, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Voitsekhovich, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' White, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Yavorskij, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' TG, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='-E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' 22 Contributorsa, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Fusion 53, 104022 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' [15] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Gorelenkov, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Pinches, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Toi, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Fusion 54, 125001 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' [16] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Heidbrink, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Zeeland, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Austin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Bierwage, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Chen, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Choi, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Lauber, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Lin, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' McKee, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Spong, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Fusion 61, 016029 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' [17] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Heidbrink, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Zeeland, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Austin, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Crocker, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Du, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' McKee, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Spong, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Fusion 61, 066031 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' [18] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Heidbrink, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Choi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Zeeland, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Austin, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Degrandchamp, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Spong, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Bierwage, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Crocker, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Du, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Lauber, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Lin, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' McKee, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Fusion 61, 106021 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' [19] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Curran, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Lauber, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Carthy, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' da Graca, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Igochine, and the ASDEX Up- grade Team, Plasma Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Fusion 54, 055001 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' [20] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Lauber, Physics Reports 533, 33 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' [21] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Chavdarovski and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Zonca, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Plasmas 21, 052506 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' [22] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Fasoli, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Brunner, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Cooper, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Graves, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='Ricci, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Sauter, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Villard, Nature Physics 12, 411 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' [23] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Bierwage and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Lauber, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Fusion 57, 116063 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' [24] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Choi, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Liu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Wei, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Nicolau, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Dong, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Zhang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Lin, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Heidbrink, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Hahm, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Fusion 61, 066007 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' [25] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Gorelenkov, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Berk, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Fredrickson, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Sharapov, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Contributors, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' A 370, 70 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' [26] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Gorelenkov, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Zeeland, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Berk, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Crocker, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Darrow, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Fredrickson, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Fu, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Heidbrink, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Menard, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Nazikian, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Plasmas 16, 056107 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' [27] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Chen and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Zonca, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Plasmas 24, 072511 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' [28] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Zonca and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Chen, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Plasmas 21, 072120 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' [29] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Zonca and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Chen, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Plasmas 21, 072121 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' [30] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Varela, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Spong, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Garcia, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Huang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Murakami, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Garofalo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Qian, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Holcomb, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Hyatt, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Ferron, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Collins, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Ren, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' McClenaghan, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Guo, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Fusion 58, 076017 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' [31] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Chen, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' White, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Rosenbluth, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' 52, 1122 (1984).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' [32] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Chen, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Plasmas 1, 1519 (1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' [33] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Tsai and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Chen, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Fluids B 5, 3284 (1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' [34] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Zonca and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Chen, Plasma Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Fusion 48, 537 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' 23 [35] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Zonca, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Buratti, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Cardinali, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Chen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='-Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Dong, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Long, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Milovanov, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Ro- manelli, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Smeulders, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Wang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Wang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Castaldo, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Cesario, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Giovannozzi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Mar- inucci, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Ridolfini, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Fusion 47, 1588 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' [36] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Ma, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Chen, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Zonca, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Li, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Qiu, Plasma Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Fusion 64, 035019 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' [37] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Lao, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' John, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Stambaugh, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Fusion 25, 1611 (1985).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' [38] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Pankin, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' McCune, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Andre, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Bateman, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Kritz, Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' 159, 157 (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' [39] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Zonca and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Chen, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Plasmas 7, 4600 (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' [40] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Zonca, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Briguglio, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Chen, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Dettrick, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Fogaccia, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Testa, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Vlad, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Plasmas 9(12), 4939 (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' [41] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Connor, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Hastie, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Taylor, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' 40, 396 (1978).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' [42] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Zonca, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Chen, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Botrugno, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Buratti, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Cardinali, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Cesario, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Ridolfini, and the JET-EFDA contributors, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Fusion 49, 085009 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' [43] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Chen, private notes (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' [44] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Rosenbluth and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Hinton, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' 80, 724 (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' [45] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Graves, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Hastie, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Hopcraft, Plasma Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Fusion 42, 1049 (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' [46] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Falessi, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Carlevaro, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Fusco, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Vlad, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Zonca, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Plasmas 26, 082502 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' [47] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Falessi, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Carlevaro, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Fusco, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Giovannozzi, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Lauber, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Vlad, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Zonca, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Plasma Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' 86, 845860501 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' [48] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Chen, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Ding, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Brower, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Finkenthal, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Muscatello, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Taussig, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' Boivin, Review of Scientific Instruments 87, 11E108 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} +page_content=' 24' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAzT4oBgHgl3EQfgPwz/content/2301.01464v1.pdf'} diff --git a/7NAzT4oBgHgl3EQf-f4D/content/tmp_files/2301.01933v1.pdf.txt b/7NAzT4oBgHgl3EQf-f4D/content/tmp_files/2301.01933v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..20bb2b93ad32f835a8f9282f11bea62f1fd9b6d7 --- /dev/null +++ b/7NAzT4oBgHgl3EQf-f4D/content/tmp_files/2301.01933v1.pdf.txt @@ -0,0 +1,1373 @@ + + +Abstract—Surface electromyogram (SEMG) decomposition +provides a promising tool for decoding and understanding neural +drive information non-invasively. In contrast to previous SEMG +decomposition methods mainly developed in offline conditions, +there are few studies on online SEMG decomposition. A novel +method for online decomposition of SEMG data is presented using +the progressive FastICA peel-off (PFP) algorithm. The online +method consists of an offline prework stage and an online +decomposition stage. More specifically, a series of separation +vectors are first initialized by the originally offline version of the +PFP algorithm from SEMG data recorded in advance. Then they +are applied to online SEMG data to extract motor unit spike trains +precisely. The performance of the proposed online SEMG +decomposition method was evaluated by both simulation and +experimental approaches. It achieved an online decomposition +accuracy of 98.53% when processing simulated SEMG data. For +decomposing experimental SEMG data, the proposed online +method was able to extract an average of 12.00 ± 3.46 MUs per +trial, with a matching rate of 90.38% compared with results from +the expert-guided offline decomposition. Our study provides a +valuable way of online decomposition of SEMG data with +advanced applications in movement control and health. + +Index Terms—Surface electromyography, motor unit, online +decomposition, progressive FastICA peel-off + +I. INTRODUCTION +lectromyogram (EMG) is an electrophysiological signal +generated by muscular activation, reflecting motor control +commands of the neuromuscular system [1]. It can be used to +analyze movement behaviors, intentions and health [2]-[4]. +Surface EMG (SEMG) refers to the EMG signals recorded by +electrodes placed on the skin surface. Due to its noninvasive +manner, SEMG has been widely applied in human-machine +interfaces [5]-[7], sports medicine [8]-[9] and rehabilitation +[10]-[12]. Ideally, an EMG signal is composed of multiple +action potentials generated by activated motor units (MUs), +transmitted and superimposed temporally and spatially at a +recording electrode [13]. Specifically, each MU consists of the +cell body and dendrites of an alpha motor neuron, the multiple + +This work was supported by the National Natural Science Foundation of +China under Grant No. 61771444. +H. Zhao and X. Zhang are with the School of Information Science and +Technology at University of Science and Technology of China, Hefei, Anhui, +230026, China (email: xuzhang90@ustc.edu.cn). +branches of its axon, and the muscle fibers that are innervated +[14]. The MU is regarded as the basic component of the +peripheral neuromuscular system to describe the neural control +of muscular contraction and movement formation [15]. +Compared with the global features such as SEMG amplitude, +the MU activities can reflect the information of neural drives to +the muscle at a microscopic level. Therefore, it is valuable to +examine the MU activities and properties. EMG decomposition +enables resolving the composite EMG signal into its constituent +MU spike trains (MUSTs) and MU action potential (MUAP) +waveforms. The availability of these individual MU activities +can provide a promising way of decoding motor neural +commands of a neurophysiological nature [16]-[22]. +Many efforts have been made toward EMG decomposition, +mainly relying on blind source separation (BSS) algorithms +which are aimed to solve the difficult math problem of +separating sources from observed signals without prior +knowledge of the source signals [23]. Besides, it brings huge +challenges to the SEMG decomposition due to its special +characteristics such as low signal-to-noise ratio, high similarity +and severe superposition of the MUAP waveforms, caused by +the low-pass filtering effect of the subcutaneous skin and fat +tissues. With the recent development of electronic and sensing +technologies, the use of high-density SEMG (HD-SEMG) by 2- +dimensional flexible electrode arrays provides abundant spatial +information simultaneously recorded from dozens or even +hundreds of SEMG channels, facilitating implementing the +BSS algorithms in general, and the SEMG decomposition in +particular [24]. Convolution kernel compensation (CKC) [25] +and progressive FastICA peel-off (PFP) [26] are both +representative HD-SEMG decomposition methods, inspired by +the advanced BSS techniques [23], [27]. The CKC estimates +and updates cross-correlation vectors between the observed +SEMG signals and MUSTs in an iterative way [23]. The PFP +applies a classic FastICA algorithm [27] to the SEMG signals +to calculate the separation vectors and introduces a “peel-off” +procedure to progressively remove the separated MUAP +waveforms from the original SEMG signals. Such a procedure +mitigates the effect of the already identified MUs on the +M. Chen and P. Zhou are with Faculty of Biomedical and Rehabilitation +Engineering, University of Health and Rehabilitation Sciences, Qingdao, +Shandong, 266024, China (email: dr.ping.zhou@outlook.com). + +Online Decomposition of Surface +Electromyogram into Individual Motor Unit +Activities Using Progressive FastICA Peel-off +Haowen Zhao, Xu Zhang, Maoqi Chen and Ping Zhou +E + + +FastICA convergence and effectively increase the number of +obtained MUs. The performance of both CKC and PFP has been +extensively validated [28]-[32]. Variations of both methods +have been developed to extract a relatively large number of +MUs at high muscle contraction levels, with successful +applications mainly in offline conditions [33]-[37]. +Considering the application prospects of SEMG in many +fields, there are substantial demands for robust online SEMG +decomposition. Glaser et al. [38] conducted a pilot study on the +real-time SEMG decomposition based on the CKC algorithm +and demonstrated its feasibility. Afterwards, more relevant +studies were reported [39]-[44]. The development of these +online decomposition algorithms mainly relies on a basic +assumption that SEMG signals are quasi-stationary, and the +MU behaviors do not change in pattern over a short period of +time. This assumption has served as a primary basis of +conventional offline SEMG decomposition [25], [26]. On this +basis, these online decomposition algorithms were always +designed to use results from an offline decomposition as prior +knowledge, thus saving computational resources and allowing +the feasibility of online signal processing. Specifically, most +previous studies conducted online SEMG decomposition using +modified versions of the CKC method, whereas the online +version of the PFP method has not been investigated yet. +Considering the advantages of the PFP method in extracting a +great many MUs with high precision, it is necessary and +promising to develop its online version. +Accordingly, this paper presents an online SEMG +decomposition method based on the PFP algorithm, evolving +the key techniques of the PFP algorithm to meet the +requirements for its real-time usability. To avoid the time- +consuming complexity from the offline decomposition methods, +the proposed method utilized a two-stage approach consisting +of an offline prework stage and an online decomposition stage. +Furthermore, an adaptive threshold selection algorithm was +developed to make it more suitable for precisely determining +each MUST while processing in real time. The performance of +the proposed online decomposition method was validated on +both simulated and experimental SEMG datasets. +II. RELATED WORK +A. SEMG Observation +Each MU has a unique and stable MUAP waveform +distribution pattern in different channels of a 2-dimensional +array, which can be used to distinguish and identify the MU. +The SEMG signal can be observed by a convolutional mixing +model expressed as [45]: +��(�) = � � ���(�)��(� − �) + ��(�) +��� +��� +� +��� + + +(1) + +where � = 1,2,3 … … � and � = 1,2, … … � , ��(�) is the � th +SEMG channel and ��(�) represents the additive noise in the +�th channel. ���(�) denotes the waveform vector of length L, +which represents the waveform of the �th MU in the �th channel. +��(�) = ∑ �(� − ��(�)) +� + is the MUST expressed as a 0-1 +impulse sequence indicating every spike firing timing at ��(�) +for the �th firing of the �th MU, whereas � is Dirac Delta +function. For each �, ��(� + 1) − ��(�) > � can be assumed. +Define the expansion vector of EMG signals and MUSTs as + ��(�) = [��(�), ��(� − 1), … , ��(�), … , ��(� − � + 1)] + and + ��(�) = [��(�), ��(� − 1), … , ��(�), … , ��(� − � + 1)]. +Thus, the equation can be rewritten in matrix form: +��(�) = ����(�) + ��(�) +(2) +where ��(�) represents noise. �� is a matrix containing all +waveform vectors ���. For the mixing model analyzed above, +the task of EMG decomposition is to find a suitable separation +matrix � +��� that consists of many separation vectors to extract the +MU firing events. As a result, the source signals of all MUs can +be estimated by ��(�) = � +�����(�). + +B. Automatic PFP (APFP) +The PFP algorithm has been automated, but it is suitable just +for offline data processing. More details of the algorithm and +the corresponding parameters can be found in [33] and the +APFP method was used in this study with the same settings as +reported in [33]. Below is a brief introduction to the APFP +method. +If a whitened observed signal � has been obtained and we +need to find an independent component � = ��� from it using +the ICA algorithm [23], [27], the following maximum negative +entropy problem needs to be optimized: +max ��(�) = [�{�(���)} − �{�(�)}]� + �. �. ℎ(�) = �{��} − 1 = �|�|� +� − 1 = 0 + +(3) + +where � is a non-polynomial function, and � is a random +variable with standard normal distribution. +The problem above can be solved using the procedure of the +fix-point algorithm [46] to obtain a series of MU source signals +and their corresponding separation vectors. The spike trains can +be precisely extracted from these source signals using the initial +threshold determined by the Otsu algorithm [47]. However, the +spikes from one source signal often do not just belong to one +MU due to heavy MUAP superimposition or high MU +synchronization levels. Thus, a valley-seeking clustering +approach [48] is used to distinguish the spikes from the same +source signal based on their morphological features. On this +basis, the spikes belonging to each cluster are most likely from +the same MU [33]. After the valley-seeking clustering approach, +the constrained FastICA algorithm [49] is performed using the +extracted and clustered spike trains as constraints to converge. +Therefore, the MU source signals can be effectively updated +and meanwhile the possible firing errors are corrected. To +assess the reliability of the constrained FastICA outputs and +their corresponding MUSTs representing true MU activities, +some metrics are employed from the perspective of the +significance of correlation constrain [49], including the +consistency of spike amplitudes and inter-spike intervals [50], +and the physiologically reasonable firing rate [51]. In the APFP +method, the correlation coefficient between the output of +constrained FastICA and the testing spike trains (denoted as ξ), +the coefficient of variation of spike amplitudes and inter-spike +intervals (denoted as ������ and ������ ), and the firing rate + + +(denoted as FR) are employed. Moreover, a two-step criterion +describing a reasonable range of the above four metrics is +employed to judge the MU reliability comprehensively [33]. +A “peel-off” procedure is performed later to subtract the +obtained MUAP waveforms from the original signals. The +MUAP waveforms of the identified MUs were estimated by a +straightforward approach following a least squares problem +[26], [52] instead of the conventional high-resolution alignment +algorithm [53]. More MUs can emerge when processing the +residual signals again with the FastICA algorithm. The +framework of the offline APFP method is summarized as +follows: +(1) Initialize the residual signal to the original EMG signal, +and make the MUST set γ empty. +(2) Apply the FastICA algorithm to the expanded residual +signal and obtain a series of source signals. +(3) Extract non-repetitive spike trains by Otsu algorithm and +use valley-seeking clustering to distinguish these spikes to +separate spike trains from different MUs. +(4) Use MUSTs obtained in step (2) as a reference signal, and +apply the constrained FastICA algorithm on the expanded +original EMG signal to detect the reliability of the MUSTs +and to correct possible erroneous or missing discharges. +(5) Judge whether the MUs obtained are reliable through +metrics calculation. Put reliable results in set γ. +(6) Estimate the waveforms of the reliable MUs, subtract the +estimated MUAP waveforms from the original signal and +update the residual signal. +(7) If no new reliable MU is found in the above steps, or the +APFP method reaches the preset termination condition, +the algorithm ends. Otherwise, go back to step (2). + +III. METHODOLOGY +A. Experimental SEMG Data Collection and Preprocessing +1) Subjects and Experiments +Eight subjects (26.13±4.29 years) without any known history +of muscular or neural disorder participated in this study. The +study was approved by the Ethics Review Board of the +University of Science and Technology of China (Hefei, China). +All subjects signed consent prior to any procedure of the +experiments. +In this work, the HD-SEMG data were recorded from +abductor pollicis brevis (APB) muscle due to its wide +explorations and applications in SEMG studies [19]-[21]. Here, +a home-made, multi-channel signal acquisition system with a +force sensor and a set of 3D-printed apparatuses was used to +collect data, as shown in Fig. 1a. The subject’s hand was placed +on the fixed 3D-printed apparatus to prevent muscular +movement interferences from the wrist and other fingers, and +the muscle force was recorded by a load cell (LDST-V-HY, +Luckly Inc., Beijing, China) connected to a ring around the +thumb. Multiple electrodes were arranged in the form of 8 rows +× 8 columns to form a 2-dimensional electrode array. Each +electrode probe had a diameter of 2 mm, and the inter-electrode +distance between consecutive electrodes was 4 mm. Each +electrode was designed in a monopolar manner relative to a +round common reference electrode placed on the back of the +tested hand. +During the experiments, subjects were asked to sit and place +the tested hand in a relaxed and comfortable way. Before data +collection, the maximum voluntary contraction (MVC) of the +thumb abduction muscle was tested and recorded. Then, in each +trial of the task performance, subjects were instructed to +perform isometric muscle contractions with the muscle force +gradually increasing from 0 to a targeted force level (quantified +by MVC percentage) in 2s and then maintained at the targeted +level for around 3s, as shown in Fig. 1b. According to this force +generation pattern, the designed force curve was shown on the +screen to facilitate the subject’s task performance in each trial. +The targeted force level in this experiment was set to 30% MVC +and the trial was repeated at least nine times to acquire a +sufficient amount of data. The force and SEMG data were +digitized via a 16-bit A/D converter (ADS1198, Texas +Instruments, TX) at a sample rate of 2 kHz, and the data were +stored into the hard disk of a computer and imported into the +MATLAB software (version R2020a, MathWorks, Natick, MA, +USA) for further analyses. + +2) Data Preprocessing +All channels of the recorded HD-SEMG signals were +inspected, and a few channels (3.75 ± 1.28 channels across all +subjects in this study) with low quality were discarded (due to +their excessive noise contamination resulting from motion +artifacts, occasional electrode drop, or environmental +interferences from surrounding electronic devices). The +channel deletion remained consistent within the EMG signals +of the same subject. The HD-SEMG signals within the +remaining channels were filtered through a 10-order +Butterworth band-pass filter to reduce possible low-frequency +or high-frequency interference. The bandwidth of the filter was +20-500Hz. Finally, the power line interference was removed +through a 50Hz second-order notch filter. The deleted channels +were not considered in the subsequent process of SEMG + +Fig. 1. The experimental setup and protocol. (a) Apparatuses for +simultaneously recording thumb abduction force by a load cell and HD- +SEMG data by a piece of 2-dimensional electrode array arranged in an 8×8 +formation. (b) The illustration of the force generation pattern with both the +designed force curve (blue line) and an actual recorded force curve (red +line) in one trial of task performance. + +Force:C8 +30%MVC +C1C57 +C642 +5Time(s)(b) +(a) +decomposition, but they were filled in by interpolation from +neighboring channels and considered during the estimation of +MUAP waveforms. In order to facilitate the data analysis, all of +the SEMG data were divided into a series of non-overlapping +data segments corresponding to the force generation task +repetitions over time. Therefore, the length of every SEMG data +segment was around 5 seconds. + +B. SEMG Data Simulation +A data simulation approach was conducted to generate HD- +SEMG data with known MU activities, which were used as the +ground-truth for validating the performance of the developed +online SEMG decomposition method. In the current study, this +approach was based on simulation models well described by +previous studies, including the motoneuron pool model [54], +the model describing the MUAP waveforms of different MUs, +and a tripole model [55] considering the generation and +extinction of the action potentials at the fiber end-plate and +tendon. +Here a cylindrical muscle with a radius of 8 mm was +simulated and the fat and skin layers of the muscle were set to +2.5 mm thickness. 120 MUs were set and distributed in parallel +in the muscle fibers. Most of the MUs had low recruitment +thresholds and a few had high thresholds. When the excitation +exceeded the threshold, every MU discharged at 8 Hz and its +firing rate increased as the excitation increased. All the relevant +parameters are listed in Table I. +The simulated SEMG signals were also set to be recorded by +a 64-channel surface electrode array arranged in an 8×8 grid +form. The inter-electrode distance was set at 4 mm for both +horizontal and vertical directions. The electrode array was +placed parallel to the muscle fiber direction and its center +electrodes were set to approximately over the innervation zones. +To be consistent with the force generation pattern of the +actual experiments, the excitation was set to increase from 0 to +a specific excitation level in the first 2 seconds, and maintained +for another 3 seconds with several repetitions. The maximum +excitation level was set to be 3%, corresponding to 33 active +MUs. In addition, zero-mean Gaussian noises were added to the +simulated EMG signals, generating three levels of SNR (signal- +to-noise ratio) at 10 dB, 20 dB and 30 dB, respectively. Thus, +we considered four noise levels, three SNR levels and the level +without any additional noise. For each noise level, 21 +repetitions were simulated to ensure data diversity, as shown in +Fig. 2. Therefore, 84 data segments (4 noise levels × 21 +repetitions) were simulated in total. +C. Online Decomposition +The overall whole block diagram summarizing the proposed +online decomposition method is described in Fig. 3. +TABLE I +PARAMETERS FOR SEMG SIMULATION + +Distribution +Mean +SD +Range +Fiber number +Uniform +70000 + +±0.5 mean +MU fiber endplate +center position +Uniform +0 + +±8 mm +Fiber endplate +position variation +Uniform +0 + +±2 mm +Half fiber length +Gaussian +40mm +4mm +±2 SD +Mean fiber +diameter for a MU +Gaussian +55μm +10μm +±2 SD +Fiber diameter +variation within a +MU +Gaussian +0 +1μm +±2 SD +ISI variation +Gaussian +0 +0.2*instant +mean ISI +±2 SD + + +Fig. 2. (a). The contraction condition of simulated signals. (b). Multi- +channel simulated SEMG signals. + + +Fig. 3. Block diagram of the proposed method for online SEMG decomposition + + +Online +Divided +extraction +extraction +data input +Separation +1vectors +4 +Whitening +calculation +Vector set +1 +and +Offlin +MUST +MUST(a) +Maximum +excitationExtending +Offline PFP +Φ= +W1, W2 -.. Wn? +data +connection +indno +decomposition +-0 2 +5 +5s +100s(b) +ChannelMUST +-- +Preprocessing +Offline prework +Online Decomposition +extraction#1#64 +Data +#1 +#2 +#3 +#20 +#21 +SegmentTime window +Peak +With full consideration of the real-time usability of the +proposed online method, a two-stage approach was designed to +avoid considerable computational complexity caused by the +repeated operation of the FastICA algorithm and the iterations +of the constrained FastICA algorithm. More specifically, the +reliable separation vectors were initialized in the offline +prework stage and saved to accelerate the subsequent online +data processing. In the online decomposition stage, the data +stream of the input EMG signals was divided into a series of +temporally overlapping windows with window length and +increment set at 1 s and 0.2 s, respectively. Both settings helped +to facilitate online processing. +During the offline prework stage, several 5-s segments of +EMG signals were separately decomposed offline using the +APFP method and all of the resultant separation vectors were +put into the set �. The quality of these vectors was evaluated by +both criteria employed in the offline APFP method [33]: if the +coefficient of variation of spike amplitudes ������ was higher +than 0.3, and the coefficient of variation of inter-spike intervals +������ was higher than 0.4, the corresponding separation vector +was considered to be low-quality and it was removed from the +set � . Furthermore, any duplicated separation vector +corresponding to the same MU was removed as well. +In the online decomposition stage, every 0.2 s of data input +was combined with 0.8 s of historical data to form a 1-s window +for decomposition. The decomposed results from consecutive +windows were connected, while their overlapping portion was +used to align the obtained MUSTs. This ensured continuity of + +Fig. 4. Illustration of the online SEMG decomposition process using the proposed method. + + +#14#15 +0 +5 +10 +15 +20Channel +Channel +#1 +#1Time(s)#64 +#64Spike extraction & ConnectionWindow sliding + vectors +X +山 = {W1.W? ... WNMUST + Experimental Muscle Force +MU +DAWDecompose sEMG signals +window by window#1 +30%Window +Source signal of MU1 +Source signal of MU2开2DecomposeOffline Prework +Online Decomposition +0.2s +the decomposition results along with the original SEMG data +stream. The SEMG data in each window were first whitened +and extended. Then, the multiplication procedure was directly +applied to the extended EMG signals with separation vectors in +set � to estimate different MU source signals, from which +individual MUSTs were consequently identified. +For extracting MUSTs from the MU source signals, the +original offline APFP method employs repeated iterations of +the constrained FastICA algorithm, involving complex +computations as described above. This process was unsuitable +for online processing and therefore it was removed to avoid +heavy computational burden. To maintain high-precision +MUST extraction, the simple amplitude-thresholding process +by the Otsu algorithm had to be updated. A new algorithm was +designed for our online PFP method. First, this algorithm needs +to determine an initial threshold that is applied to each source +signal, using the Otsu algorithm in the same way as conducted +in the offline APFP method. Then, a group of spikes beyond +this threshold is detected and the corresponding amplitudes can +be ranked from small to large. Next, a series of successively +increasing thresholds that are a little higher than these +amplitudes are adopted to estimate a series of different spike +trains. Each resultant spike train can be further evaluated by +both ������ and ������ metrics, and the spike train with the +minimal summation of both metrics is finally considered the +most appropriate MUST. This algorithm for adaptive threshold +selection was termed the successive multi-threshold Otsu +algorithm. +A k-means clustering algorithm was usually used in some +offline decomposition methods [36]-[37] for extracting MUSTs +from the source signals. It was also implemented in this study +as an alternative threshold selection algorithm, in comparison +to the successive multi-threshold Otsu algorithm used in our +method. By applying the k-means clustering algorithm, all +sample amplitudes of the source signal time series can be +classified into 2-4 groups (2 in this work), so that the group with +the largest amplitudes of samples is selected as the extracted +MUST. +After the spike trains of all MUs were appropriately detected, +they were connected over windows to form the resultant MUST +for each MU, and its MUAP waveforms that spanned over all +channels were correspondingly estimated. Fig. 4 illustrates an +example of the online decomposition results. The pseudocode +of the proposed online decomposition method is shown in +Algorithm 1. + +D. Performance Evaluation +For processing the experimental SEMG data, the proposed +online decomposition method was conducted in a user-specific +manner. Four segments were used in the offline prework stage +and the remaining 4 segments were processed in the online +decomposition stage. For processing the simulated SEMG data, +the first segment was used in the offline prework stage and the +remaining 20 segments were processed in the online +decomposition stage. All SEMG segments tested in the online +decomposition stage were sequentially arranged in the form of +a data stream to be processed continuously using our proposed +method. For comparison purposes, all of the SEMG segments +to be processed online was also decomposed by the offline +APFP method as well. +To evaluate the performance of online decomposition and +assess the decomposition results more comprehensively, we +Algorithm 1 The proposed online decomposition +method +1: +Decompose the SEMG signals offline. Extract +the MUSTs and calculate the corresponding +separation vectors. +2: +Remove the duplicated separation vectors and +vectors that are not well-decomposed. +3: +Save all the separation vectors ��, ��, ��…�� +for the online decomposition stage. +4: +while Acquiring SEMG signals do +5: + Load and extend the EMG signals (��). +6: + for j = 1; j < N + 1; j ++ do +7: + Calculate the source signal, �� = �� +���. +8: + Estimate the initial threshold through the +Otsu algorithm and extract the spike train. +9: + Successively increase the threshold and +extract a series of spike trains ����, ����, ����… +10: + Find the spike train with the lowest +������ and ������ as the �th MUST ���. +11: + end for +12: + Connect the MUSTs over the sliding +windows. +13: +end while + +Fig. 5. The results for decomposing simulated SEMG data in terms of MR(a), FDR(b) and FNR(c) averaged over all data segments using the offline APFP +method, the proposed online PFP method and the online PFP method with k-means clustering at four noise levels, respectively. The error bar represents +standard deviations. N in the horizontal axis denotes the condition without any additional noise. + +0.10.05 +1N +30 +20 +10 +N +30 +20 +10 +N +30 +20 +10 +SNR (dB) +SNR (dB) +SNR (dB) The offline APFP method + The online PFP method with k-means clustering + The proposed online PFP method(a) +(0) +() +MR(%) +FDR +FNR +001 +0.250.3 +80 +0.2 +used a series of metrics: matching rate (MR) can be calculated +as [33]: +�� = +2 ∙ ������� +������� + ���������� + +(4) +where ������� denotes the number of firing events of the online +decomposition results, and ���������� denotes the number of +the reference spike trains. In the simulated data, the reference +spike train indicates the ground-truth firing events. However, +the actual MUSTs are not known a priori in the experimental +data. Therefore, the decomposition results of the experimental +data processed by the offline APFP method were used to define +���������� . ������� indicates the number of common +discharges appearing in both the online decomposition result +and the reference. The MR measures the matching degree and +it is able to quantify the precision of an online decomposition +method. + +Fig. 6. A representative example of validating the decomposition results from the online PFP method in terms of all decomposed MUSTs (in blue) with +respect to the reference (in red) derived from summarized offline decomposition results, using a data segment from one subject. The position of the black +dot indicates the missing or fault discharges and MR values are computed and shown on the right side of these spike trains. + +Fig. 7. Two MUAPs of matched MUs with time-varying waveform shapes. Here we illustrate 64 electrode channels arranged in an 8×8 grid form. Blue and +red lines indicate the MUAP shapes from online PFP and the reference of offline decomposition, respectively. + + +Fig. 8. The relationship between the matching rate and the composite +decomposability index. + + +T Online PFP +10 +MR (%)111194.87111 Online PFP +600μv +The reference (Offline decomposition +30msnumb +10098.95MU1 +MU2 +1 +2 +3 +4 +5 +6 +7 +8 +1 +2 +3 +4 +5 +6 +7 +81002 +23 +3 +4 +45 +5 +67 +8 +80.95 +Matching Rate +0.9 +0.85 +0.8 +0 +10 +20 +30 +40 +50 +Composite Decomposability Index0 +1 +2 +4 +5Time(s) +Besides MR, both false negative rate (FNR) and false +discovery rate (FDR) were used to reveal the cause of the error +discharges. They are defined as + ��� = ���������� − ������� +���������� + + +(5) + ��� = ������� − ������� +������� + + + +They count the proportion of the number of unmatched +discharges to the total number of their respective discharges. +Specifically, the FNR measures the rate of “missing” discharges +with respect to the reference, and the FDR quantifies the rate of +“faulty” discharges appearing in the online decomposition +results. Generally speaking, the MR of a reliable MUST is close +to 1 but the FNR and FDR are close to 0. +For a more comprehensive view of the decomposition results, +we also calculated the mean discharge rate (MDR) and the +coefficient of variation (CoV) of the online identified MUSTs +with respect to the reference spike trains. It should be noted that +the CoV refers to the coefficient of variation of the inter-spike +intervals ������ to better understand the MU firing behaviors. +In addition, we calculated the decomposability index (DI) for +each common MU of experimental EMG data to precisely +quantify the proposed method’s performance [56]: + �� = min {‖���‖, ‖��� − ��∗�‖} +�� +��� + + +(6) + +where ��� is the MUAP of the �th MU in the �th channel and +��∗� is the MUAP most similar to ��� among the other +MUAPs in the � th channel. �� +��� is the root mean square +amplitude (RMS) of the � th channel and the operator ‖∙‖ +denotes the Euclidean norm. The DI measures the separation +between ��� and the template of MUAP nearest to it (or the +baseline), normalized by the standard deviation of the noise +component (interference plus baseline noise) projected along +their vector difference. The overall decomposability of the �th +MU was measured by the composite DI (CDI), defined as the +norm of the individual DIs [56]. +For developing a real-time decomposition method, it is +necessary to evaluate the processing time delay which is +expected to be as short as possible. The time delay for +processing one single time window was recorded, and all these +time delay values were averaged across all windows and all +subjects to indicate the computational complexity. All of the +algorithms were implemented on a desktop computer with an +Intel Core i5-10400 processor (2.90 GHz) and 16 GB of +memory. +IV. RESULTS +A. Results of Simulated Data +As an offline decomposition method for validation, 21 MUs +were identified using offline APFP and the number was 22 +using online PFP when no additional noise was added. Further, +the number of MUs correctly decomposed using online PFP +decreased to 11, 7, and 6 when noise was added at 30 dB, 20 dB +and 10 dB SNR, respectively. +The results for decomposing simulated SEMG data are +reported in Fig. 5. As compared with the offline APFP method, +the proposed online PFP method achieved comparable +performance in terms of a high MR over 90%, and a low FNR +below 0.05. The proposed online PFP method had a fluctuated +and relatively higher FDR than the offline APFP method under +three SNR levels. Specifically, a decreasing trend of the MR +was found from 99.29% to 94.13% for the offline APFP method +and from 98.53% to 92.79% for the online PFP method, +respectively, when the noise was successively added to generate +four noise levels. The ANOVAs revealed no significant +difference in either MR, FDR or FNR, between the offline +APFP method and the proposed online PFP method (p > 0.05). +When both threshold selection algorithms were compared, it +was evidently found that the successive multi-threshold Otsu +algorithm in the proposed online PFP method significantly +outperformed the K-means clustering algorithm in terms of +higher MR (p = 0.025) and lower FNR (p =0.022). Both +algorithms did not exhibit a significant difference in the FDR +metrics (p = 0.273). +Table II reports both MDR and CoV values calculated for all +common MUs between the decomposition results achieved by +the proposed online method and the ground truth. The ANOVA +revealed no difference in MDR (p = 0.217) or CoV (p = 0.105) +at no presence of noise. However, the MDR and CoV of online +decomposition results became significantly different from those +of the ground-truth (p < 0.05) when the noises were added. +B. Results of Experimental Data +When implementing online decomposition of experimental +data, the offline decomposition method was applied to establish +the reference for validation, and 10.31±1.79 MUs were +obtained, averaged across all subjects. +Fig. 6 is an example of an online decomposition result using +the proposed method, showing the decomposed MUSTs with +respect to the reference. It can be observed that almost all the +MU discharges derived from the online PFP method are well +matched with those in the reference, with sporadic missing or +erroneous ones. Fig. 7 illustrates the MUAP waveforms of two +matched MUs derived from both the online PFP method and the +reference, which demonstrate a very consistent waveform shape +in each channel and almost the same distribution pattern across +the electrode array. Fig. 8 plots the relationship between the +matching rate and composite decomposability index (CDI), +which displays the overall trend of the matching rates varying +TABLE II +COMPARISON OF MDR AND COV OF THE SIMULATED EMG SIGNALS + +SNR 10dB +Online PFP/ +Ground-truth +SNR 10dB +Online PFP/ +Ground-truth +SNR 30dB +Online PFP/ +Ground-truth +No adding noise +Online PFP/ +Ground-truth +MDR +9.86±1.99 +8.77±0.18 +9.55±1.54 +8.75±0.23 +10.47±1.81 +8.74±0.22 +8.77±0.51 +8.70±0.18 +CoV +0.245±0.053 +0.199±0.003 +0.257±0.032 +0.202±0.005 +0.231±0.044 +0.201±0.005 +0.211±0.024 +0.199±0.007 + + + +with the CDIs. It contains the common MUs of all of the +collected SEMG segments. +Table III reports both the number of MUs decomposed by the +online PFP method and the number of common MUs matched +those in the reference (offline decomposition) for 8 subjects, +respectively. An average of 12.00±3.46 MUs were successfully +identified by the online PFP method, with an average of +6.69±1.84 MUs correctly matched. Besides, three metrics are +also computed from those common MUs and reported in Table +III. Averaged over all data segments to be decomposed and all +subjects, the MR was (90.38±2.80) %, the FDR was +0.091±0.022, and the FNR was 0.089±0.041. The estimated +MDR (p = 0.872) and CoV (p = 0.503) of online decomposition +results were not significantly different from the offline +decomposition reference. + +C. Time Delay +The time delay for decomposing a 1-s window of SEMG data +using the proposed method in the online decomposition stage +was 0.084±0.028 s, averaged over all data segments and all +subjects; it was less than a 0.2-s time increment. For +comparison purposes, the offline APFP method costs 60.07 ± +9.82 s to decompose SEMG data in a single time window, much +longer than that of the proposed online decomposition method. +V. DISCUSSION +As a promising SEMG decomposition method, the PFP +algorithm has been reported recently and, therefore, it is +necessary and promising to develop an online version. This +study sought to propose an online SEMG decomposition +method based on the PFP algorithm. The results of both +simulated and experimental SEMG data analyses demonstrated +the feasibility of the proposed online PFP method in +decomposing a large number of MUs with high precision in the +context of isometric muscle contractions. Our study offers a +valuable tool for online SEMG decomposition with great +applications in biomechanics and rehabilitation. +In the results of processing simulated data, the proposed +online PFP method decomposed a similar number of MUs as +the offline APFP method, illustrating comparable performance. +Due to the use of initial separation vectors provided by the +APFP method in the offline prework stage, the proposed online +PFP method is expected to inherit a good capability of +decomposing a great number of MUs from its original offline +version. In terms of MR, the proposed online PFP method got a +slightly lower value compared with the offline APFP method. +This can be explained by the fact that the source signals were +calculated by directly multiplying previously initialized +separation vectors with the SEMG signals for the purpose of +real-time processing. In addition, the MUSTs were estimated +without the examination of iterative constrained FastICA, thus +increasing the negative influence of noise. The result +demonstrates that online decomposition was speeded up at the +cost of a little bit of decrease in precision. This is the main and +common difficulty in generalizing an offline decomposition +method to its online version [38]-[43]. However, it has been +found that the MDR and CoV of online decomposition were +significantly different from those of the ground-truth when the +noise was added. This can be partly explained by the limitations +of +the +online +decomposition +method +such +as +MU +synchronization [26] and firing events drift [33] that previous +studies have faced. +When some noises were successively added to EMG signals +to be decomposed, both the number of correctly identified MUs +and the precision of determining their firing timings were +reported to decrease substantially. This could partly explain that +the decrease of SNR resulted in more serious noise interference +to some small MUs and thus caused a negative influence on the +calculation of separation vectors as well as the performance of +the online decomposition method. On the other hand, it became +much harder to precisely extract MUSTs from source signals in +the online decomposition stage at a low SNR level, reflecting +the decline of the MR. As a consequence, it can be inferred that +the quality of SEMG signals significantly influenced the +performance of the decomposition method, as reported in [33], +[40]. +It is worth mentioning that the proposed online PFP method +introduced a progressive multi-thresholding process for +extracting MUSTs. The successive multi-threshold Otsu +TABLE III +SUMMARY OF DECOMPOSITION RESULTS FOR EXPERIMENTAL EMG SIGNALS. +Subject +Number of motor units + +MDR (Hz) + +CoV (%) +MR (%) +FDR +FNR +The +reference +Online PFP + +The +reference +Online +PFP + +The +reference +Online +PFP +All +Matched +1 +12.75±1.50 +19 +9.00±1.41 +19.76±5.08 19.71±4.43 +22.92±7.71 +24.31±8.53 +92.06±5.91 +0.084±0.082 +0.051±0.057 +2 +9.50±1.29 +8 +4.50±0.58 +22.00±4.76 20.63±4.00 +27.44±6.59 +22.46±5.95 +89.92±7.21 +0.093±0.075 +0.106±0.086 +3 +9.00±0.82 +14 +6.00±0.81 +15.79±3.22 14.65±3.30 +23.44±3.78 +25.44±4.59 +93.20±6.02 +0.065±0.068 +0.067±0.074 +4 +11.00±1.41 +13 +8.75±1.71 +20.15±3.93 21.28±3.84 +24.08±6.98 +24.67±6.25 +91.17±3.35 +0.076±0.033 +0.056±0.029 +5 +8.50±0.57 +9 +5.50±0.58 +20.29±3.99 20.62±3.00 +26.09±4.19 +28.48±4.70 +85.18±4.04 +0.116±0.051 +0.175±0.068 +6 +9.50±1.29 +10 +6.25±1.71 +20.35±4.25 19.67±4.30 +23.87±3.05 +24.18±3.73 +91.51±6.45 +0.084±0.071 +0.082±0.076 +7 +11.75±1.71 +11 +7.00±0.82 +23.03±3.60 24.66±3.94 +24.46±3.54 +24.82±2.78 +87.26±5.47 +0.131±0.073 +0.108±0.028 +8 +10.50±1.29 +12 +6.50±1.73 +18.57±2.72 18.73±1.86 +18.74±2.96 +19.41±1.66 +92.70±4.26 +0.080±0.058 +0.064±0.040 +Average 10.31±1.79 12.00±3.46 +6.69±1.84 +19.99±2.18 19.99±2.79 +23.88±2.55 +24.22±2.56 +90.38±2.80 +0.091±0.022 +0.089±0.041 + + + +algorithm outperformed the conventional k-means clustering +algorithm especially in the condition of noise interference, +proving the potential to extract more precise discharges at low +SNR levels. The successive multi-threshold algorithm based on +the Otsu algorithm was inspired from the common Otsu +algorithm [48] used in the offline APFP method [33]. It was +able to successively increase multiple thresholds to overcome +the effect of noise interferences and find the most appropriate +one to extract MUSTs that followed the physiological +properties of MUs. The successive multi-threshold Otsu +algorithm takes consideration into the interval and waveform +information to ensure the result to be much more reliable, +depending on ������ and ������ . By contrast, the k-means +clustering algorithm only focuses on the amplitude information +of EMG source signals. As a result, it makes it much more +difficult to remove the noise interferences and leads to +decomposition performance degradation. The proposed online +PFP method replaced the complex iterative calculation of +constrained FastICA with the successive multi-threshold Otsu +algorithm +to +extract +MUSTs, +showing +a +significant +improvement in reducing the calculation complexity while +maintaining its high precision. +To evaluate the real-time performance, this study recorded +the processing time of online decomposition. The time delay +was effectively reduced from 60 seconds for the offline APFP +method to less than 0.08 seconds for the online decomposition. +The acceleration of data processing is attributed to reasons in +two respects. The first is that the repeated iteration of FastICA +was put in the offline prework stage, which initialized the +separation vectors for online decomposition. On the other hand, +some complex calculation procedures were adaptively +simplified. For example, the constrained FastICA algorithm in +the APFP method was replaced with the successive multi- +threshold algorithm, as discussed above. +In the experimental SEMG data, a large number of MUs +decomposed by offline PFP can be correctly identified with +high precision in the online decomposition process, +demonstrating that the separation vectors used in the online +decomposition process were comprehensive and precise. In +addition, the MDR and CoV of online decomposition showed +no significant difference with the offline reference. These +findings indicate that the performance of the online +decomposition method is very close to that of the original +offline method, proving the feasibility and effectiveness of the +proposed online PFP method. In addition, it illustrates that the +advantages of the offline APFP method were still maintained in +the proposed online decomposition method. +There are still some limitations in this work. First, the online +decomposition process relied too much on the separation +vectors provided by the offline prework, proving the feasibility +that the separation vectors obtained from offline decomposition +can be used for online decomposition. However, the conditions +of muscle contraction change over time and the initialization +process needs to update the separation vectors, which has not +been validated in this work. In other words, the online process +was verifying whether the MUs corresponding to the separate +vectors were activated and the newly recruited MUs couldn’t be +captured. Moreover, the initial MU information and spike drift +needs to be corrected over time. Second, the experimental EMG +data were collected only from isometric contraction and most +muscle contractions in daily life are non-isometric and dynamic. +More contraction patterns will be added to the experimental +data for analysis. Third, the peel-off procedure needs to be +adopted in a real-time way to find more MUs and fully take +advantage of the offline PFP method. Further research will be +devoted to overcoming the limitations above. +VI. CONCLUSION +A new online SEMG decomposition method based on the +Progressive FastICA Peel-off procedure was proposed in this +paper, including offline prework and online decomposition +process. The proposed decomposition method took advantage +of offline PFP algorithms and demonstrated high precision with +the most identified MUs both on simulated and experimental +EMG signals. These results offer a new tool for precisely +identifying individual MU activities in a real-time way with the +potential applications of high-density EMG as a neural interface +in the fields of biomechanics, sports and rehabilitation. +REFERENCES +[1] +C. J. De Luca, “Electromyography,” Encyclopedia of Medical Devices +and Instrumentation, 2006. +[2] +T. Pistohl, C. Cipriani, A. Jackson, and K. Nazarpour, “Abstract and +proportional myoelectric control for multi-fingered hand prostheses,” +Ann. Biomed. Eng., vol. 41, no. 12, pp. 2687-2698, 2013. +[3] +K. Kiguchi and Y. Hayashi, “An EMG-based control for an upper-limb +power-assist exoskeleton robot,” IEEE Trans. Syst. Man, Cybern. Part B +Cybern., vol. 42, no. 4, pp. 1064-1071, 2012. +[4] +R. C. Dutton, W. D. Smith, H. L. Bennett, S. Archer, and N. T. Smith, +“Craniofacial electromyogram activation response: Another indicator of +anesthetic depth,” J. Clin. Monit. Comput., vol. 14, no. 1, pp. 5-17,1998. +[5] +D. Farina, et al., “Man/machine interface based on the discharge timings +of spinal motor neurons after targeted muscle reinnervation,” Nature +Biomedical Engineering, vol. 1, no. 2, pp. 1-12, 2017. +[6] +C. Castellini and P. Van Der Smagt, “Surface EMG in advanced hand +prosthetics,” Biol. Cybern., vol. 100, pp. 35-47, no. 1, 2009. +[7] +M. Mulas, M. Folgheraiter, and G. Gini, “An EMG-controlled +exoskeleton for hand rehabilitation,” in Proceedings of the 2005 IEEE +9th International Conference on Rehabilitation Robotics, 2005, pp. 371- +374. +[8] +D. A. Gabriel, G. Kamen, and G. Frost, “Neural Adaptations to Resistive +Exercise,” Sport. Med., vol. 36, no. 2, pp. 133-149, 2006. +[9] +P. A. Karthick, D. M. Ghosh, and S. Ramakrishnan, “Surface +electromyography based muscle fatigue detection using high-resolution +time-frequency methods and machine learning algorithms,” Comput. +Methods Programs Biomed, vol. 154, pp. 45-56, 2018. +[10] M. Atzori, M. Cognolato, and H. Müller, “Deep learning with +convolutional neural networks applied to electromyography data: A +resource for the classification of movements for prosthetic hands,” Front. +Neurorobot., vol. 10, SEP 7, 2016. +[11] A. D. Vigotsky, I. Halperin, G. J. Lehman, G. S. Trajano, and T. M. +Vieira, “Interpreting signal amplitudes in surface electromyography +studies in sport and rehabilitation sciences,” Frontiers in Physiology, vol. +8, JAN 4, 2018. +[12] F. Zhang, P. Li, ZG Hou, Z. Lu, Y. Chen, Q. Li, and M. Tan, “SEMG- +based continuous estimation of joint angles of human legs by using BP +neural network,” Neurocomputing, vol. 78, no. 1, pp. 139-148, 2012. +[13] D. Stashuk, “EMG signal decomposition: How can it be accomplished +and used?,” J. Electromyogr. Kinesiol., vol. 11, no. 3, pp. 151-173, 2001. +[14] P. Konrad, “The ABC of EMG A Practical Introduction to Kinesiological +Electromyography,” Noraxon INC. USA. A, no. April, 2005. +[15] M. Weinberger and J. O. Dostrovsky, “Motor Unit,” Oxford: Academic +Press, pp. 204–206, 2010. +[16] X. Zhang, G. Zhu, M. Chen, X. Chen, X. Chen, and P. Zhou, “Muscle + + +Force Estimation Based on Neural Drive Information from Individual +Motor Units,” IEEE Trans. Neural Syst. Rehabil. Eng., vol. 28, no. 12, +pp. 3148-3157, 2020. +[17] C. Dai, Y. Zheng, and X. Hu, “Estimation of muscle force based on neural +drive in a hemispheric stroke survivor,” Front. Neurol., vol. 9, MAR 23, +2018. +[18] D. Farina et al., “The extraction of neural information from the surface +EMG for the control of upper-limb prostheses: Emerging avenues and +challenges,” IEEE Trans. Neural Syst. Rehabil. Eng., vol. 22, no. 4, pp. +797-809, 2014. +[19] X. Li, X. Zhang, X. Tang, M. Chen, X. Chen, X. Chen, and A. Liu, +“Decoding muscle force from individual motor unit activities using a +twitch force model and hybrid neural networks,” Biomed. Signal Process. +Control, vol. 72, p. 103297, 2022. +[20] X. Tang, X. Zhang, M. Chen, X. Chen and X. Chen, “Decoding Muscle +Force From Motor Unit Firings,” IEEE Trans. Neural Syst. Rehabil. Eng., +vol. 29, pp. 2484-2495, 2021. +[21] F. Xu, Y. Zheng, and X. Hu, “Estimation of joint kinematics and fingertip +forces using motoneuron firing activities: A preliminary report,” in +International IEEE/EMBS Conference on Neural Engineering, NER, +2021, pp. 1035-1038. +[22] M. Chen, X. Zhang, and P. Zhou, “Automatic Multichannel +Intramuscular Electromyogram Decomposition: Progressive FastICA +Peel-Off and Performance Validation,” IEEE Trans. Neural Syst. Rehabil. +Eng., vol. 27, no. 1, pp. 76-84, 2019. +[23] P. Comon, C. Jutten, “Handbook of Blind Source Separation: +Independent component analysis and applications,” Academic press, +2010. +[24] B. G. Lapatki, J. P. Van Dijk, I. E. Jonas, M. J. Zwarts, and D. F. +Stegeman, “A thin, flexible multielectrode grid for high-density surface +EMG,” J. Appl. Physiol., vol. 96, no. 1, pp. 327-336, 2004. +[25] A. Holobar and D. Zazula, “Multichannel blind source separation using +convolution Kernel compensation,” IEEE Trans. Signal Process., vol. 55, +no. 9, pp. 4487-4496, 2007. +[26] M. Chen and P. Zhou, “A novel framework based on FastICA for high +density surface EMG decomposition,” IEEE Trans. Neural Syst. Rehabil. +Eng., vol. 24, no. 1, pp. 117-127, 2016. +[27] A. Hyvärinen and E. Oja, “Independent component analysis: Algorithms +and applications,” Neural Networks, vol. 13, no. 4–5, pp. 411-430, 2000. +[28] A. Holobar, M. A. Minetto, A. Botter, F. Negro, and D. Farina, +“Experimental analysis of accuracy in the identification of motor unit +spike trains from high-density surface EMG,” IEEE Trans. Neural Syst. +Rehabil. Eng., vol. 18, no. 3, pp. 221-229, 2010. +[29] H. R. Marateb, K. C. McGill, A. Holobar, Z. C. Lateva, M. Mansourian, +and R. Merletti, “Accuracy assessment of CKC high-density surface +EMG decomposition in biceps femoris muscle,” J. Neural Eng., vol. 8, +no. 6, p. 066002, 2011. +[30] M. Chen, X. Zhang, Z. Lu, X. Li, and P. Zhou, “Two-Source Validation +of Progressive FastICA Peel-Off for Automatic Surface EMG +Decomposition in Human First Dorsal Interosseous Muscle,” Int. J. +Neural Syst., vol. 28, no. 9, p. 1850019, 2018. +[31] M. Chen, X. Zhang, and P. Zhou, “A novel validation approach for high- +density surface EMG decomposition in motor neuron disease,” IEEE +Trans. Neural Syst. Rehabil. Eng., vol. 26, no. 6, pp. 1161-1168, 2018. +[32] M. Chen, A. Holobar, X. Zhang, and P. Zhou, “Progressive FastICA Peel- +Off and Convolution Kernel Compensation Demonstrate High +Agreement for High Density Surface EMG Decomposition,” Neural +Plast., vol. 2016, p. 3489540, 2016. +[33] M. Chen, X. Zhang, X. Chen, and P. Zhou, “Automatic Implementation +of Progressive FastICA Peel-Off for High Density Surface EMG +Decomposition,” IEEE Trans. Neural Syst. Rehabil. Eng., vol. 26, no. 1, +pp. 144-152, 2018. +[34] V. Glaser, A. Holobar, and D. Zazula, “Decomposition of Synthetic +Surface Electromyograms Using Sequential Convolution Kernel +Compensation,” WSEAS - Trans. Biol. Biomed., vol. 7, no. 7, pp. 143- +152, 2008. +[35] A. Holobar and D. Zazula, “Gradient convolution kernel compensation +applied to surface electromyograms,” in Lecture Notes in Computer +Science (including subseries Lecture Notes in Artificial Intelligence and +Lecture Notes in Bioinformatics), vol. 4666, pp. 617-624, 2007. +[36] F. Negro, S. Muceli, A. M. Castronovo, A. Holobar, and D. Farina, +“Multi-channel intramuscular and surface EMG decomposition by +convolutive blind source separation,” J. Neural Eng., vol. 13, no. 2, p. +026027, 2016. +[37] Y. Ning, X. Zhu, S. Zhu, and Y. Zhang, “Surface EMG decomposition +based on K-means clustering and convolution kernel compensation,” +IEEE J. Biomed. Heal. Informatics, vol. 19, no. 2, pp. 471-477, 2015. +[38] V. Glaser, A. Holobar, and D. Zazula, “Real-time motor unit +identification from high-density surface EMG,” IEEE Trans. Neural Syst. +Rehabil. Eng., vol. 21, no. 6, pp. 949-958, 2013. +[39] C. Chen, Y. Yu, X. Sheng, and X. Zhu, “Analysis of motor unit activities +during multiple motor tasks by real-time EMG decomposition: +Perspective for myoelectric control,” in Proceedings of the Annual +International Conference of the IEEE Engineering in Medicine and +Biology Society, EMBS, 2020, pp. 4791-4794. +[40] C. Chen, S. Ma, X. Sheng, D. Farina, and X. Zhu, “Adaptive Real-Time +Identification of Motor Unit Discharges from Non-Stationary High- +Density Surface Electromyographic Signals,” IEEE Trans. Biomed. Eng., +vol. 67, no. 12, pp. 3501-3509, 2020. +[41] Y. Zheng and X. Hu, “Real-time isometric finger extension force +estimation based on motor unit discharge information,” J. Neural Eng., +vol. 16, no. 6, p. 066006, 2019. +[42] D. Y. Barsakcioglu and D. Farina, “A real-time surface EMG +decomposition system for non-invasive human-machine interfaces,” in +IEEE Biomedical Circuits and Systems Conference (BioCAS), Cleveland, +OH, USA, 2018, pp. 459-462. +[43] Y. Zheng and X. Hu, “Adaptive Real-time Decomposition of +Electromyogram During Sustained Muscle Activation: A Simulation +Study,” IEEE Trans. Biomed. Eng., vol. 69, no. 2, pp. 645-653, 2022. +[44] C. Chen, Y. Yu, X. Sheng, D. Farina, and X. Zhu, “Simultaneous and +proportional control of wrist and hand movements by decoding motor +unit discharges in real time,” J. Neural Eng., vol. 18, no. 5, p. 056010, +2021. +[45] F. J. Theis and G. A. García, “On the use of sparse signal decomposition +in the analysis of multi-channel surface electromyograms,” Signal +Processing, vol. 86, no. 3, pp. 603-623, 2006. +[46] A. Hyvärinen, “Fast and robust fixed-point algorithms for independent +component analysis,” IEEE Trans. Neural Networks, vol. 10, no. 3, pp. +626-634, 1999. +[47] N. Otsu, “A threshold selection method from gray-level histograms,” +IEEE Trans Syst Man Cybern, vol. SMC-9, no. 1, pp. 62-66, 1979. +[48] Simao M, C. Zhang, X. Zhang, M. Q. Zhang, and Y. Li, “Neighbor +number, valley seeking and clustering,” Pattern Recognit. Lett., vol. 28, +no. 2, pp. 173–180, 2007. +[49] C. J. James and O. J. Gibson, “Temporally constrained ICA: An +application to artifact rejection in electromagnetic brain signal analysis,” +IEEE Trans. Biomed. Eng., vol. 50, no. 9, pp. 1108-1116, 2003. +[50] E. Martinez-Valdes, C. M. Laine, D. Falla, F. Mayer, and D. Farina, +“High-density surface electromyography provides reliable estimates of +motor unit behavior,” Clin. Neurophysiol., vol. 127, no. 6, pp. 2534-2541, +2016. +[51] A. Fuglsang-Frederiksen and J. Rønager, “The motor unit firing rate and +the power spectrum of EMG in humans,” Electroencephalogr. Clin. +Neurophysiol., vol. 70, no. 1, pp. 68-72, 1988. +[52] J. W. Pillow, J. Shlens, E. J. Chichilnisky, and E. P. Simoncelli, “A model- +based spike sorting algorithm for removing correlation artifacts in multi- +neuron recordings,” PLoS One, vol. 8, no. 5, p. e62123, 2013. +[53] K. C. McGill and L. J. Dorfman, “High-Resolution Alignment of +Sampled Waveforms,” IEEE Trans. Biomed. Eng., vol. BME-31, no.6, +pp. 462-468, 1984. +[54] A. J. Fuglevand, D. A. Winter, and A. E. Patla, “Models of recruitment +and rate coding organization in motor-unit pools,” J Neurophysiol, vol. +70, no. 6, pp. 2470-88, Dec, 1993. +[55] R. Merletti, L. Lo Conte, E. Avignone, and P. Guglielminotti, “Modeling +of surface myoelectric signals. I. model implementation”, IEEE +Transactions on Biomedical Engineering, vol. 46, no. 7, pp. 810–820, +1999. +[56] J. R. Florestal, P. A. Mathieu, and K. C. McGill, “Automatic +decomposition of multichannel intramuscular EMG signals,” J. +Electromyogr. Kinesiol., vol. 19, no. 1, pp. 1-9, 2009. + diff --git a/7NAzT4oBgHgl3EQf-f4D/content/tmp_files/load_file.txt b/7NAzT4oBgHgl3EQf-f4D/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c8351f24c7fd6556e9acb10d8936b8a4da13be5e --- /dev/null +++ b/7NAzT4oBgHgl3EQf-f4D/content/tmp_files/load_file.txt @@ -0,0 +1,1122 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf,len=1121 +page_content='\uf020 Abstract—Surface electromyogram (SEMG) decomposition provides a promising tool for decoding and understanding neural drive information non-invasively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' In contrast to previous SEMG decomposition methods mainly developed in offline conditions, there are few studies on online SEMG decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' A novel method for online decomposition of SEMG data is presented using the progressive FastICA peel-off (PFP) algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' The online method consists of an offline prework stage and an online decomposition stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' More specifically, a series of separation vectors are first initialized by the originally offline version of the PFP algorithm from SEMG data recorded in advance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Then they are applied to online SEMG data to extract motor unit spike trains precisely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' The performance of the proposed online SEMG decomposition method was evaluated by both simulation and experimental approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' It achieved an online decomposition accuracy of 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='53% when processing simulated SEMG data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' For decomposing experimental SEMG data, the proposed online method was able to extract an average of 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='00 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='46 MUs per trial, with a matching rate of 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='38% compared with results from the expert-guided offline decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Our study provides a valuable way of online decomposition of SEMG data with advanced applications in movement control and health.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Index Terms—Surface electromyography, motor unit, online decomposition, progressive FastICA peel-off I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' INTRODUCTION lectromyogram (EMG) is an electrophysiological signal generated by muscular activation, reflecting motor control commands of the neuromuscular system [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' It can be used to analyze movement behaviors, intentions and health [2]-[4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Surface EMG (SEMG) refers to the EMG signals recorded by electrodes placed on the skin surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Due to its noninvasive manner, SEMG has been widely applied in human-machine interfaces [5]-[7], sports medicine [8]-[9] and rehabilitation [10]-[12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Ideally, an EMG signal is composed of multiple action potentials generated by activated motor units (MUs), transmitted and superimposed temporally and spatially at a recording electrode [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Specifically, each MU consists of the cell body and dendrites of an alpha motor neuron, the multiple This work was supported by the National Natural Science Foundation of China under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 61771444.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Zhao and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Zhang are with the School of Information Science and Technology at University of Science and Technology of China, Hefei, Anhui, 230026, China (email: xuzhang90@ustc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' branches of its axon, and the muscle fibers that are innervated [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' The MU is regarded as the basic component of the peripheral neuromuscular system to describe the neural control of muscular contraction and movement formation [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Compared with the global features such as SEMG amplitude, the MU activities can reflect the information of neural drives to the muscle at a microscopic level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Therefore, it is valuable to examine the MU activities and properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' EMG decomposition enables resolving the composite EMG signal into its constituent MU spike trains (MUSTs) and MU action potential (MUAP) waveforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' The availability of these individual MU activities can provide a promising way of decoding motor neural commands of a neurophysiological nature [16]-[22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Many efforts have been made toward EMG decomposition, mainly relying on blind source separation (BSS) algorithms which are aimed to solve the difficult math problem of separating sources from observed signals without prior knowledge of the source signals [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Besides, it brings huge challenges to the SEMG decomposition due to its special characteristics such as low signal-to-noise ratio, high similarity and severe superposition of the MUAP waveforms, caused by the low-pass filtering effect of the subcutaneous skin and fat tissues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' With the recent development of electronic and sensing technologies, the use of high-density SEMG (HD-SEMG) by 2- dimensional flexible electrode arrays provides abundant spatial information simultaneously recorded from dozens or even hundreds of SEMG channels, facilitating implementing the BSS algorithms in general, and the SEMG decomposition in particular [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Convolution kernel compensation (CKC) [25] and progressive FastICA peel-off (PFP) [26] are both representative HD-SEMG decomposition methods, inspired by the advanced BSS techniques [23], [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' The CKC estimates and updates cross-correlation vectors between the observed SEMG signals and MUSTs in an iterative way [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' The PFP applies a classic FastICA algorithm [27] to the SEMG signals to calculate the separation vectors and introduces a “peel-off” procedure to progressively remove the separated MUAP waveforms from the original SEMG signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Such a procedure mitigates the effect of the already identified MUs on the M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Chen and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Zhou are with Faculty of Biomedical and Rehabilitation Engineering, University of Health and Rehabilitation Sciences, Qingdao, Shandong, 266024, China (email: dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='ping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='zhou@outlook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='com).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Online Decomposition of Surface Electromyogram into Individual Motor Unit Activities Using Progressive FastICA Peel-off Haowen Zhao, Xu Zhang, Maoqi Chen and Ping Zhou E FastICA convergence and effectively increase the number of obtained MUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' The performance of both CKC and PFP has been extensively validated [28]-[32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Variations of both methods have been developed to extract a relatively large number of MUs at high muscle contraction levels, with successful applications mainly in offline conditions [33]-[37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Considering the application prospects of SEMG in many fields, there are substantial demands for robust online SEMG decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Glaser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' [38] conducted a pilot study on the real-time SEMG decomposition based on the CKC algorithm and demonstrated its feasibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Afterwards, more relevant studies were reported [39]-[44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' The development of these online decomposition algorithms mainly relies on a basic assumption that SEMG signals are quasi-stationary, and the MU behaviors do not change in pattern over a short period of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' This assumption has served as a primary basis of conventional offline SEMG decomposition [25], [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' On this basis, these online decomposition algorithms were always designed to use results from an offline decomposition as prior knowledge, thus saving computational resources and allowing the feasibility of online signal processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Specifically, most previous studies conducted online SEMG decomposition using modified versions of the CKC method, whereas the online version of the PFP method has not been investigated yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Considering the advantages of the PFP method in extracting a great many MUs with high precision, it is necessary and promising to develop its online version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Accordingly, this paper presents an online SEMG decomposition method based on the PFP algorithm, evolving the key techniques of the PFP algorithm to meet the requirements for its real-time usability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' To avoid the time- consuming complexity from the offline decomposition methods, the proposed method utilized a two-stage approach consisting of an offline prework stage and an online decomposition stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Furthermore, an adaptive threshold selection algorithm was developed to make it more suitable for precisely determining each MUST while processing in real time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' The performance of the proposed online decomposition method was validated on both simulated and experimental SEMG datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' RELATED WORK A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' SEMG Observation Each MU has a unique and stable MUAP waveform distribution pattern in different channels of a 2-dimensional array, which can be used to distinguish and identify the MU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' The SEMG signal can be observed by a convolutional mixing model expressed as [45]: ��(�) = � � ���(�)��(� − �) + ��(�) ��� ��� � ��� (1) where � = 1,2,3 … … � and � = 1,2, … … � , ��(�) is the � th SEMG channel and ��(�) represents the additive noise in the �th channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' ���(�) denotes the waveform vector of length L, which represents the waveform of the �th MU in the �th channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' ��(�) = ∑ �(� − ��(�)) � is the MUST expressed as a 0-1 impulse sequence indicating every spike firing timing at ��(�) for the �th firing of the �th MU, whereas � is Dirac Delta function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' For each �, ��(� + 1) − ��(�) > � can be assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Define the expansion vector of EMG signals and MUSTs as ��(�) = [��(�), ��(� − 1), … , ��(�), … , ��(� − � + 1)] and ��(�) = [��(�), ��(� − 1), … , ��(�), … , ��(� − � + 1)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Thus, the equation can be rewritten in matrix form: ��(�) = ����(�) + ��(�) (2) where ��(�) represents noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' �� is a matrix containing all waveform vectors ���.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' For the mixing model analyzed above, the task of EMG decomposition is to find a suitable separation matrix � ��� that consists of many separation vectors to extract the MU firing events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' As a result, the source signals of all MUs can be estimated by ��(�) = � �����(�).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Automatic PFP (APFP) The PFP algorithm has been automated, but it is suitable just for offline data processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' More details of the algorithm and the corresponding parameters can be found in [33] and the APFP method was used in this study with the same settings as reported in [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Below is a brief introduction to the APFP method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' If a whitened observed signal � has been obtained and we need to find an independent component � = ��� from it using the ICA algorithm [23], [27], the following maximum negative entropy problem needs to be optimized: max ��(�) = [�{�(���)} − �{�(�)}]� �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' ℎ(�) = �{��} − 1 = �|�|� � − 1 = 0 (3) where � is a non-polynomial function, and � is a random variable with standard normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' The problem above can be solved using the procedure of the fix-point algorithm [46] to obtain a series of MU source signals and their corresponding separation vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' The spike trains can be precisely extracted from these source signals using the initial threshold determined by the Otsu algorithm [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' However, the spikes from one source signal often do not just belong to one MU due to heavy MUAP superimposition or high MU synchronization levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Thus, a valley-seeking clustering approach [48] is used to distinguish the spikes from the same source signal based on their morphological features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' On this basis, the spikes belonging to each cluster are most likely from the same MU [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' After the valley-seeking clustering approach, the constrained FastICA algorithm [49] is performed using the extracted and clustered spike trains as constraints to converge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Therefore, the MU source signals can be effectively updated and meanwhile the possible firing errors are corrected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' To assess the reliability of the constrained FastICA outputs and their corresponding MUSTs representing true MU activities, some metrics are employed from the perspective of the significance of correlation constrain [49], including the consistency of spike amplitudes and inter-spike intervals [50], and the physiologically reasonable firing rate [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' In the APFP method, the correlation coefficient between the output of constrained FastICA and the testing spike trains (denoted as ξ), the coefficient of variation of spike amplitudes and inter-spike intervals (denoted as ������ and ������ ), and the firing rate (denoted as FR) are employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Moreover, a two-step criterion describing a reasonable range of the above four metrics is employed to judge the MU reliability comprehensively [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' A “peel-off” procedure is performed later to subtract the obtained MUAP waveforms from the original signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' The MUAP waveforms of the identified MUs were estimated by a straightforward approach following a least squares problem [26], [52] instead of the conventional high-resolution alignment algorithm [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' More MUs can emerge when processing the residual signals again with the FastICA algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' The framework of the offline APFP method is summarized as follows: (1) Initialize the residual signal to the original EMG signal, and make the MUST set γ empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' (2) Apply the FastICA algorithm to the expanded residual signal and obtain a series of source signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' (3) Extract non-repetitive spike trains by Otsu algorithm and use valley-seeking clustering to distinguish these spikes to separate spike trains from different MUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' (4) Use MUSTs obtained in step (2) as a reference signal, and apply the constrained FastICA algorithm on the expanded original EMG signal to detect the reliability of the MUSTs and to correct possible erroneous or missing discharges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' (5) Judge whether the MUs obtained are reliable through metrics calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Put reliable results in set γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' (6) Estimate the waveforms of the reliable MUs, subtract the estimated MUAP waveforms from the original signal and update the residual signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' (7) If no new reliable MU is found in the above steps, or the APFP method reaches the preset termination condition, the algorithm ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Otherwise, go back to step (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' METHODOLOGY A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Experimental SEMG Data Collection and Preprocessing 1) Subjects and Experiments Eight subjects (26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='13±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='29 years) without any known history of muscular or neural disorder participated in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' The study was approved by the Ethics Review Board of the University of Science and Technology of China (Hefei, China).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' All subjects signed consent prior to any procedure of the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' In this work, the HD-SEMG data were recorded from abductor pollicis brevis (APB) muscle due to its wide explorations and applications in SEMG studies [19]-[21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Here, a home-made, multi-channel signal acquisition system with a force sensor and a set of 3D-printed apparatuses was used to collect data, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' The subject’s hand was placed on the fixed 3D-printed apparatus to prevent muscular movement interferences from the wrist and other fingers, and the muscle force was recorded by a load cell (LDST-V-HY, Luckly Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=', Beijing, China) connected to a ring around the thumb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Multiple electrodes were arranged in the form of 8 rows × 8 columns to form a 2-dimensional electrode array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Each electrode probe had a diameter of 2 mm, and the inter-electrode distance between consecutive electrodes was 4 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Each electrode was designed in a monopolar manner relative to a round common reference electrode placed on the back of the tested hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' During the experiments, subjects were asked to sit and place the tested hand in a relaxed and comfortable way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Before data collection, the maximum voluntary contraction (MVC) of the thumb abduction muscle was tested and recorded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Then, in each trial of the task performance, subjects were instructed to perform isometric muscle contractions with the muscle force gradually increasing from 0 to a targeted force level (quantified by MVC percentage) in 2s and then maintained at the targeted level for around 3s, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' According to this force generation pattern, the designed force curve was shown on the screen to facilitate the subject’s task performance in each trial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' The targeted force level in this experiment was set to 30% MVC and the trial was repeated at least nine times to acquire a sufficient amount of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' The force and SEMG data were digitized via a 16-bit A/D converter (ADS1198, Texas Instruments, TX) at a sample rate of 2 kHz, and the data were stored into the hard disk of a computer and imported into the MATLAB software (version R2020a, MathWorks, Natick, MA, USA) for further analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 2) Data Preprocessing All channels of the recorded HD-SEMG signals were inspected, and a few channels (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='75 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='28 channels across all subjects in this study) with low quality were discarded (due to their excessive noise contamination resulting from motion artifacts, occasional electrode drop, or environmental interferences from surrounding electronic devices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' The channel deletion remained consistent within the EMG signals of the same subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' The HD-SEMG signals within the remaining channels were filtered through a 10-order Butterworth band-pass filter to reduce possible low-frequency or high-frequency interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' The bandwidth of the filter was 20-500Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Finally, the power line interference was removed through a 50Hz second-order notch filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' The deleted channels were not considered in the subsequent process of SEMG Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' The experimental setup and protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' (a) Apparatuses for simultaneously recording thumb abduction force by a load cell and HD- SEMG data by a piece of 2-dimensional electrode array arranged in an 8×8 formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' (b) The illustration of the force generation pattern with both the designed force curve (blue line) and an actual recorded force curve (red line) in one trial of task performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Force:C8 30%MVC C1C57 C642 5Time(s)(b) (a) decomposition, but they were filled in by interpolation from neighboring channels and considered during the estimation of MUAP waveforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' In order to facilitate the data analysis, all of the SEMG data were divided into a series of non-overlapping data segments corresponding to the force generation task repetitions over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Therefore, the length of every SEMG data segment was around 5 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' SEMG Data Simulation A data simulation approach was conducted to generate HD- SEMG data with known MU activities, which were used as the ground-truth for validating the performance of the developed online SEMG decomposition method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' In the current study, this approach was based on simulation models well described by previous studies, including the motoneuron pool model [54], the model describing the MUAP waveforms of different MUs, and a tripole model [55] considering the generation and extinction of the action potentials at the fiber end-plate and tendon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Here a cylindrical muscle with a radius of 8 mm was simulated and the fat and skin layers of the muscle were set to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='5 mm thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 120 MUs were set and distributed in parallel in the muscle fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Most of the MUs had low recruitment thresholds and a few had high thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' When the excitation exceeded the threshold, every MU discharged at 8 Hz and its firing rate increased as the excitation increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' All the relevant parameters are listed in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' The simulated SEMG signals were also set to be recorded by a 64-channel surface electrode array arranged in an 8×8 grid form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' The inter-electrode distance was set at 4 mm for both horizontal and vertical directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' The electrode array was placed parallel to the muscle fiber direction and its center electrodes were set to approximately over the innervation zones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' To be consistent with the force generation pattern of the actual experiments, the excitation was set to increase from 0 to a specific excitation level in the first 2 seconds, and maintained for another 3 seconds with several repetitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' The maximum excitation level was set to be 3%, corresponding to 33 active MUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' In addition, zero-mean Gaussian noises were added to the simulated EMG signals, generating three levels of SNR (signal- to-noise ratio) at 10 dB, 20 dB and 30 dB, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Thus, we considered four noise levels, three SNR levels and the level without any additional noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' For each noise level, 21 repetitions were simulated to ensure data diversity, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Therefore, 84 data segments (4 noise levels × 21 repetitions) were simulated in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Online Decomposition The overall whole block diagram summarizing the proposed online decomposition method is described in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' TABLE I PARAMETERS FOR SEMG SIMULATION Distribution Mean SD Range Fiber number Uniform 70000 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='5 mean MU fiber endplate center position Uniform 0 ±8 mm Fiber endplate position variation Uniform 0 ±2 mm Half fiber length Gaussian 40mm 4mm ±2 SD Mean fiber diameter for a MU Gaussian 55μm 10μm ±2 SD Fiber diameter variation within a MU Gaussian 0 1μm ±2 SD ISI variation Gaussian 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='2*instant mean ISI ±2 SD Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' The contraction condition of simulated signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Multi- channel simulated SEMG signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Block diagram of the proposed method for online SEMG decomposition Online Divided extraction extraction data input Separation 1vectors 4 Whitening calculation Vector set 1 and Offlin MUST MUST(a) Maximum excitationExtending Offline PFP Φ= W1, W2 -.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='. Wn?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' data connection indno decomposition 0 2 5 5s 100s(b) ChannelMUST -- Preprocessing Offline prework Online Decomposition extraction#1#64 Data #1 #2 #3 #20 #21 SegmentTime window Peak With full consideration of the real-time usability of the proposed online method, a two-stage approach was designed to avoid considerable computational complexity caused by the repeated operation of the FastICA algorithm and the iterations of the constrained FastICA algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' More specifically, the reliable separation vectors were initialized in the offline prework stage and saved to accelerate the subsequent online data processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' In the online decomposition stage, the data stream of the input EMG signals was divided into a series of temporally overlapping windows with window length and increment set at 1 s and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='2 s, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Both settings helped to facilitate online processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' During the offline prework stage, several 5-s segments of EMG signals were separately decomposed offline using the APFP method and all of the resultant separation vectors were put into the set �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' The quality of these vectors was evaluated by both criteria employed in the offline APFP method [33]: if the coefficient of variation of spike amplitudes ������ was higher than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='3, and the coefficient of variation of inter-spike intervals ������ was higher than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='4, the corresponding separation vector was considered to be low-quality and it was removed from the set � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Furthermore, any duplicated separation vector corresponding to the same MU was removed as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' In the online decomposition stage, every 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='2 s of data input was combined with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='8 s of historical data to form a 1-s window for decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' The decomposed results from consecutive windows were connected, while their overlapping portion was used to align the obtained MUSTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' This ensured continuity of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Illustration of the online SEMG decomposition process using the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' #14#15 0 5 10 15 20Channel Channel #1 #1Time(s)#64 #64Spike extraction & ConnectionWindow sliding vectors X 山 = {W1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='W?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' WNMUST Experimental Muscle Force MU DAWDecompose sEMG signals window by window#1 30%Window Source signal of MU1 Source signal of MU2开2DecomposeOffline Prework Online Decomposition 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='2s the decomposition results along with the original SEMG data stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' The SEMG data in each window were first whitened and extended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Then, the multiplication procedure was directly applied to the extended EMG signals with separation vectors in set � to estimate different MU source signals, from which individual MUSTs were consequently identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' For extracting MUSTs from the MU source signals, the original offline APFP method employs repeated iterations of the constrained FastICA algorithm, involving complex computations as described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' This process was unsuitable for online processing and therefore it was removed to avoid heavy computational burden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' To maintain high-precision MUST extraction, the simple amplitude-thresholding process by the Otsu algorithm had to be updated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' A new algorithm was designed for our online PFP method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' First, this algorithm needs to determine an initial threshold that is applied to each source signal, using the Otsu algorithm in the same way as conducted in the offline APFP method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Then, a group of spikes beyond this threshold is detected and the corresponding amplitudes can be ranked from small to large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Next, a series of successively increasing thresholds that are a little higher than these amplitudes are adopted to estimate a series of different spike trains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Each resultant spike train can be further evaluated by both ������ and ������ metrics, and the spike train with the minimal summation of both metrics is finally considered the most appropriate MUST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' This algorithm for adaptive threshold selection was termed the successive multi-threshold Otsu algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' A k-means clustering algorithm was usually used in some offline decomposition methods [36]-[37] for extracting MUSTs from the source signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' It was also implemented in this study as an alternative threshold selection algorithm, in comparison to the successive multi-threshold Otsu algorithm used in our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' By applying the k-means clustering algorithm, all sample amplitudes of the source signal time series can be classified into 2-4 groups (2 in this work), so that the group with the largest amplitudes of samples is selected as the extracted MUST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' After the spike trains of all MUs were appropriately detected, they were connected over windows to form the resultant MUST for each MU, and its MUAP waveforms that spanned over all channels were correspondingly estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 4 illustrates an example of the online decomposition results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' The pseudocode of the proposed online decomposition method is shown in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Performance Evaluation For processing the experimental SEMG data, the proposed online decomposition method was conducted in a user-specific manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Four segments were used in the offline prework stage and the remaining 4 segments were processed in the online decomposition stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' For processing the simulated SEMG data, the first segment was used in the offline prework stage and the remaining 20 segments were processed in the online decomposition stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' All SEMG segments tested in the online decomposition stage were sequentially arranged in the form of a data stream to be processed continuously using our proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' For comparison purposes, all of the SEMG segments to be processed online was also decomposed by the offline APFP method as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' To evaluate the performance of online decomposition and assess the decomposition results more comprehensively, we Algorithm 1 The proposed online decomposition method 1: Decompose the SEMG signals offline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Extract the MUSTs and calculate the corresponding separation vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 2: Remove the duplicated separation vectors and vectors that are not well-decomposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 3: Save all the separation vectors ��, ��, ��…�� for the online decomposition stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 4: while Acquiring SEMG signals do 5: Load and extend the EMG signals (��).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 6: for j = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' j < N + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' j ++ do 7: Calculate the source signal, �� = �� ���.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 8: Estimate the initial threshold through the Otsu algorithm and extract the spike train.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 9: Successively increase the threshold and extract a series of spike trains ����, ����, ����… 10: Find the spike train with the lowest ������ and ������ as the �th MUST ���.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 11: end for 12: Connect the MUSTs over the sliding windows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 13: end while Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' The results for decomposing simulated SEMG data in terms of MR(a), FDR(b) and FNR(c) averaged over all data segments using the offline APFP method, the proposed online PFP method and the online PFP method with k-means clustering at four noise levels, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' The error bar represents standard deviations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' N in the horizontal axis denotes the condition without any additional noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='05 1N 30 20 10 N 30 20 10 N 30 20 10 SNR (dB) SNR (dB) SNR (dB) The offline APFP method The online PFP method with k-means clustering The proposed online PFP method(a) (0) () MR(%) FDR FNR 001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='3 80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='2 used a series of metrics: matching rate (MR) can be calculated as [33]: �� = 2 ∙ ������� ������� + ���������� (4) where ������� denotes the number of firing events of the online decomposition results, and ���������� denotes the number of the reference spike trains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' In the simulated data, the reference spike train indicates the ground-truth firing events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' However, the actual MUSTs are not known a priori in the experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Therefore, the decomposition results of the experimental data processed by the offline APFP method were used to define ���������� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' ������� indicates the number of common discharges appearing in both the online decomposition result and the reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' The MR measures the matching degree and it is able to quantify the precision of an online decomposition method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' A representative example of validating the decomposition results from the online PFP method in terms of all decomposed MUSTs (in blue) with respect to the reference (in red) derived from summarized offline decomposition results, using a data segment from one subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' The position of the black dot indicates the missing or fault discharges and MR values are computed and shown on the right side of these spike trains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Two MUAPs of matched MUs with time-varying waveform shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Here we illustrate 64 electrode channels arranged in an 8×8 grid form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Blue and red lines indicate the MUAP shapes from online PFP and the reference of offline decomposition, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' The relationship between the matching rate and the composite decomposability index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' T Online PFP 10 MR (%)111194.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='87111 Online PFP 600μv The reference (Offline decomposition 30msnumb 10098.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='95MU1 MU2 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 81002 23 3 4 45 5 67 8 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='95 Matching Rate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='8 0 10 20 30 40 50 Composite Decomposability Index0 1 2 4 5Time(s) Besides MR, both false negative rate (FNR) and false discovery rate (FDR) were used to reveal the cause of the error discharges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' They are defined as ��� = ���������� − ������� ���������� (5) ��� = ������� − ������� ������� They count the proportion of the number of unmatched discharges to the total number of their respective discharges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Specifically, the FNR measures the rate of “missing” discharges with respect to the reference, and the FDR quantifies the rate of “faulty” discharges appearing in the online decomposition results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Generally speaking, the MR of a reliable MUST is close to 1 but the FNR and FDR are close to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' For a more comprehensive view of the decomposition results, we also calculated the mean discharge rate (MDR) and the coefficient of variation (CoV) of the online identified MUSTs with respect to the reference spike trains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' It should be noted that the CoV refers to the coefficient of variation of the inter-spike intervals ������ to better understand the MU firing behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' In addition, we calculated the decomposability index (DI) for each common MU of experimental EMG data to precisely quantify the proposed method’s performance [56]: �� = min {‖���‖, ‖��� − ��∗�‖} �� ��� (6) where ��� is the MUAP of the �th MU in the �th channel and ��∗� is the MUAP most similar to ��� among the other MUAPs in the � th channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' �� ��� is the root mean square amplitude (RMS) of the � th channel and the operator ‖∙‖ denotes the Euclidean norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' The DI measures the separation between ��� and the template of MUAP nearest to it (or the baseline), normalized by the standard deviation of the noise component (interference plus baseline noise) projected along their vector difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' The overall decomposability of the �th MU was measured by the composite DI (CDI), defined as the norm of the individual DIs [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' For developing a real-time decomposition method, it is necessary to evaluate the processing time delay which is expected to be as short as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' The time delay for processing one single time window was recorded, and all these time delay values were averaged across all windows and all subjects to indicate the computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' All of the algorithms were implemented on a desktop computer with an Intel Core i5-10400 processor (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='90 GHz) and 16 GB of memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' RESULTS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Results of Simulated Data As an offline decomposition method for validation, 21 MUs were identified using offline APFP and the number was 22 using online PFP when no additional noise was added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Further, the number of MUs correctly decomposed using online PFP decreased to 11, 7, and 6 when noise was added at 30 dB, 20 dB and 10 dB SNR, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' The results for decomposing simulated SEMG data are reported in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' As compared with the offline APFP method, the proposed online PFP method achieved comparable performance in terms of a high MR over 90%, and a low FNR below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' The proposed online PFP method had a fluctuated and relatively higher FDR than the offline APFP method under three SNR levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Specifically, a decreasing trend of the MR was found from 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='29% to 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='13% for the offline APFP method and from 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='53% to 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='79% for the online PFP method, respectively, when the noise was successively added to generate four noise levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' The ANOVAs revealed no significant difference in either MR, FDR or FNR, between the offline APFP method and the proposed online PFP method (p > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='05).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' When both threshold selection algorithms were compared, it was evidently found that the successive multi-threshold Otsu algorithm in the proposed online PFP method significantly outperformed the K-means clustering algorithm in terms of higher MR (p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='025) and lower FNR (p =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Both algorithms did not exhibit a significant difference in the FDR metrics (p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='273).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Table II reports both MDR and CoV values calculated for all common MUs between the decomposition results achieved by the proposed online method and the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' The ANOVA revealed no difference in MDR (p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='217) or CoV (p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='105) at no presence of noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' However, the MDR and CoV of online decomposition results became significantly different from those of the ground-truth (p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='05) when the noises were added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Results of Experimental Data When implementing online decomposition of experimental data, the offline decomposition method was applied to establish the reference for validation, and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='31±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='79 MUs were obtained, averaged across all subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 6 is an example of an online decomposition result using the proposed method, showing the decomposed MUSTs with respect to the reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' It can be observed that almost all the MU discharges derived from the online PFP method are well matched with those in the reference, with sporadic missing or erroneous ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 7 illustrates the MUAP waveforms of two matched MUs derived from both the online PFP method and the reference, which demonstrate a very consistent waveform shape in each channel and almost the same distribution pattern across the electrode array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 8 plots the relationship between the matching rate and composite decomposability index (CDI), which displays the overall trend of the matching rates varying TABLE II COMPARISON OF MDR AND COV OF THE SIMULATED EMG SIGNALS SNR 10dB Online PFP/ Ground-truth SNR 10dB Online PFP/ Ground-truth SNR 30dB Online PFP/ Ground-truth No adding noise Online PFP/ Ground-truth MDR 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='86±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='99 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='77±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='18 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='55±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='54 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='75±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='23 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='47±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='81 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='74±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='22 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='77±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='51 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='70±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='18 CoV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='245±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='053 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='199±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='257±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='032 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='202±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='231±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='044 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='201±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='211±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='024 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='199±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='007 with the CDIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' It contains the common MUs of all of the collected SEMG segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Table III reports both the number of MUs decomposed by the online PFP method and the number of common MUs matched those in the reference (offline decomposition) for 8 subjects, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' An average of 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='00±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='46 MUs were successfully identified by the online PFP method, with an average of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='69±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='84 MUs correctly matched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Besides, three metrics are also computed from those common MUs and reported in Table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Averaged over all data segments to be decomposed and all subjects, the MR was (90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='38±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='80) %, the FDR was 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='091±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='022, and the FNR was 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='089±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='041.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' The estimated MDR (p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='872) and CoV (p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='503) of online decomposition results were not significantly different from the offline decomposition reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Time Delay The time delay for decomposing a 1-s window of SEMG data using the proposed method in the online decomposition stage was 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='084±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='028 s, averaged over all data segments and all subjects;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' it was less than a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='2-s time increment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' For comparison purposes, the offline APFP method costs 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='07 ± 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='82 s to decompose SEMG data in a single time window, much longer than that of the proposed online decomposition method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' DISCUSSION As a promising SEMG decomposition method, the PFP algorithm has been reported recently and, therefore, it is necessary and promising to develop an online version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' This study sought to propose an online SEMG decomposition method based on the PFP algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' The results of both simulated and experimental SEMG data analyses demonstrated the feasibility of the proposed online PFP method in decomposing a large number of MUs with high precision in the context of isometric muscle contractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Our study offers a valuable tool for online SEMG decomposition with great applications in biomechanics and rehabilitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' In the results of processing simulated data, the proposed online PFP method decomposed a similar number of MUs as the offline APFP method, illustrating comparable performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Due to the use of initial separation vectors provided by the APFP method in the offline prework stage, the proposed online PFP method is expected to inherit a good capability of decomposing a great number of MUs from its original offline version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' In terms of MR, the proposed online PFP method got a slightly lower value compared with the offline APFP method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' This can be explained by the fact that the source signals were calculated by directly multiplying previously initialized separation vectors with the SEMG signals for the purpose of real-time processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' In addition, the MUSTs were estimated without the examination of iterative constrained FastICA, thus increasing the negative influence of noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' The result demonstrates that online decomposition was speeded up at the cost of a little bit of decrease in precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' This is the main and common difficulty in generalizing an offline decomposition method to its online version [38]-[43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' However, it has been found that the MDR and CoV of online decomposition were significantly different from those of the ground-truth when the noise was added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' This can be partly explained by the limitations of the online decomposition method such as MU synchronization [26] and firing events drift [33] that previous studies have faced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' When some noises were successively added to EMG signals to be decomposed, both the number of correctly identified MUs and the precision of determining their firing timings were reported to decrease substantially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' This could partly explain that the decrease of SNR resulted in more serious noise interference to some small MUs and thus caused a negative influence on the calculation of separation vectors as well as the performance of the online decomposition method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' On the other hand, it became much harder to precisely extract MUSTs from source signals in the online decomposition stage at a low SNR level, reflecting the decline of the MR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' As a consequence, it can be inferred that the quality of SEMG signals significantly influenced the performance of the decomposition method, as reported in [33], [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' It is worth mentioning that the proposed online PFP method introduced a progressive multi-thresholding process for extracting MUSTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' The successive multi-threshold Otsu TABLE III SUMMARY OF DECOMPOSITION RESULTS FOR EXPERIMENTAL EMG SIGNALS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Subject Number of motor units MDR (Hz) CoV (%) MR (%) FDR FNR The reference Online PFP The reference Online PFP The reference Online PFP All Matched 1 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='75±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='50 19 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='00±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='41 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='76±5.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='091±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='022 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='089±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='041 algorithm outperformed the conventional k-means clustering algorithm especially in the condition of noise interference, proving the potential to extract more precise discharges at low SNR levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' The successive multi-threshold algorithm based on the Otsu algorithm was inspired from the common Otsu algorithm [48] used in the offline APFP method [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' It was able to successively increase multiple thresholds to overcome the effect of noise interferences and find the most appropriate one to extract MUSTs that followed the physiological properties of MUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' The successive multi-threshold Otsu algorithm takes consideration into the interval and waveform information to ensure the result to be much more reliable, depending on ������ and ������ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' By contrast, the k-means clustering algorithm only focuses on the amplitude information of EMG source signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' As a result, it makes it much more difficult to remove the noise interferences and leads to decomposition performance degradation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' The proposed online PFP method replaced the complex iterative calculation of constrained FastICA with the successive multi-threshold Otsu algorithm to extract MUSTs, showing a significant improvement in reducing the calculation complexity while maintaining its high precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' To evaluate the real-time performance, this study recorded the processing time of online decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' The time delay was effectively reduced from 60 seconds for the offline APFP method to less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='08 seconds for the online decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' The acceleration of data processing is attributed to reasons in two respects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' The first is that the repeated iteration of FastICA was put in the offline prework stage, which initialized the separation vectors for online decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' On the other hand, some complex calculation procedures were adaptively simplified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' For example, the constrained FastICA algorithm in the APFP method was replaced with the successive multi- threshold algorithm, as discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' In the experimental SEMG data, a large number of MUs decomposed by offline PFP can be correctly identified with high precision in the online decomposition process, demonstrating that the separation vectors used in the online decomposition process were comprehensive and precise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' In addition, the MDR and CoV of online decomposition showed no significant difference with the offline reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' These findings indicate that the performance of the online decomposition method is very close to that of the original offline method, proving the feasibility and effectiveness of the proposed online PFP method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' In addition, it illustrates that the advantages of the offline APFP method were still maintained in the proposed online decomposition method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' There are still some limitations in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' First, the online decomposition process relied too much on the separation vectors provided by the offline prework, proving the feasibility that the separation vectors obtained from offline decomposition can be used for online decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' However, the conditions of muscle contraction change over time and the initialization process needs to update the separation vectors, which has not been validated in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' In other words, the online process was verifying whether the MUs corresponding to the separate vectors were activated and the newly recruited MUs couldn’t be captured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Moreover, the initial MU information and spike drift needs to be corrected over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Second, the experimental EMG data were collected only from isometric contraction and most muscle contractions in daily life are non-isometric and dynamic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' More contraction patterns will be added to the experimental data for analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Third, the peel-off procedure needs to be adopted in a real-time way to find more MUs and fully take advantage of the offline PFP method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Further research will be devoted to overcoming the limitations above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' CONCLUSION A new online SEMG decomposition method based on the Progressive FastICA Peel-off procedure was proposed in this paper, including offline prework and online decomposition process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' The proposed decomposition method took advantage of offline PFP algorithms and demonstrated high precision with the most identified MUs both on simulated and experimental EMG signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' These results offer a new tool for precisely identifying individual MU activities in a real-time way with the potential applications of high-density EMG as a neural interface in the fields of biomechanics, sports and rehabilitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' REFERENCES [1] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' De Luca, “Electromyography,” Encyclopedia of Medical Devices and Instrumentation, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' [2] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Pistohl, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Cipriani, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Jackson, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Nazarpour, “Abstract and proportional myoelectric control for multi-fingered hand prostheses,” Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Biomed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 41, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 12, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 2687-2698, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' [3] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Kiguchi and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Hayashi, “An EMG-based control for an upper-limb power-assist exoskeleton robot,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Man, Cybern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Part B Cybern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 42, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 1064-1071, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' [4] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Dutton, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Smith, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Bennett, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Archer, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Smith, “Craniofacial electromyogram activation response: Another indicator of anesthetic depth,” J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Clin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Monit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 14, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 5-17,1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' [5] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Farina, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=', “Man/machine interface based on the discharge timings of spinal motor neurons after targeted muscle reinnervation,” Nature Biomedical Engineering, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 1, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 1-12, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' [6] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Castellini and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Van Der Smagt, “Surface EMG in advanced hand prosthetics,” Biol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Cybern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 100, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 35-47, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 1, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' [7] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Mulas, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Folgheraiter, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Gini, “An EMG-controlled exoskeleton for hand rehabilitation,” in Proceedings of the 2005 IEEE 9th International Conference on Rehabilitation Robotics, 2005, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 371- 374.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' [8] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Gabriel, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Kamen, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Frost, “Neural Adaptations to Resistive Exercise,” Sport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Med.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 36, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 133-149, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' [9] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Karthick, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Ghosh, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Ramakrishnan, “Surface electromyography based muscle fatigue detection using high-resolution time-frequency methods and machine learning algorithms,” Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Methods Programs Biomed, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 154, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 45-56, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' [10] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Atzori, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Cognolato, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Müller, “Deep learning with convolutional neural networks applied to electromyography data: A resource for the classification of movements for prosthetic hands,” Front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Neurorobot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 10, SEP 7, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' [11] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Vigotsky, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Halperin, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Lehman, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Trajano, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Vieira, “Interpreting signal amplitudes in surface electromyography studies in sport and rehabilitation sciences,” Frontiers in Physiology, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 8, JAN 4, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' [12] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Zhang, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Li, ZG Hou, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Lu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Chen, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Li, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Tan, “SEMG- based continuous estimation of joint angles of human legs by using BP neural network,” Neurocomputing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 78, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 139-148, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' [13] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Stashuk, “EMG signal decomposition: How can it be accomplished and used?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=',” J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Electromyogr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Kinesiol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 11, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 151-173, 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' [14] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Konrad, “The ABC of EMG A Practical Introduction to Kinesiological Electromyography,” Noraxon INC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' A, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' April, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' [15] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Weinberger and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Dostrovsky, “Motor Unit,” Oxford: Academic Press, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 204–206, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' [16] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Zhang, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Zhu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Chen, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Chen, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Chen, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Zhou, “Muscle Force Estimation Based on Neural Drive Information from Individual Motor Units,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Neural Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Rehabil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 28, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 12, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 3148-3157, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' [17] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Dai, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Zheng, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Hu, “Estimation of muscle force based on neural drive in a hemispheric stroke survivor,” Front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Neurol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 9, MAR 23, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' [18] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Farina et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=', “The extraction of neural information from the surface EMG for the control of upper-limb prostheses: Emerging avenues and challenges,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Neural Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Rehabil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 22, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 797-809, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' [19] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Li, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Zhang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Tang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Chen, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Chen, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Chen, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Liu, “Decoding muscle force from individual motor unit activities using a twitch force model and hybrid neural networks,” Biomed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Signal Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Control, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 72, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 103297, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' [20] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Tang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Zhang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Chen, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Chen and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Chen, “Decoding Muscle Force From Motor Unit Firings,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Neural Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Rehabil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 29, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 2484-2495, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' [21] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Xu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Zheng, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Hu, “Estimation of joint kinematics and fingertip forces using motoneuron firing activities: A preliminary report,” in International IEEE/EMBS Conference on Neural Engineering, NER, 2021, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 1035-1038.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' [22] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Chen, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Zhang, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Zhou, “Automatic Multichannel Intramuscular Electromyogram Decomposition: Progressive FastICA Peel-Off and Performance Validation,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Neural Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Rehabil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 27, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 76-84, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' [23] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Comon, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Jutten, “Handbook of Blind Source Separation: Independent component analysis and applications,” Academic press, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' [24] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Lapatki, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Van Dijk, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Jonas, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Zwarts, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Stegeman, “A thin, flexible multielectrode grid for high-density surface EMG,” J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Physiol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 96, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 327-336, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' [25] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Holobar and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Zazula, “Multichannel blind source separation using convolution Kernel compensation,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Signal Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 55, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 9, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 4487-4496, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' [26] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Chen and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Zhou, “A novel framework based on FastICA for high density surface EMG decomposition,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Neural Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Rehabil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 24, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 117-127, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' [27] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Hyvärinen and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Oja, “Independent component analysis: Algorithms and applications,” Neural Networks, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 13, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 4–5, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 411-430, 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' [28] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Holobar, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Minetto, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Botter, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Negro, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Farina, “Experimental analysis of accuracy in the identification of motor unit spike trains from high-density surface EMG,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Neural Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Rehabil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 18, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 221-229, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' [29] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Marateb, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' McGill, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Holobar, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Lateva, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Mansourian, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Merletti, “Accuracy assessment of CKC high-density surface EMG decomposition in biceps femoris muscle,” J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Neural Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 8, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 6, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 066002, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' [30] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Chen, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Zhang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Lu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Li, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Zhou, “Two-Source Validation of Progressive FastICA Peel-Off for Automatic Surface EMG Decomposition in Human First Dorsal Interosseous Muscle,” Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Neural Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 28, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 9, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 1850019, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' [31] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Chen, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Zhang, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Zhou, “A novel validation approach for high- density surface EMG decomposition in motor neuron disease,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Neural Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Rehabil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 26, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 6, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 1161-1168, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' [32] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Chen, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Holobar, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Zhang, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Zhou, “Progressive FastICA Peel- Off and Convolution Kernel Compensation Demonstrate High Agreement for High Density Surface EMG Decomposition,” Neural Plast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 2016, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 3489540, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' [33] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Chen, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Zhang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Chen, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Zhou, “Automatic Implementation of Progressive FastICA Peel-Off for High Density Surface EMG Decomposition,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Neural Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Rehabil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 26, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 144-152, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' [34] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Glaser, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Holobar, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Zazula, “Decomposition of Synthetic Surface Electromyograms Using Sequential Convolution Kernel Compensation,” WSEAS - Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Biol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Biomed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 7, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 7, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 143- 152, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' [35] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Holobar and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Zazula, “Gradient convolution kernel compensation applied to surface electromyograms,” in Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 4666, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 617-624, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' [36] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Negro, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Muceli, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Castronovo, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Holobar, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Farina, “Multi-channel intramuscular and surface EMG decomposition by convolutive blind source separation,” J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Neural Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 13, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 2, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 026027, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' [37] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Ning, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Zhu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Zhu, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Zhang, “Surface EMG decomposition based on K-means clustering and convolution kernel compensation,” IEEE J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Biomed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Heal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Informatics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 19, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 471-477, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' [38] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Glaser, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Holobar, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Zazula, “Real-time motor unit identification from high-density surface EMG,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Neural Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Rehabil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 21, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 6, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 949-958, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' [39] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Chen, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Yu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Sheng, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Zhu, “Analysis of motor unit activities during multiple motor tasks by real-time EMG decomposition: Perspective for myoelectric control,” in Proceedings of the Annual International Conference of the IEEE Engineering in Medicine and Biology Society, EMBS, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 4791-4794.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' [40] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Chen, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Ma, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Sheng, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Farina, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Zhu, “Adaptive Real-Time Identification of Motor Unit Discharges from Non-Stationary High- Density Surface Electromyographic Signals,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Biomed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 67, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 12, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 3501-3509, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' [41] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Zheng and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Hu, “Real-time isometric finger extension force estimation based on motor unit discharge information,” J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Neural Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 16, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 6, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 066006, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' [42] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Barsakcioglu and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Farina, “A real-time surface EMG decomposition system for non-invasive human-machine interfaces,” in IEEE Biomedical Circuits and Systems Conference (BioCAS), Cleveland, OH, USA, 2018, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 459-462.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' [43] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Zheng and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Hu, “Adaptive Real-time Decomposition of Electromyogram During Sustained Muscle Activation: A Simulation Study,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Biomed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 69, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 645-653, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' [44] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Chen, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Yu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Sheng, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Farina, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Zhu, “Simultaneous and proportional control of wrist and hand movements by decoding motor unit discharges in real time,” J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Neural Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 18, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 5, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 056010, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' [45] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Theis and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' García, “On the use of sparse signal decomposition in the analysis of multi-channel surface electromyograms,” Signal Processing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 86, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 603-623, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' [46] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Hyvärinen, “Fast and robust fixed-point algorithms for independent component analysis,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Neural Networks, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 10, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 626-634, 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' [47] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Otsu, “A threshold selection method from gray-level histograms,” IEEE Trans Syst Man Cybern, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' SMC-9, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 62-66, 1979.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' [48] Simao M, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Zhang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Zhang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Zhang, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Li, “Neighbor number, valley seeking and clustering,” Pattern Recognit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 28, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 173–180, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' [49] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' James and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Gibson, “Temporally constrained ICA: An application to artifact rejection in electromagnetic brain signal analysis,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Biomed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 50, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 9, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 1108-1116, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' [50] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Martinez-Valdes, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Laine, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Falla, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Mayer, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Farina, “High-density surface electromyography provides reliable estimates of motor unit behavior,” Clin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Neurophysiol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 127, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 6, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 2534-2541, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' [51] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Fuglsang-Frederiksen and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Rønager, “The motor unit firing rate and the power spectrum of EMG in humans,” Electroencephalogr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Clin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Neurophysiol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 70, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 68-72, 1988.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' [52] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Pillow, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Shlens, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Chichilnisky, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Simoncelli, “A model- based spike sorting algorithm for removing correlation artifacts in multi- neuron recordings,” PLoS One, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 8, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 5, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' e62123, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' [53] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' McGill and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Dorfman, “High-Resolution Alignment of Sampled Waveforms,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Biomed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' BME-31, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content='6, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 462-468, 1984.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' [54] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Fuglevand, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Winter, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Patla, “Models of recruitment and rate coding organization in motor-unit pools,” J Neurophysiol, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 70, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 6, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 2470-88, Dec, 1993.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' [55] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Merletti, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Lo Conte, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Avignone, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Guglielminotti, “Modeling of surface myoelectric signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' model implementation”, IEEE Transactions on Biomedical Engineering, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 46, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 7, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 810–820, 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' [56] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Florestal, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Mathieu, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' McGill, “Automatic decomposition of multichannel intramuscular EMG signals,” J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Electromyogr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' Kinesiol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 19, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} +page_content=' 1-9, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAzT4oBgHgl3EQf-f4D/content/2301.01933v1.pdf'} diff --git 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b/B9E1T4oBgHgl3EQfVwR_/content/tmp_files/2301.03106v1.pdf.txt @@ -0,0 +1,461 @@ +(1+1) dimensional scalar field theory on q-deformed +space +Poula Tadros +Department of Applied Physics, Aalto University School of Science, FI-00076 +Aalto, Finland. +email:poulatadros9@gmail.com +Abstract +We study scalar field theory in one space and one time dimensions on +a q-deformed space with static background. We write the Lagrangian and +the equation of motion and solve it to the first order in q − 1 where q is +the deformation parameter of the space. +1 +Introduction +Non-commutative geometry was first introduced in string theory in ref- +erence [1], where it was shown that the coordinates of the endpoints of +strings on D-branes in the presence of a Neveu-Schwartz field are non- +commutative. Non-commutative field theories have also been defined, as +they can be derived from string theories and have interesting features, as +described in references [2] and [3]. +The introduction of non-commutative spacetime in field theory is mo- +tivated by the Heisenberg uncertainty principle in quantum mechanics, +which states that at small distance scales, there is a large uncertainty in +momentum measurement. This means that energy can reach very high +values in a small spatial distance, approaching the Planck scale. However, +according to the general theory of relativity, high energy in a small spatial +distance creates a black hole, which prevents the position from being fully +certain. To reconcile these two phenomena, it is necessary to introduce +non-commutativity in spacetime, which implies non locality in the theory. +This is explained in references [4] and [5]. +In this paper we study (1+1) dimensional classical scalar field theory +with static spacetime on a q-deformed space, we present both analytical +and numerical analysis of the resulting theory. In section 2, we review +the some types of non-commutativity on space times and motivate the +choice of q-deformation non-commutativity as the subject of the study . +In section 3. we study the scalar field theory on q-deformed space time, we +write the Lagrangian and deduce the equation of motion, we also truncate +the equation of motion to the linear order in q −1 and solved the resulting +equation. In section 4, we study the numerical solutions of the truncated +equation of motion showing that the solutions grow exponentially with x +and t meaning that the equation is stiff and there are instabilities in the +theory. In section 5, we conclude the study and suggest topics for further +research. +1 +arXiv:2301.03106v1 [hep-th] 8 Jan 2023 + +2 +Types of non-commutativity +Here, we briefly review three of the most popular types of non-commutativity +relations and justify our motivation to use the q-deformation type +1. Canonical non-commutativity It is the simplest type which used in +physics literature, it was introduced in [6], it is defined by imposing +the following commutation relations +[xµ, xν] = iθµν, +where xµ are the spacetime coordinates and θµν is a constant, anti- +symmetric matrix. +The idea of canonical non-commutativity involves smearing the struc- +ture of space-time in a particular way, regardless of the specific +mathematical details of the space. +In order to incorporate non- +commutative geometry capturing the mathematical structures on +the manifold, it is necessary to consider more complex forms of non- +commutativity beyond just this basic version. +2. Lie-type non commutativity +In this case the coordinates has a Lie algebra structure i.e. +the +commutation relations can capture a Lie algebra structures [7]. The +commutation relations are given by +[xµ, xν] = if µν +ρ xρ, +where f µν +ρ +are the structure constants of the defined Lie algebra. +However, this type is not useful because Lie structures are rigid i.e. +any small deformation of a Lie algebra is isomorphic to the original +Lie algebra. +3. q-deformations +A solution to the rigidity problem for Lie algebras is to replace Lie +group with a flexible structure called quantum groups [8-10]. The +term quantum group used in this context refers to the deformations +of the universal enveloping algebra of a given group, these objects +have Hopf algebra structures which are flexible structures unlike Lie +groups and algebras. +The commutation relations are given by +xµxν = 1 +q Rµν +στxσxτ, +where q is a parameter and Rµν +στ is the R-matrix of the quantum +group defined on the space. +In this space a Lie algebra is replaced by a non-commutative Hopf +algebra with deformation parameter q. The resulting space is de- +formed according to the Lie group on the space and on the parame- +ter q, this is the simplest way to deform a space time while capturing +the full algebraic structure of the space. +2 + +3 +Lagrangian and the equation of motion +We begin with the Lagrangian of the scalar field on the commutative man- +ifold then introduce non-commutativity by replacing the derivatives by +Jackson derivatives, since the symmetry group is U(1), the deformations +of its universal enveloping algebra gives a commutative algebra. Thus, we +do not have to worry about defining a product of functions on the new +space. The Lagrangian is then +L = 1 +2∂µφ∂µφ − 1 +2m2φ2 → Lq = DµqφDµ +q φ − m2φ2, +where µ = 0, 1 with x0 = t and x1 = x. +Assuming the field is defined everywhere and is infinitely differentiable +and the deformations are small i.e. q ≈ 1, we can relate the theory on +the non commutative topological space to the theory on the commuta- +tive manifold (i.e. transforming the non-commutative theory back to the +commutative manifold) using the formulae +Dxq(f(x)) = ∂xf + +∞ +� +k=1 +(q − 1)k +(k + 1)! xkf (k+1)(x), +where f (k) is the k-th ordinary derivative of f with respect to x. +Dtq(f(x)) = ∂tf + +∞ +� +k=1 +(q − 1)k +(k + 1)! xkf (k+1)(x), +where f [k] is the k-th ordinary derivative of f with respect to t. +The resulting Lagrangian on the commutative manifold is +Lq = 1 +2∂φ∂φ − 1 +2m2φ2 + 2∂φ +∞ +� +k=1 +(q − 1)k +(k + 1)! xkφ(k+1) ++ +∞ +� +l,m=1 +(q − 1)(l+m) +(m + 1)!(l + 1)!xk+lφ(l+1)φ(m+1) + (x → t). +where (x → t) means the same terms but with x replaced by t includ- +ing in the derivatives. +The Lagrangian has an infinite series of derivatives, in this case the +Euler-Lagrange equation will be +∂Lq +∂φ + +∞ +� +k=1 +(−1)k dk +dxk ( ∂Lq +∂φ(k) ) + +∞ +� +k=1 +(−1)k dk +dtk ( ∂Lq +∂φ[k] ) = 0, +(1) +where k = 2, 3, .... +The Lagrangian is clearly non local as expected from a non-commutative +theory. +The derivatives of the Lagrangian are given by +∂Lq +∂φ = −mφ, +3 + +∂Lq +∂(∂φ) = ∂xφ + 2 +∞ +� +n=1 +(q − 1)n +(n + 1)! xnφ(n+1) +(2) +→ d +dx( ∂Lq +∂(∂xφ)) += ∂x∂xφ + 2 +∞ +� +n=1 +(n(q − 1)n +(n + 1)! xn−1φ(n+1)) + 2 +∞ +� +n=1 +((q − 1)n +(n + 1)! xnφ(n+2)), (3) +∂Lq +∂(φ(k)) = 2(q − 1)k−1xk−1 +k! +∞ +� +n=0 +(q − 1)n +(n + 1)! xnφ(n+1) +→ d +dx( +∂Lq +∂(φ(k))) += 2(q − 1)k−1x2k−1 +k! +∞ +� +m,n=0 +� +k +m +� +(q − 1)n +(n + 1)! +(n + k + 1)! +(n + 2k − m − 1)!xn−mφ(n+k+1), +(4) +with similar formulae for derivatives with respect to t. +Putting all together from (2), (3), (4) in (1) we get +−∂µ∂µφ − m2φ − 2 +∞ +� +n=1 +n(q − 1)n +(n + 1)! xn−1φ(n+1) − 2 +∞ +� +n=1 +(q − 1)n +(n + 1)! xnφ(n+2) ++ +∞ +� +k=2 +(−1)k 2(q − 1)k−1x2k−1 +k! +∞ +� +n=0 +k +� +m=0 +� +k +m +� +(q − 1)n(n + k − 1)! +(n + 1)!(n + 2k − m − 1)!xn−mφ(n+k+1) ++ (x → t) = 0. +(5) +This is a partial differential equation of infinite order with variable +coefficients. +If we consider only small deformations i.e. q ≈ 1, then we can only +keep terms up to the linear order in q − 1, the first order equation will be +−∂µ∂µφ−m2φ−(q−1)[φ(2)+xφ(3)− x3 +6 φ(3)−x2φ(3)−xφ3+(x → t)] = 0. +This equation is a stiff equation i.e. it is numerically unstable, this +may indicate an instability in the theory due to the linear approximation +used, but as seen from the full equation of motion the full theory is stable. +The solution is φ = F(t)G(x) where +F(t) = c1eiAt/√q+c2e−iAt/√q+(q−1)eiAt/√q +2iA√q [ iA +24q t4+(iA3 +3 − +1 +12√q − i +8A)t3 ++(A + q +2q +− A√q +2 +− +i +8A)t2 + (i(A + q) +2A√q +− iqA +4 ++ +√q +8A2 )t +4 + ++(A + q +4A2 ++ q +√ +A +4 ++ +iq +16A3 )] + O((q − 1)2), +G(x) = c3eikx/√q+c4e−ikx/√q+(q−1)eikx/√q +2ik√q [ ik +24q x4+(ik3 +3 − +1 +12√q − i +8A)x3 ++(k + q +2q +− k√q +2 +− i +8k )x2 + (i(k + q) +2k√q +− iqk +4 ++ +√q +8k2 )x ++(k + q +4k2 ++ q +√ +k +4 ++ +iq +16k3 )] + O((q − 1)2), +where c1, c2, c3, c4, A are normalisation constants and k = ± +√ +A2 + m2. +When q = 1, it reduces to the solution to the Klein Gordon equation as +expected. +4 +Numerical results +Here, we present numerical solutions to the equation of motion to the +first order in q − 1, we focus on G(x) only since the remaining part is +similar. The solutions are exponentially growing in time establishing that +the equation of motion was stiff. +We set c3 = c4 = k = 1, A2 = 1 +2 and we plot the solution for different +values of the parameter q. +Figure 1: At q − 1 = 0.1 the solution grows exponentially with |x|. This is a +feature of a stiff equation with unstable numerical solution. In the vicinity of +x = 0 it is close to the usual Klein-Gordon solution but as we go further it +becomes more and more distant +5 + +200000 +100000 +0 +100000 +-200000 +100 +75 +50 +25 +25 +50 +75 +100Figure 2: At q − 1 = 0.001 the solution still grows exponentially but slower. +Figure 3: At q − 1 = 10−6 the solution resembles the Klein-Gordon solution up +to |x| = 50 then decays for a bit but eventually blows up. +Figure 4: At q − 1 = 10−9 the solution has the same behaviour as the previous +graph but the decay happens at larger |x|, all smaller q − 1 values follow this +pattern. +6 + +2000 +1500 +1000 - +500 +0 +500 +1000 +1500 +2000 +100 +7550 +-25 +0 +25 +50 +75 +10030 +20 +10 +0 +10 +20 +30 +-200 +150 +100 +50 +0 +50 +100 +150 +200E +2 +1 +0 +-1 +-2 +-3 +600 +400 +200 +0 +200 +400 +600The above results shows an instability in the theory leading to di- +vergent solutions to the equations of motion as x → ∞. To remove the +instability we must add infinite terms corresponding to an infinite series +of higher derivatives i.e. we have to consider the full theory. However, +this approximation gives us an intuition on how the q-deformation affects +the space, small q-deformations beside leading to non local effects appear +to affect the space irregularly with only small effects locally. +5 +Conclusion and outlook +In conclusion, we showed that defining a field theory on a q-deformed +space leads to an infinite series of higher derivatives in the Lagrangian +even with static background. In the case presented the algebra was com- +mutative so no new product of functions is needed. We also demonstrated +that any approximation or truncation to the theory will lead to stiff equa- +tions of motion resulting from instabilities in the theory. +While we made a progress in the field, much more is to be studied, fu- +ture research in this direction should focus on defining more complicated +theories on q-deformed spaces with non-commutative function algebras +and with dynamical spacetimes, also to define higher spin fields on such +space and study the new symmetries of the theories as well as the types +of instabilities arise if the Lagrangian is truncated. +Acknowledgments +We would like to thank Dr.Ivan Kolar for the useful discussions on the +topic +References +[1] Seiberg, N. and Witten, E. (1999) “String theory and noncommu- +tative geometry,” Journal of High Energy Physics, 1999(09), pp. +032–032. +[2] Szabo, R. (2003) “Quantum field theory on noncommutative spaces,” +Physics Reports, 378(4), pp. 207–299. +[3] Sheikh-Jabbari, M.M. (1999) “Super Yang-Mills theory on noncom- +mutative torus from open strings interactions,” Physics Letters B, +450(1-3), pp. 119–125. +[4] Doplicher, S., Fredenhagen, K. and Roberts, J.E. (1995) “The quan- +tum structure of spacetime at the Planck scale and Quantum Fields,” +Communications in Mathematical Physics, 172(1), pp. 187–220. +[5] Ahluwalia, D.V. (1994) “Quantum measurement, gravitation, and +locality,” Physics Letters B, 339(4), pp. 301–303. +[6] C. S. Chu and P. M. Ho, Noncommutative open string and D-brane, +Nucl. Phys. B 550, 151 (1999) [hep-th/9812219]. +7 + +[7] B. Jurco, S. Schraml, P. Schupp and J. Wess, Enveloping alge- +bra valued gauge transformations for non-Abelian gauge groups on +non-commutative spaces, Eur. +Phys. +J. C17, 521 (2000) [hep- +th/0006246]. +[8] Chaichian, M. and Demichev, A.P. Introduction to quantum groups. +Singapore: World Scientific (1996). +[9] Bonneau, P. et al. (2004) “Quantum groups and deformation quanti- +zation: Explicit approaches and implicit aspects,” Journal of Math- +ematical Physics, 45(10), pp. 3703–3741. +[10] A. Klimyk and K. Schmudgen, Quantum Groups and Their Repre- +sentations, Springer (1997). +8 + diff --git a/B9E1T4oBgHgl3EQfVwR_/content/tmp_files/load_file.txt b/B9E1T4oBgHgl3EQfVwR_/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..80cd00bee4504c9e7a9b16415dcf2edf9a0bfc0f --- /dev/null +++ b/B9E1T4oBgHgl3EQfVwR_/content/tmp_files/load_file.txt @@ -0,0 +1,157 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf,len=156 +page_content='(1+1) dimensional scalar field theory on q-deformed space Poula Tadros Department of Applied Physics, Aalto University School of Science, FI-00076 Aalto, Finland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' email:poulatadros9@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content='com Abstract We study scalar field theory in one space and one time dimensions on a q-deformed space with static background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' We write the Lagrangian and the equation of motion and solve it to the first order in q − 1 where q is the deformation parameter of the space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' 1 Introduction Non-commutative geometry was first introduced in string theory in ref- erence [1], where it was shown that the coordinates of the endpoints of strings on D-branes in the presence of a Neveu-Schwartz field are non- commutative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' Non-commutative field theories have also been defined, as they can be derived from string theories and have interesting features, as described in references [2] and [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' The introduction of non-commutative spacetime in field theory is mo- tivated by the Heisenberg uncertainty principle in quantum mechanics, which states that at small distance scales, there is a large uncertainty in momentum measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' This means that energy can reach very high values in a small spatial distance, approaching the Planck scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' However, according to the general theory of relativity, high energy in a small spatial distance creates a black hole, which prevents the position from being fully certain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' To reconcile these two phenomena, it is necessary to introduce non-commutativity in spacetime, which implies non locality in the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' This is explained in references [4] and [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' In this paper we study (1+1) dimensional classical scalar field theory with static spacetime on a q-deformed space, we present both analytical and numerical analysis of the resulting theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' In section 2, we review the some types of non-commutativity on space times and motivate the choice of q-deformation non-commutativity as the subject of the study .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' In section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' we study the scalar field theory on q-deformed space time, we write the Lagrangian and deduce the equation of motion, we also truncate the equation of motion to the linear order in q −1 and solved the resulting equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' In section 4, we study the numerical solutions of the truncated equation of motion showing that the solutions grow exponentially with x and t meaning that the equation is stiff and there are instabilities in the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' In section 5, we conclude the study and suggest topics for further research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content='03106v1 [hep-th] 8 Jan 2023 2 Types of non-commutativity Here, we briefly review three of the most popular types of non-commutativity relations and justify our motivation to use the q-deformation type 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' Canonical non-commutativity It is the simplest type which used in physics literature, it was introduced in [6], it is defined by imposing the following commutation relations [xµ, xν] = iθµν, where xµ are the spacetime coordinates and θµν is a constant, anti- symmetric matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' The idea of canonical non-commutativity involves smearing the struc- ture of space-time in a particular way, regardless of the specific mathematical details of the space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' In order to incorporate non- commutative geometry capturing the mathematical structures on the manifold, it is necessary to consider more complex forms of non- commutativity beyond just this basic version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' Lie-type non commutativity In this case the coordinates has a Lie algebra structure i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' the commutation relations can capture a Lie algebra structures [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' The commutation relations are given by [xµ, xν] = if µν ρ xρ, where f µν ρ are the structure constants of the defined Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' However, this type is not useful because Lie structures are rigid i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' any small deformation of a Lie algebra is isomorphic to the original Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' q-deformations A solution to the rigidity problem for Lie algebras is to replace Lie group with a flexible structure called quantum groups [8-10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' The term quantum group used in this context refers to the deformations of the universal enveloping algebra of a given group, these objects have Hopf algebra structures which are flexible structures unlike Lie groups and algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' The commutation relations are given by xµxν = 1 q Rµν στxσxτ, where q is a parameter and Rµν στ is the R-matrix of the quantum group defined on the space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' In this space a Lie algebra is replaced by a non-commutative Hopf algebra with deformation parameter q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' The resulting space is de- formed according to the Lie group on the space and on the parame- ter q, this is the simplest way to deform a space time while capturing the full algebraic structure of the space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' 2 3 Lagrangian and the equation of motion We begin with the Lagrangian of the scalar field on the commutative man- ifold then introduce non-commutativity by replacing the derivatives by Jackson derivatives, since the symmetry group is U(1), the deformations of its universal enveloping algebra gives a commutative algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' Thus, we do not have to worry about defining a product of functions on the new space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' The Lagrangian is then L = 1 2∂µφ∂µφ − 1 2m2φ2 → Lq = DµqφDµ q φ − m2φ2, where µ = 0, 1 with x0 = t and x1 = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' Assuming the field is defined everywhere and is infinitely differentiable and the deformations are small i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' q ≈ 1, we can relate the theory on the non commutative topological space to the theory on the commuta- tive manifold (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' transforming the non-commutative theory back to the commutative manifold) using the formulae Dxq(f(x)) = ∂xf + ∞ � k=1 (q − 1)k (k + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' xkf (k+1)(x), where f (k) is the k-th ordinary derivative of f with respect to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' Dtq(f(x)) = ∂tf + ∞ � k=1 (q − 1)k (k + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' xkf (k+1)(x), where f [k] is the k-th ordinary derivative of f with respect to t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' The resulting Lagrangian on the commutative manifold is Lq = 1 2∂φ∂φ − 1 2m2φ2 + 2∂φ ∞ � k=1 (q − 1)k (k + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' xkφ(k+1) + ∞ � l,m=1 (q − 1)(l+m) (m + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' (l + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content='xk+lφ(l+1)φ(m+1) + (x → t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' where (x → t) means the same terms but with x replaced by t includ- ing in the derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' The Lagrangian has an infinite series of derivatives, in this case the Euler-Lagrange equation will be ∂Lq ∂φ + ∞ � k=1 (−1)k dk dxk ( ∂Lq ∂φ(k) ) + ∞ � k=1 (−1)k dk dtk ( ∂Lq ∂φ[k] ) = 0, (1) where k = 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content='. The Lagrangian is clearly non local as expected from a non-commutative theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' The derivatives of the Lagrangian are given by ∂Lq ∂φ = −mφ, 3 ∂Lq ∂(∂φ) = ∂xφ + 2 ∞ � n=1 (q − 1)n (n + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' xnφ(n+1) (2) → d dx( ∂Lq ∂(∂xφ)) = ∂x∂xφ + 2 ∞ � n=1 (n(q − 1)n (n + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' xn−1φ(n+1)) + 2 ∞ � n=1 ((q − 1)n (n + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' xnφ(n+2)), (3) ∂Lq ∂(φ(k)) = 2(q − 1)k−1xk−1 k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' ∞ � n=0 (q − 1)n (n + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' xnφ(n+1) → d dx( ∂Lq ∂(φ(k))) = 2(q − 1)k−1x2k−1 k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' ∞ � m,n=0 � k m � (q − 1)n (n + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' (n + k + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' (n + 2k − m − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content='xn−mφ(n+k+1), (4) with similar formulae for derivatives with respect to t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' Putting all together from (2), (3), (4) in (1) we get −∂µ∂µφ − m2φ − 2 ∞ � n=1 n(q − 1)n (n + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' xn−1φ(n+1) − 2 ∞ � n=1 (q − 1)n (n + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' xnφ(n+2) + ∞ � k=2 (−1)k 2(q − 1)k−1x2k−1 k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' ∞ � n=0 k � m=0 � k m � (q − 1)n(n + k − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' (n + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' (n + 2k − m − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content='xn−mφ(n+k+1) + (x → t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' (5) This is a partial differential equation of infinite order with variable coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' If we consider only small deformations i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' q ≈ 1, then we can only keep terms up to the linear order in q − 1, the first order equation will be −∂µ∂µφ−m2φ−(q−1)[φ(2)+xφ(3)− x3 6 φ(3)−x2φ(3)−xφ3+(x → t)] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' This equation is a stiff equation i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' it is numerically unstable, this may indicate an instability in the theory due to the linear approximation used, but as seen from the full equation of motion the full theory is stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' The solution is φ = F(t)G(x) where F(t) = c1eiAt/√q+c2e−iAt/√q+(q−1)eiAt/√q 2iA√q [ iA 24q t4+(iA3 3 − 1 12√q − i 8A)t3 +(A + q 2q − A√q 2 − i 8A)t2 + (i(A + q) 2A√q − iqA 4 + √q 8A2 )t 4 +(A + q 4A2 + q √ A 4 + iq 16A3 )] + O((q − 1)2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' G(x) = c3eikx/√q+c4e−ikx/√q+(q−1)eikx/√q 2ik√q [ ik 24q x4+(ik3 3 − 1 12√q − i 8A)x3 +(k + q 2q − k√q 2 − i 8k )x2 + (i(k + q) 2k√q − iqk 4 + √q 8k2 )x +(k + q 4k2 + q √ k 4 + iq 16k3 )] + O((q − 1)2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' where c1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' c2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' c3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' c4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' A are normalisation constants and k = ± √ A2 + m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' When q = 1, it reduces to the solution to the Klein Gordon equation as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' 4 Numerical results Here, we present numerical solutions to the equation of motion to the first order in q − 1, we focus on G(x) only since the remaining part is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' The solutions are exponentially growing in time establishing that the equation of motion was stiff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' We set c3 = c4 = k = 1, A2 = 1 2 and we plot the solution for different values of the parameter q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' Figure 1: At q − 1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content='1 the solution grows exponentially with |x|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' This is a feature of a stiff equation with unstable numerical solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' In the vicinity of x = 0 it is close to the usual Klein-Gordon solution but as we go further it becomes more and more distant 5 200000 100000 0 100000 200000 100 75 50 25 25 50 75 100Figure 2: At q − 1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content='001 the solution still grows exponentially but slower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' Figure 3: At q − 1 = 10−6 the solution resembles the Klein-Gordon solution up to |x| = 50 then decays for a bit but eventually blows up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' Figure 4: At q − 1 = 10−9 the solution has the same behaviour as the previous graph but the decay happens at larger |x|, all smaller q − 1 values follow this pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' 6 2000 1500 1000 - 500 0 500 1000 1500 2000 100 7550 25 0 25 50 75 10030 20 10 0 10 20 30 200 150 100 50 0 50 100 150 200E 2 1 0 1 2 3 600 400 200 0 200 400 600The above results shows an instability in the theory leading to di- vergent solutions to the equations of motion as x → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' To remove the instability we must add infinite terms corresponding to an infinite series of higher derivatives i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' we have to consider the full theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' However, this approximation gives us an intuition on how the q-deformation affects the space, small q-deformations beside leading to non local effects appear to affect the space irregularly with only small effects locally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' 5 Conclusion and outlook In conclusion, we showed that defining a field theory on a q-deformed space leads to an infinite series of higher derivatives in the Lagrangian even with static background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' In the case presented the algebra was com- mutative so no new product of functions is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' We also demonstrated that any approximation or truncation to the theory will lead to stiff equa- tions of motion resulting from instabilities in the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' While we made a progress in the field, much more is to be studied, fu- ture research in this direction should focus on defining more complicated theories on q-deformed spaces with non-commutative function algebras and with dynamical spacetimes, also to define higher spin fields on such space and study the new symmetries of the theories as well as the types of instabilities arise if the Lagrangian is truncated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' Acknowledgments We would like to thank Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content='Ivan Kolar for the useful discussions on the topic References [1] Seiberg, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' and Witten, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' (1999) “String theory and noncommu- tative geometry,” Journal of High Energy Physics, 1999(09), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' 032–032.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' [2] Szabo, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' (2003) “Quantum field theory on noncommutative spaces,” Physics Reports, 378(4), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' 207–299.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' [3] Sheikh-Jabbari, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' (1999) “Super Yang-Mills theory on noncom- mutative torus from open strings interactions,” Physics Letters B, 450(1-3), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' 119–125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' [4] Doplicher, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=', Fredenhagen, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' and Roberts, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' (1995) “The quan- tum structure of spacetime at the Planck scale and Quantum Fields,” Communications in Mathematical Physics, 172(1), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' 187–220.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' [5] Ahluwalia, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' (1994) “Quantum measurement, gravitation, and locality,” Physics Letters B, 339(4), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' 301–303.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' [6] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' Chu and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' Ho, Noncommutative open string and D-brane, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' B 550, 151 (1999) [hep-th/9812219].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' 7 [7] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' Jurco, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' Schraml, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' Schupp and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' Wess, Enveloping alge- bra valued gauge transformations for non-Abelian gauge groups on non-commutative spaces, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' C17, 521 (2000) [hep- th/0006246].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' [8] Chaichian, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' and Demichev, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' Introduction to quantum groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' Singapore: World Scientific (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' [9] Bonneau, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' (2004) “Quantum groups and deformation quanti- zation: Explicit approaches and implicit aspects,” Journal of Math- ematical Physics, 45(10), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' 3703–3741.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' [10] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' Klimyk and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' Schmudgen, Quantum Groups and Their Repre- sentations, Springer (1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} +page_content=' 8' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'} diff --git a/CNE3T4oBgHgl3EQfUQpw/content/tmp_files/2301.04449v1.pdf.txt b/CNE3T4oBgHgl3EQfUQpw/content/tmp_files/2301.04449v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..f030b92f198f45f76ec0d0afe8a9b8b063adba33 --- /dev/null +++ b/CNE3T4oBgHgl3EQfUQpw/content/tmp_files/2301.04449v1.pdf.txt @@ -0,0 +1,2398 @@ +Diving Deep into Modes of Fact Hallucinations in Dialogue Systems +Souvik Das Sougata Saha Rohini K. Srihari +{souvikda, sougatas, rohini}@buffalo.edu +Department of Computer Science and Engineering, University at Buffalo, NY. +Abstract +Knowledge Graph(KG) grounded conversa- +tions often use large pre-trained models and +usually suffer from fact hallucination. +Fre- +quently entities with no references in knowl- +edge sources and conversation history are in- +troduced into responses, thus hindering the +flow of the conversation—existing work at- +tempt to overcome this issue by tweaking the +training procedure or using a multi-step refin- +ing method. However, minimal effort is put +into constructing an entity-level hallucination +detection system, which would provide fine- +grained signals that control fallacious content +while generating responses. +As a first step +to address this issue, we dive deep to iden- +tify various modes of hallucination in KG- +grounded chatbots through human feedback +analysis. +Secondly, we propose a series of +perturbation strategies to create a synthetic +dataset named FADE (FActual Dialogue Hal- +lucination DEtection Dataset)1. +Finally, we +conduct comprehensive data analyses and cre- +ate multiple baseline models for hallucination +detection to compare against human-verified +data and already established benchmarks. +1 +Introduction +Knowledge-grounded conversational models often +use large pre-trained models (Radford et al., 2019; +Brown et al., 2020). These models are notorious for +producing responses that do not comply with the +provided knowledge; this phenomenon is known +as hallucination (Dziri et al., 2022b; Rashkin et al., +2021b). Faithfulness to the supplementary knowl- +edge is one of the prime designing factors in these +knowledge-grounded chatbots. If a response is +unfaithful to some given knowledge, it becomes +uninformative and risks jeopardizing the flow of +the conversation. Despite retaining strong linguis- +tics abilities, these large language models(LM) in- +adequately comprehend and present facts during +1https://github.com/souvikdgp16/FADE +conversations. LMs are trained to emulate distribu- +tional properties of data that intensify its hallucina- +tory attributes during test time. +Figure 1: Hallucination manifested by generated responses +using GPT2(Radford et al., 2019) trained on KG triples can +be more nuanced. +On the one hand, many prior works (Wiseman +et al., 2017; Parikh et al., 2020; Tuan et al., 2019) +have suggested training these models on external +data to ensure faithfulness may lead to a source- +reference divergence problem, where the reference +contains additional factual information. +To ad- +dress this problem holistically, Dziri et al. has +proposed a two-step generate-then-refine approach +by augmenting conventional dialogue generation +with a different refinement stage enabling the di- +alogue system to correct potential hallucinations +by querying the KG. Also, this work employs a +token-level hallucination classifier trained on a syn- +thetic dataset constructed using two perturbation +strategies 2. Though this method has clear benefits, +the hallucination perturbation strategies proposed +in this work might fail to capture some of the sub- +tle attributions of a factual generative model. As +illustrated in Figure 1, neural models can inject hal- +lucinated entities into responses that are present in +the k-hop KG and are deceptively similar to what +is expected. Also, if we cannot detect these elusive +hallucinations beforehand, it will cause a cascad- +ing effect and amplify hallucinations in subsequent +turns (See and Manning, 2021). +2(1) Extrinsic perturbation: Dziri et al. have swapped an +entity with a different entity of the same type and not present in +1-hop subgraph. (2) Intrinsic perturbation: they have swapped +an entity with its object or vice versa, taken from the golden +1-hop subgraph. +arXiv:2301.04449v1 [cs.CL] 11 Jan 2023 + +Path(s): T, :['Outlander', 'written_by','Diana Gabaldon'] [Gold] +T, :['Outlander','publication_date','1st June'] [Retrieved from +1-hop KG] +T, :['Outlander', 'published_by','Dell Publishing'l [Retrieved from +1-hop KG] +History: ['Do you like the book Outlander ?'] +GPT2 Response: “I've never read it, but I know it was written by Dell +PublishingOn the other hand, relying on human annotations +is challenging due to error-prone collection proto- +cols and human ignorance to complete the tasks +with care (Smith et al., 2022). Prior research (Dziri +et al., 2022c) shows that knowledge-grounded con- +versational benchmarks contain hallucinations pro- +moted by a design framework that encourages infor- +mativeness over faithfulness. As studied by Dziri +et al., when the annotators are asked to identify +hallucination in a response, there is a high chance +of error due to lack of incentive, personal bias, or +poor attention to the provided knowledge. +See and Manning have studied different short- +comings in a real-time neural model. In this work, +based on some of the findings of See and Manning, +like repetitive and unclear utterances promoting +hallucination, we extend the already defined modes +of hallucinations (Maynez et al., 2020; Dziri et al., +2021a). Our contributions to this work are three- +fold: +• We extend fact hallucination in KG-grounded +dialogue systems into eight categories. To +understand the degree to which our defined +classes exist in real-life data, we conduct a sys- +tematic human evaluation of data generated +by a state-of-the-art neural generator. +• Since human annotation is expensive and of- +ten inaccurate, we design a series of novel +perturbation strategies to simulate the de- +fined ways of fact hallucinations and build +a set of synthetic datasets collectively named +as FADE (FActual Dialogue Hallucination +DEtection Dataset). +• We create multiple pre-trained model-based +baselines and compare the performances on +several constituent and mixed datasets. To +assess our dataset’s generalization capability, +we perform zero-shot inference on BEGIN +(Dziri et al., 2021b), and FaithDial (Dziri et al., +2022a) datasets, which encompasses all cate- +gories of hallucinated responses. +2 +Different Modes of Hallucination in +KG-grounded Dialogue Systems +2.1 +Background +We focus on the task of detecting halluci- +nated spans in dialogues that are factually +grounded on factoids derived from multi-relational +graphs G = (V, E, R), termed as Knowledge- +Graphs(KG). Each KG consists of an directed edge +triples t = ⟨[SBJ], [PRE], [OBJ]⟩, where +[SBJ], [OBJ] ∈ V are nodes denoting subject +and object entities and [PRE] ∈ R is a predicate +which can be understood as a relation type. Primar- +ily, a neural dialogue system is guilty of generating +hallucinated text when a valid path in the k-hop +sub-graph Gk +c ∈ G of the original KG anchored +around a context entity c does not support it. +Our study extends the work of (Dziri et al., +2021a) where they specifically explore two broad +circumstances – extrinsic and intrinsic to the pro- +vided KG, under which LMs are likely to exhibit +unfaithful behavior. Though this categorization is +beneficial for detecting hallucinations, these cate- +gories can be further subdivided into subcategories, +which are described in §2.3. +2.2 +Base Dataset +We use OpenDialKG (Moon et al., 2019), a +crowded-sourced English dialogue dataset where +two workers are paired to chat about a particular +topic(mainly movie, music, sport, and book). We +use this dataset for training a GPT2-based model +for generating data for human feedback analysis +and creating the perturbed datasets. More details +about the dataset can be found in §C +2.3 +Definitions +We define below several categories of fact halluci- +nation, comprehensive illustrations of each types +are provided in Figure 2. In addition we have in- +cluded detailed descriptions of each definitions in +§A +(a) (Extrinsic-Soft). An extrinsic-soft hallucina- +tion corresponds to an utterance that brings a new +span of text which is similar to the expected span +but does not correspond to a valid triple in Gk +c . +(b) (Extrinsic-Hard). An extrinsic-hard halluci- +nation corresponds to an utterance that brings a +new span of text which is different from the expected +span and does not correspond to a valid triple in +Gk +c . +(c) (Extrinsic-Grouped). An extrinsic-grouped +hallucination corresponds to an utterance that +brings a new span of text which is different from the +expected span but is of a specific predefined type +and does not correspond to a valid triple in Gk +c . +(d) (Intrinsic-Soft). An intrinsic-soft hallucina- +tion corresponds to an utterance that misuses any +triple in Gk +c such that there is no direct path be- +tween the entities but they are similar to each other. +(e) (Intrinsic-Hard). An intrinsic-hard hallucina- +tion corresponds to an utterance that misuses any + +Figure 2: Illustration of our defined categories of fact hallucinations in KG-grounded dialogue systems +triple in Gk +c such that there is no direct path be- +tween the entities and they are not related in any +form. +(f) (Intrinsic-Repetitive). An intrinsic-repetitive +hallucination corresponds to an utterance that ei- +ther misuses [SBJ] or [OBJ] in Gk +c such that +there is no direct path between the entities but the +entity has previously occurred in conversational +history.. +(g) (History Corrupted- Intrinsic/ Extrinsic). A +history corrupted(intrinsic/extrinsic) hallucination +corresponds to an utterance that is subjected to +intrinsic or extrinsic hallucination which is influ- +enced by hallucinated entities in conversational +history. +2.4 +Human Feedback Analysis +To study the extent to which the previously de- +scribed modes of hallucination exist in a real-world +system, we did human feedback analysis on re- +sponses generated using a GPT2-based generative +model fine-tuned on OpenDialKG as described +by Dziri et al.. We sampled 200 responses each +from four different decoding strategies, Greedy, +Beam Search, and Nucleus Sampling, with a prob- +ability of 0.9 and 0.5. +For each dialogue in- +stance, we crowd-source human judgment by solic- +iting evaluations from 2 different annotators(with +a high approval rating) from Amazon Mechanical +Turk(AMT)(Details in §B). One computer science +graduate student additionally verified the Human +Intelligence Task (HITS). For examples where hal- +lucination was present, we asked the workers to +identify the type of hallucination(examples of dif- +ferent types of hallucinations were shown in the +GPT2-KG +Greedy +Beam Search +Nucleus 0.9 +Nucleus 0.5 +Extrinsic-Soft +10.91 +8.8 +15.5 +14.77 +Extrinsic-Hard +3.45 +4.22 +8.3 +9.8 +Extrinsic-Grouped +1.12 +1 +0.44 +1.6 +History Corrupted-Extrinsic +3.3 +3.1 +2.33 +1.1 +Intrinsic-Soft +1.2 +1.38 +0.8 +0.3 +Intrinsic-Hard +0.2 +0.8 +1.1 +2 +Intrinsic-Repetitive +0.2 +0.8 +1.8 +4 +History Corrupted-Intrinsic +0.7 +0.5 +1.33 +3.3 +Extrinsic Total +18.78 +17.12 +26.57 +27.27 +Intrinsic Total +2.3 +3.48 +5.03 +9.6 +Total +21.08 +20.6 +31.6 +36.87 +Table 1: Fine-grain human feedback analysis +instruction). The result of the human feedback is +exhibited in Table 1. We rejected 21% of the HITS +because of poor quality; we reported the average +Krippendorf alpha coefficient to be 0.74 on the +remaining annotations, indicating a moderate to +a high agreement. Using Table 1 we made these +observations: +• Extrinsic-soft hallucination is the dominant +form of hallucination. Also, this bolsters our +prior observation that LMs generate entities +similar to the golden entity. +• Comparatively less amount of hallucinations +was seen in responses generated using beam +search decoding scheme, though the percent- +age of extrinsic-hard hallucination was higher +than greedy decoding. +• Intrinsic-hard hallucination appears to be the +least among all types. This suggests LM will +always try to learn something from the given +KG triples; generating something dissimilar +will have a very low probability. +3 +Dataset Creation +FADE is a collection of datasets consisting of com- +ponent datasets created using several perturbations + +Anchor Entity(c) +HISTORY +GOLDEN RESPONSE. +HISTORY (CORRUPTED) +1-Hop KG +A: Could you recommend movies +B: Christopher Nolan was the director . +A: Could you recommend movies similar to +similar to The Dark Knight ? +He also directed Insomnia and +The Dark Knight ? +Inception. +B: The sequel to [The Dark Knight -→ The +B: The sequel to Batman Begins is The +GOLD TRIPLE(S) +Dark Knight Rises(Int.)] [The Dark Knight - +Dark Knight . +Spider-Man(Ext.)] is Batman Begins . +['The Dark Knight', 'directed_by', +A: Okay . Who is the director of The +Christopher Nolan'] +A: Okay . Who is the director of The Dark +Dark Knight and any other movies from +DarkKnight +['Christopher Nolan', 'is-a', 'Film +Knight and any other movies from him not +him not related to Batman ? +director'l +related to Batman ? +Perturbed Entity +PERTURBED RESPONSE(Soft) +PERTURBED RESPONSE(Soft) +PERTURBED RESPONSE(Intrinsic) +B:Steven Spielberg was the director +B: The Dark Knight RisesWas the +B: The Dark Knight Rises was the +He also directed insomnia and +director . He also directed insomnia +director . He also directed insomnia +inception +and inception +and inception. +PERTURBED RESPONSE(Hard) +PERTURBED RESPONSE(Hard) +PERTURBED RESPONSE(Extrinsic) +B: Joe Biden was the director . He also +B: United States of America was the +B: Steven Spielberg was the director . +directedinsomnia and inception +director . He also directed insomnia +He also directed insomnia and +and inception +inception : +PERTURBED RESPONSE(Grouped) +PERTURBEDRESPONSE(Repetitive) +B: Warner Bros. was the director . He +B: Batman Begins was the director . He +also directed insomnia and inception +also directed insomnia and inception +(a) Extrinsic Hallucination Types +(b) Intrinsic Hallucination Types +(c) History Corrupt Hallucination TypesHallucination Type +Index Type +Selection Criteria +Soft +Same as original entity +ei with max document score +Hard +Same as original entity +ei with min document score +Grouped +Same as one predefined type, selected randomly +same as soft +Table 2: Extrinsic hallucination perturbed entity selection +criteria +and a set of mixed datasets constructed using the +component datasets. +3.1 +Perturbation Strategies +Extrinsic Hallucination All the entities present in +OpenDialKG undergo a indexing process. At first, +using Spacy we determine the named entity type 3 +for each entity, and create BM25 indexes4 for each +entity type. Each KG triple corresponding to an en- +tity is represented in this format – "[SBJ] [PRE] +[OBJ]" and denoted as ti. Now, for an entity(ei) +we create a document di = concat(t1, t2, ..tn), n +is the number of KG-triples for that entity. Af- +ter this, we index di and ei in the index corre- +sponding to the entity type. During the perturba- +tion process, we retrieve all the KG-triples for the +entity we want to perturb and form 3 queries for +each triple by permuting ([SBJ],[PRE],[OBJ]). +Then based on the type of extrinsic halluci- +nation, we query the indices to get the docu- +ment scores in the following way: +scores = +average({BM25(qi, dj)}i∈(s,r,o),j∈(0,n)), the se- +lection criteria of the perturbed entities are pro- +vided in table 2. +The groups for extrinsic-grouped hallucination +are mentioned in Table 10. During the selection +process, we iteratively check whether the perturbed +entity exists in the conversation history, matches +with the actual entity, and has appeared in the 1-hop +sub-graph of the original entity. If an occurrence is +found, we proceed to the following best entity. +Intrinsic Hallucination Here, we dynamically +create a BM25 index and index all the KG triples +in the 1-hop sub-graph of the original entity. Again, +a KG triple is represented in the same fashion as in +extrinsic hallucination – "[SBJ] [PRE] [OBJ]". +The goal here is to select entities that are similar +or dissimilar to the original entities and present in +the 1-hop graph. To achieve that, we follow a hy- +brid triple retrieval approach to score each triple +associated with the original entity. First, we use the +final hidden layer of a pre-trained GPT2 to obtain +initial embeddings for each node in Gk +c (for details, +check §D.3). A query is formed by using Equa- +3https://spacy.io/api/entityrecognizer +4https://solr.apache.org/ +Hallucination Type +Selection Criteria +Soft +[SUB] or [OBJ] with max triple score +Hard +[SUB] or [OBJ] with min triple score +Repetitive +same as soft, should be occurring in the conversation history +Table 3: Intrinsic hallucination perturbed entity selection cri- +teria +tion 1 each triple in Gk +c is scored using a similarity +scoring system as described in Equation 3. +q = +� +i∈{s,r,o} +ε +p(qi) + ε vqi +(1) +Here ε is a free term parameter (§D.2), p(qi) is +unigram probability of the query term and vqi is the +embedding for each query term(here query terms +are [SBJ], [PRE] ,[OBJ] of the original entity). +ni = +ε +p(s) + ε vs + +ε +q(r) + ε vr + +ε +p(o) + ε vo +(2) +ni in Equation 2 represents a triple embedding +in Gk +c , when q(r) represents the rarity of the rela- +tionship term in the subgraph, high occurrence is +penalized, rest terms are analogous to Equation 1. +EntitySimilarity(Q, t) = cos(q, ni) +(3) +Now, we query the BM25 index that we have +created before with a simple query using the orig- +inal triple: "[SBJ] [PRE] [OBJ]" and get the +score for each of the triple(t). Finally, we get the +final scores using Equation 4. +Score(Q, t) = βEntitySimilarity(Q, t) ++(1 − β)BM25(Q, t) +(4) +Here 0 < β < 1. +We select the perturbed entities based on the +scores and selection criteria as defined in Table 3. +Like extrinsic hallucinations, we iteratively filter +the best-scored entity until it does not match the +original entity or appears in history. +History Corrupted Hallucination Conversa- +tional history is corrupted using intrinsic or extrin- +sic corruption strategy. We select the last k turns of +the conversation and randomly perturb the entities. +We also ensure that at least 50% of the previous k +turns are corrupted. +3.2 +Dataset Analysis +Below we provide data statistics and character- +ize the composition and properties of the datasets +that are generated using our proposed perturbation +strategies. + +Type +Perturbed +Non-perturbed +Turn with +perturbation>2 +soft +12752 +64634 +558 +hard +8540 +68872 +8254 +grouped +22858 +54542 +11296 +history-corrupt +8534 +68878 +8247 +Table 4: Extrinsic hallucination data statistics +Type +Perturbed +Non-perturbed +Turn with +perturbation>2 +soft +18560 +58558 +5 +hard +18605 +58534 +6 +repetitive +9712 +67560 +0 +history-corrupt +18597 +58542 +6 +Table 5: Intrinsic hallucination data statistics +3.2.1 +Data Statistics +Table 4 and 5 shows the statistics of datasets created +using different perturbation strategies. The base +dataset contains 77,430 data points. However, the +perturbed turns in each of these datasets are quite +low in comparison. This low number is because +not every entity in an utterance has a valid KG path. +For extrinsic hallucination, ∼12,000 to ∼23,000 +utterances were perturbed, and ∼550 to ∼11,300 +utterances have multiple perturbations. The num- +ber of perturbed data points for intrinsic hallucina- +tion is less than extrinsic(∼9,000 to ∼18,000). The +number of utterances with multiple perturbations +is negligible due to the many checks the perturbed +entities go through(for example, whether the KG +path is present, has already occurred or not, etc.) +To train and evaluate models, we vary the size of +the train split in this range of 10% to 30%5 with a +step of 2.5%, keeping in mind to avoid overfitting. +The remaining data is split into equal halves for +validation and testing. +3.2.2 +Parsing Features +In Figure 3 we show the top 10 Named Entity +Recognition(NER) tags as identified by the Spacy +library in extrinsic hallucinations. For extrinsic- +soft hallucination, most NER tags are of type PER- +SON. This corresponds to the fact that the original +entities in the base dataset are primarily related to +movies, books, and music. In extrinsic-soft halluci- +nation, the associated PERSON name is changed +to a closely affiliated person, or a movie name is +changed to its director’s name. In contrast, the dis- +tribution of NER tags is uniform for extrinsic-hard +hallucination. Figure 4 and 5 shows the top-10 rela- +tions of the perturbed entity with the original entity +in both intrinsic-soft and hard hallucinations and +the corresponding value in their counterparts. In +intrinsic-soft hallucination, more relevant relations +are selected like "release year", "starred actors", +"written by", etc. On the other hand, in intrin- +5sequential split +Figure 3: NER distribution in Extrinsic-soft and hard halluci- +nation +sic hard hallucination, more unusual relations like +"Country of Origin", and "Country of Nationality" +were among the top relations. +Figure 4: Top 10 relation in perturbed KG triples in intrinsic- +soft hallucination +Figure 5: Top 10 relation in perturbed KG triples in intrinsic- +hard hallucination +3.3 +Mixing Datasets +Since in actual data, all kinds of hallucinations are +expected to occur. We mix the previously con- +structed datasets in specific proportions to create a +more challenging dataset. Table 11 shows the dif- +ferent mixing ratios for four types of datasets is as +follows: Observed: We try to mimic the observed +data, which is shown in §2.4, we take an average of +percentages in for all the decoding strategies. Bal- +anced: Goal here is to create a balanced dataset +between hallucinated and non-hallucinated turns, +each type of hallucination is also balanced. Extin- +sic+: In this scenario, we increase the percentages + +PERSON +1%1% +4% +5% +Extrinsic Soft +■ORG +2%1%4% +2% R +2% +6% +DATE +7% +Extrinsic Hard +GPE +39% +11% +■CARDINAL +■WORK OF +18% +ART +■NORP +12% +65% +EVENT +ORDINAL +23% +■LOCrelease_year +3% +2%2% +3% +Intrinsic Soft +21% +starred actors +4% +12% +8% +written_by +Intrinsic Hard +5% +13% +21% +has_genre +11% + is-a +AdaptedFrom +4% +18% +Gender +15% +16% +in_language +23% + Subject +18% +ProducedbyCountry of origin +Ihas_genre +Intrinsic Soft +21% +5% +starred actors +26% +7% +17% +Intrinsic Hard +Country of +7% +nationality +Iwritten_by +8% +15% +- Produced by +3% +8% +Original language +3% +23% +13% +10% +AwardWon +10% +in_language +22% +2% +■release_yearof extrinsic-soft, hard, and grouped by a factor of +2, 1.5, and 1.5, respectively. Intrinsic+: here we +increase the percentages of intrinsic-soft, hard and +repetitive by a factor of 1.5. More details in §D.4. +3.4 +Human Verification +To verify whether our proposed perturbation strate- +gies inject hallucinations in the original data, we +randomly sample 150 examples from each of the +mixed dataset’s test splits. Subsequently, these sam- +ples were randomly ordered to form a consolidated +sample of 600 data points annotated by at least +three AMT workers, with the same setting as de- +scribed in §2.4. Additionally, the graduate student +verified where the hallucinations adhere to the per- +turbation norms. Krippendorff’s alpha were 0.88 +and 0.76 among workers, and workers with per- +turbed data(average), indicating a very high agree- +ment. Since our perturbation strategies are purely +deterministic, we kept a large-scale human verifi- +cation of the automatically annotated data outside +the scope of this work. We create a human-verified +dataset of 500 samples, 300 taken from this set and +200 from the human feedback study 2.4. +4 +Task +To identify utterances that contain hallucinations +and to locate the entities of concern. We create two +tasks: +1. Utterance classification: Given the dialog +history D, knowledge triples Kn and the cur- +rent utterance xn+1 we classify xn+1 is hallu- +cinated or not. +2. Token classification: Given D, Kn and xn+1, +we need to perform sequence labelling on +xn+1 and identify the hallucinated spans. +5 +Baseline Models +As an initial effort toward tackling the suggested +hallucination detection task, we create several +baseline detection models based on pre-trained +transformer models, including BERT, XLNet, and +RoBERTa. These transformer-based models repre- +sent the state-of-the-art and can potentially better +leverage context or embedded world knowledge +to detect self-contradictory or anti-commonsense +content. +For training the utterance classifier, given D, Kn +and xn+1, we fine tune a pre-trained model M to +predict binary hallucinated label y for xn+1 . Here, +D and Kn are considered as sequence A with token +type ids as 0 and xn+1 is considered as sequence B +with token type ids as 1. During inference, from the +last hidden states H ∈ Rl×h (h, l are hidden size +and sequence length, respectively), then we obtain +the representation w ∈ Rh by max pooling(i.e., +w = max_pool(H)). We then pass w through +a MLP layer with a tanh activation to get the bi- +nary label y ∈ {0, 1}. During training time, we +fine-tune the model using cross entropy objective +between the predicted labels and the actual labels. +Similarly, for training the sequence classifier, we +fine-tune a pre-trained model Ms. At first, we +encode D, Kn and xn+1 using Ms to get the last +hidden states H ∈ Rl×h, (h, l are hidden size and +sequence length, respectively). Instead of doing +a binary classification of each token, we adopt a +BILOU encoding scheme. The hidden states are +passed through an MLP layer with a tanh activa- +tion to get the 5-way label y ∈ {B, I, L, O, U}. +During training time, we fine-tune the model us- +ing a cross-entropy objective between the predicted +and actual labels. +6 +Experimental Setup +Baseline configurations we experiment with +a +variety +of +pre-trained +models +via +Hug- +ging Face Transformers, including BERT-base- +uncased(110M), RoBERTa-base(125M) and XL- +Net-base-cased(110M). Though using large or +medium versions of these models will produce bet- +ter results, we refrain from using those models as +scaling large models in production is costly. More +details about training parameters can be found in +§E +We also experimented with model architecture as +follows: (i) Varied the length of the history (ii) Ex- +perimented with max/ mean pooling. (iii) Whether +to concatenate the hidden states corresponding to +Kn with the hidden states corresponding to xn+1 +before passing them through the MLP layer. (iv) +Using a CRF layer instead of MLP for predicting +labels in the sequence tagger. The best configu- +ration uses 4 turns of conversational history, max +pooling, it does not concatenate hidden states of +Kn with hidden states of xn+1 and uses a 2-layer +MLP. +Evaluation metrics We evaluate the baselines +with formal classification metrics, including preci- +sion, recall, and F1 for the hallucination sequence +tagger. For the utterance-level hallucination classi- +fier, we report accuracy, precision, recall, F1, and + +Dataset +Best Model +Token Level +Utterance Level +F1 +P +R +F1 +P +R +G-Mean(↑) +BSS(↓) +AUC +extrinsic-grouped +BERT(base-uncased) +80.69 +80.56 +80.82 +91.30 +91.80 +90.81 +93.58 +5.29 +93.62 +extrinsic-hard +XLNet(base-cased) +72.12 +71.98 +72.25 +87.36 +87.13 +87.60 +92.80 +2.93 +92.96 +extrinsic-history-corrupt +XLNet(base-cased) +72.38 +72.35 +72.40 +88.10 +87.86 +88.34 +93.24 +2.75 +93.38 +extrinsic-soft +BERT(base-uncased) +64.09 +69.22 +59.67 +74.80 +81.96 +68.80 +81.62 +8.03 +82.81 +intrinsic-hard +XLNet(base-cased) +84.44 +85.08 +83.81 +90.88 +92.88 +88.97 +93.24 +4.48 +93.34 +intrinsic-history-corrupt +XLNet(base-cased) +83.67 +82.27 +85.11 +91.30 +91.86 +90.74 +93.97 +4.34 +94.02 +intrinsic-repetitive +RoBERTa(base) +82.70 +82.76 +82.64 +88.01 +89.51 +86.55 +92.31 +3.15 +92.50 +intrinsic-soft +RoBERTa(base) +78.80 +80.19 +77.45 +87.10 +90.54 +83.92 +90.26 +6.22 +90.50 +Table 6: Test benchmark (numbers in percentages (%)) for component datasets, models trained on 25% of the total dataset. +Dataset +Best Model +Token Level +Utterance Level +F1 +P +R +F1 +P +R +G-Mean(↑) +BSS(↓) +AUC +balanced +RoBERTa-base +73.41 +68.75 +78.74 +88.24 +83.85 +93.12 +86.21 +13.14 +86.47 +observed +XLNet(base-cased) +63.44 +57.98 +70.03 +77.71 +71.05 +85.73 +85.40 +14.73 +85.40 +intrinsic+ +RoBERTa-base +75.05 +71.11 +79.44 +90.16 +86.52 +94.12 +84.51 +12.78 +85.00 +extrinsic+ +XLNet(base-cased) +75.59 +70.79 +81.10 +90.75 +86.77 +95.11 +83.21 +12.65 +83.95 +Table 7: Test benchmark (numbers in percentages (%)) for mixed datasets, models trained on 25% of the total dataset. +AUC (Area Under Curve) for ROC. We also use +the G-Mean metric (Espíndola and Ebecken, 2005), +which measures the geographic mean of sensitiv- +ity and specificity. We also employ the Brier Skill +Score (BSS) metric (Center, 2005), which com- +putes the mean squared error between the reference +distribution and the hypothesis probabilities. +7 +Results and Discussion +Baseline performance Table 6 and Table 7 show +the baseline performance for the component +datasets and mixed datasets. In both the settings, +the utterance level hallucination classifier performs +better than the token tagger in terms of F1. It can be +inferred from Table 6 that, on average, it is compar- +atively easier to detect intrinsic hallucinations than +extrinsic hallucinations; due to grounding on exter- +nal knowledge, which indicates the validity of our +perturbation techniques. However, comparing the +occurrence statistics from Table 1, it is noticed that +extrinsic-soft hallucination, which has the least F1 +score among all types, has the highest occurrences. +In extrinsic-grouped and extrinsic-soft hallucina- +tions, it is interesting that BERT performs better +than the other pre-trained models. Now for mixed +datasets, we ran inference on the test set of ob- +served dataset, as expected F1 scores(for utterance +classifier and token level tagger) of the observed +dataset are low as compared to other datasets due +to high percentage of extrinsic-soft hallucination. +Among other mixed datasets, the XLNet model +fine-tuned on extrinsic+ dataset performs best in +terms of F1 scores. +Performance on human-verified data We test +the best performing models fine-tuned on our +mixed datasets on human-verified data as de- +Fine-tuned on +Pretrain Model +F1 +(Utterance-level) +F1 +(Token-level) +MNLI +RoBERTa-large +12.5 +- +BEGIN +RoBERTa-large +15.4 +- +FaithDial +RoBERTa-large +22.1 +- +Intrin-Extrin(Dziri et al., 2021a) +RoBERTa-large +83.81 +68.2 +balanced +RoBERTa-base +92.27 +78.61 +observed +XLNet(base-cased) +90.15 +70.27 +extrinsic+ +XLNet(base-cased) +93.97* +85.7* +intrinsic+ +RoBERTa-base +93.01 +84.33 +Table 8: Performance of several benchmark models and mod- +els trained on FADE on the 500 human-verified data( *p-value +< 0.001)) +Fine-tuned on +Model +BEGIN +FaithDial +MNLI(3-way)(Dziri et al., 2021b) +T5 +49.5 +- +MNLI(Dziri et al., 2022a) +RoBERTa-large +61.1 +81.6 +intrinsic_hard +RoBERTa-base +37.12 +51.34 +intrinsic_history_corrupt +RoBERTa-base +43.23 +63.11 +intrinsic_hard +RoBERTa-large +44.42 +64.1 +intrinsic_history_corrupt +RoBERTa-large +55.11 +71.43 +Table 9: Zero-sort inference F1 scores on BEGIN and Faith- +Dial benchmarks using utterance classification models trained +on FADE +scribed in §3.4. Using the existing benchmark and +baseline models, we also perform a zero-shot in- +ference on the human-verified data. From Table +8, it is clear that the models fine-tuned on existing +benchmark data cannot understand fact hallucina- +tion, especially when entities are misplaced. On the +other hand, models trained on our datasets have F1 +scores over 90% and outperform the current base- +line by 10.16% and 17.5% in the two tasks using +a pre-trained model with fewer parameters. This +suggests that identifying abrupt fact hallucination +is more challenging than other types of halluci- +nation(like presenting more data than expected), +which are more commonly exhibited in the bench- +mark datasets. +Generalisability We make zero-shot inference +on BEGIN and FaithDial datasets’ test splits. To +make a fair comparison with the benchmark mod- +els, we further fine-tune roberta-large model +on our datasets. Table 9 shows that F1 scores ob- +tained from our best models underperform the best + +Figure 6: Positive and negative model predictions +Figure 7: Generalisation capability of RoBERTa-large model +fine-tuned using multiple splits of intrinsic-history-corrupt +dataset +performing baseline by 6% in BEGIN dataset and +10.17% in the FaithDial dataset. Even though the +performance is low, we have to understand that +the benchmark datasets contain hallucinations that +are fundamentally very different from fact hallu- +cinations. Also, we notice that models trained on +intrinsic hallucination perform the best because the +hallucinatory responses in the benchmark dataset +do not deviate much from the evidence. To estimate +how much training data is optimum for generalis- +ability, we ran inference on benchmark datasets +using models fine-tuned to 10% to 30% (with a +step of 2.5%) data in train split. As shown in Fig- +ure 7 approximately 25% is found to be optimum. +Model Predictions We visualized the predic- +tions on different datasets in Figure 6. Our models +were able to easily identify the hallucinated entities +as shown in Figure 6a here "The Departed" is a +movie in which "Mark Wahlberg" has acted but is +not related to the movie discussed in the context, +i.e., "The Italian Job". Similarly, predictions made +on the FaithDial dataset(Figure 6c) show that our +models could produce accurate predictions when +the response is generating something that is not +expected, but the hallucination has similarities with +the evidence. Our model sometimes fails to under- +stand when the history is convoluted(Figure 6b)). +8 +Related Work +Hallucination in Dialogue Systems Hallucination +in knowledge-grounded dialogue generation sys- +tem is an emerging area of research (Roller et al., +2021; Mielke et al., 2020; Shuster et al., 2021; +Rashkin et al., 2021b; Dziri et al., 2021a). Prior +work addressed this issue by conditioning genera- +tion on control tokens (Rashkin et al., 2021b), by +training a token level hallucination critic to identify +troublesome entities and rectify them (Dziri et al., +2021a) or by augmenting a generative model with +a knowledge retrieval mechanism (Shuster et al., +2021). Though beneficial, these models are trained +on noisy training data (Dziri et al., 2022b) which +can amplify the hallucinations further. Closest to +our work (Dziri et al., 2021a) has created a hallu- +cination critic using extrinsic-intrinsic corruption +strategies. In contrast, we create more fine-grained +corruption strategies so that hallucinated data mim- +ics the attributions of a neural chat module. +Hallucination Evaluation Recently several +benchmarks have been introduced, such as BE- +GIN(Dziri et al., 2021b), DialFact(Gupta et al., +2022), FaithDial(Dziri et al., 2022a) and At- +tributable to Identified Sources (AIS) (Rashkin +et al., 2021a) framework. Though these methods +can serve as a decent benchmarking system, their +performance in detecting entity-level hallucination +is unknown. In this work, we further contribute to +this problem by proposing an entity-level halluci- +nation detector trained on data created by various +fine-grained perturbation strategies. +9 +Conclusion +In this work, we have analyzed the modes of entity- +level fact hallucination, which is an open problem +in KG-grounded dialogue systems. Through a hu- +man feedback analysis, we demonstrate that these +KG-grounded neural generators manifest more nu- +anced hallucinations than straightforward studied +approaches. We have proposed fine-grained per- +turbation strategies to create a dataset that mimics +the real-world observations and create a series of +datasets collectively known as FADE. Our entity- +level hallucination detection model can predict hal- + +Knowledge Triples: ['Mike Zimmer', 'Sport coached', 'American +football'j] +Evidence: Dylan's Candy Bar is a chain of boutique candy +History: ['Can you tell me some information about the Minnesota +shops and candy supplier currently located in New York City +Knowledge Triples: ['The Italian Job', 'starred_actors', +'Mark Wahlberg'J] +Vikings?','TheMinnesota Vikingsarecoached byMikeZimmerand +East Hampton, New York; Los Angeles, Chicago and Miami +apart of theNational Football League.Not a bigfan though.,'Me +Beach, as well as in wholesale venues around the globe. +either . Which team do you like ?', 'My most favorite American +History: ['Do you knows who stars in The Italian Job ?'] +Football team is the Seattle Seahawks , I meant I was not a big fan of +History: ["I love candy, what's a good brand?"] +the Mlnnesota Vikings . Do you like American Football ?'l +Response: Certainly! it stars Seth Green and The +Departed. are you familiar with either? +Response:Ido like Mike Zimmer.i like the washington redskins +Response: I don't know how good they are, but Dylan's +vikings have our old qb , kurt cousins . +Candy Bar has a chain of candy shops in various cities. +Tagged Response(RoBERTa.) +): Certainly! it stars +trinsic_sofi +Seth Green and The Departed. Are you familiar with either? +Tagged Response(RoBERTa., +ve): I do like Mike Zimmer . i +):Hallucination +like the washington redskins . vikings' have our old qb , kurt cousins . +a)Intrinsic-soft:Correct +(c) FaithDial: CorrectFaithDial +BEGIN +60 +score +40 +20 +0 +10 +15 +20 +25 +30 +Training Data Size (%)lucinated entities with an F1 score of 75.59% and +classify whether an utterance is hallucinated or not +with an F1 score of 90.75%. Our models can gener- +alize well when zero-shot predictions are made on +benchmarks like BEGIN and FaithDial, indicating +our perturbation strategies’ robustness. This work +can be extended by devising more sophisticated per- +turbation mechanisms, which can simulate other +types of hallucinations. +Limitations +The major limitations of this work are as follows: +• The token-level hallucination classifier and +utterance-level hallucination classifier can +have contradictory results; however, this hap- +pens in a small percentage of data. +• Models trained on extrinsic datasets do not +generalize well on the benchmark datasets, as +the benchmark dataset contains hallucination +mostly related to the evidence provided. +Acknowledgements +We thank the anonymous reviewers for provid- +ing valuable feedback on our manuscript. This +work is partly supported by NSF grant number IIS- +2214070. The content in this paper is solely the +responsibility of the authors and does not neces- +sarily represent the official views of the funding +entity. +References +Hannah Bast, Florian Bäurle, Björn Buchhold, and El- +mar Haußmann. 2014. Easy access to the freebase +dataset. +In Proceedings of the 23rd International +Conference on World Wide Web, WWW ’14 Com- +panion, page 95–98, New York, NY, USA. Associa- +tion for Computing Machinery. +Tom B. Brown, Benjamin Mann, Nick Ryder, Melanie +Subbiah, Jared Kaplan, Prafulla Dhariwal, Arvind +Neelakantan, Pranav Shyam, Girish Sastry, Amanda +Askell, +Sandhini Agarwal, +Ariel Herbert-Voss, +Gretchen Krueger, Tom Henighan, Rewon Child, +Aditya Ramesh, Daniel M. Ziegler, Jeffrey Wu, +Clemens Winter, Christopher Hesse, Mark Chen, +Eric Sigler, Mateusz Litwin, Scott Gray, Benjamin +Chess, Jack Clark, Christopher Berner, Sam Mc- +Candlish, Alec Radford, Ilya Sutskever, and Dario +Amodei. 2020. Language models are few-shot learn- +ers. +NOAA-CIRES Climate Diagnostics Center. 2005. +Brier skill scores, rocs, and economic value dia- +grams can report false skill. +Nouha Dziri, Ehsan Kamalloo, Sivan Milton, Os- +mar Zaiane, Mo Yu, Edoardo M. Ponti, and Siva +Reddy. 2022a. Faithdial: A faithful benchmark for +information-seeking dialogue. +Nouha Dziri, Andrea Madotto, Osmar Zaïane, and +Avishek Joey Bose. 2021a. Neural path hunter: Re- +ducing hallucination in dialogue systems via path +grounding. In Proceedings of the 2021 Conference +on Empirical Methods in Natural Language Process- +ing, pages 2197–2214, Online and Punta Cana, Do- +minican Republic. Association for Computational +Linguistics. +Nouha Dziri, Sivan Milton, Mo Yu, Osmar Zaiane, and +Siva Reddy. 2022b. On the origin of hallucinations +in conversational models: Is it the datasets or the +models? +Nouha Dziri, Sivan Milton, Mo Yu, Osmar Zaiane, and +Siva Reddy. 2022c. On the origin of hallucinations +in conversational models: Is it the datasets or the +models? +Nouha Dziri, Hannah Rashkin, Tal Linzen, and David +Reitter. 2021b. +Evaluating groundedness in dia- +logue systems: The begin benchmark. +Rogério P Espíndola and Nelson FF Ebecken. 2005. +On extending f-measure and g-mean metrics to +multi-class problems. +WIT Transactions on Infor- +mation and Communication Technologies, 35. +Prakhar Gupta, Chien-Sheng Wu, Wenhao Liu, and +Caiming Xiong. 2022. DialFact: A benchmark for +fact-checking in dialogue. +In Proceedings of the +60th Annual Meeting of the Association for Compu- +tational Linguistics (Volume 1: Long Papers), pages +3785–3801, Dublin, Ireland. Association for Com- +putational Linguistics. +Joshua Maynez, Shashi Narayan, Bernd Bohnet, and +Ryan McDonald. 2020. On faithfulness and factu- +ality in abstractive summarization. In Proceedings +of the 58th Annual Meeting of the Association for +Computational Linguistics, pages 1906–1919, On- +line. Association for Computational Linguistics. +Sabrina J. Mielke, Arthur Szlam, Y-Lan Boureau, and +Emily Dinan. 2020. Linguistic calibration through +metacognition: aligning dialogue agent responses +with expected correctness. +Seungwhan Moon, Pararth Shah, Anuj Kumar, and Ra- +jen Subba. 2019. OpenDialKG: Explainable conver- +sational reasoning with attention-based walks over +knowledge graphs. In Proceedings of the 57th An- +nual Meeting of the Association for Computational +Linguistics, pages 845–854, Florence, Italy. Associ- +ation for Computational Linguistics. +Ankur Parikh, Xuezhi Wang, Sebastian Gehrmann, +Manaal Faruqui, Bhuwan Dhingra, Diyi Yang, and +Dipanjan Das. 2020. ToTTo: A controlled table-to- +text generation dataset. In Proceedings of the 2020 + +Conference on Empirical Methods in Natural Lan- +guage Processing (EMNLP), pages 1173–1186, On- +line. Association for Computational Linguistics. +Alec Radford, Jeffrey Wu, Rewon Child, David Luan, +Dario Amodei, Ilya Sutskever, et al. 2019. +Lan- +guage models are unsupervised multitask learners. +OpenAI blog, 1(8):9. +Hannah Rashkin, Vitaly Nikolaev, Matthew Lamm, +Michael Collins, Dipanjan Das, Slav Petrov, Gau- +rav Singh Tomar, Iulia Turc, and David Reitter. +2021a. +Measuring attribution in natural language +generation models. +Hannah Rashkin, David Reitter, Gaurav Singh Tomar, +and Dipanjan Das. 2021b. Increasing faithfulness +in knowledge-grounded dialogue with controllable +features. In Proceedings of the 59th Annual Meet- +ing of the Association for Computational Linguistics +and the 11th International Joint Conference on Nat- +ural Language Processing (Volume 1: Long Papers), +pages 704–718, Online. Association for Computa- +tional Linguistics. +Stephen Roller, Emily Dinan, Naman Goyal, Da Ju, +Mary Williamson, Yinhan Liu, Jing Xu, Myle Ott, +Eric Michael Smith, Y-Lan Boureau, and Jason We- +ston. 2021. +Recipes for building an open-domain +chatbot. In Proceedings of the 16th Conference of +the European Chapter of the Association for Compu- +tational Linguistics: Main Volume, pages 300–325, +Online. Association for Computational Linguistics. +Abigail See and Christopher Manning. 2021. Under- +standing and predicting user dissatisfaction in a neu- +ral generative chatbot. In Proceedings of the 22nd +Annual Meeting of the Special Interest Group on Dis- +course and Dialogue, pages 1–12, Singapore and +Online. Association for Computational Linguistics. +Kurt Shuster, Spencer Poff, Moya Chen, Douwe Kiela, +and Jason Weston. 2021. +Retrieval augmentation +reduces hallucination in conversation. In Findings +of the Association for Computational Linguistics: +EMNLP 2021, pages 3784–3803, Punta Cana, Do- +minican Republic. Association for Computational +Linguistics. +Eric Smith, Orion Hsu, Rebecca Qian, Stephen Roller, +Y-Lan Boureau, and Jason Weston. 2022. +Hu- +man evaluation of conversations is an open prob- +lem: comparing the sensitivity of various methods +for evaluating dialogue agents. In Proceedings of +the 4th Workshop on NLP for Conversational AI, +pages 77–97, Dublin, Ireland. Association for Com- +putational Linguistics. +Yi-Lin Tuan, Yun-Nung Chen, and Hung-yi Lee. +2019. DyKgChat: Benchmarking dialogue genera- +tion grounding on dynamic knowledge graphs. In +Proceedings of the 2019 Conference on Empirical +Methods in Natural Language Processing and the +9th International Joint Conference on Natural Lan- +guage Processing (EMNLP-IJCNLP), pages 1855– +1865, Hong Kong, China. Association for Computa- +tional Linguistics. +Sam Wiseman, Stuart Shieber, and Alexander Rush. +2017. +Challenges in data-to-document generation. +In Proceedings of the 2017 Conference on Empiri- +cal Methods in Natural Language Processing, pages +2253–2263, Copenhagen, Denmark. Association for +Computational Linguistics. +Thomas Wolf, Lysandre Debut, Victor Sanh, Julien +Chaumond, Clement Delangue, Anthony Moi, Pier- +ric Cistac, Tim Rault, Remi Louf, Morgan Funtow- +icz, Joe Davison, Sam Shleifer, Patrick von Platen, +Clara Ma, Yacine Jernite, Julien Plu, Canwen Xu, +Teven Le Scao, Sylvain Gugger, Mariama Drame, +Quentin Lhoest, and Alexander Rush. 2020. Trans- +formers: State-of-the-art natural language process- +ing. In Proceedings of the 2020 Conference on Em- +pirical Methods in Natural Language Processing: +System Demonstrations, pages 38–45, Online. Asso- +ciation for Computational Linguistics. +A +Definition Details +Figure 8: Extrinsic Hallucination +(a) (Extrinsic-Soft). An extrinsic-soft hallucina- +tion corresponds to an utterance that brings a new +span of text which is similar to the expected span +but does not correspond to a valid triple in Gk +c . +A hallucination is considered extrinsic when +knowledge is injected which is not authentically +captured by Gk +c . However, the injected knowledge + +HISTORY +A: Could you recommend movies similar to +the The Dark Knight ? +B: The sequel to Batman Begins is The Dark +Knight. +A: Okay . Who is the director of The Dark +Knight and any other movies from him not +related to Batman ? +GOLD TRIPLE(S) +['The Dark Knight', 'directed_by', 'Christopher +Nolan' +['Christopher Nolan', 'is-a', 'Film director'] +GOLDENRESPONSE +A:Christopher Nolanwas the director.He +also directed Insomnia and Inception . +CORRUPTEDRESPONSE(Soft) +A:Steven Spielbergwas the director .He also +directed insomnia and inception. +CORRUPTED RESPONSE(Hard) +A:Joe Bidenwas the director .He also +directed insomniaand inception. +CORRUPTEDRESPONSE(Grouped) +A:Warner Bros.was the director . He also +directed insomnia and inception .Group +Definition +Groups +1 +A person, organization, political party, or part +of a religious group can be related to each other. +"PERSON", "ORG", "NORP +2 +Location, building, airports, infrastructure +elements, countries, cities, and states can be interrelated +"LOC", "GPE", "FAC" +3 +A product, work of art, or law can be interrelated. +"PRODUCT", "WORK_OF_ART", "LAW" +Table 10: Defined groups for extrinsic-grouped hallucination +is similar to the expected entity. Identifying this +type of hallucination can be challenging due to +the high similarity between the injected and gold +knowledge. For example, in Figure 8 the dialogue +sample contains an extrinsic-soft hallucination as +the entity in response – "Steven Spielberg" is simi- +lar to "Christopher Nolan", and it is not supported +within 1-hop sub-graph. +(b) (Extrinsic-Hard). An extrinsic-hard hallucina- +tion corresponds to an utterance that brings a new +span of text which is different from the expected +span and does not correspond to a valid triple in +Gk +c . +An extrinsic-hard hallucination occurs when in- +jected knowledge is dissimilar to the expected en- +tity and is not supported within the 1-hop sub-graph. +It is easier to detect extrinsic-hard than extrinsic- +soft as the entities are fundamentally different from +the entities present in the 1-hop sub-graph. How- +ever, the entity type is retained, like an entity with a +type "person" will be replaced by the same type of +entity. Figure 8 shows an example of extrinsic-hard +hallucination, where the golden entity "Christopher +Nolan" is replaced by a different category of entity, +"Joe Biden", but the type of entity is retained. +(c) (Extrinsic-Grouped). An extrinsic-grouped hal- +lucination corresponds to an utterance that brings +a new span of text which is different from the ex- +pected span but is of a specific predefined type and +does not correspond to a valid triple in Gk +c . +Like an extrinsic-hard hallucination, extrinsic- +grouped hallucination introduces an entity that is +functionally different from the original entity and +not supported by the 1-hop sub-graph. The only +difference is that the corrupted entity is not of the +same type; instead, it is replaced by an entity of +a similar type, defined in Table 10. For example, +Figure 8 shows "Christopher Nolan" which is of +type "person" is replaced by "Warner Bros." of +type "organization". Here, the types "person" and +"organization" are placed in the same group. +(d) (Intrinsic-Soft). An intrinsic-soft hallucination +corresponds to an utterance that misuses any triple +Figure 9: Intrinsic Hallucination +in Gk +c such that there is no direct path between the +entities, but they are similar to each other. +Intrinsic hallucinations occur when the KG +triples are misused, especially in intrinsic-soft hal- +lucination an entity is selected from Gk +c which is +very similar or closely related to the original entity. +For example, in Figure 9, "Christopher Nolan" is +replaced with "The Dark Knight Rises" which is +retrieved from the 1-hop sub-graph and has close re- +lation with the original entity "Christopher Nolan". +(e) (Intrinsic-Hard). An intrinsic-hard hallucina- +tion corresponds to an utterance that misuses any +triple in Gk +c such that there is no direct path be- +tween the entities, and they are not related in any +form. +Similar to intrinsic-soft hallucination, it also mis- +uses the information in KG triples. However, the +similarity of the corrupted entity with the original +entity is relatively tiny. For example, in Figure +9, "Christopher Nolan" is replaced with "United +States of America". Although the corrupted en- +tity is drawn from Gk +c , it is very different from the +original entity. +(f) (Intrinsic-Repetitive). An intrinsic-repetitive +hallucination corresponds to an utterance that ei- +ther misuse [SBJ] or [OBJ] in Gk +c such that there +is no direct path between the entities but the entity + +HISTORY +A: Could you recommend movies similar to +the The Dark Knight ? +B:The sequel toBatman Beginsis The Dark +Knight . +A: Okay . Who is the director of The Dark +Knight and any other movies from him not +related to Batman ? +GOLD TRIPLE(S) +['The Dark Knight', 'directed_by','Christophel +Nolan'] +['Christopher Nolan', "is-a','Film director' +GOLDEN RESPONSE +A: Christopher Nolan was the director . He +also directed Insomnia and Inception . +CORRUPTEDRESPONSE(Soft) +A: The Dark Knight Riseswas the director +He also directed insomnia and inception . +CORRUPTED RESPONSE(Hard) +A: United States of America was the director . +He also directed insomnia and inception . +CORRUPTEDRESPONSE(Repetitive) +A:Batman Beginswas the director . He also +directed insomnia and inception .has previously occurred in conversational history.. +An entity from the conversational history is of- +ten repeated in the current utterances, which corre- +sponds to intrinsic-repetitive hallucination. Here, +an entity from the history which also occurs in Gk +c +and of high relatedness, is swapped with the origi- +nal entity. Figure 9 shows "Batman Begins" which +is supported by Gk +c is replaced with "Christopher +Nolan". +Figure 10: History Corrupted Hallucination +(g) (History Corrupted- Intrinsic/ Extrinsic). A +history corrupted(intrinsic/extrinsic) hallucination +corresponds to an utterance subjected to intrin- +sic or extrinsic hallucination influenced by halluci- +nated entities in conversational history. +Sometimes conversational agents are driven into +a perplexed state, and we can witness hallucina- +tions in most turns. So, this hallucinated history +can trigger hallucination in the current utterance. +This phenomenon can be seen both in extrinsic and +intrinsic forms of hallucination. Figure 10 depicts +extrinsic/intrinsic hallucination occurring in his- +tory – "The Dark Knight" is changed to "The Dark +Knight Rises" for intrinsic hallucination; similarly, +"The Dark Knight" is changed to "Spider-Man" +for extrinsic hallucination. Hallucinations in the +current utterance happen as described in previous +sections. +B +AMT Instructions +We present the screenshot of the annotation inter- +face in Figure 12, 12 and 13. Workers were paid an +average of $7-8 per hour across all tasks. We agree +that this annotation process has a high learning +curve. Even workers with high approval rates made +errors in the initial rounds of annotation. A grad- +uate computer science student manually verified +randomly selected samples and provided feedback +to the workers. Feedback was given to the workers, +especially when they selected the same answers +for ten consecutive HITS. After sending feedback +three times, all spammed HITS were discarded. +C +OpenDialKG +We use OpenDialKG (Moon et al., 2019), a +crowded-sourced English dialogue dataset where +two workers are paired together to chat about a par- +ticular topic. The first speaker is requested to start +the conversation about a given entity. The second +speaker is assigned to write an accurate response +based on facts extracted from an existing KG, Free- +base (Bast et al., 2014). The facts represent paths +from the KG that are either 1-hop or 2-hop from the +initial entity. Once the second speaker responds, +the first speaker continues discussing the topic en- +gagingly, and new multi-hop facts from the KG are +shown to the second speaker. The dialogue can +be considered as traversing multiple paths in the +KG. However, not all utterances within the same +conversation are grounded on facts from the KG. +The second speaker can decide not to select a path +from the KG to form an answer and instead forms +a "chit-chat" response. Overall, the dataset consists +of four domains: movie, music, sport, and book, +where each second speaker’s utterance is annotated +with paths from the KG. The KG corresponds to an +extensive subgraph extracted from Freebase with +∼ 1.2M triples (subject, predicate, object), ∼ 101k +distinct entities, and 1357 distinct relations. We use +77,430 data points in the dataset for constructing +FADE. +D +Perturbation Hyper-parameters +D.1 +Search Index Details +We use Solr in case of extrinsic hallucination. +We use the BM25 index, defined by the class +solr.BM25SimilarityFactory. We man- +ually labeled 50 data points(for the entity type +PERSON) for tuning the indexes through grid + +HISTORY (CORRUPTED) +A:Couldyou recommendmovies similarto +the The Dark Knight ? +B: The sequel to [The Dark Knight →The Dark +Knight Rises(Int.)] [The Dark Knight → +Spider-Man(Ext.)] is Batman Begins . +A: Okay . Who is the director of The Dark +Knight and any other movies from him not +related to Batman ? +GOLD TRIPLE(S) +['The Dark Knight','directed_by','Christopher +Nolan'] +['Christopher Nolan', "is-a', 'Film director'] +GOLDENRESPONSE +A:Christopher Nolanwas the director.He +also directed Insomnia and Inception +CORRUPTED RESPONSE(Intrinsic) +A:United States of Americawas the director. +He also directed insomnia and inception . +CORRUPTED RESPONSE(Extrinsic) +A:Joe Bidenwasthedirector.He also +directed insomnia and inception .Figure 11: Annotation interface for human feedback analysis(Instructions, part 1) +Figure 12: Annotation interface for human feedback analysis(Instructions, part 2) + +Please state if the response contains irrelevant phrase(s) or not. If yes, +then, please select its type and note down the phrase +We have provided you with some knowledge paths and conversational history. In the given response, +phrase. Examples of each type of error are provided below: +Conversation history: +Speaker A: Could you recommend movies similar to The Dark Knight? +Speaker B: The sequel to Batman Begins is The Dark Knight. +Speaker A: Okay . Who is the director of The Dark Knight and any other movies from him not related to Batman? +Knowledge paths: +Path 1: ['The Dark Knight', 'directed_by', 'Christopher Nolan'] +Path 2: ['Christopher Nolan', 'is-a', 'Film director'] +Golden Response(this is for reference, it does not appear in the real data): +Speaker B: Christopher Nolan was the director. He also directed Insomnia and Inception. +Extrinsic Hallucinations: +Extrinsic soft: When an irrelevant phrase is introduced which is similar to the expected phrase but the +phrase does not appear in the knowledge paths. For example, +Speaker B: Steven Spielberg was the director. He also directed Insomnia and Inception +Extrinsic hard: When an irrelevant phrase is introduced which is not similar to the expected phrase and the +phrase does not appear in the knowledge paths. For example, +Speaker B: Joe Biden was the director. He also directed Insomnia and Inception. +Extrinsic grouped: When an irrelevant phrase is introduced which is related to the expected phrase but the +phrasedoesnotappearintheknowledgepaths.Forexample, +SpeakerB:WarnerBros_wasthedirector.HealsodirectedInsomniaandInception +Valid relations: +· +Aperson,organization,political party,or part of a religiousgroup can be related to each other +Location, building, airports, infrastructure elements, countries, cities, and states can be interrelated. +Aproduct,workofart,orlaw canbeinterrelated.Intrinsic Hallucinations: +Intrinsic soft: When an irrelevant phrase is introduced which is similar to the expected phrase and the +Speaker B: The Dark Knight was the director. He also directed Insomnia and Inception. +Intrinsic hard: When an irrelevant phrase is introduced which is not similar to the expected phrase and the +phrase does appear/ or is related to the knowledge paths. For example, +Speaker B: United States of America was the director. He also directed Insomnia and Inception. +(Christopher Nolan is a citizen of the United States of America) +Intrinsic repetitive: When an irrelevant phrase is introduced which is related to the expected phrase, +appears in conversational history, and the phrase does appear/ or is related to the knowledge paths. For +example, +SpeakerB:BatmanBegins_wasthedirector.Healsodirected InsomniaandInception +History Corrupted Hallucinations: +Corrupted Conversation history: +Speaker A: Could you recommend movies similar to The Dark Knight? +Speaker B: The sequel to The Dark Knight Rises is Spider-Man. +Speaker A: Okay . Who is the director of The Dark Knight and any other movies from him not related to Batman? +Now consider this conversation history, if you look closely, the second turn is corrupted with irrelevant +entities. +History corrupt intrinsic: When an irrelevant phrase is introduced which is of any type of intrinsic +hallucination AND the conversation history is corrupted. For example, +Speaker B: The Dark Knight was the director. He also directed Insomnia and Inception. +History corrupt extrinsic: When an irrelevant phrase is introduced which is of any type of intrinsic +hallucination AND the conversation history is corrupted. For example, +SpeakerB:WarnerBroswasthedirector.He also directed Insomnia and Inception.Figure 13: Annotation interface for human feedback analysis(example annotation, workers were ask to find up to 3 spans if +hallucinations are found in the data) +search. Grid-search conditions were as follows: +b was varied from 0.3 to 0.9 with a step of +0.1 and k1 was varied from 0.8 to 2.0 with a +step of 0.2. Following grid search, an optimum +MAP score of 0.789 was found, with b = 0.9 +and k1= 1.6. For the dynamic indexes that were +created in the case of intrinsic hallucination, we +use the python library https://github.com/ +dorianbrown/rank_bm25 with default con- +figurations. +D.2 +Free parameter & β optimization +We use a free term weight parameter(ε) in in- +trinsic hallucination to represent the queries and +nodes. Similar to extrinsic hallucination we man- +ually annotated 50 data-points and ran grid search +for ε ∈ {10−i, 2 × 10−i; i ∈ {1, 5}}, and found +ε = 2×10−4 to be the optimum value. We used the +same technique for optimizing β, and the search +space ranged from 0.1 to 0.7 with a step of 0.05. +D.3 +KG embeddings +We follow the same approach (Dziri et al., 2021a) +for generating the KG embeddings. OpenDialKG +Dataset Type +Ext-Soft(%) +Ext-Hard(%) +Ext-Grp(%) +Int-Soft(%) +Int-Hard(%) +Int-Rep(%) +HC-Ext(%) +HC-Int(%) +N-Halluc(%) +Observed +12.495 +6.4425 +1.04 +0.92 +1.025 +1.7 +2.4575 +1.4575 +72.4625 +Balanced +6.25 +6.25 +6.25 +6.25 +6.25 +6.25 +6.25 +6.25 +50 +Extrinsic+ +12.5 +9.375 +9.375 +6.25 +6.25 +6.25 +6.25 +6.25 +37.5 +Intrinsic+ +6.25 +6.25 +6.25 +9.375 +9.375 +9.375 +6.25 +6.25 +40.625 +Table 11: Mixing ratios for different datasets +triples are also represented using a textual term +called "render". For the triples containing this term, +we pass it through to GPT2 and then extract hidden +state representations for each entity’s word piece +and finally obtain a final representation by applying +a MaxPool over the hidden representations. For +entity mentions not described in “render”, we get +their representations directly from the last hidden +states in GPT2. +D.4 +Mixing Ratios +Mixing ratios for creating the mixed datasets are +defined in Table 11. Perturbed and non-perturbed +samples are drawn randomly from component +datasets. +E +Implementation Details +The utterance and token level classifier are imple- +mented using the Pytorch Huggingface Transform- +ers library (Wolf et al., 2020). The following con- + +Now complete the following task: +Knowledge paths: +Path 1: ['Gautam Gambhir', 'is-a', 'Athlete'] +Path 2: ['Athlete', '~is-a', 'Venus Williams']] +Conversation history: +Speaker A: What do you think about Gautam Gambhir Indian cricketer ? +Response: +Speaker B: to be honest, I don't really know anything about him. I'm more of a tennis fan . one of my favorite players is Gautam +Gambhir +Does the response contain irrelevant phrase(s)? +O Yes +O No +If yes, then write down the irrelevant phrases(s) and select their type(up to 3): +irrelavant phrase(s) +Type: +O extrinsic_soft +Oextrinsic_hard +Oextrinsic_grouped +O intrinsic_soft +O intrinsic_hard +Oextrinsic_history_corrupt +O intrinsic_history_corrupt +irrelavant phrase(s) +Type: +O extrinsic_soft +O extrinsic_hard +Oextrinsic_grouped +O intrinsic_soft +O intrinsic_hard +Oextrinsic_history_corrupt +O intrinsic_history_corrupt +irrelavant phrase(s) +Type: +O extrinsic_soft +O extrinsic_hard +O extrinsic_grouped +O intrinsic_soft +O intrinsic_hard +Oextrinsic_history_corrupt +O intrinsic_history_corrupt +SubmitHyperparameter +Value +train_batch_size +12 +gradient_accumulation_steps +2 +num_train_epochs +4(Token)/10(Utt) +weight_decay +0.01 +warmup_proportion +0.1 +learning_rate +1e-5 +adam_epsilon +1e-8 +max_grad_norm +1 +eval_batch_size +18 +Table 12: RoBERTa-base hyper parameters +Hyperparameter +Value +train_batch_size +12 +gradient_accumulation_steps +2 +num_train_epochs +4(Token)/10(Utt) +weight_decay +0.01 +warmup_proportion +0.1 +learning_rate +2e-5 +adam_epsilon +1.5e-8 +max_grad_norm +1 +eval_batch_size +18 +Table 13: RoBERTa-large hyper parameters +figuration were found to be best performing for +each models, as shown in Table 12, 13, 14 and +15. The models were trained in a single NVIDIA +A5000 GPU, the average running time for the base +models were 2.5 hours, and for the large model was +∼ 5 hours. +F +Supplementary results +We report metrics for all the models trained using +25% of the dataset, for component datasets in Table +16 and mixed datasets in Table 17. +Hyperparameter +Value +train_batch_size +12 +gradient_accumulation_steps +2 +num_train_epochs +4(Token)/10(Utt) +weight_decay +0.01 +warmup_proportion +0.1 +learning_rate +5e-5 +adam_epsilon +1e-8 +max_grad_norm +1 +eval_batch_size +18 +Table 14: BERT-base-uncased hyper parameters +Hyperparameter +Value +train_batch_size +12 +gradient_accumulation_steps +2 +num_train_epochs +4(Token)/10(Utt) +weight_decay +0.01 +warmup_proportion +0.1 +learning_rate +5e-5 +adam_epsilon +1e-8 +max_grad_norm +1 +eval_batch_size +18 +Table 15: XLNet-base hyper parameters + +Dataset +Best Model +Token Level +Utterance Level +F1 +P +R +F1 +P +R +G-Mean +BSS +AUC +extrinsic_hard +roberta-base +0.70613382 +0.68956357 +0.72352004 +0.86181139 +0.83985441 +0.88494727 +0.93029609 +0.03277357 +0.93145803 +extrinsic_grouped +roberta-base +0.7986706 +0.77534593 +0.82344214 +0.90499405 +0.89090483 +0.91953606 +0.93487266 +0.0589842 +0.93500056 +intrinsic_hard +roberta-base +0.84409519 +0.84717262 +0.84104004 +0.90789771 +0.92741563 +0.8891844 +0.93192336 +0.04522725 +0.9329505 +intrinsic_soft +roberta-base +0.78797921 +0.80193163 +0.774504 +0.87102229 +0.90540109 +0.83915877 +0.90255779 +0.06217348 +0.90495271 +intrinsic_repetitive +roberta-base +0.82702178 +0.82759578 +0.82644857 +0.88005638 +0.89506881 +0.86553923 +0.92305012 +0.03146957 +0.92496078 +intrinsic_history_corrupt +roberta-base +0.83406626 +0.82763636 +0.84059684 +0.90857229 +0.92340555 +0.89420804 +0.93381877 +0.04511612 +0.93469609 +extrinsic_history_corrupt +roberta-base +0.72010547 +0.71212516 +0.72826667 +0.87400219 +0.85486834 +0.89401217 +0.93612638 +0.02971357 +0.93711831 +extrinsic_soft +roberta-base +0.60045426 +0.60811376 +0.59298532 +0.72017689 +0.74873563 +0.69371672 +0.81231271 +0.09344656 +0.82245014 +extrinsic_hard +bert-base-uncased +0.71146832 +0.72259569 +0.70067846 +0.88285121 +0.88299233 +0.88271013 +0.93232489 +0.02705296 +0.93371925 +extrinsic_grouped +bert-base-uncased +0.80688364 +0.8056026 +0.80816875 +0.91302235 +0.9180408 +0.90805848 +0.93577473 +0.05285693 +0.93619772 +intrinsic_hard +bert-base-uncased +0.83328471 +0.82308025 +0.84374538 +0.91416629 +0.92395896 +0.90457903 +0.93917074 +0.04259417 +0.93983215 +intrinsic_soft +bert-base-uncased +0.75277325 +0.79087205 +0.71817644 +0.85483616 +0.91836735 +0.79952621 +0.88349437 +0.06794858 +0.88790364 +intrinsic_repetitive +bert-base-uncased +0.7481198 +0.71392596 +0.78575388 +0.84134941 +0.82295256 +0.86058758 +0.91436157 +0.04330141 +0.9160416 +intrinsic_history_corrupt +bert-base-uncased +0.82318199 +0.8229997 +0.82336435 +0.90891209 +0.9316067 +0.8872969 +0.93164021 +0.04459424 +0.93274826 +extrinsic_history_corrupt +bert-base-uncased +0.67029785 +0.69294369 +0.64908533 +0.87358552 +0.88214169 +0.86519372 +0.92312672 +0.02886461 +0.9250663 +extrinsic_soft +bert-base-uncased +0.64089366 +0.6922167 +0.59665579 +0.7480315 +0.81958894 +0.68796592 +0.81616138 +0.08033967 +0.82810534 +extrinsic_hard +xlnet-base-cased +0.72115512 +0.71982018 +0.72249502 +0.8736255 +0.8712651 +0.87599872 +0.92800607 +0.02926739 +0.92954989 +extrinsic_grouped +xlnet-base-cased +0.78452923 +0.77288925 +0.79652518 +0.89920345 +0.8915677 +0.90697112 +0.92895654 +0.06212166 +0.92922301 +intrinsic_hard +xlnet-base-cased +0.84443122 +0.85082459 +0.83813322 +0.90878914 +0.92875867 +0.88966027 +0.93238499 +0.04477944 +0.93341088 +intrinsic_soft +xlnet-base-cased +0.76722735 +0.80484632 +0.73296801 +0.85379657 +0.90941058 +0.80459259 +0.88491991 +0.06892207 +0.88892969 +intrinsic_repetitive +xlnet-base-cased +0.7941989 +0.79135701 +0.79706127 +0.86978508 +0.88154897 +0.85833102 +0.91820183 +0.03428001 +0.9202899 +intrinsic_history_corrupt +xlnet-base-cased +0.83667247 +0.82269807 +0.85112982 +0.91298209 +0.91864812 +0.90738552 +0.9396723 +0.04337198 +0.94024672 +extrinsic_history_corrupt +xlnet-base-cased +0.72378159 +0.72354039 +0.72402294 +0.88100942 +0.87862377 +0.88340807 +0.93239789 +0.02749991 +0.93375627 +extrinsic_soft +xlnet-base-cased +0.60896216 +0.63207547 +0.58747961 +0.73844753 +0.79296016 +0.69094782 +0.81535862 +0.08484401 +0.82655921 +Table 16: All models benchmark (numbers in fractions) for component datasets, models trained on 25% of the total dataset. +Dataset +Best Model +Token Level +Utterance Level +F1 +P +R +F1 +P +R +G-Mean +BSS +AUC +balanced +roberta-base +0.73405875 +0.68751809 +0.78735795 +0.882424 +0.83853553 +0.93116042 +0.86213807 +0.131385 +0.86469621 +observed +roberta-base +0.62554537 +0.59004757 +0.66558773 +0.77904114 +0.73266454 +0.83168565 +0.85077041 +0.14126728 +0.85098938 +extrinsic_plus +roberta-base +0.74849152 +0.71339648 +0.78721816 +0.90921175 +0.87804878 +0.94266814 +0.84332203 +0.12278872 +0.84855698 +intrinsic_plus +roberta-base +0.75045075 +0.71112613 +0.79437919 +0.90157054 +0.86518353 +0.94115257 +0.84511316 +0.12778319 +0.85001331 +balanced +bert-base-uncased +0.6570643 +0.57930535 +0.7589345 +0.85119497 +0.78285516 +0.9326075 +0.81309735 +0.17265032 +0.82075474 +observed +bert-base-uncased +0.59965325 +0.52847854 +0.6929832 +0.76124302 +0.67629046 +0.87060443 +0.84589508 +0.16352531 +0.84624573 +extrinsic_plus +bert-base-uncased +0.72993044 +0.663004 +0.81188563 +0.90179749 +0.84940317 +0.96108049 +0.8086632 +0.1365238 +0.82074909 +intrinsic_plus +bert-base-uncased +0.71653573 +0.65640721 +0.78879093 +0.89301716 +0.84373548 +0.94841293 +0.82126564 +0.14130552 +0.82978853 +balanced +xlnet-base-cased +0.71863497 +0.66214437 +0.78566356 +0.87222741 +0.81850039 +0.93350331 +0.84619893 +0.14481173 +0.85028143 +observed +xlnet-base-cased +0.63436089 +0.57976023 +0.70031519 +0.77706573 +0.71053723 +0.85733951 +0.8540217 +0.14730018 +0.85402812 +extrinsic_plus +xlnet-base-cased +0.75593757 +0.7079124 +0.81095307 +0.90747949 +0.86768256 +0.95110254 +0.83209459 +0.12649216 +0.8395401 +intrinsic_plus +xlnet-base-cased +0.74488988 +0.68995602 +0.80932808 +0.90141776 +0.85748704 +0.95009285 +0.83869733 +0.129222 +0.84522772 +Table 17: All model benchmark (numbers in fractiom) for mixed datasets, models trained on 25% of the total dataset. + diff --git a/CNE3T4oBgHgl3EQfUQpw/content/tmp_files/load_file.txt b/CNE3T4oBgHgl3EQfUQpw/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..617bd32b865c2f401e147794a09d93006c369a56 --- /dev/null +++ b/CNE3T4oBgHgl3EQfUQpw/content/tmp_files/load_file.txt @@ -0,0 +1,1303 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf,len=1302 +page_content='Diving Deep into Modes of Fact Hallucinations in Dialogue Systems Souvik Das Sougata Saha Rohini K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Srihari {souvikda, sougatas, rohini}@buffalo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='edu Department of Computer Science and Engineering, University at Buffalo, NY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Abstract Knowledge Graph(KG) grounded conversa- tions often use large pre-trained models and usually suffer from fact hallucination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Fre- quently entities with no references in knowl- edge sources and conversation history are in- troduced into responses, thus hindering the flow of the conversation—existing work at- tempt to overcome this issue by tweaking the training procedure or using a multi-step refin- ing method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' However, minimal effort is put into constructing an entity-level hallucination detection system, which would provide fine- grained signals that control fallacious content while generating responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' As a first step to address this issue, we dive deep to iden- tify various modes of hallucination in KG- grounded chatbots through human feedback analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Secondly, we propose a series of perturbation strategies to create a synthetic dataset named FADE (FActual Dialogue Hal- lucination DEtection Dataset)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Finally, we conduct comprehensive data analyses and cre- ate multiple baseline models for hallucination detection to compare against human-verified data and already established benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' 1 Introduction Knowledge-grounded conversational models often use large pre-trained models (Radford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' These models are notorious for producing responses that do not comply with the provided knowledge;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' this phenomenon is known as hallucination (Dziri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=', 2022b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Rashkin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=', 2021b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Faithfulness to the supplementary knowl- edge is one of the prime designing factors in these knowledge-grounded chatbots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' If a response is unfaithful to some given knowledge, it becomes uninformative and risks jeopardizing the flow of the conversation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Despite retaining strong linguis- tics abilities, these large language models(LM) in- adequately comprehend and present facts during 1https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='com/souvikdgp16/FADE conversations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' LMs are trained to emulate distribu- tional properties of data that intensify its hallucina- tory attributes during test time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Figure 1: Hallucination manifested by generated responses using GPT2(Radford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=', 2019) trained on KG triples can be more nuanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' On the one hand, many prior works (Wiseman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Parikh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Tuan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=', 2019) have suggested training these models on external data to ensure faithfulness may lead to a source- reference divergence problem, where the reference contains additional factual information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' To ad- dress this problem holistically, Dziri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' has proposed a two-step generate-then-refine approach by augmenting conventional dialogue generation with a different refinement stage enabling the di- alogue system to correct potential hallucinations by querying the KG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Also, this work employs a token-level hallucination classifier trained on a syn- thetic dataset constructed using two perturbation strategies 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Though this method has clear benefits, the hallucination perturbation strategies proposed in this work might fail to capture some of the sub- tle attributions of a factual generative model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' As illustrated in Figure 1, neural models can inject hal- lucinated entities into responses that are present in the k-hop KG and are deceptively similar to what is expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Also, if we cannot detect these elusive hallucinations beforehand, it will cause a cascad- ing effect and amplify hallucinations in subsequent turns (See and Manning, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' 2(1) Extrinsic perturbation: Dziri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' have swapped an entity with a different entity of the same type and not present in 1-hop subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' (2) Intrinsic perturbation: they have swapped an entity with its object or vice versa, taken from the golden 1-hop subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='04449v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content="CL] 11 Jan 2023 Path(s): T, :['Outlander', 'written_by','Diana Gabaldon'] [Gold] T, :['Outlander','publication_date','1st June'] [Retrieved from 1-hop KG] T, :['Outlander', 'published_by','Dell Publishing'l [Retrieved from 1-hop KG] History: ['Do you like the book Outlander ?" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content="'] GPT2 Response: “I've never read it, but I know it was written by Dell PublishingOn the other hand, relying on human annotations is challenging due to error-prone collection proto- cols and human ignorance to complete the tasks with care (Smith et al." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Prior research (Dziri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=', 2022c) shows that knowledge-grounded con- versational benchmarks contain hallucinations pro- moted by a design framework that encourages infor- mativeness over faithfulness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' As studied by Dziri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=', when the annotators are asked to identify hallucination in a response, there is a high chance of error due to lack of incentive, personal bias, or poor attention to the provided knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' See and Manning have studied different short- comings in a real-time neural model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' In this work, based on some of the findings of See and Manning, like repetitive and unclear utterances promoting hallucination, we extend the already defined modes of hallucinations (Maynez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Dziri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=', 2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Our contributions to this work are three- fold: We extend fact hallucination in KG-grounded dialogue systems into eight categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' To understand the degree to which our defined classes exist in real-life data, we conduct a sys- tematic human evaluation of data generated by a state-of-the-art neural generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Since human annotation is expensive and of- ten inaccurate, we design a series of novel perturbation strategies to simulate the de- fined ways of fact hallucinations and build a set of synthetic datasets collectively named as FADE (FActual Dialogue Hallucination DEtection Dataset).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' We create multiple pre-trained model-based baselines and compare the performances on several constituent and mixed datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' To assess our dataset’s generalization capability, we perform zero-shot inference on BEGIN (Dziri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=', 2021b), and FaithDial (Dziri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=', 2022a) datasets, which encompasses all cate- gories of hallucinated responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' 2 Different Modes of Hallucination in KG-grounded Dialogue Systems 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='1 Background We focus on the task of detecting halluci- nated spans in dialogues that are factually grounded on factoids derived from multi-relational graphs G = (V, E, R), termed as Knowledge- Graphs(KG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Each KG consists of an directed edge triples t = ⟨[SBJ], [PRE], [OBJ]⟩, where [SBJ], [OBJ] ∈ V are nodes denoting subject and object entities and [PRE] ∈ R is a predicate which can be understood as a relation type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Primar- ily, a neural dialogue system is guilty of generating hallucinated text when a valid path in the k-hop sub-graph Gk c ∈ G of the original KG anchored around a context entity c does not support it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Our study extends the work of (Dziri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=', 2021a) where they specifically explore two broad circumstances – extrinsic and intrinsic to the pro- vided KG, under which LMs are likely to exhibit unfaithful behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Though this categorization is beneficial for detecting hallucinations, these cate- gories can be further subdivided into subcategories, which are described in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='2 Base Dataset We use OpenDialKG (Moon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=', 2019), a crowded-sourced English dialogue dataset where two workers are paired to chat about a particular topic(mainly movie, music, sport, and book).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' We use this dataset for training a GPT2-based model for generating data for human feedback analysis and creating the perturbed datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' More details about the dataset can be found in §C 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='3 Definitions We define below several categories of fact halluci- nation, comprehensive illustrations of each types are provided in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' In addition we have in- cluded detailed descriptions of each definitions in §A (a) (Extrinsic-Soft).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' An extrinsic-soft hallucina- tion corresponds to an utterance that brings a new span of text which is similar to the expected span but does not correspond to a valid triple in Gk c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' (b) (Extrinsic-Hard).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' An extrinsic-hard halluci- nation corresponds to an utterance that brings a new span of text which is different from the expected span and does not correspond to a valid triple in Gk c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' (c) (Extrinsic-Grouped).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' An extrinsic-grouped hallucination corresponds to an utterance that brings a new span of text which is different from the expected span but is of a specific predefined type and does not correspond to a valid triple in Gk c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' (d) (Intrinsic-Soft).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' An intrinsic-soft hallucina- tion corresponds to an utterance that misuses any triple in Gk c such that there is no direct path be- tween the entities but they are similar to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' (e) (Intrinsic-Hard).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' An intrinsic-hard hallucina- tion corresponds to an utterance that misuses any Figure 2: Illustration of our defined categories of fact hallucinations in KG-grounded dialogue systems triple in Gk c such that there is no direct path be- tween the entities and they are not related in any form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' (f) (Intrinsic-Repetitive).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' An intrinsic-repetitive hallucination corresponds to an utterance that ei- ther misuses [SBJ] or [OBJ] in Gk c such that there is no direct path between the entities but the entity has previously occurred in conversational history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='. (g) (History Corrupted- Intrinsic/ Extrinsic).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' A history corrupted(intrinsic/extrinsic) hallucination corresponds to an utterance that is subjected to intrinsic or extrinsic hallucination which is influ- enced by hallucinated entities in conversational history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='4 Human Feedback Analysis To study the extent to which the previously de- scribed modes of hallucination exist in a real-world system, we did human feedback analysis on re- sponses generated using a GPT2-based generative model fine-tuned on OpenDialKG as described by Dziri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='. We sampled 200 responses each from four different decoding strategies, Greedy, Beam Search, and Nucleus Sampling, with a prob- ability of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='9 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' For each dialogue in- stance, we crowd-source human judgment by solic- iting evaluations from 2 different annotators(with a high approval rating) from Amazon Mechanical Turk(AMT)(Details in §B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' One computer science graduate student additionally verified the Human Intelligence Task (HITS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' For examples where hal- lucination was present, we asked the workers to identify the type of hallucination(examples of dif- ferent types of hallucinations were shown in the GPT2-KG Greedy Beam Search Nucleus 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='9 Nucleus 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='5 Extrinsic-Soft 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='91 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='8 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='5 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='77 Extrinsic-Hard 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='45 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='22 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='3 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='8 Extrinsic-Grouped 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='12 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='44 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='6 History Corrupted-Extrinsic 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='33 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='1 Intrinsic-Soft 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='3 Intrinsic-Hard 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='1 2 Intrinsic-Repetitive 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='8 4 History Corrupted-Intrinsic 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='33 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='3 Extrinsic Total 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='78 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='12 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='57 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='27 Intrinsic Total 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='48 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='03 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='6 Total 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='08 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='6 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='6 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='87 Table 1: Fine-grain human feedback analysis instruction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' The result of the human feedback is exhibited in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' We rejected 21% of the HITS because of poor quality;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' we reported the average Krippendorf alpha coefficient to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='74 on the remaining annotations, indicating a moderate to a high agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Using Table 1 we made these observations: Extrinsic-soft hallucination is the dominant form of hallucination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Also, this bolsters our prior observation that LMs generate entities similar to the golden entity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Comparatively less amount of hallucinations was seen in responses generated using beam search decoding scheme, though the percent- age of extrinsic-hard hallucination was higher than greedy decoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Intrinsic-hard hallucination appears to be the least among all types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' This suggests LM will always try to learn something from the given KG triples;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' generating something dissimilar will have a very low probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' 3 Dataset Creation FADE is a collection of datasets consisting of com- ponent datasets created using several perturbations Anchor Entity(c) HISTORY GOLDEN RESPONSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' HISTORY (CORRUPTED) 1-Hop KG A: Could you recommend movies B: Christopher Nolan was the director .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' A: Could you recommend movies similar to similar to The Dark Knight ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' He also directed Insomnia and The Dark Knight ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Inception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' B: The sequel to [The Dark Knight -→ The B: The sequel to Batman Begins is The GOLD TRIPLE(S) Dark Knight Rises(Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=')] [The Dark Knight - Dark Knight .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Spider-Man(Ext.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=')] is Batman Begins .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=" ['The Dark Knight', 'directed_by', A: Okay ." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=" Who is the director of The Christopher Nolan'] A: Okay ." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=" Who is the director of The Dark Dark Knight and any other movies from DarkKnight ['Christopher Nolan', 'is-a', 'Film Knight and any other movies from him not him not related to Batman ?" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=" director'l related to Batman ?" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Perturbed Entity PERTURBED RESPONSE(Soft) PERTURBED RESPONSE(Soft) PERTURBED RESPONSE(Intrinsic) B:Steven Spielberg was the director B: The Dark Knight RisesWas the B: The Dark Knight Rises was the He also directed insomnia and director .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' He also directed insomnia director .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' He also directed insomnia inception and inception and inception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' PERTURBED RESPONSE(Hard) PERTURBED RESPONSE(Hard) PERTURBED RESPONSE(Extrinsic) B: Joe Biden was the director .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' He also B: United States of America was the B: Steven Spielberg was the director .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' directedinsomnia and inception director .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' He also directed insomnia He also directed insomnia and and inception inception : PERTURBED RESPONSE(Grouped) PERTURBEDRESPONSE(Repetitive) B: Warner Bros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' was the director .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' He B: Batman Begins was the director .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' He also directed insomnia and inception also directed insomnia and inception (a) Extrinsic Hallucination Types (b) Intrinsic Hallucination Types (c) History Corrupt Hallucination TypesHallucination Type Index Type Selection Criteria Soft Same as original entity ei with max document score Hard Same as original entity ei with min document score Grouped Same as one predefined type,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' selected randomly same as soft Table 2: Extrinsic hallucination perturbed entity selection criteria and a set of mixed datasets constructed using the component datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='1 Perturbation Strategies Extrinsic Hallucination All the entities present in OpenDialKG undergo a indexing process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' At first, using Spacy we determine the named entity type 3 for each entity, and create BM25 indexes4 for each entity type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Each KG triple corresponding to an en- tity is represented in this format – "[SBJ] [PRE] [OBJ]" and denoted as ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Now, for an entity(ei) we create a document di = concat(t1, t2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='.tn), n is the number of KG-triples for that entity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Af- ter this, we index di and ei in the index corre- sponding to the entity type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' During the perturba- tion process, we retrieve all the KG-triples for the entity we want to perturb and form 3 queries for each triple by permuting ([SBJ],[PRE],[OBJ]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Then based on the type of extrinsic halluci- nation, we query the indices to get the docu- ment scores in the following way: scores = average({BM25(qi, dj)}i∈(s,r,o),j∈(0,n)), the se- lection criteria of the perturbed entities are pro- vided in table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' The groups for extrinsic-grouped hallucination are mentioned in Table 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' During the selection process, we iteratively check whether the perturbed entity exists in the conversation history, matches with the actual entity, and has appeared in the 1-hop sub-graph of the original entity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' If an occurrence is found, we proceed to the following best entity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Intrinsic Hallucination Here, we dynamically create a BM25 index and index all the KG triples in the 1-hop sub-graph of the original entity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Again, a KG triple is represented in the same fashion as in extrinsic hallucination – "[SBJ] [PRE] [OBJ]".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' The goal here is to select entities that are similar or dissimilar to the original entities and present in the 1-hop graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' To achieve that, we follow a hy- brid triple retrieval approach to score each triple associated with the original entity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' First, we use the final hidden layer of a pre-trained GPT2 to obtain initial embeddings for each node in Gk c (for details, check §D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' A query is formed by using Equa- 3https://spacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='io/api/entityrecognizer 4https://solr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='apache.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='org/ Hallucination Type Selection Criteria Soft [SUB] or [OBJ] with max triple score Hard [SUB] or [OBJ] with min triple score Repetitive same as soft, should be occurring in the conversation history Table 3: Intrinsic hallucination perturbed entity selection cri- teria tion 1 each triple in Gk c is scored using a similarity scoring system as described in Equation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' q = � i∈{s,r,o} ε p(qi) + ε vqi (1) Here ε is a free term parameter (§D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='2), p(qi) is unigram probability of the query term and vqi is the embedding for each query term(here query terms are [SBJ], [PRE] ,[OBJ] of the original entity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' ni = ε p(s) + ε vs + ε q(r) + ε vr + ε p(o) + ε vo (2) ni in Equation 2 represents a triple embedding in Gk c , when q(r) represents the rarity of the rela- tionship term in the subgraph, high occurrence is penalized, rest terms are analogous to Equation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' EntitySimilarity(Q, t) = cos(q, ni) (3) Now, we query the BM25 index that we have created before with a simple query using the orig- inal triple: "[SBJ] [PRE] [OBJ]" and get the score for each of the triple(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Finally, we get the final scores using Equation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Score(Q, t) = βEntitySimilarity(Q, t) +(1 − β)BM25(Q, t) (4) Here 0 < β < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' We select the perturbed entities based on the scores and selection criteria as defined in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Like extrinsic hallucinations, we iteratively filter the best-scored entity until it does not match the original entity or appears in history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' History Corrupted Hallucination Conversa- tional history is corrupted using intrinsic or extrin- sic corruption strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' We select the last k turns of the conversation and randomly perturb the entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' We also ensure that at least 50% of the previous k turns are corrupted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='2 Dataset Analysis Below we provide data statistics and character- ize the composition and properties of the datasets that are generated using our proposed perturbation strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Type Perturbed Non-perturbed Turn with perturbation>2 soft 12752 64634 558 hard 8540 68872 8254 grouped 22858 54542 11296 history-corrupt 8534 68878 8247 Table 4: Extrinsic hallucination data statistics Type Perturbed Non-perturbed Turn with perturbation>2 soft 18560 58558 5 hard 18605 58534 6 repetitive 9712 67560 0 history-corrupt 18597 58542 6 Table 5: Intrinsic hallucination data statistics 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='1 Data Statistics Table 4 and 5 shows the statistics of datasets created using different perturbation strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' The base dataset contains 77,430 data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' However, the perturbed turns in each of these datasets are quite low in comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' This low number is because not every entity in an utterance has a valid KG path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' For extrinsic hallucination, ∼12,000 to ∼23,000 utterances were perturbed, and ∼550 to ∼11,300 utterances have multiple perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' The num- ber of perturbed data points for intrinsic hallucina- tion is less than extrinsic(∼9,000 to ∼18,000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' The number of utterances with multiple perturbations is negligible due to the many checks the perturbed entities go through(for example, whether the KG path is present, has already occurred or not, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=') To train and evaluate models, we vary the size of the train split in this range of 10% to 30%5 with a step of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='5%, keeping in mind to avoid overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' The remaining data is split into equal halves for validation and testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='2 Parsing Features In Figure 3 we show the top 10 Named Entity Recognition(NER) tags as identified by the Spacy library in extrinsic hallucinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' For extrinsic- soft hallucination, most NER tags are of type PER- SON.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' This corresponds to the fact that the original entities in the base dataset are primarily related to movies, books, and music.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' In extrinsic-soft halluci- nation, the associated PERSON name is changed to a closely affiliated person, or a movie name is changed to its director’s name.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' In contrast, the dis- tribution of NER tags is uniform for extrinsic-hard hallucination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Figure 4 and 5 shows the top-10 rela- tions of the perturbed entity with the original entity in both intrinsic-soft and hard hallucinations and the corresponding value in their counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' In intrinsic-soft hallucination, more relevant relations are selected like "release year", "starred actors", "written by", etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' On the other hand, in intrin- 5sequential split Figure 3: NER distribution in Extrinsic-soft and hard halluci- nation sic hard hallucination, more unusual relations like "Country of Origin", and "Country of Nationality" were among the top relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Figure 4: Top 10 relation in perturbed KG triples in intrinsic- soft hallucination Figure 5: Top 10 relation in perturbed KG triples in intrinsic- hard hallucination 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='3 Mixing Datasets Since in actual data, all kinds of hallucinations are expected to occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' We mix the previously con- structed datasets in specific proportions to create a more challenging dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Table 11 shows the dif- ferent mixing ratios for four types of datasets is as follows: Observed: We try to mimic the observed data, which is shown in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='4, we take an average of percentages in for all the decoding strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Bal- anced: Goal here is to create a balanced dataset between hallucinated and non-hallucinated turns, each type of hallucination is also balanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Extin- sic+: In this scenario,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' we increase the percentages ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='PERSON ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='1%1% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='4% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='5% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='Extrinsic Soft ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='■ORG ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='2%1%4% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='2% R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='2% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='6% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='DATE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='7% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='Extrinsic Hard ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='GPE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='39% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='11% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='■CARDINAL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='■WORK OF ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='18% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='ART ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='■NORP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='12% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='65% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='EVENT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='ORDINAL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='23% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='■LOCrelease_year ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='3% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='2%2% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='3% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='Intrinsic Soft ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='21% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='starred actors ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='4% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='12% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='8% ' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='3% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='8% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='Original language ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='3% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='23% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='13% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='10% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='AwardWon ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='10% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='in_language ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='22% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='2% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='■release_yearof extrinsic-soft,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' hard,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' and grouped by a factor of 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='5, and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Intrinsic+: here we increase the percentages of intrinsic-soft, hard and repetitive by a factor of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' More details in §D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='4 Human Verification To verify whether our proposed perturbation strate- gies inject hallucinations in the original data, we randomly sample 150 examples from each of the mixed dataset’s test splits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Subsequently, these sam- ples were randomly ordered to form a consolidated sample of 600 data points annotated by at least three AMT workers, with the same setting as de- scribed in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Additionally, the graduate student verified where the hallucinations adhere to the per- turbation norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Krippendorff’s alpha were 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='88 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='76 among workers, and workers with per- turbed data(average), indicating a very high agree- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Since our perturbation strategies are purely deterministic, we kept a large-scale human verifi- cation of the automatically annotated data outside the scope of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' We create a human-verified dataset of 500 samples, 300 taken from this set and 200 from the human feedback study 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' 4 Task To identify utterances that contain hallucinations and to locate the entities of concern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' We create two tasks: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Utterance classification: Given the dialog history D, knowledge triples Kn and the cur- rent utterance xn+1 we classify xn+1 is hallu- cinated or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Token classification: Given D, Kn and xn+1, we need to perform sequence labelling on xn+1 and identify the hallucinated spans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' 5 Baseline Models As an initial effort toward tackling the suggested hallucination detection task, we create several baseline detection models based on pre-trained transformer models, including BERT, XLNet, and RoBERTa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' These transformer-based models repre- sent the state-of-the-art and can potentially better leverage context or embedded world knowledge to detect self-contradictory or anti-commonsense content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' For training the utterance classifier, given D, Kn and xn+1, we fine tune a pre-trained model M to predict binary hallucinated label y for xn+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Here, D and Kn are considered as sequence A with token type ids as 0 and xn+1 is considered as sequence B with token type ids as 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' During inference, from the last hidden states H ∈ Rl×h (h, l are hidden size and sequence length, respectively), then we obtain the representation w ∈ Rh by max pooling(i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=', w = max_pool(H)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' We then pass w through a MLP layer with a tanh activation to get the bi- nary label y ∈ {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' During training time, we fine-tune the model using cross entropy objective between the predicted labels and the actual labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Similarly, for training the sequence classifier, we fine-tune a pre-trained model Ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' At first, we encode D, Kn and xn+1 using Ms to get the last hidden states H ∈ Rl×h, (h, l are hidden size and sequence length, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Instead of doing a binary classification of each token, we adopt a BILOU encoding scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' The hidden states are passed through an MLP layer with a tanh activa- tion to get the 5-way label y ∈ {B, I, L, O, U}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' During training time, we fine-tune the model us- ing a cross-entropy objective between the predicted and actual labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' 6 Experimental Setup Baseline configurations we experiment with a variety of pre-trained models via Hug- ging Face Transformers, including BERT-base- uncased(110M), RoBERTa-base(125M) and XL- Net-base-cased(110M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Though using large or medium versions of these models will produce bet- ter results, we refrain from using those models as scaling large models in production is costly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' More details about training parameters can be found in §E We also experimented with model architecture as follows: (i) Varied the length of the history (ii) Ex- perimented with max/ mean pooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' (iii) Whether to concatenate the hidden states corresponding to Kn with the hidden states corresponding to xn+1 before passing them through the MLP layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' (iv) Using a CRF layer instead of MLP for predicting labels in the sequence tagger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' The best configu- ration uses 4 turns of conversational history, max pooling, it does not concatenate hidden states of Kn with hidden states of xn+1 and uses a 2-layer MLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Evaluation metrics We evaluate the baselines with formal classification metrics, including preci- sion, recall, and F1 for the hallucination sequence tagger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' For the utterance-level hallucination classi- fier, we report accuracy, precision, recall, F1, and Dataset Best Model Token Level Utterance Level F1 P R F1 P R G-Mean(↑) BSS(↓) AUC extrinsic-grouped BERT(base-uncased) 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='69 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='56 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='82 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='30 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='80 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='81 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='58 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='29 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='62 extrinsic-hard XLNet(base-cased) 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='12 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='98 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='25 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='36 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='13 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='60 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='80 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='93 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='96 extrinsic-history-corrupt XLNet(base-cased) 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='38 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='35 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='40 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='10 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='86 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='34 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='24 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='75 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='38 extrinsic-soft BERT(base-uncased) 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='09 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='22 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='67 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='80 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='96 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='80 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='62 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='03 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='81 intrinsic-hard XLNet(base-cased) 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='44 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='08 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='81 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='88 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='88 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='97 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='24 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='48 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='34 intrinsic-history-corrupt XLNet(base-cased) 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='67 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='27 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='11 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='30 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='86 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='74 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='97 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='34 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='02 intrinsic-repetitive RoBERTa(base) 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='70 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='76 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='64 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='01 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='51 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='55 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='31 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='15 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='50 intrinsic-soft RoBERTa(base) 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='80 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='19 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='45 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='10 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='54 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='92 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='26 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='22 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='50 Table 6: Test benchmark (numbers in percentages (%)) for component datasets, models trained on 25% of the total dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Dataset Best Model Token Level Utterance Level F1 P R F1 P R G-Mean(↑) BSS(↓) AUC balanced RoBERTa-base 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='41 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='75 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='74 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='24 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='85 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='12 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='21 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='14 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='47 observed XLNet(base-cased) 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='44 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='98 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='03 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='71 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='05 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='73 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='40 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='73 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='40 intrinsic+ RoBERTa-base 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='05 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='11 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='44 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='16 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='52 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='12 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='51 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='78 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='00 extrinsic+ XLNet(base-cased) 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='59 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='79 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='10 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='75 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='77 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='11 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='21 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='65 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='95 Table 7: Test benchmark (numbers in percentages (%)) for mixed datasets, models trained on 25% of the total dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' AUC (Area Under Curve) for ROC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' We also use the G-Mean metric (Espíndola and Ebecken, 2005), which measures the geographic mean of sensitiv- ity and specificity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' We also employ the Brier Skill Score (BSS) metric (Center, 2005), which com- putes the mean squared error between the reference distribution and the hypothesis probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' 7 Results and Discussion Baseline performance Table 6 and Table 7 show the baseline performance for the component datasets and mixed datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' In both the settings, the utterance level hallucination classifier performs better than the token tagger in terms of F1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' It can be inferred from Table 6 that, on average, it is compar- atively easier to detect intrinsic hallucinations than extrinsic hallucinations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' due to grounding on exter- nal knowledge, which indicates the validity of our perturbation techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' However, comparing the occurrence statistics from Table 1, it is noticed that extrinsic-soft hallucination, which has the least F1 score among all types, has the highest occurrences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' In extrinsic-grouped and extrinsic-soft hallucina- tions, it is interesting that BERT performs better than the other pre-trained models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Now for mixed datasets, we ran inference on the test set of ob- served dataset, as expected F1 scores(for utterance classifier and token level tagger) of the observed dataset are low as compared to other datasets due to high percentage of extrinsic-soft hallucination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Among other mixed datasets, the XLNet model fine-tuned on extrinsic+ dataset performs best in terms of F1 scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Performance on human-verified data We test the best performing models fine-tuned on our mixed datasets on human-verified data as de- Fine-tuned on Pretrain Model F1 (Utterance-level) F1 (Token-level) MNLI RoBERTa-large 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='5 BEGIN RoBERTa-large 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='4 FaithDial RoBERTa-large 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='1 Intrin-Extrin(Dziri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=', 2021a) RoBERTa-large 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='81 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='2 balanced RoBERTa-base 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='27 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='61 observed XLNet(base-cased) 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='15 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='27 extrinsic+ XLNet(base-cased) 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='97* 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='7* intrinsic+ RoBERTa-base 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='01 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='33 Table 8: Performance of several benchmark models and mod- els trained on FADE on the 500 human-verified data( *p-value < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='001)) Fine-tuned on Model BEGIN FaithDial MNLI(3-way)(Dziri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=', 2021b) T5 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='5 MNLI(Dziri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=', 2022a) RoBERTa-large 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='1 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='6 intrinsic_hard RoBERTa-base 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='12 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='34 intrinsic_history_corrupt RoBERTa-base 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='23 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='11 intrinsic_hard RoBERTa-large 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='42 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='1 intrinsic_history_corrupt RoBERTa-large 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='11 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='43 Table 9: Zero-sort inference F1 scores on BEGIN and Faith- Dial benchmarks using utterance classification models trained on FADE scribed in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Using the existing benchmark and baseline models, we also perform a zero-shot in- ference on the human-verified data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' From Table 8, it is clear that the models fine-tuned on existing benchmark data cannot understand fact hallucina- tion, especially when entities are misplaced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' On the other hand, models trained on our datasets have F1 scores over 90% and outperform the current base- line by 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='16% and 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='5% in the two tasks using a pre-trained model with fewer parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' This suggests that identifying abrupt fact hallucination is more challenging than other types of halluci- nation(like presenting more data than expected), which are more commonly exhibited in the bench- mark datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Generalisability We make zero-shot inference on BEGIN and FaithDial datasets’ test splits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' To make a fair comparison with the benchmark mod- els, we further fine-tune roberta-large model on our datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Table 9 shows that F1 scores ob- tained from our best models underperform the best Figure 6: Positive and negative model predictions Figure 7: Generalisation capability of RoBERTa-large model fine-tuned using multiple splits of intrinsic-history-corrupt dataset performing baseline by 6% in BEGIN dataset and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='17% in the FaithDial dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Even though the performance is low, we have to understand that the benchmark datasets contain hallucinations that are fundamentally very different from fact hallu- cinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Also, we notice that models trained on intrinsic hallucination perform the best because the hallucinatory responses in the benchmark dataset do not deviate much from the evidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' To estimate how much training data is optimum for generalis- ability, we ran inference on benchmark datasets using models fine-tuned to 10% to 30% (with a step of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='5%) data in train split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' As shown in Fig- ure 7 approximately 25% is found to be optimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Model Predictions We visualized the predic- tions on different datasets in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Our models were able to easily identify the hallucinated entities as shown in Figure 6a here "The Departed" is a movie in which "Mark Wahlberg" has acted but is not related to the movie discussed in the context, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=', "The Italian Job".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Similarly, predictions made on the FaithDial dataset(Figure 6c) show that our models could produce accurate predictions when the response is generating something that is not expected, but the hallucination has similarities with the evidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Our model sometimes fails to under- stand when the history is convoluted(Figure 6b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' 8 Related Work Hallucination in Dialogue Systems Hallucination in knowledge-grounded dialogue generation sys- tem is an emerging area of research (Roller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Mielke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Shuster et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Rashkin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=', 2021b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Dziri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=', 2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Prior work addressed this issue by conditioning genera- tion on control tokens (Rashkin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=', 2021b), by training a token level hallucination critic to identify troublesome entities and rectify them (Dziri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=', 2021a) or by augmenting a generative model with a knowledge retrieval mechanism (Shuster et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Though beneficial, these models are trained on noisy training data (Dziri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=', 2022b) which can amplify the hallucinations further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Closest to our work (Dziri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=', 2021a) has created a hallu- cination critic using extrinsic-intrinsic corruption strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' In contrast, we create more fine-grained corruption strategies so that hallucinated data mim- ics the attributions of a neural chat module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Hallucination Evaluation Recently several benchmarks have been introduced, such as BE- GIN(Dziri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=', 2021b), DialFact(Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=', 2022), FaithDial(Dziri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=', 2022a) and At- tributable to Identified Sources (AIS) (Rashkin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=', 2021a) framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Though these methods can serve as a decent benchmarking system, their performance in detecting entity-level hallucination is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' In this work, we further contribute to this problem by proposing an entity-level halluci- nation detector trained on data created by various fine-grained perturbation strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' 9 Conclusion In this work, we have analyzed the modes of entity- level fact hallucination, which is an open problem in KG-grounded dialogue systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Through a hu- man feedback analysis, we demonstrate that these KG-grounded neural generators manifest more nu- anced hallucinations than straightforward studied approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' We have proposed fine-grained per- turbation strategies to create a dataset that mimics the real-world observations and create a series of datasets collectively known as FADE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=" Our entity- level hallucination detection model can predict hal- Knowledge Triples: ['Mike Zimmer', 'Sport coached', 'American football'j] Evidence: Dylan's Candy Bar is a chain of boutique candy History: ['Can you tell me some information about the Minnesota shops and candy supplier currently located in New York City Knowledge Triples: ['The Italian Job', 'starred_actors', 'Mark Wahlberg'J] Vikings?" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=" ','TheMinnesota Vikingsarecoached byMikeZimmerand East Hampton, New York;" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Los Angeles, Chicago and Miami apart of theNational Football League.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='Not a bigfan though.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=",'Me Beach, as well as in wholesale venues around the globe." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' either .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Which team do you like ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=" ', 'My most favorite American History: ['Do you knows who stars in The Italian Job ?" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='\'] Football team is the Seattle Seahawks , I meant I was not a big fan of History: ["I love candy, what\'s a good brand?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='"] the Mlnnesota Vikings .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Do you like American Football ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=" 'l Response: Certainly!" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' it stars Seth Green and The Departed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' are you familiar with either?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Response:Ido like Mike Zimmer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content="i like the washington redskins Response: I don't know how good they are, but Dylan's vikings have our old qb , kurt cousins ." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Candy Bar has a chain of candy shops in various cities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Tagged Response(RoBERTa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=') ): Certainly!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' it stars trinsic_sofi Seth Green and The Departed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Are you familiar with either?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Tagged Response(RoBERTa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=', ve): I do like Mike Zimmer .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' i ):Hallucination like the washington redskins .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=" vikings' have our old qb , kurt cousins ." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' a)Intrinsic-soft:Correct (c) FaithDial: CorrectFaithDial BEGIN 60 score 40 20 0 10 15 20 25 30 Training Data Size (%)lucinated entities with an F1 score of 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='59% and classify whether an utterance is hallucinated or not with an F1 score of 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='75%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Our models can gener- alize well when zero-shot predictions are made on benchmarks like BEGIN and FaithDial, indicating our perturbation strategies’ robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' This work can be extended by devising more sophisticated per- turbation mechanisms, which can simulate other types of hallucinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Limitations The major limitations of this work are as follows: The token-level hallucination classifier and utterance-level hallucination classifier can have contradictory results;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' however, this hap- pens in a small percentage of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Models trained on extrinsic datasets do not generalize well on the benchmark datasets, as the benchmark dataset contains hallucination mostly related to the evidence provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Acknowledgements We thank the anonymous reviewers for provid- ing valuable feedback on our manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' This work is partly supported by NSF grant number IIS- 2214070.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' The content in this paper is solely the responsibility of the authors and does not neces- sarily represent the official views of the funding entity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' References Hannah Bast, Florian Bäurle, Björn Buchhold, and El- mar Haußmann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Easy access to the freebase dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' In Proceedings of the 23rd International Conference on World Wide Web, WWW ’14 Com- panion, page 95–98, New York, NY, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Associa- tion for Computing Machinery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Tom B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Brown, Benjamin Mann, Nick Ryder, Melanie Subbiah, Jared Kaplan, Prafulla Dhariwal, Arvind Neelakantan, Pranav Shyam, Girish Sastry, Amanda Askell, Sandhini Agarwal, Ariel Herbert-Voss, Gretchen Krueger, Tom Henighan, Rewon Child, Aditya Ramesh, Daniel M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Ziegler, Jeffrey Wu, Clemens Winter, Christopher Hesse, Mark Chen, Eric Sigler, Mateusz Litwin, Scott Gray, Benjamin Chess, Jack Clark, Christopher Berner, Sam Mc- Candlish, Alec Radford, Ilya Sutskever, and Dario Amodei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Language models are few-shot learn- ers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' NOAA-CIRES Climate Diagnostics Center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Brier skill scores, rocs, and economic value dia- grams can report false skill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Nouha Dziri, Ehsan Kamalloo, Sivan Milton, Os- mar Zaiane, Mo Yu, Edoardo M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Ponti, and Siva Reddy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' 2022a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Faithdial: A faithful benchmark for information-seeking dialogue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Nouha Dziri, Andrea Madotto, Osmar Zaïane, and Avishek Joey Bose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' 2021a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Neural path hunter: Re- ducing hallucination in dialogue systems via path grounding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' In Proceedings of the 2021 Conference on Empirical Methods in Natural Language Process- ing, pages 2197–2214, Online and Punta Cana, Do- minican Republic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Nouha Dziri, Sivan Milton, Mo Yu, Osmar Zaiane, and Siva Reddy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' 2022b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' On the origin of hallucinations in conversational models: Is it the datasets or the models?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Nouha Dziri, Sivan Milton, Mo Yu, Osmar Zaiane, and Siva Reddy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' 2022c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' On the origin of hallucinations in conversational models: Is it the datasets or the models?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Nouha Dziri, Hannah Rashkin, Tal Linzen, and David Reitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' 2021b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Evaluating groundedness in dia- logue systems: The begin benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Rogério P Espíndola and Nelson FF Ebecken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' On extending f-measure and g-mean metrics to multi-class problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' WIT Transactions on Infor- mation and Communication Technologies, 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Prakhar Gupta, Chien-Sheng Wu, Wenhao Liu, and Caiming Xiong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' DialFact: A benchmark for fact-checking in dialogue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' In Proceedings of the 60th Annual Meeting of the Association for Compu- tational Linguistics (Volume 1: Long Papers), pages 3785–3801, Dublin, Ireland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Association for Com- putational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Joshua Maynez, Shashi Narayan, Bernd Bohnet, and Ryan McDonald.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' On faithfulness and factu- ality in abstractive summarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' In Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics, pages 1906–1919, On- line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Sabrina J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Mielke, Arthur Szlam, Y-Lan Boureau, and Emily Dinan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Linguistic calibration through metacognition: aligning dialogue agent responses with expected correctness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Seungwhan Moon, Pararth Shah, Anuj Kumar, and Ra- jen Subba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' OpenDialKG: Explainable conver- sational reasoning with attention-based walks over knowledge graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' In Proceedings of the 57th An- nual Meeting of the Association for Computational Linguistics, pages 845–854, Florence, Italy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Associ- ation for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Ankur Parikh, Xuezhi Wang, Sebastian Gehrmann, Manaal Faruqui, Bhuwan Dhingra, Diyi Yang, and Dipanjan Das.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' ToTTo: A controlled table-to- text generation dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' In Proceedings of the 2020 Conference on Empirical Methods in Natural Lan- guage Processing (EMNLP), pages 1173–1186, On- line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Alec Radford, Jeffrey Wu, Rewon Child, David Luan, Dario Amodei, Ilya Sutskever, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Lan- guage models are unsupervised multitask learners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' OpenAI blog, 1(8):9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Hannah Rashkin, Vitaly Nikolaev, Matthew Lamm, Michael Collins, Dipanjan Das, Slav Petrov, Gau- rav Singh Tomar, Iulia Turc, and David Reitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' 2021a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Measuring attribution in natural language generation models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Hannah Rashkin, David Reitter, Gaurav Singh Tomar, and Dipanjan Das.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' 2021b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Increasing faithfulness in knowledge-grounded dialogue with controllable features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' In Proceedings of the 59th Annual Meet- ing of the Association for Computational Linguistics and the 11th International Joint Conference on Nat- ural Language Processing (Volume 1: Long Papers), pages 704–718, Online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Association for Computa- tional Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Stephen Roller, Emily Dinan, Naman Goyal, Da Ju, Mary Williamson, Yinhan Liu, Jing Xu, Myle Ott, Eric Michael Smith, Y-Lan Boureau, and Jason We- ston.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Recipes for building an open-domain chatbot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' In Proceedings of the 16th Conference of the European Chapter of the Association for Compu- tational Linguistics: Main Volume, pages 300–325, Online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Abigail See and Christopher Manning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Under- standing and predicting user dissatisfaction in a neu- ral generative chatbot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' In Proceedings of the 22nd Annual Meeting of the Special Interest Group on Dis- course and Dialogue, pages 1–12, Singapore and Online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Kurt Shuster, Spencer Poff, Moya Chen, Douwe Kiela, and Jason Weston.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Retrieval augmentation reduces hallucination in conversation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' In Findings of the Association for Computational Linguistics: EMNLP 2021, pages 3784–3803, Punta Cana, Do- minican Republic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Eric Smith, Orion Hsu, Rebecca Qian, Stephen Roller, Y-Lan Boureau, and Jason Weston.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Hu- man evaluation of conversations is an open prob- lem: comparing the sensitivity of various methods for evaluating dialogue agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' In Proceedings of the 4th Workshop on NLP for Conversational AI, pages 77–97, Dublin, Ireland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Association for Com- putational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Yi-Lin Tuan, Yun-Nung Chen, and Hung-yi Lee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' DyKgChat: Benchmarking dialogue genera- tion grounding on dynamic knowledge graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' In Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Lan- guage Processing (EMNLP-IJCNLP), pages 1855– 1865, Hong Kong, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Association for Computa- tional Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Sam Wiseman, Stuart Shieber, and Alexander Rush.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Challenges in data-to-document generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' In Proceedings of the 2017 Conference on Empiri- cal Methods in Natural Language Processing, pages 2253–2263, Copenhagen, Denmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Thomas Wolf, Lysandre Debut, Victor Sanh, Julien Chaumond, Clement Delangue, Anthony Moi, Pier- ric Cistac, Tim Rault, Remi Louf, Morgan Funtow- icz, Joe Davison, Sam Shleifer, Patrick von Platen, Clara Ma, Yacine Jernite, Julien Plu, Canwen Xu, Teven Le Scao, Sylvain Gugger, Mariama Drame, Quentin Lhoest, and Alexander Rush.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Trans- formers: State-of-the-art natural language process- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' In Proceedings of the 2020 Conference on Em- pirical Methods in Natural Language Processing: System Demonstrations, pages 38–45, Online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Asso- ciation for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' A Definition Details Figure 8: Extrinsic Hallucination (a) (Extrinsic-Soft).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' An extrinsic-soft hallucina- tion corresponds to an utterance that brings a new span of text which is similar to the expected span but does not correspond to a valid triple in Gk c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' A hallucination is considered extrinsic when knowledge is injected which is not authentically captured by Gk c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' However, the injected knowledge HISTORY A: Could you recommend movies similar to the The Dark Knight ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' B: The sequel to Batman Begins is The Dark Knight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' A: Okay .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Who is the director of The Dark Knight and any other movies from him not related to Batman ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=" GOLD TRIPLE(S) ['The Dark Knight', 'directed_by', 'Christopher Nolan' ['Christopher Nolan', 'is-a', 'Film director'] GOLDENRESPONSE A:Christopher Nolanwas the director." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='He also directed Insomnia and Inception .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' CORRUPTEDRESPONSE(Soft) A:Steven Spielbergwas the director .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='He also directed insomnia and inception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' CORRUPTED RESPONSE(Hard) A:Joe Bidenwas the director .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='He also directed insomniaand inception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' CORRUPTEDRESPONSE(Grouped) A:Warner Bros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='was the director .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' He also directed insomnia and inception .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='Group Definition Groups 1 A person, organization, political party, or part of a religious group can be related to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' "PERSON", "ORG", "NORP 2 Location, building, airports, infrastructure elements, countries, cities, and states can be interrelated "LOC", "GPE", "FAC" 3 A product, work of art, or law can be interrelated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' "PRODUCT", "WORK_OF_ART", "LAW" Table 10: Defined groups for extrinsic-grouped hallucination is similar to the expected entity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Identifying this type of hallucination can be challenging due to the high similarity between the injected and gold knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' For example, in Figure 8 the dialogue sample contains an extrinsic-soft hallucination as the entity in response – "Steven Spielberg" is simi- lar to "Christopher Nolan", and it is not supported within 1-hop sub-graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' (b) (Extrinsic-Hard).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' An extrinsic-hard hallucina- tion corresponds to an utterance that brings a new span of text which is different from the expected span and does not correspond to a valid triple in Gk c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' An extrinsic-hard hallucination occurs when in- jected knowledge is dissimilar to the expected en- tity and is not supported within the 1-hop sub-graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' It is easier to detect extrinsic-hard than extrinsic- soft as the entities are fundamentally different from the entities present in the 1-hop sub-graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' How- ever, the entity type is retained, like an entity with a type "person" will be replaced by the same type of entity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Figure 8 shows an example of extrinsic-hard hallucination, where the golden entity "Christopher Nolan" is replaced by a different category of entity, "Joe Biden", but the type of entity is retained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' (c) (Extrinsic-Grouped).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' An extrinsic-grouped hal- lucination corresponds to an utterance that brings a new span of text which is different from the ex- pected span but is of a specific predefined type and does not correspond to a valid triple in Gk c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Like an extrinsic-hard hallucination, extrinsic- grouped hallucination introduces an entity that is functionally different from the original entity and not supported by the 1-hop sub-graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' The only difference is that the corrupted entity is not of the same type;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' instead, it is replaced by an entity of a similar type, defined in Table 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' For example, Figure 8 shows "Christopher Nolan" which is of type "person" is replaced by "Warner Bros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='" of type "organization".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Here, the types "person" and "organization" are placed in the same group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' (d) (Intrinsic-Soft).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' An intrinsic-soft hallucination corresponds to an utterance that misuses any triple Figure 9: Intrinsic Hallucination in Gk c such that there is no direct path between the entities, but they are similar to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Intrinsic hallucinations occur when the KG triples are misused, especially in intrinsic-soft hal- lucination an entity is selected from Gk c which is very similar or closely related to the original entity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' For example, in Figure 9, "Christopher Nolan" is replaced with "The Dark Knight Rises" which is retrieved from the 1-hop sub-graph and has close re- lation with the original entity "Christopher Nolan".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' (e) (Intrinsic-Hard).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' An intrinsic-hard hallucina- tion corresponds to an utterance that misuses any triple in Gk c such that there is no direct path be- tween the entities, and they are not related in any form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Similar to intrinsic-soft hallucination, it also mis- uses the information in KG triples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' However, the similarity of the corrupted entity with the original entity is relatively tiny.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' For example, in Figure 9, "Christopher Nolan" is replaced with "United States of America".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Although the corrupted en- tity is drawn from Gk c , it is very different from the original entity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' (f) (Intrinsic-Repetitive).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' An intrinsic-repetitive hallucination corresponds to an utterance that ei- ther misuse [SBJ] or [OBJ] in Gk c such that there is no direct path between the entities but the entity HISTORY A: Could you recommend movies similar to the The Dark Knight ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' B:The sequel toBatman Beginsis The Dark Knight .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' A: Okay .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Who is the director of The Dark Knight and any other movies from him not related to Batman ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' GOLD TRIPLE(S) [\'The Dark Knight\', \'directed_by\',\'Christophel Nolan\'] [\'Christopher Nolan\', "is-a\',\'Film director\' GOLDEN RESPONSE A: Christopher Nolan was the director .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' He also directed Insomnia and Inception .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' CORRUPTEDRESPONSE(Soft) A: The Dark Knight Riseswas the director He also directed insomnia and inception .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' CORRUPTED RESPONSE(Hard) A: United States of America was the director .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' He also directed insomnia and inception .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' CORRUPTEDRESPONSE(Repetitive) A:Batman Beginswas the director .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' He also directed insomnia and inception .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='has previously occurred in conversational history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='. An entity from the conversational history is of- ten repeated in the current utterances, which corre- sponds to intrinsic-repetitive hallucination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Here, an entity from the history which also occurs in Gk c and of high relatedness, is swapped with the origi- nal entity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Figure 9 shows "Batman Begins" which is supported by Gk c is replaced with "Christopher Nolan".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Figure 10: History Corrupted Hallucination (g) (History Corrupted- Intrinsic/ Extrinsic).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' A history corrupted(intrinsic/extrinsic) hallucination corresponds to an utterance subjected to intrin- sic or extrinsic hallucination influenced by halluci- nated entities in conversational history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Sometimes conversational agents are driven into a perplexed state, and we can witness hallucina- tions in most turns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' So, this hallucinated history can trigger hallucination in the current utterance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' This phenomenon can be seen both in extrinsic and intrinsic forms of hallucination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Figure 10 depicts extrinsic/intrinsic hallucination occurring in his- tory – "The Dark Knight" is changed to "The Dark Knight Rises" for intrinsic hallucination;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' similarly, "The Dark Knight" is changed to "Spider-Man" for extrinsic hallucination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Hallucinations in the current utterance happen as described in previous sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' B AMT Instructions We present the screenshot of the annotation inter- face in Figure 12, 12 and 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Workers were paid an average of $7-8 per hour across all tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' We agree that this annotation process has a high learning curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Even workers with high approval rates made errors in the initial rounds of annotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' A grad- uate computer science student manually verified randomly selected samples and provided feedback to the workers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Feedback was given to the workers, especially when they selected the same answers for ten consecutive HITS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' After sending feedback three times, all spammed HITS were discarded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' C OpenDialKG We use OpenDialKG (Moon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=', 2019), a crowded-sourced English dialogue dataset where two workers are paired together to chat about a par- ticular topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' The first speaker is requested to start the conversation about a given entity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' The second speaker is assigned to write an accurate response based on facts extracted from an existing KG, Free- base (Bast et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=', 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' The facts represent paths from the KG that are either 1-hop or 2-hop from the initial entity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Once the second speaker responds, the first speaker continues discussing the topic en- gagingly, and new multi-hop facts from the KG are shown to the second speaker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' The dialogue can be considered as traversing multiple paths in the KG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' However, not all utterances within the same conversation are grounded on facts from the KG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' The second speaker can decide not to select a path from the KG to form an answer and instead forms a "chit-chat" response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Overall, the dataset consists of four domains: movie, music, sport, and book, where each second speaker’s utterance is annotated with paths from the KG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' The KG corresponds to an extensive subgraph extracted from Freebase with ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='2M triples (subject, predicate, object), ∼ 101k distinct entities, and 1357 distinct relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' We use 77,430 data points in the dataset for constructing FADE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' D Perturbation Hyper-parameters D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='1 Search Index Details We use Solr in case of extrinsic hallucination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' We use the BM25 index, defined by the class solr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='BM25SimilarityFactory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' We man- ually labeled 50 data points(for the entity type PERSON) for tuning the indexes through grid HISTORY (CORRUPTED) A:Couldyou recommendmovies similarto the The Dark Knight ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' B: The sequel to [The Dark Knight →The Dark Knight Rises(Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=')] [The Dark Knight → Spider-Man(Ext.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=')] is Batman Begins .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' A: Okay .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Who is the director of The Dark Knight and any other movies from him not related to Batman ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' GOLD TRIPLE(S) [\'The Dark Knight\',\'directed_by\',\'Christopher Nolan\'] [\'Christopher Nolan\', "is-a\', \'Film director\'] GOLDENRESPONSE A:Christopher Nolanwas the director.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='He also directed Insomnia and Inception CORRUPTED RESPONSE(Intrinsic) A:United States of Americawas the director.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' He also directed insomnia and inception .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' CORRUPTED RESPONSE(Extrinsic) A:Joe Bidenwasthedirector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='He also directed insomnia and inception .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='Figure 11: Annotation interface for human feedback analysis(Instructions, part 1) Figure 12: Annotation interface for human feedback analysis(Instructions, part 2) Please state if the response contains irrelevant phrase(s) or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' If yes, then, please select its type and note down the phrase We have provided you with some knowledge paths and conversational history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' In the given response, phrase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Examples of each type of error are provided below: Conversation history: Speaker A: Could you recommend movies similar to The Dark Knight?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Speaker B: The sequel to Batman Begins is The Dark Knight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Speaker A: Okay .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Who is the director of The Dark Knight and any other movies from him not related to Batman?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=" Knowledge paths: Path 1: ['The Dark Knight', 'directed_by', 'Christopher Nolan'] Path 2: ['Christopher Nolan', 'is-a', 'Film director'] Golden Response(this is for reference, it does not appear in the real data): Speaker B: Christopher Nolan was the director." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' He also directed Insomnia and Inception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Extrinsic Hallucinations: Extrinsic soft: When an irrelevant phrase is introduced which is similar to the expected phrase but the phrase does not appear in the knowledge paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' For example, Speaker B: Steven Spielberg was the director.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' He also directed Insomnia and Inception Extrinsic hard: When an irrelevant phrase is introduced which is not similar to the expected phrase and the phrase does not appear in the knowledge paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' For example, Speaker B: Joe Biden was the director.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' He also directed Insomnia and Inception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Extrinsic grouped: When an irrelevant phrase is introduced which is related to the expected phrase but the phrasedoesnotappearintheknowledgepaths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='Forexample, SpeakerB:WarnerBros_wasthedirector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='HealsodirectedInsomniaandInception Valid relations: Aperson,organization,political party,or part of a religiousgroup can be related to each other Location, building, airports, infrastructure elements, countries, cities, and states can be interrelated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Aproduct,workofart,orlaw canbeinterrelated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='Intrinsic Hallucinations: Intrinsic soft: When an irrelevant phrase is introduced which is similar to the expected phrase and the Speaker B: The Dark Knight was the director.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' He also directed Insomnia and Inception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Intrinsic hard: When an irrelevant phrase is introduced which is not similar to the expected phrase and the phrase does appear/ or is related to the knowledge paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' For example, Speaker B: United States of America was the director.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' He also directed Insomnia and Inception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' (Christopher Nolan is a citizen of the United States of America) Intrinsic repetitive: When an irrelevant phrase is introduced which is related to the expected phrase, appears in conversational history, and the phrase does appear/ or is related to the knowledge paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' For example, SpeakerB:BatmanBegins_wasthedirector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='Healsodirected InsomniaandInception History Corrupted Hallucinations: Corrupted Conversation history: Speaker A: Could you recommend movies similar to The Dark Knight?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Speaker B: The sequel to The Dark Knight Rises is Spider-Man.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Speaker A: Okay .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Who is the director of The Dark Knight and any other movies from him not related to Batman?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Now consider this conversation history, if you look closely, the second turn is corrupted with irrelevant entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' History corrupt intrinsic: When an irrelevant phrase is introduced which is of any type of intrinsic hallucination AND the conversation history is corrupted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' For example, Speaker B: The Dark Knight was the director.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' He also directed Insomnia and Inception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' History corrupt extrinsic: When an irrelevant phrase is introduced which is of any type of intrinsic hallucination AND the conversation history is corrupted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' For example, SpeakerB:WarnerBroswasthedirector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='He also directed Insomnia and Inception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='Figure 13: Annotation interface for human feedback analysis(example annotation, workers were ask to find up to 3 spans if hallucinations are found in the data) search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Grid-search conditions were as follows: b was varied from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='3 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='9 with a step of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='1 and k1 was varied from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='8 to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='0 with a step of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Following grid search, an optimum MAP score of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='789 was found, with b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='9 and k1= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' For the dynamic indexes that were created in the case of intrinsic hallucination, we use the python library https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='com/ dorianbrown/rank_bm25 with default con- figurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='2 Free parameter & β optimization We use a free term weight parameter(ε) in in- trinsic hallucination to represent the queries and nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Similar to extrinsic hallucination we man- ually annotated 50 data-points and ran grid search for ε ∈ {10−i, 2 × 10−i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' i ∈ {1, 5}}, and found ε = 2×10−4 to be the optimum value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' We used the same technique for optimizing β, and the search space ranged from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='1 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='7 with a step of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='3 KG embeddings We follow the same approach (Dziri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=', 2021a) for generating the KG embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' OpenDialKG Dataset Type Ext-Soft(%) Ext-Hard(%) Ext-Grp(%) Int-Soft(%) Int-Hard(%) Int-Rep(%) HC-Ext(%) HC-Int(%) N-Halluc(%) Observed 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='495 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='4425 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='92 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='025 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='4575 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='4575 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='4625 Balanced 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='25 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='25 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='25 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='25 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='25 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='25 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='25 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='25 50 Extrinsic+ 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='375 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='375 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='25 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='25 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='25 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='25 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='25 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='5 Intrinsic+ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='25 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='25 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='25 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='375 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='375 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='375 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='25 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='25 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='625 Table 11: Mixing ratios for different datasets triples are also represented using a textual term called "render".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' For the triples containing this term, we pass it through to GPT2 and then extract hidden state representations for each entity’s word piece and finally obtain a final representation by applying a MaxPool over the hidden representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' For entity mentions not described in “render”, we get their representations directly from the last hidden states in GPT2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='4 Mixing Ratios Mixing ratios for creating the mixed datasets are defined in Table 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Perturbed and non-perturbed samples are drawn randomly from component datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' E Implementation Details The utterance and token level classifier are imple- mented using the Pytorch Huggingface Transform- ers library (Wolf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=" The following con- Now complete the following task: Knowledge paths: Path 1: ['Gautam Gambhir', 'is-a', 'Athlete'] Path 2: ['Athlete', '~is-a', 'Venus Williams']] Conversation history: Speaker A: What do you think about Gautam Gambhir Indian cricketer ?" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=" Response: Speaker B: to be honest, I don't really know anything about him." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=" I'm more of a tennis fan ." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' one of my favorite players is Gautam Gambhir Does the response contain irrelevant phrase(s)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' O Yes O No If yes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' then write down the irrelevant phrases(s) and select their type(up to 3): ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='irrelavant phrase(s) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='Type: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='O extrinsic_soft ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='Oextrinsic_hard ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='Oextrinsic_grouped ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='O intrinsic_soft ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='O intrinsic_hard ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='Oextrinsic_history_corrupt ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='O intrinsic_history_corrupt ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='irrelavant phrase(s) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='Type: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='O extrinsic_soft ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='O extrinsic_hard ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='Oextrinsic_grouped ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='O intrinsic_soft ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='O intrinsic_hard ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='Oextrinsic_history_corrupt ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='O intrinsic_history_corrupt ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='irrelavant phrase(s) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='Type: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='O extrinsic_soft ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='O extrinsic_hard ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='O extrinsic_grouped ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='O intrinsic_soft ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='O intrinsic_hard ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='Oextrinsic_history_corrupt ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='O intrinsic_history_corrupt ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='SubmitHyperparameter ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='Value ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='train_batch_size ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='gradient_accumulation_steps ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='num_train_epochs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='4(Token)/10(Utt) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='weight_decay ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='01 warmup_proportion 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='1 learning_rate 1e-5 adam_epsilon 1e-8 max_grad_norm 1 eval_batch_size 18 Table 12: RoBERTa-base hyper parameters Hyperparameter Value train_batch_size 12 gradient_accumulation_steps 2 num_train_epochs 4(Token)/10(Utt) weight_decay 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='01 warmup_proportion 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='1 learning_rate 2e-5 adam_epsilon 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='5e-8 max_grad_norm 1 eval_batch_size 18 Table 13: RoBERTa-large hyper parameters figuration were found to be best performing for each models, as shown in Table 12, 13, 14 and 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' The models were trained in a single NVIDIA A5000 GPU, the average running time for the base models were 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='5 hours, and for the large model was ∼ 5 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' F Supplementary results We report metrics for all the models trained using 25% of the dataset, for component datasets in Table 16 and mixed datasets in Table 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content=' Hyperparameter Value train_batch_size 12 gradient_accumulation_steps 2 num_train_epochs 4(Token)/10(Utt) weight_decay 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='01 warmup_proportion 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='1 learning_rate 5e-5 adam_epsilon 1e-8 max_grad_norm 1 eval_batch_size 18 Table 14: BERT-base-uncased hyper parameters Hyperparameter Value train_batch_size 12 gradient_accumulation_steps 2 num_train_epochs 4(Token)/10(Utt) weight_decay 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='01 warmup_proportion 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='1 learning_rate 5e-5 adam_epsilon 1e-8 max_grad_norm 1 eval_batch_size 18 Table 15: XLNet-base hyper parameters Dataset Best Model Token Level Utterance Level F1 P R F1 P R G-Mean BSS AUC extrinsic_hard roberta-base 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='70613382 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='68956357 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='72352004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNE3T4oBgHgl3EQfUQpw/content/2301.04449v1.pdf'} +page_content='86181139 0.' metadata={'source': 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a/CtFJT4oBgHgl3EQfAiym/content/tmp_files/2301.11421v1.pdf.txt b/CtFJT4oBgHgl3EQfAiym/content/tmp_files/2301.11421v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..a4f2fdedfe685664be88ee8cfaa6fa40501cea1e --- /dev/null +++ b/CtFJT4oBgHgl3EQfAiym/content/tmp_files/2301.11421v1.pdf.txt @@ -0,0 +1,887 @@ +Non-invasive and noise-robust light focusing using confocal wavefront +shaping +Dror Aizik, Anat Levin +Department of Electrical and Computer Engineering, Technion, Haifa, Israel +Abstract +One of the hardest barriers to our ability to see inside +biological tissue is the fact that light is highly aberrated +when scattering by the tissue sub-components. One of +the promising approaches for overcoming such aberrations +is wavefront-shaping, where one modulates the incoming +and/or the outgoing wavefront in a way that will allow +it to focus into one spot despite scattering in the tissue. +Wavefront modulations are specific to the tissue sample +being imaged and need to be estimated based on a non- +invasive feedback from a camera collecting back-scattered +light. Such modulations have been successively estimated +using feedback from strong fluorescent beads which have +been manually added to a sample. +However, in a real +biomedical application, such feedback should be provided +by the fluorescent components of the tissue itself, whose +emission is orders of magnitude lower than the one pro- +vided by beads. +When such a low number of photons +is spread over multiple sensor pixels, the image is highly +susceptible to noise, and the feedback signal required for +previous algorithms cannot be detected. +In this work we suggest a wavefront shaping approach +that works with a confocal modulation of both illumina- +tion and imaging arms. The advantage of this approach is +that as part of the optimization, aberrations are corrected +in the optics, before the detector. Hence the low photon +budget can be directed into a single sensor spot and de- +tected with high SNR. We derive the optimization prob- +lem from mathematical principles and show why it favors +modulations that actually correct the aberrations and fo- +cus all light into one spot. We successfully demonstrate +wavefront-shaping correction on EGFP neurons sliced out +of a mouse brain, despite scattering through thick tissue. +1 +Introduction +One of the hardest barriers to light-based approaches to +tissue imaging is the fact that light is heavily scattered +due to variations in the refractive index of tissue struc- +tures. A promising approach for overcoming the scatter- +ing challenge is a wavefront-shaping based correction. By +using a spatial light modulator (SLM) device, one can +reshape the coherent wavefront illuminating the sample, +such that its aberration is conjugate to the aberration +that will happen inside the tissue. When such a wavefront +propagates through the sample, all incoming light can be +brought (focused) into a small spot. In the same way, by +using a wavefront modulation element between the tissue +and the sensor, we can correct the outgoing wavefront, +so that light photons emerging from a single target point +are brought into a single sensor point, despite the tissue +aberration. The main advantage of this approach results +from the fact that unlike ballistic-filtering approaches, all +light photons are used. +Earlier wavefront shaping approaches, formally known +as adaptive optics [5, 12, 15], were first used to correct +modest aberrations, for example due to imperfect optics +or refractive index variations in the tissue [6,16,31]. More +recently, wavefront shaping techniques [11, 13, 37] have +shown that it is possible to focus light through thick, +highly-scattering layers [28–30,35]. +Despite the large potential of the idea, finding the de- +sired shape of the modulation correction is rather chal- +lenging. The desired modulation varies between different +tissue samples and even varies spatially between different +positions of the same tissue. For thick tissue, the modu- +lation is a complex pattern containing a large number of +free modes. +Earlier proof-of-concept demonstrations have used a +validation camera behind the tissue to provide feed- +back to the algorithm [7, 8, 24, 29, 30, 35], and other +approaches have relied on the existence of a guiding- +star [10, 13, 14, 18–20, 27, 28, 32–34]. +In the absence of +such a guiding star, and when only non-invasive feed- +back is available, determining whatever a wavefront has +focused inside the tissue is not straightforward. The dif- +ficulty results from the fact that even if we can find an +illumination wavefront that actually focuses into a small +spot inside the tissue, the light back-scattering from this +1 +arXiv:2301.11421v1 [physics.optics] 26 Jan 2023 + +spot is aberrated again on its way to the camera, forming +yet another scattered pattern. +The simplest way to evaluate whatever a wavefront +modulation has focused is to use multi-photon fluores- +cence feedback. In this way, the light emitted from a flu- +orescence spot is a non-linear function of the excitation +intensity arriving it, so when all light is focused into a sin- +gle spot the total emission energy is maximized [15, 18]. +However, obtaining feedback using single-photon fluores- +cence is highly desired as the process is significantly sim- +pler and cheaper than the multi-photon one. The single- +photon case cannot be evaluated using the simple score +function applied in the multi-photon case, since the emis- +sion energy is a linear function of the excitation energy +and thus the amount of emission energy does not increase +when all excitation is focused into a spot. +Recently, progress has been made on non-invasive wave- +front shaping using single-photon feedback [1, 3]. First, +Boniface et al. [3] have suggested that one can evalu- +ate whatever an incoming wavefront modulation has fo- +cused by computing the variance of the emitted speckle +pattern. More recently, Aizik et al. [1] has suggested a +rapid approach that can find a wavefront shaping mod- +ulation using a small number of iterative phase conjuga- +tion iterations. Both approaches were only demonstrated +when the fluorescent feedback was provided by synthetic +fluorescent beads, which emit a relatively strong signal. +However, the ultimate goal is to apply wavefront shap- +ing modulation using feedback from biological samples, +such as neurons. The signal emitted from such samples +is orders of magnitude weaker than the one provided by +fluorescent beads, and bleaching is reached much earlier. +Both algorithms [1,3] inherently assume that the speckle +pattern emitted from a single fluorescent spot can be mea- +sured. However, the number of fluorescent photons emit- +ted from a neuron spot is so low that when these photons +are aberrated and spread over multiple sensor pixels, no +speckle pattern can be observed and one can mostly mea- +sure noise, see visualization in Fig. 2. With such a low +photon count no speckle variance can be estimated as is +required by [3], and no phase retrieval process can be ro- +bustly carried as in [1]. +This work proposes a wavefront shaping framework +that can apply in low-light scenarios and use feedback +from realistic biological data. To this end, we propose to +use a simultaneous wavefront shaping modulation both +on the incoming excitation wavefront and on the outgo- +ing emitted light. The advantage is that since scattered +photons are corrected in the optical path and we attempt +to bring all photons emitted from a single spot into a sin- +gle detector, we can measure them with a much higher +signal-to-noise (SNR) ratio. +To quantify the quality of a candidate modulation cor- +rection, we do not attempt to maximize the total energy +emitted from the target. Rather, we seek to maximize the +energy of the corrected wavefront in a single pixel. We +show that despite the fact that we use linear single-photon +fluorescence, due to the double correction on both illumi- +nation and imaging arms, our score function scales non- +linearly with the intensity arriving at the fluorescence tar- +get. Thus, the returning energy at a single pixel is max- +imized by a focusing modulation that manages to bring +all light into a single spot. We show that effectively, this +score function is equivalent to the one used by previous +two-photon fluorescence wavefront-shaping work [18]. +2 +Problem formulation +Imaging setup: +In Fig. 1 we visualize a wavefront- +shaping imaging setup. A laser beam illuminates a tis- +sue sample via a microscope objective. A phase SLM in +the illumination arm modulates the illumination pattern. +We wish to image a fluorescent target at the back of the +tissue layer. The light returning from the target is col- +lected via the same objective, and reflected at a dichroic +beam-splitter. A second phase SLM at the imaging arm +modulates the emitted light. Lastly, the modulated light +is measured by the front main camera. In our setup the +SLMs (holoeye-pluto) are placed in the Fourier plane of +the system. +The setup includes a second validation camera behind +the tissue sample to assess focusing quality and image an +undistorted reference of the target. While earlier research +demonstrations of wavefront-shaping use this camera to +provide feedback to the algorithm, we emphasize that our +goal in this research is to develop non-invasive techniques +that can only use feedback by the main (front) camera. +The validation camera cannot provide any input to the +algorithm. +We note that some research on wavefront shaping and +adaptive optics modulates only the illumination or imag- +ing arms. For generality, our problem formulation below +will consider modulations at both arms. +Image formation model: Consider a set of K fluores- +cent particles inside a sample, and denote their positions +by o1, . . . , oK. We assume the SLM in the illumination +arm is illuminated with a spatially uniform plane wave +and use the SLM to display a complex 2D electric field +that we denote by u. Although u is a 2D field, we reshape +it as a 1D vector. We also use ν to denote a K ×1 vector +of the field propagating through the sample at each of the +K fluorescent sources. +The relation between u and ν is linear and can be +described as a multiplication by a (very large) matrix +ν = T iu. T i is the incoming transmission matrix de- +scribing coherent light propagation in the tissue. We note +that T i is specific to the tissue sample being tested, and +2 + +Fig. 1: Our wavefront correction fluorescent microscope setup: A laser beam is exciting fluorescent beads at the back of a +tissue layer, and fluorescent emission is scattered again through the tissue, reflects at a dichroic beam-splitter and is collected +by a main (front) camera. We place two SLMs in the Fourier planes of both illumination and imaging arms to allow reshaping +these wavefronts. A validation camera views the beads at the back of the tissue directly. This camera is not actually used +by the algorithm, and is only assessing its success. +LP=linear polarizer, BS=beam-splitter, DBS=dichroic beam-splitter, +BPF=bandpass filter, L1 . . . L7=lenses, Obj=Objective. +different tissue samples are described by very different +transmission matrices. +For thick tissue T i can be an +arbitrarily complex matrix incorporating multiple scat- +tering events in the tissue. Likewise, the light returning +from the particles to the SLM of the imaging arm can +be described as T oν, where T o is the back-propagation +transmission matrix. +The propagation of light from the illumination SLM +plane to the particles and back to the SLM of the imag- +ing arm is then modeled using the combined transmission +matrix +T a ≡ T o · T i. +(1) +We denote by ζ the wavefront placed on the SLM of +the imaging arm, and by D(ζ) a diagonal matrix with ζ +on its diagonal. We denote by F the Fourier transform of +the wavefront from the SLM plane to the camera sensor +plane. +In the fluorescent case, the emissions from different +points are incoherent. The recorded intensity can be ex- +pressed as +I = +� +k +|FD(ζ)T o +↓,k|2|νk|2α, +(2) +where |νk|2 is the energy of the excitation light arriving +at particle ok, and T o +↓,k is the k-th column of T o, so that +FD(ζ)T o +↓,k is the wavefront arriving to the sensor from +ok. +Since light emitted from different fluorescent par- +ticles changes phase incoherently, effectively the sensor +sums the intensity of the wavefronts emitted by differ- +ent particles, and their phases do not interfere. In (2) α +denotes the type of fluorescent excitation. The simplest +case α = 1 is known as single-photon fluorescence where +the emission is linear in the excitation energy |νk|2. In +two-photon fluorescence, α = 2, namely the emission is +proportional to the squared excitation. +Linear fluores- +cence is significantly simpler and cheaper to achieve, but +as we explain below, a non-linear (two-photon) fluores- +cence feedback simplifies modulation estimation. +Phase conjugation: +For coherent imaging, the +Helmholtz reciprocity principle leads to wave conjuga- +tion, namely if we record the wavefront emitted from a +source point inside the tissue and play it back in the re- +verse direction, the wavefront will focus at the same point. +This implies that the returning transmission matrix is the +transpose of the incoming one [25]: +T o = T i⊤. +(3) +Note that this is just a transpose and not the Hermi- +tian (conjugate) transpose. In the fluorescent case, we +should note that T i, T o describe propagation at a dif- +ferent wavelength hence they cannot be completely the +same. However, for linear single-photon fluorescence the +excitation and emission wavelengths are relatively similar +and we still assume T o ≈ T i⊤. +Normalization: we assume for simplicity that our trans- +mission matrices are normalized such that every column +or row has a unit energy, that is for every k +� +x +|T o +x,k|2 = 1, +� +x +|T i +k,x|2 = 1. +(4) +3 + +Illumination +SLM +Validation Arm +Imaging Arm +Laser +Illumination ArmThis means that the total amount of energy that can ar- +rive to particle ok or emerge from it is fixed. As the laser +energy is fixed, we also assume w.l.o.g. that all illumina- +tion vectors have a unit norm ∥u∥ = 1. As propagation +through the tissue does not generate new energy, every +incoming vector u should satisfy ∥T iu∥ ≤ 1 and thus the +energy at the target is also bounded +� +k +|νk|2 ≤ 1. +(5) +3 +Scoring wavefront shaping mod- +ulations +The first challenge when coming to design a wavefront +shaping modulation is coming up with a score function +that can actually evaluate the focusing quality facilitated +by a candidate modulation mask, using a noninvasive +feedback alone. We start by reviewing scores that were +previously introduced in the literature and then propose +our new, noise-robust confocal score. +Image quality scores: Modulation evaluation is a sim- +pler task when the same modulation can correct a suffi- +ciently large isoplanatic image region. This assumption +was made by adaptive optics research [2, 4, 5, 12, 15] and +also by wavefront shaping approaches [9,26,36]. When the +same modulation can correct a large image region, one of- +ten evaluates the quality of the resulting image, either in +terms of contrast [15], sharpness, or variance [36]. How- +ever, for thick tissue, wavefront shaping correction can +vary quickly between nearby pixels, and a modulation +may only explain a very local region. This case makes +the above image quality scores less applicable, as inher- +ently they evaluate the quality of an image region rather +then a pixel. For spatially varying modulations, ideally, +we would like to be able to evaluate the success of the +modulation based on a per-pixel criteria. +The total intensity score: Consider a configuration +where we only try to correct the illumination arm, and +the SLM in the imaging arm of Fig. 1 is not used (equiv- +alently, D(ζ) in (2) is the identity matrix). The easiest +score that was considered in the literature [15,18] is just +the total intensity measured over the entire sensor plane. +Using (2) and (4) it is easy to show that this total inten- +sity score reduces to +MTI(u) ≡ +� +x +I(x) = +� +k +|νk|2α. +(6) +Since the energy at the target is bounded (see (5)), for the +case α > 1 this score is maximized when ν is a one-hot +vector, which equals 1 at a single entry and zero at all +the others. Therefore, in two-photon fluorescence finding +a good modulation is easy. If we manage to modulate the +illumination such that it focuses all the excitation energy +in a single spot, the emitted power is maximized. +Two-photon fluorescence is however more expensive +and harder to implement, and solutions that can use +a single-photon excitation feedback are highly desired. +However, in the single-photon case where α = 1, (6) re- +duces to the total power in ν, MTI(u) = � +k |νk|2, and +since this power is fixed, the same amount of energy re- +turns whether we spread the excitation power over mul- +tiple fluorescence sources or bring all of it into one spot. +Therefore, wavefront shaping using single-photon fluores- +cence has remained an open challenge in the literature +until recently. +The variance maximization score: Following on a +setup that modulates only the illumination and not the +imaging arm, Boniface et al. [3] have recently suggested +that to evaluate focusing with linear single-photon feed- +back, one should maximize the variance of the intensity +measured by the sensor. The idea is that if we manage to +focus all the excitation light at a single spot, the emitted +light scattered through the tissue will generate a highly +varying speckle pattern on the sensor plane. If the excita- +tion is not focused, multiple sources emit simultaneously. +The light emitted by these sources sums incoherently, and +hence the variance of the speckle pattern on the sensor +decays. A short calculation shows +MVar(u) ≡ Var[I] ≡ +≡ 1 +n +� +x +|I(x)|2 − +� +1 +n +� +x +I(x) +�2 += +� +k +|νk|4, +(7) +where n is the number of image pixels. Hence, as before, +the score is a non-linear function of the power at differ- +ent fluorescent particles and is maximized by a one-hot +vector. +This score was an important advance of the state-of- +the-art, but it may be hard to evaluate it with suffi- +cient noise robustness using weak biological sources. To +demonstrate this, Fig. 2 visualizes two types of fluores- +cent emissions, when excitation light is correctly focused +into a single spot. Fig. 2(a) demonstrates an invitrogen +bead (ThermoFisher Fluo-Spheres dark red) that was at- +tached to a chicken breast tissue layer. This is a strong +source, and a clear speckle pattern is imaged. The au- +thors of [3] have demonstrated their approach on similar +beads. However, biological samples are often significantly +weaker than such beads. For example, in Fig. 2(c) we +image EGFP neurons sliced out of a mouse brain. The +fluorescent emission here is orders of magnitudes weaker, +and the amount of laser power we can apply before the +neuron bleaches is also limited. One can see that rather +than a real speckle pattern, we mostly image noise. The +variance of this image is dominated by the noise variance +rather than the actual speckle variance. +4 + +(a) scattering, bead +(b) focusing, bead +(c) scattering, neuron +(d) focusing, neuron +Fig. 2: +Types of fluorescent data: +(a,b) emission from invitrogen fluorescent microspheres (excitation/emission at +640/680nm). +A single bead is excited and the emitted light scatters through the tissue to generate a wide speckle pat- +tern in (a). In (b) we use an aberration correction in the imaging arm so that the sensor measures a sharp spot. With such +synthetic sources we can image a speckle pattern at high SNR, but this is not always the case with real biological samples. +For example, (c,d) demonstrate fluorescent emission from EGFP neurons (excitation/emission at 490/510nm), which is orders +of magnitude weaker. When the aberrated wavefront propagates to the sensor a limited number of photons are spread over +multiple pixels and noise is dominant. +In (d) we have applied aberration correction in the optics and as all photons are +collected by a single pixel, SNR is drastically improved. Note that images (c,d) are taken under equal exposure and equal +excitation power. +Confocal energy score: In this research we suggest a +new score for evaluating a wavefront shaping modulation. +While the previous score corrected only the illumination +arm, we suggest to put the same correction at both illu- +mination and imaging arms. +The idea is that if we find a modulation focusing all ex- +citation light into one spot, due to reciprocity, the same +modulation also corrects the emitted light, bringing all of +it into a single sensor pixel (assuming the excitation and +emission wavelengths are sufficiently similar). To score +the focusing quality of each modulation we will use the +intensity at the central pixel, rather than the total inten- +sity throughout the sensor. +Assuming the central pixel is measuring the DC compo- +nent of the Fourier transformation from the SLM plane to +the image plane, the central row of Psens is just a simple +averaging. Thus we can express its value as the product +of the SLM modulation (at the imaging arm) with the +outgoing transmission matrix: +F0,→D(ζ)T o +↓,k = ζT T o +↓,k. +(8) +When the same modulation u is used in both illumination +and imaging arm, we can express the energy of the central +pixel as: +MConf(u) ≡ I(0) = += +� +k +|uT T o +↓,k|2|T i +k,→u|2 = +� +k +|νk|4. +(9) +As before, this score favors one-hot ν vectors and the +score is maximized when all light is focused at a single +spot. +While this score is equivalent to the variance maxi- +mization score above, it is significantly less susceptible to +noise. This is due to the fact that the small number of +photons we have at hand are collected by one detector, +rather than being spread over multiple pixels. Fig. 2(c- +d) shows the images emitted from a single neural spot +with and without modulation in the imaging arm, and +the significant noise reduction. +In this work we have explicitly optimized the confo- +cal score ((9)) using standard Hadamard basis optimiza- +tion [23], detailed in the supplement. This optimization +is significantly slower than [1]. As emission is very weak, +the fact that the SLM correction is applied before imaging +helps collect all photons at one sensor pixel and improve +SNR. +Iterative phase conjugation: Recently [1] has pro- +posed an incoherent iterative phase conjugation algorithm +that can rapidly estimate a wavefront shaping modula- +tion. This algorithm is not explicitly maximizing a cost +function. It uses fast power iteration to seek an excitation +wavefront which is an eigenvector of the transmission ma- +trix of the tissue, although the definition of transmission +matrices with incoherent light is a bit challenging. In- +tuitively the confocal cost of (9) is maximized when the +wavefront T o +↓,k emerging from the system is correlated +with the illuminating wavefront u. This means that the +optimum can also be thought of as an eigenvector. The +algorithm of [1] was successfully applied on fluorescent +beads, which are both strong and sparse. In this research +we aim to apply the confocal score of (9) to real bio- +logical data such as EGFP neurons. This data is signif- +icantly weaker, and also the fluorescent target exhibits +a continuous area rather than sparse isolated dots. The +algorithm of [1] heavily relies on the existence of some +speckle variation in the input image due to the need to +retrieve the phase of the wavefront arriving the sensor +from the measured intensity. Thus, it does not directly +apply to continuous fluorescent sources, where not much +speckle variation can be measured. +5 + +10μm0.644 +0.53 +0.416 +0.302 +0.188 +0.07410 μm8.264 +6.649 +5.034 +3.419 +1.804 +0.18910gem0.207 +0.171 +0.135 +0.099 +0.063 +0.02710 gem1.23 +0.992 +0.753 +0.515 +0.276 +0.0384 +Results +We image slices of mice brain with EGFP neurons, ex- +cited at 488nm and imaged at 510nm. The mouse brain +slices are 50nm thick, so neurons exhibit some 3D varia- +tion and modest scattering. For more challenging scatter- +ing we place these slices behind a layer of chicken breast +tissue (200−300µm thick) or parafilm. We image fluores- +cent emission with a sensitive sCMOS sensor prime BSI +express. +In Fig. 3 we visualize some results of our algorithm. +Before starting the optimization we focus the objective +such that the excitation light observed by the validation +camera illuminates the smallest possible area. Fig. 3(a) +shows an image of this excitation pattern from the val- +idation camera behind the tissue. As can be observed, +the tissue exhibits significant scattering. We made our +best attempts to reduce the diameter of this pattern by +adjusting the distance of the objective and the sample, +but even at the best focus position, the light scatters to +cover a wide sample area. In Fig. 3(b) we visualize the +excitation light after optimizing the wavefront shaping +modulation, which is nicely focused into a sharp spot. In +Fig. 3(c-d) we have placed a band-pass filter on the val- +idation camera to show the emission light. Before opti- +mization a wide area is excited and we can see the neuron +shape. At the end of the optimization a single point is +excited. In Fig. 3(e-f) we visualize the views of the front +main camera, providing the sole input our algorithm can +access. Before optimization the emitted light is scattered +over a wide sensor area. As a low number of photons is +spread over multiple sensor pixels, the captured imaged +is very noisy. Despite this low SNR, at the end of the +optimization the aberration is corrected and all the pho- +tons are brought into a single sensor pixel, leading to a +high quality image where noise is much less visible, see +Fig. 3(f). In Fig. 3(g) we demonstrate the actual point +spread function of the tissue aberration. For that we have +used the correction only at the illumination arm and fo- +cused the illumination to excite a single spot. We used a +blank SLM at the imaging arm so the emitted light is not +corrected. One can see that the aberration of a single flu- +orescent target is not negligible. We emphasize that each +of the images in Fig. 3 is normalized so that its maximum +is 1, but clearly the spot at the focused images received a +much higher number of photons than the wide scattering +images of unfocused light, despite the fact that all images +were captured under equal exposure and equal excitation +power. +In Fig. 4 we use the recovered wavefront shaping mod- +ulation to image a wide area rather than a single spot. +For that we excite a wide area and use a correction only +at the imaging arm. Due to the memory effect [17, 22], +the same modulation can allow us to image a small lo- +cal patch rather than a single spot. With the correction, +the neuron is observed with a much higher contrast and +even the axons (thin lines around the neuron) emerging +from it, whose emission is much weaker, can be partially +observed. As our SLMs are placed in the Fourier plane +of the system and not at a plane conjugate to the sam- +ple itself as suggested by [21], we tilt-shift the modulated +pattern to image a somewhat wider area, as explained +in [1]. +5 +Discussion +In this research we have analyzed score functions for wave- +front shaping correction using non-invasive feedback at +the absence of a guiding star. Obtaining such feedback is +challenging, because even if excitation light is corrected +and focused at a single object spot, the light returning to +the sensor is undergoing another aberration process while +propagating through the tissue, leading to yet another +scattered pattern. Moreover, real biological fluorescent +sources are weak emitting a limited number of photons. +When these photons are spread over multiple sensor pix- +els the detectable signal is highly contaminated by noise. +To assess focusing quality, we need a score function that +can measure a non-linear function of the light emitted by +different sources. This is naturally achieved when using +two-photon fluorescent feedback, but is harder to achieve +with linear fluorescence. We show that by using a confo- +cal correction at both illumination and imaging arms we +can measure such a non-linear feedback, which is maxi- +mized when all excitation light is brought into one spot. +Moreover, the fact that our system uses a correction of +the emitted light as part of the optical path allows us to +bring the limited number of emitted photons into a single +sensor spot, facilitating a high SNR measurement. +The drawback of our current approach is that it uses +a slow Hadamard basis optimization [23]. +In our cur- +rent implementation it takes about 30 min to optimize +for one modulation pattern. Some of this can be largely +optimized by better hardware such as a faster SLM. How- +ever, due to the large number of iterations required by the +simple coordinate descent optimization, this approach is +inherently slower than the power iterations of [1]. We are +exploring ways to accelerate our current optimization by +extending the phase retrieval framework of [1] to model +the full image formation model of the incoherent case. +References +[1] Aizik, D., Gkioulekas, I., and Levin, A. Fluo- +rescent wavefront shaping using incoherent iterative +phase conjugation. Optica 9, 7 (Jul 2022), 746–754. +6 + +(a) Valid. laser +(b) Valid. laser +(c) Valid. fluor. +(d) Valid. fluor. +(e) Main fluor. +(f) Main fluor. +(g) Main fluor. +No modulation +With modulation +No modulation +With modulation +No modulation +With modulation +Point aberration +Fig. 3: Wavefront shaping results: we visualize views from the validation and main cameras, at the beginning of the algorithm +where no correction is applied, compared to the modulated image at the end of the optimization. (a-b) The excitation light as +viewed by the validation camera at the back of the tissue. Due to significant scattering, at the beginning a wide speckle pattern +is generated, but after optimization, the modulated wavefront is brought into a single spot. (c-d) By placing a band-pass filter +on the validation camera we visualize the emitted light with and without correction. (e-f) Views of the emitted light at the +main front camera with and without correction. Note that this is the only input used by our algorithm. Without correction, +light is scattered over a wide image area and is being measured with a very low SNR. A sharp clean spot can be imaged when +the limited number of photons is corrected in the optical path and brought into a single sensor pixel. (g) By correcting the +emission such that a single spot is excited and leaving the imaging path uncorrected, we can visualize the actual aberration +of a single fluorescent point source. +[2] Antonello, J., Barbotin, A., Chong, E. Z., +Rittscher, J., and Booth, M. J. +Multi-scale +sensorless adaptive optics: application to stimulated +emission depletion microscopy. Opt. Express 28, 11 +(May 2020), 16749–16763. +[3] Boniface, A., Blochet, B., Dong, J., and Gi- +gan, S. +Noninvasive light focusing in scattering +media using speckle variance optimization. Optica +(2019). +[4] Bonora, S., and Zawadzki, R. J. Wavefront sen- +sorless modal deformable mirror correction in adap- +tive optics: optical coherence tomography. Opt. Lett. +38, 22 (Nov 2013), 4801–4804. +[5] Booth, M. Adaptive optical microscopy: The on- +going quest for a perfect image. Light: Science and +Applications 3 (04 2014), e165. +[6] Booth, M. J., Neil, M. A. A., Juˇskaitis, R., +and Wilson, T. Adaptive aberration correction in +a confocal microscope. Proceedings of the National +Academy of Sciences 99, 9 (2002), 5788–5792. +[7] Chen, Y., Sharma, M. K., Sabharwal, A., +Veeraraghavan, A., and Sankaranarayanan, +A. C. +3PointTM: Faster measurement of high- +dimensional transmission matrices. In Euro. Conf. +Computer Vision (ECCV) (2020). +[8] Conkey, D. B., Brown, A. N., Caravaca- +Aguirre, A. M., and Piestun, R. Genetic algo- +rithm optimization for focusing through turbid me- +dia in noisy environments. Opt. Express (2012). +[9] Daniel, A., Oron, D., and Silberberg, Y. Light +focusing through scattering media via linear fluores- +cence variance maximization, and its application for +fluorescence imaging. Opt. Express (2019). +[10] Fiolka, R., Si, K., and Cui, M. Complex wave- +front corrections for deep tissue focusing using low +coherence backscattered light. Opt. Express 20, 15 +(Jul 2012), 16532–16543. +[11] Gigan, S., Katz, O., et al. Roadmap on wave- +front shaping and deep imaging in complex media. +arXiv preprint arXiv:2111.14908 (2021). +7 + +10μm10 μm10 μm10μm10μm0.176 +0.147 +0.119 +0.09 +0.061 +0.03210μm2.002 +1.614 +1.225 +0.837 +0.448 +0.0610 μm0.276 +0.228 +0.179 +0.131 +0.082 +0.03410μm10μm10 μm10μm10 μm0.222 +0.184 +0.146 +0.108 +0.07 +0.03210μm3.191 +2.57 +1.948 +1.326 +0.704 +0.08210μm0.436 +0.356 +0.277 +0.197 +0.117 +0.03710μm10 μm10μm10μm10μm0.265 +0.218 +0.171 +0.125 +0.078 +0.03110μm1.004 +0.812 +0.62 +0.427 +0.235 +0.04310μm0.271 +0.223 +0.176 +0.128 +0.081 +0.034(a) Uncorrected +(b) Corrected +(c) Reference +Main Camera +Main Camera +Validation Camera +Fig. 4: Wide area imaging: We use an unmodulated illumination to excite a wide fluorescent region, but place the recovered +modulation the imaging SLM to correct the emitted light. We compare this to an undistorted reference viewed by the validation +camera, and to the initial uncorrected image of the main camera. Notice how the neuron contrast is improved and even some +of the axons are revealed. +[12] Hampson, +K., +Turcotte, +R., +Miller, +D., +Kurokawa, K., Males, J., Ji, N., and Booth, +M. Adaptive optics for high-resolution imaging. Na- +ture Reviews Methods Primers 1 (10 2021), 68. +[13] Horstmeyer, R., Ruan, H., and Yang, C. +Guidestar-assisted wavefront-shaping methods for +focusing light into biological tissue. Nature Photon- +ics (2015). +[14] Jang, J., Lim, J., Yu, H., Choi, H., Ha, J., +Park, J.-H., Oh, W.-Y., Jang, W., Lee, S., and +Park, Y. Complex wavefront shaping for optimal +depth-selective focusing in optical coherence tomog- +raphy. Opt. Express 21, 3 (Feb 2013), 2890–2902. +[15] Ji, N. Adaptive optical fluorescence microscopy. Na- +ture Methods 14 (03 2017), 374–380. +[16] Ji, N., Milkie, D., and Betzig, E. Adaptive op- +tics via pupil segmentation for high-resolution imag- +ing in biological tissues. Nature methods 7 (02 2010), +141–7. +[17] Judkewitz, B., Horstmeyer, R., Vellekoop, +I., and Yang, C. +Translation correlations in +anisotropically scattering media. +Nature Physics +(2014). +[18] Katz, O., Small, E., Guan, Y., and Silber- +berg, Y. Noninvasive nonlinear focusing and imag- +ing through strongly scattering turbid layers. Optica +1, 3 (Sep 2014), 170–174. +[19] Kong, F., Silverman, R. H., Liu, L., Chit- +nis, P. V., Lee, K. K., and Chen, Y. C. +Photoacoustic-guided convergence of light through +optically diffusive media. +Opt. Lett. 36, 11 (Jun +2011), 2053–2055. +8 + +10μm10 μm10 μm10μm10μm10μm10μm10μm10μm[20] Liu, T.-L., et al. Observing the cell in its native +state: Imaging subcellular dynamics in multicellular +organisms. Science 360, 6386 (2018). +[21] Mertz, J., Paudel, H., and Bifano, T. G. Field +of view advantage of conjugate adaptive optics in +microscopy applications. Applied Optics (2015). +[22] Osnabrugge, +G., +Horstmeyer, +R., +Pa- +padopoulos, +I. +N., +Judkewitz, +B., +and +Vellekoop, I. M. Generalized optical memory ef- +fect. Optica (2017). +[23] Popoff, S. M., Aubry, A., Lerosey, G., Fink, +M., Boccara, A. C., and Gigan, S. Exploiting +the time-reversal operator for adaptive optics, selec- +tive focusing, and scattering pattern analysis. Phys. +Rev. Lett. 107 (Dec 2011), 263901. +[24] Popoff, S. M., Lerosey, G., Carminati, R., +Fink, M., Boccara, A. C., and Gigan, S. Mea- +suring the transmission matrix in optics: +An ap- +proach to the study and control of light propagation +in disordered media. Phys. Rev. Lett. (2010). +[25] Rotter, S., and Gigan, S. Light fields in complex +media: Mesoscopic scattering meets wave control. +Rev. Mod. Phys. (2017). +[26] Stern, G., and Katz, O. +Noninvasive focusing +through scattering layers using speckle correlations. +Opt. Lett. (2019). +[27] Tang, J., Germain, R. N., and Cui, M. +Su- +perpenetration optical microscopy by iterative mul- +tiphoton adaptive compensation technique. Proceed- +ings of the National Academy of Sciences 109, 22 +(2012), 8434–8439. +[28] Vellekoop, I. M., Cui, M., and Yang, C. Digi- +tal optical phase conjugation of fluorescence in turbid +tissue. Applied Physics Letters (2012). +[29] Vellekoop, I. M., Lagendijk, A., and Mosk, +A. P. Exploiting disorder for perfect focusing. Na- +ture Photonics (2010). +[30] Vellekoop, I. M., and Mosk, A. P. Focusing co- +herent light through opaque strongly scattering me- +dia. Opt. Lett. (2007). +[31] Verstraete, H. R. G. W., Wahls, S., Kalk- +man, J., and Verhaegen, M. Model-based sensor- +less wavefront aberration correction in optical coher- +ence tomography. Opt. Lett. 40, 24 (Dec 2015), 5722– +5725. +[32] Wang, C., Liu, R., Milkie, D., Sun, W., Tan, +Z., Kerlin, A., Chen, T.-W., Kim, D., and Ji, +N. +Multiplexed aberration measurement for deep +tissue imaging in vivo. Nature methods 11 (08 2014). +[33] Wang, Y. M., Judkewitz, B., DiMarzio, C. A., +and Yang, C. Deep-tissue focal fluorescence imag- +ing with digitally time-reversed ultrasound-encoded +light. Nature Communications 3 (2012). +[34] Xu, X., Liu, H., and Wang, L. Time-reversed ul- +trasonically encoded optical focusing into scattering +media. Nature Photonics 5 (2011), 154–157. +[35] Yaqoob, Z., Psaltis, D., Feld, M., and Yang, +C. Optical phase conjugation for turbidity suppres- +sion in biological samples. Nature photonics (2008). +[36] Yeminy, T., and Katz, O. Guidestar-free image- +guided wavefront-shaping. Science Advances 7, 21 +(2021), eabf5364. +[37] Yu, H., Park, J., Lee, K., Yoon, J., Kim, K., +Lee, S., and Park, Y. Recent advances in wave- +front shaping techniques for biomedical applications. +Current Applied Physics 15, 5 (2015), 632–641. +9 + diff --git a/CtFJT4oBgHgl3EQfAiym/content/tmp_files/load_file.txt b/CtFJT4oBgHgl3EQfAiym/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..0a5cc9a6828cdcb85ecbbad128239a3102621e48 --- /dev/null +++ b/CtFJT4oBgHgl3EQfAiym/content/tmp_files/load_file.txt @@ -0,0 +1,616 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf,len=615 +page_content='Non-invasive and noise-robust light focusing using confocal wavefront shaping Dror Aizik, Anat Levin Department of Electrical and Computer Engineering, Technion, Haifa, Israel Abstract One of the hardest barriers to our ability to see inside biological tissue is the fact that light is highly aberrated when scattering by the tissue sub-components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' One of the promising approaches for overcoming such aberrations is wavefront-shaping, where one modulates the incoming and/or the outgoing wavefront in a way that will allow it to focus into one spot despite scattering in the tissue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Wavefront modulations are specific to the tissue sample being imaged and need to be estimated based on a non- invasive feedback from a camera collecting back-scattered light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Such modulations have been successively estimated using feedback from strong fluorescent beads which have been manually added to a sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' However, in a real biomedical application, such feedback should be provided by the fluorescent components of the tissue itself, whose emission is orders of magnitude lower than the one pro- vided by beads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' When such a low number of photons is spread over multiple sensor pixels, the image is highly susceptible to noise, and the feedback signal required for previous algorithms cannot be detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' In this work we suggest a wavefront shaping approach that works with a confocal modulation of both illumina- tion and imaging arms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' The advantage of this approach is that as part of the optimization, aberrations are corrected in the optics, before the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Hence the low photon budget can be directed into a single sensor spot and de- tected with high SNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' We derive the optimization prob- lem from mathematical principles and show why it favors modulations that actually correct the aberrations and fo- cus all light into one spot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' We successfully demonstrate wavefront-shaping correction on EGFP neurons sliced out of a mouse brain, despite scattering through thick tissue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' 1 Introduction One of the hardest barriers to light-based approaches to tissue imaging is the fact that light is heavily scattered due to variations in the refractive index of tissue struc- tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' A promising approach for overcoming the scatter- ing challenge is a wavefront-shaping based correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' By using a spatial light modulator (SLM) device, one can reshape the coherent wavefront illuminating the sample, such that its aberration is conjugate to the aberration that will happen inside the tissue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' When such a wavefront propagates through the sample, all incoming light can be brought (focused) into a small spot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' In the same way, by using a wavefront modulation element between the tissue and the sensor, we can correct the outgoing wavefront, so that light photons emerging from a single target point are brought into a single sensor point, despite the tissue aberration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' The main advantage of this approach results from the fact that unlike ballistic-filtering approaches, all light photons are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Earlier wavefront shaping approaches, formally known as adaptive optics [5, 12, 15], were first used to correct modest aberrations, for example due to imperfect optics or refractive index variations in the tissue [6,16,31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' More recently, wavefront shaping techniques [11, 13, 37] have shown that it is possible to focus light through thick, highly-scattering layers [28–30,35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Despite the large potential of the idea, finding the de- sired shape of the modulation correction is rather chal- lenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' The desired modulation varies between different tissue samples and even varies spatially between different positions of the same tissue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' For thick tissue, the modu- lation is a complex pattern containing a large number of free modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Earlier proof-of-concept demonstrations have used a validation camera behind the tissue to provide feed- back to the algorithm [7, 8, 24, 29, 30, 35], and other approaches have relied on the existence of a guiding- star [10, 13, 14, 18–20, 27, 28, 32–34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' In the absence of such a guiding star, and when only non-invasive feed- back is available, determining whatever a wavefront has focused inside the tissue is not straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' The dif- ficulty results from the fact that even if we can find an illumination wavefront that actually focuses into a small spot inside the tissue, the light back-scattering from this 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content='11421v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content='optics] 26 Jan 2023 spot is aberrated again on its way to the camera, forming yet another scattered pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' The simplest way to evaluate whatever a wavefront modulation has focused is to use multi-photon fluores- cence feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' In this way, the light emitted from a flu- orescence spot is a non-linear function of the excitation intensity arriving it, so when all light is focused into a sin- gle spot the total emission energy is maximized [15, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' However, obtaining feedback using single-photon fluores- cence is highly desired as the process is significantly sim- pler and cheaper than the multi-photon one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' The single- photon case cannot be evaluated using the simple score function applied in the multi-photon case, since the emis- sion energy is a linear function of the excitation energy and thus the amount of emission energy does not increase when all excitation is focused into a spot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Recently, progress has been made on non-invasive wave- front shaping using single-photon feedback [1, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' First, Boniface et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' [3] have suggested that one can evalu- ate whatever an incoming wavefront modulation has fo- cused by computing the variance of the emitted speckle pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' More recently, Aizik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' [1] has suggested a rapid approach that can find a wavefront shaping mod- ulation using a small number of iterative phase conjuga- tion iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Both approaches were only demonstrated when the fluorescent feedback was provided by synthetic fluorescent beads, which emit a relatively strong signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' However, the ultimate goal is to apply wavefront shap- ing modulation using feedback from biological samples, such as neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' The signal emitted from such samples is orders of magnitude weaker than the one provided by fluorescent beads, and bleaching is reached much earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Both algorithms [1,3] inherently assume that the speckle pattern emitted from a single fluorescent spot can be mea- sured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' However, the number of fluorescent photons emit- ted from a neuron spot is so low that when these photons are aberrated and spread over multiple sensor pixels, no speckle pattern can be observed and one can mostly mea- sure noise, see visualization in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' With such a low photon count no speckle variance can be estimated as is required by [3], and no phase retrieval process can be ro- bustly carried as in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' This work proposes a wavefront shaping framework that can apply in low-light scenarios and use feedback from realistic biological data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' To this end, we propose to use a simultaneous wavefront shaping modulation both on the incoming excitation wavefront and on the outgo- ing emitted light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' The advantage is that since scattered photons are corrected in the optical path and we attempt to bring all photons emitted from a single spot into a sin- gle detector, we can measure them with a much higher signal-to-noise (SNR) ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' To quantify the quality of a candidate modulation cor- rection, we do not attempt to maximize the total energy emitted from the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Rather, we seek to maximize the energy of the corrected wavefront in a single pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' We show that despite the fact that we use linear single-photon fluorescence, due to the double correction on both illumi- nation and imaging arms, our score function scales non- linearly with the intensity arriving at the fluorescence tar- get.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Thus, the returning energy at a single pixel is max- imized by a focusing modulation that manages to bring all light into a single spot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' We show that effectively, this score function is equivalent to the one used by previous two-photon fluorescence wavefront-shaping work [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' 2 Problem formulation Imaging setup: In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' 1 we visualize a wavefront- shaping imaging setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' A laser beam illuminates a tis- sue sample via a microscope objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' A phase SLM in the illumination arm modulates the illumination pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' We wish to image a fluorescent target at the back of the tissue layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' The light returning from the target is col- lected via the same objective, and reflected at a dichroic beam-splitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' A second phase SLM at the imaging arm modulates the emitted light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Lastly, the modulated light is measured by the front main camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' In our setup the SLMs (holoeye-pluto) are placed in the Fourier plane of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' The setup includes a second validation camera behind the tissue sample to assess focusing quality and image an undistorted reference of the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' While earlier research demonstrations of wavefront-shaping use this camera to provide feedback to the algorithm, we emphasize that our goal in this research is to develop non-invasive techniques that can only use feedback by the main (front) camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' The validation camera cannot provide any input to the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' We note that some research on wavefront shaping and adaptive optics modulates only the illumination or imag- ing arms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' For generality, our problem formulation below will consider modulations at both arms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Image formation model: Consider a set of K fluores- cent particles inside a sample, and denote their positions by o1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' , oK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' We assume the SLM in the illumination arm is illuminated with a spatially uniform plane wave and use the SLM to display a complex 2D electric field that we denote by u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Although u is a 2D field, we reshape it as a 1D vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' We also use ν to denote a K ×1 vector of the field propagating through the sample at each of the K fluorescent sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' The relation between u and ν is linear and can be described as a multiplication by a (very large) matrix ν = T iu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' T i is the incoming transmission matrix de- scribing coherent light propagation in the tissue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' We note that T i is specific to the tissue sample being tested, and 2 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' 1: Our wavefront correction fluorescent microscope setup: A laser beam is exciting fluorescent beads at the back of a tissue layer, and fluorescent emission is scattered again through the tissue, reflects at a dichroic beam-splitter and is collected by a main (front) camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' We place two SLMs in the Fourier planes of both illumination and imaging arms to allow reshaping these wavefronts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' A validation camera views the beads at the back of the tissue directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' This camera is not actually used by the algorithm, and is only assessing its success.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' LP=linear polarizer, BS=beam-splitter, DBS=dichroic beam-splitter, BPF=bandpass filter, L1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' L7=lenses, Obj=Objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' different tissue samples are described by very different transmission matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' For thick tissue T i can be an arbitrarily complex matrix incorporating multiple scat- tering events in the tissue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Likewise, the light returning from the particles to the SLM of the imaging arm can be described as T oν, where T o is the back-propagation transmission matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' The propagation of light from the illumination SLM plane to the particles and back to the SLM of the imag- ing arm is then modeled using the combined transmission matrix T a ≡ T o · T i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' (1) We denote by ζ the wavefront placed on the SLM of the imaging arm, and by D(ζ) a diagonal matrix with ζ on its diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' We denote by F the Fourier transform of the wavefront from the SLM plane to the camera sensor plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' In the fluorescent case, the emissions from different points are incoherent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' The recorded intensity can be ex- pressed as I = � k |FD(ζ)T o ↓,k|2|νk|2α, (2) where |νk|2 is the energy of the excitation light arriving at particle ok, and T o ↓,k is the k-th column of T o, so that FD(ζ)T o ↓,k is the wavefront arriving to the sensor from ok.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Since light emitted from different fluorescent par- ticles changes phase incoherently, effectively the sensor sums the intensity of the wavefronts emitted by differ- ent particles, and their phases do not interfere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' In (2) α denotes the type of fluorescent excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' The simplest case α = 1 is known as single-photon fluorescence where the emission is linear in the excitation energy |νk|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' In two-photon fluorescence, α = 2, namely the emission is proportional to the squared excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Linear fluores- cence is significantly simpler and cheaper to achieve, but as we explain below, a non-linear (two-photon) fluores- cence feedback simplifies modulation estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Phase conjugation: For coherent imaging, the Helmholtz reciprocity principle leads to wave conjuga- tion, namely if we record the wavefront emitted from a source point inside the tissue and play it back in the re- verse direction, the wavefront will focus at the same point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' This implies that the returning transmission matrix is the transpose of the incoming one [25]: T o = T i⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' (3) Note that this is just a transpose and not the Hermi- tian (conjugate) transpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' In the fluorescent case, we should note that T i, T o describe propagation at a dif- ferent wavelength hence they cannot be completely the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' However, for linear single-photon fluorescence the excitation and emission wavelengths are relatively similar and we still assume T o ≈ T i⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Normalization: we assume for simplicity that our trans- mission matrices are normalized such that every column or row has a unit energy, that is for every k � x |T o x,k|2 = 1, � x |T i k,x|2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' (4) 3 Illumination SLM Validation Arm Imaging Arm Laser Illumination ArmThis means that the total amount of energy that can ar- rive to particle ok or emerge from it is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' As the laser energy is fixed, we also assume w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' that all illumina- tion vectors have a unit norm ∥u∥ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' As propagation through the tissue does not generate new energy, every incoming vector u should satisfy ∥T iu∥ ≤ 1 and thus the energy at the target is also bounded � k |νk|2 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' (5) 3 Scoring wavefront shaping mod- ulations The first challenge when coming to design a wavefront shaping modulation is coming up with a score function that can actually evaluate the focusing quality facilitated by a candidate modulation mask, using a noninvasive feedback alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' We start by reviewing scores that were previously introduced in the literature and then propose our new, noise-robust confocal score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Image quality scores: Modulation evaluation is a sim- pler task when the same modulation can correct a suffi- ciently large isoplanatic image region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' This assumption was made by adaptive optics research [2, 4, 5, 12, 15] and also by wavefront shaping approaches [9,26,36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' When the same modulation can correct a large image region, one of- ten evaluates the quality of the resulting image, either in terms of contrast [15], sharpness, or variance [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' How- ever, for thick tissue, wavefront shaping correction can vary quickly between nearby pixels, and a modulation may only explain a very local region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' This case makes the above image quality scores less applicable, as inher- ently they evaluate the quality of an image region rather then a pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' For spatially varying modulations, ideally, we would like to be able to evaluate the success of the modulation based on a per-pixel criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' The total intensity score: Consider a configuration where we only try to correct the illumination arm, and the SLM in the imaging arm of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' 1 is not used (equiv- alently, D(ζ) in (2) is the identity matrix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' The easiest score that was considered in the literature [15,18] is just the total intensity measured over the entire sensor plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Using (2) and (4) it is easy to show that this total inten- sity score reduces to MTI(u) ≡ � x I(x) = � k |νk|2α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' (6) Since the energy at the target is bounded (see (5)), for the case α > 1 this score is maximized when ν is a one-hot vector, which equals 1 at a single entry and zero at all the others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Therefore, in two-photon fluorescence finding a good modulation is easy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' If we manage to modulate the illumination such that it focuses all the excitation energy in a single spot, the emitted power is maximized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Two-photon fluorescence is however more expensive and harder to implement, and solutions that can use a single-photon excitation feedback are highly desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' However, in the single-photon case where α = 1, (6) re- duces to the total power in ν, MTI(u) = � k |νk|2, and since this power is fixed, the same amount of energy re- turns whether we spread the excitation power over mul- tiple fluorescence sources or bring all of it into one spot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Therefore, wavefront shaping using single-photon fluores- cence has remained an open challenge in the literature until recently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' The variance maximization score: Following on a setup that modulates only the illumination and not the imaging arm, Boniface et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' [3] have recently suggested that to evaluate focusing with linear single-photon feed- back, one should maximize the variance of the intensity measured by the sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' The idea is that if we manage to focus all the excitation light at a single spot, the emitted light scattered through the tissue will generate a highly varying speckle pattern on the sensor plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' If the excita- tion is not focused, multiple sources emit simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' The light emitted by these sources sums incoherently, and hence the variance of the speckle pattern on the sensor decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' A short calculation shows MVar(u) ≡ Var[I] ≡ ≡ 1 n � x |I(x)|2 − � 1 n � x I(x) �2 = � k |νk|4, (7) where n is the number of image pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Hence, as before, the score is a non-linear function of the power at differ- ent fluorescent particles and is maximized by a one-hot vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' This score was an important advance of the state-of- the-art, but it may be hard to evaluate it with suffi- cient noise robustness using weak biological sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' To demonstrate this, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' 2 visualizes two types of fluores- cent emissions, when excitation light is correctly focused into a single spot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' 2(a) demonstrates an invitrogen bead (ThermoFisher Fluo-Spheres dark red) that was at- tached to a chicken breast tissue layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' This is a strong source, and a clear speckle pattern is imaged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' The au- thors of [3] have demonstrated their approach on similar beads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' However, biological samples are often significantly weaker than such beads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' For example, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' 2(c) we image EGFP neurons sliced out of a mouse brain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' The fluorescent emission here is orders of magnitudes weaker, and the amount of laser power we can apply before the neuron bleaches is also limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' One can see that rather than a real speckle pattern, we mostly image noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' The variance of this image is dominated by the noise variance rather than the actual speckle variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' 4 (a) scattering, bead (b) focusing, bead (c) scattering, neuron (d) focusing, neuron Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' 2: Types of fluorescent data: (a,b) emission from invitrogen fluorescent microspheres (excitation/emission at 640/680nm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' A single bead is excited and the emitted light scatters through the tissue to generate a wide speckle pat- tern in (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' In (b) we use an aberration correction in the imaging arm so that the sensor measures a sharp spot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' With such synthetic sources we can image a speckle pattern at high SNR, but this is not always the case with real biological samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' For example, (c,d) demonstrate fluorescent emission from EGFP neurons (excitation/emission at 490/510nm), which is orders of magnitude weaker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' When the aberrated wavefront propagates to the sensor a limited number of photons are spread over multiple pixels and noise is dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' In (d) we have applied aberration correction in the optics and as all photons are collected by a single pixel, SNR is drastically improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Note that images (c,d) are taken under equal exposure and equal excitation power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Confocal energy score: In this research we suggest a new score for evaluating a wavefront shaping modulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' While the previous score corrected only the illumination arm, we suggest to put the same correction at both illu- mination and imaging arms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' The idea is that if we find a modulation focusing all ex- citation light into one spot, due to reciprocity, the same modulation also corrects the emitted light, bringing all of it into a single sensor pixel (assuming the excitation and emission wavelengths are sufficiently similar).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' To score the focusing quality of each modulation we will use the intensity at the central pixel, rather than the total inten- sity throughout the sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Assuming the central pixel is measuring the DC compo- nent of the Fourier transformation from the SLM plane to the image plane, the central row of Psens is just a simple averaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Thus we can express its value as the product of the SLM modulation (at the imaging arm) with the outgoing transmission matrix: F0,→D(ζ)T o ↓,k = ζT T o ↓,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' (8) When the same modulation u is used in both illumination and imaging arm, we can express the energy of the central pixel as: MConf(u) ≡ I(0) = = � k |uT T o ↓,k|2|T i k,→u|2 = � k |νk|4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' (9) As before, this score favors one-hot ν vectors and the score is maximized when all light is focused at a single spot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' While this score is equivalent to the variance maxi- mization score above, it is significantly less susceptible to noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' This is due to the fact that the small number of photons we have at hand are collected by one detector, rather than being spread over multiple pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' 2(c- d) shows the images emitted from a single neural spot with and without modulation in the imaging arm, and the significant noise reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' In this work we have explicitly optimized the confo- cal score ((9)) using standard Hadamard basis optimiza- tion [23], detailed in the supplement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' This optimization is significantly slower than [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' As emission is very weak, the fact that the SLM correction is applied before imaging helps collect all photons at one sensor pixel and improve SNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Iterative phase conjugation: Recently [1] has pro- posed an incoherent iterative phase conjugation algorithm that can rapidly estimate a wavefront shaping modula- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' This algorithm is not explicitly maximizing a cost function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' It uses fast power iteration to seek an excitation wavefront which is an eigenvector of the transmission ma- trix of the tissue, although the definition of transmission matrices with incoherent light is a bit challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' In- tuitively the confocal cost of (9) is maximized when the wavefront T o ↓,k emerging from the system is correlated with the illuminating wavefront u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' This means that the optimum can also be thought of as an eigenvector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' The algorithm of [1] was successfully applied on fluorescent beads, which are both strong and sparse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' In this research we aim to apply the confocal score of (9) to real bio- logical data such as EGFP neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' This data is signif- icantly weaker, and also the fluorescent target exhibits a continuous area rather than sparse isolated dots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' The algorithm of [1] heavily relies on the existence of some speckle variation in the input image due to the need to retrieve the phase of the wavefront arriving the sensor from the measured intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Thus, it does not directly apply to continuous fluorescent sources, where not much speckle variation can be measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' 5 10μm0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content='644 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content='034 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content='419 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content='804 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content='18910gem0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content='207 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content='171 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content='135 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content='099 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content='063 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content='02710 gem1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content='992 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content='753 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content='515 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content='276 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content='0384 Results We image slices of mice brain with EGFP neurons, ex- cited at 488nm and imaged at 510nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' The mouse brain slices are 50nm thick, so neurons exhibit some 3D varia- tion and modest scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' For more challenging scatter- ing we place these slices behind a layer of chicken breast tissue (200−300µm thick) or parafilm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' We image fluores- cent emission with a sensitive sCMOS sensor prime BSI express.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' 3 we visualize some results of our algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Before starting the optimization we focus the objective such that the excitation light observed by the validation camera illuminates the smallest possible area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' 3(a) shows an image of this excitation pattern from the val- idation camera behind the tissue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' As can be observed, the tissue exhibits significant scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' We made our best attempts to reduce the diameter of this pattern by adjusting the distance of the objective and the sample, but even at the best focus position, the light scatters to cover a wide sample area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' 3(b) we visualize the excitation light after optimizing the wavefront shaping modulation, which is nicely focused into a sharp spot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' 3(c-d) we have placed a band-pass filter on the val- idation camera to show the emission light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Before opti- mization a wide area is excited and we can see the neuron shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' At the end of the optimization a single point is excited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' 3(e-f) we visualize the views of the front main camera, providing the sole input our algorithm can access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Before optimization the emitted light is scattered over a wide sensor area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' As a low number of photons is spread over multiple sensor pixels, the captured imaged is very noisy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Despite this low SNR, at the end of the optimization the aberration is corrected and all the pho- tons are brought into a single sensor pixel, leading to a high quality image where noise is much less visible, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' 3(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' 3(g) we demonstrate the actual point spread function of the tissue aberration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' For that we have used the correction only at the illumination arm and fo- cused the illumination to excite a single spot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' We used a blank SLM at the imaging arm so the emitted light is not corrected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' One can see that the aberration of a single flu- orescent target is not negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' We emphasize that each of the images in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' 3 is normalized so that its maximum is 1, but clearly the spot at the focused images received a much higher number of photons than the wide scattering images of unfocused light, despite the fact that all images were captured under equal exposure and equal excitation power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' 4 we use the recovered wavefront shaping mod- ulation to image a wide area rather than a single spot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' For that we excite a wide area and use a correction only at the imaging arm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Due to the memory effect [17, 22], the same modulation can allow us to image a small lo- cal patch rather than a single spot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' With the correction, the neuron is observed with a much higher contrast and even the axons (thin lines around the neuron) emerging from it, whose emission is much weaker, can be partially observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' As our SLMs are placed in the Fourier plane of the system and not at a plane conjugate to the sam- ple itself as suggested by [21], we tilt-shift the modulated pattern to image a somewhat wider area, as explained in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' 5 Discussion In this research we have analyzed score functions for wave- front shaping correction using non-invasive feedback at the absence of a guiding star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Obtaining such feedback is challenging, because even if excitation light is corrected and focused at a single object spot, the light returning to the sensor is undergoing another aberration process while propagating through the tissue, leading to yet another scattered pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Moreover, real biological fluorescent sources are weak emitting a limited number of photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' When these photons are spread over multiple sensor pix- els the detectable signal is highly contaminated by noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' To assess focusing quality, we need a score function that can measure a non-linear function of the light emitted by different sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' This is naturally achieved when using two-photon fluorescent feedback, but is harder to achieve with linear fluorescence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' We show that by using a confo- cal correction at both illumination and imaging arms we can measure such a non-linear feedback, which is maxi- mized when all excitation light is brought into one spot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Moreover, the fact that our system uses a correction of the emitted light as part of the optical path allows us to bring the limited number of emitted photons into a single sensor spot, facilitating a high SNR measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' The drawback of our current approach is that it uses a slow Hadamard basis optimization [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' In our cur- rent implementation it takes about 30 min to optimize for one modulation pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Some of this can be largely optimized by better hardware such as a faster SLM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' How- ever, due to the large number of iterations required by the simple coordinate descent optimization, this approach is inherently slower than the power iterations of [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' We are exploring ways to accelerate our current optimization by extending the phase retrieval framework of [1] to model the full image formation model of the incoherent case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' References [1] Aizik, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', Gkioulekas, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', and Levin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Fluo- rescent wavefront shaping using incoherent iterative phase conjugation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Optica 9, 7 (Jul 2022), 746–754.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' 6 (a) Valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' laser (b) Valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' laser (c) Valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' fluor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' (d) Valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' fluor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' (e) Main fluor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' (f) Main fluor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' (g) Main fluor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' No modulation With modulation No modulation With modulation No modulation With modulation Point aberration Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' 3: Wavefront shaping results: we visualize views from the validation and main cameras, at the beginning of the algorithm where no correction is applied, compared to the modulated image at the end of the optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' (a-b) The excitation light as viewed by the validation camera at the back of the tissue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Due to significant scattering, at the beginning a wide speckle pattern is generated, but after optimization, the modulated wavefront is brought into a single spot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' (c-d) By placing a band-pass filter on the validation camera we visualize the emitted light with and without correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' (e-f) Views of the emitted light at the main front camera with and without correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Note that this is the only input used by our algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Without correction, light is scattered over a wide image area and is being measured with a very low SNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' A sharp clean spot can be imaged when the limited number of photons is corrected in the optical path and brought into a single sensor pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' (g) By correcting the emission such that a single spot is excited and leaving the imaging path uncorrected, we can visualize the actual aberration of a single fluorescent point source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' [2] Antonello, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', Barbotin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', Chong, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', Rittscher, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', and Booth, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Multi-scale sensorless adaptive optics: application to stimulated emission depletion microscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Express 28, 11 (May 2020), 16749–16763.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' [3] Boniface, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', Blochet, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', Dong, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', and Gi- gan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Noninvasive light focusing in scattering media using speckle variance optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Optica (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' [4] Bonora, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', and Zawadzki, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Wavefront sen- sorless modal deformable mirror correction in adap- tive optics: optical coherence tomography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' 38, 22 (Nov 2013), 4801–4804.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' [5] Booth, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Adaptive optical microscopy: The on- going quest for a perfect image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Light: Science and Applications 3 (04 2014), e165.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' [6] Booth, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', Neil, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', Juˇskaitis, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', and Wilson, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Adaptive aberration correction in a confocal microscope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Proceedings of the National Academy of Sciences 99, 9 (2002), 5788–5792.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' [7] Chen, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', Sharma, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', Sabharwal, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', Veeraraghavan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', and Sankaranarayanan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' 3PointTM: Faster measurement of high- dimensional transmission matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' In Euro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Computer Vision (ECCV) (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' [8] Conkey, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', Brown, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', Caravaca- Aguirre, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', and Piestun, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Genetic algo- rithm optimization for focusing through turbid me- dia in noisy environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Express (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' [9] Daniel, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', Oron, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', and Silberberg, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Light focusing through scattering media via linear fluores- cence variance maximization, and its application for fluorescence imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Express (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' [10] Fiolka, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', Si, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', and Cui, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Complex wave- front corrections for deep tissue focusing using low coherence backscattered light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Express 20, 15 (Jul 2012), 16532–16543.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' [11] Gigan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', Katz, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Roadmap on wave- front shaping and deep imaging in complex media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' arXiv preprint arXiv:2111.' metadata={'source': 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+page_content=' Adaptive optics for high-resolution imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Na- ture Reviews Methods Primers 1 (10 2021), 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' [13] Horstmeyer, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', Ruan, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', and Yang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Guidestar-assisted wavefront-shaping methods for focusing light into biological tissue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Nature Photon- ics (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' [14] Jang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', Lim, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', Yu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', Choi, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', Ha, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', Park, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', Oh, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', Jang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', Lee, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', and Park, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Complex wavefront shaping for optimal depth-selective focusing in optical coherence tomog- raphy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Express 21, 3 (Feb 2013), 2890–2902.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' [15] Ji, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Adaptive optical fluorescence microscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Na- ture Methods 14 (03 2017), 374–380.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' [16] Ji, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', Milkie, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', and Betzig, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Adaptive op- tics via pupil segmentation for high-resolution imag- ing in biological tissues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Nature methods 7 (02 2010), 141–7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' [17] Judkewitz, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', Horstmeyer, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', Vellekoop, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', and Yang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Translation correlations in anisotropically scattering media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Nature Physics (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' [18] Katz, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', Small, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', Guan, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', and Silber- berg, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Noninvasive nonlinear focusing and imag- ing through strongly scattering turbid layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Optica 1, 3 (Sep 2014), 170–174.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' [19] Kong, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', Silverman, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', Liu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', Chit- nis, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', Lee, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', and Chen, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Photoacoustic-guided convergence of light through optically diffusive media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' 36, 11 (Jun 2011), 2053–2055.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' 8 10μm10 μm10 μm10μm10μm10μm10μm10μm10μm[20] Liu, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Observing the cell in its native state: Imaging subcellular dynamics in multicellular organisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Science 360, 6386 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' [21] Mertz, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', Paudel, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', and Bifano, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Field of view advantage of conjugate adaptive optics in microscopy applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Applied Optics (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' [22] Osnabrugge, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', Horstmeyer, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', Pa- padopoulos, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', Judkewitz, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', and Vellekoop, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Generalized optical memory ef- fect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Optica (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' [23] Popoff, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', Aubry, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', Lerosey, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', Fink, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', Boccara, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', and Gigan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Exploiting the time-reversal operator for adaptive optics, selec- tive focusing, and scattering pattern analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' 107 (Dec 2011), 263901.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' [24] Popoff, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', Lerosey, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', Carminati, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', Fink, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', Boccara, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', and Gigan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Mea- suring the transmission matrix in optics: An ap- proach to the study and control of light propagation in disordered media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' [25] Rotter, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', and Gigan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Light fields in complex media: Mesoscopic scattering meets wave control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' [26] Stern, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', and Katz, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Noninvasive focusing through scattering layers using speckle correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' [27] Tang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', Germain, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', and Cui, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Su- perpenetration optical microscopy by iterative mul- tiphoton adaptive compensation technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Proceed- ings of the National Academy of Sciences 109, 22 (2012), 8434–8439.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' [28] Vellekoop, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', Cui, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', and Yang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Digi- tal optical phase conjugation of fluorescence in turbid tissue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Applied Physics Letters (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' [29] Vellekoop, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', Lagendijk, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', and Mosk, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Exploiting disorder for perfect focusing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Na- ture Photonics (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' [30] Vellekoop, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', and Mosk, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Focusing co- herent light through opaque strongly scattering me- dia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' [31] Verstraete, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', Wahls, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', Kalk- man, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', and Verhaegen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Model-based sensor- less wavefront aberration correction in optical coher- ence tomography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' 40, 24 (Dec 2015), 5722– 5725.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' [32] Wang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', Liu, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', Milkie, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', Sun, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', Tan, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', Kerlin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', Chen, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', Kim, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', and Ji, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Multiplexed aberration measurement for deep tissue imaging in vivo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Nature methods 11 (08 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' [33] Wang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', Judkewitz, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', DiMarzio, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', and Yang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Deep-tissue focal fluorescence imag- ing with digitally time-reversed ultrasound-encoded light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Nature Communications 3 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' [34] Xu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', Liu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', and Wang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Time-reversed ul- trasonically encoded optical focusing into scattering media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Nature Photonics 5 (2011), 154–157.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' [35] Yaqoob, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', Psaltis, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', Feld, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', and Yang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Optical phase conjugation for turbidity suppres- sion in biological samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Nature photonics (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' [36] Yeminy, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', and Katz, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Guidestar-free image- guided wavefront-shaping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Science Advances 7, 21 (2021), eabf5364.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' [37] Yu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', Park, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', Lee, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', Yoon, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', Kim, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', Lee, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=', and Park, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Recent advances in wave- front shaping techniques for biomedical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' Current Applied Physics 15, 5 (2015), 632–641.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} +page_content=' 9' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtFJT4oBgHgl3EQfAiym/content/2301.11421v1.pdf'} diff --git a/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf b/D9E1T4oBgHgl3EQf-QZ6/content/2301.03566v1.pdf new file mode 100644 index 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subsolar-mass black hole trigger +from the second observing run of Advanced LIGO +Gonzalo Morr´as,1 Jos´e Francisco Nu˜no Siles,1 Alexis Men´endez-V´azquez,2 Christos +Karathanasis,2 Katarina Martinovic,3 Khun Sang Phukon,4, 5, 6, 7 Sebastien Clesse,8 Juan +Garc´ıa-Bellido,1 Mario Mart´ınez,2, 9 Ester Ruiz Morales,1, 10 and Mairi Sakellariadou3 +1Instituto de F´ısica Te´orica UAM/CSIC, Universidad Aut´onoma de Madrid, Cantoblanco 28049 Madrid, Spain +2Institut de F´ısica d’Altes Energies (IFAE), Barcelona Institute of Science and Technology, E-08193 Barcelona, Spain +3Theoretical Particle Physics and Cosmology Group, +Physics +Department, +King’s College London, +University +of London, +Strand, +London +WC2R +2LS, +UK +4Nikhef - National Institute for Subatomic Physics, +Science Park, 1098 XG Amsterdam, The Netherlands +5Institute for High-Energy Physics, University of Amsterdam, +Science Park, 1098 XG Amsterdam, The Netherlands +6Institute for Gravitational and Subatomic Physics, Utrecht University, +Princetonplein 1, 3584 CC Utrecht, The Netherlands +7School of Physics and Astronomy and Institute for Gravitational Wave Astronomy, +University of Birmingham, Edgbaston, Birmingham, B15 9TT, United Kingdom +8Service de Physique Th´eorique, Universit´e Libre de Bruxelles (ULB), +Boulevard du Triomphe, CP225, B-1050 Brussels, Belgium +9Instituci´o Catalana de Recerca i Estudis Avan¸cats (ICREA), Barcelona, Spain +10Departamento de F´ısica, ETSIDI, Universidad Polit´ecnica de Madrid, 28012 Madrid, Spain +(Dated: January 30, 2023) +We perform an exhaustive follow-up analysis of a subsolar-mass black hole candidate from the +second observing run of Advanced LIGO, reported by Phukon et al. in 2021. The origin of this trigger +is unclear, because the reported signal-to-noise ratio (SNR) of 8.6 and inverse false alarm rate of +about 0.5 yr are too low to claim a gravitational-wave origin, but large enough to be intriguing. When +using more precise waveforms, extending the frequency range down to 20 Hz, removing a prominent +blip glitch and marginalizing over all the model parameters, we find that the network signal-to-noise +ratio posterior distribution lies mostly below the search value, with the 90% confidence interval +being 7.94+0.70 +−1.05. If one assumes that the signal comes from a real gravitational-wave merger event, +we find a light component m2 = 0.76+0.50 +−0.14M⊙, suggesting a compact object of mass below one +solar mass at 83.8% confidence level. +Such a low mass for a compact object would suggest an +unexpectedly light neutron star or a black hole of primordial or exotic origin. The primary mass +would be m1 = 4.71+1.57 +−2.18M⊙, likely in the lower mass gap, for a mass ratio of q =0.16+0.34 +−0.06, at a +distance of DL =124+82 +−48Mpc. The improved sensitivity of the next observing runs would make it +possible to observe similar signals with a higher SNR and to distinguish a sub-solar mass component. +I. +INTRODUCTION +The development of gravitational wave (GW) astron- +omy, with about 90 binary black hole (BBH) coalescence +events detected so far [1–6] by the LIGO-Virgo-KAGRA +(LVK) collaboration [7], is driving a true revolution in +astrophysics and cosmology. As the number of detected +events grows with successive observing catalogs, prop- +erties of the progenitors seem to challenge prior expec- +tations for a population of astrophysical objects. +Re- +cent examples are BBH events like GW190521 [8, 9] with +its most massive component in the pair-instability mass +gap [10], as well as events like GW190814 which has a +very low mass ratio and a secondary in the lower mass +gap [11]. Evidence for misaligned spins in the black hole +population has been found [12], suggesting a dynamical +binary formation. +The frequency range of the LIGO [13] and Virgo [14] +detectors makes them sensitive to compact object bina- +ries with masses below 1M⊙. +There is no compelling +stellar evolution model that can produce neutron stars +or black holes below 1 M⊙. +Therefore, the detection +of a subsolar-mass (SSM) black hole directly points to +a new black hole formation mechanism operating in the +Universe, an alternative to the astrophysical evolution +and collapse of ordinary matter. Primordial black holes +(PBHs) are natural candidates since they can be pro- +duced with a wide mass spectrum in the early Universe +through the collapse of highly overdense regions [15]. An +SSM compact object detection provides the cleanest sig- +nature for a PBH, though there are some proposals of +dark matter with exotic properties that could also pro- +duce subsolar-mass objects [16–29]. +Before the advent of GW astronomy, the only way to +detect SSM black holes was via X-ray binaries [30] or +microlensing [31]. At present, some hints of the existence +of such light black holes come from microlensing events +towards the bulge [32], from Andromeda [33] and lensed +quasars [34, 35], although the mass, the nature and the +abundance of the lenses remain uncertain. +Complementary to these astrophysical searches, com- +arXiv:2301.11619v1 [gr-qc] 27 Jan 2023 + +2 +pact binary coalescences (CBCs) with at least one sub- +solar component have been searched for in the first (O1), +second (O2) and third (O3) observing runs of LVK, with- +out convincing evidence [36–40]. Nevertheless, a further +search for SSM black holes with low mass ratio in the +O2 data has recently revealed four potential candidate +events1 (we refer the reader to Table I of [41]) with a +false alarm rate smaller than 2 yr−1. +In this paper, we follow up this search and perform pa- +rameter inference of the four events. Our primary goal is +to further investigate these SSM triggers using the stan- +dard parameter estimation (PE) methods. These allow +us to extend the frequency range of the search and use +more accurate waveforms including spin precession, and +higher order modes, as well as the merger and ringdown +phases. We also can visually inspect the quality of the +data and subtract non-gaussianities using standard tools +such as BayesWave [42–44]. +We focus on the third candidate event reported in Ta- +ble I of [41], observed by both LIGO Hanford and LIGO +Livingston interferometers. It is the most significant two +detector trigger of the search and the only one having +significant support for an SSM component after further +inspection with PE. We analyse in detail the data and +perform a careful PE around this trigger, observed on +April 1st 2017 and referred here as SSM170401. +Fur- +thermore, we discuss the impact of a prominent glitch +removal. As a by-product, the PE allows us to infer the +component masses, spins, distance and sky locations. In +particular, we infer the probability of an SSM compo- +nent, if one assumes that the signal is coming from a +BBH merger event. +II. +PROPERTIES OF THE CANDIDATE +To obtain the properties of the candidate we interpret +the signal as coming from the coalescence of two com- +pact objects. The signal was found in data taken on 2017 +April 1, 01:43:34 UTC during the O2 LIGO-Virgo observ- +ing run. This candidate was not reported by any of the +LVK searches, both generic [3] and SSM specific [45], but +it was reported as part of a search for SSM candidates +in asymmetric binaries using the GstLAL pipeline [41]. +The trigger had detector frame masses of 4.897 M⊙ and +0.7795 M⊙, with a false-alarm-rate (FAR) of 0.4134 yr−1 +and a combined network signal-to-noise ratio (SNR) of +∼ 8.67. +The strain in Hanford presents a glitch 14 s before co- +alescence, as shown in Fig. 1. The search presented in +Ref. [41] uses templates starting at f=45 Hz. The loud- +est template, in this case, is only 10 s long and so should +be unaffected by the glitch. However, PE was performed +1 Three additional subsolar-mass triggers were recently reported +in [40] which could be the object of future analysis. +FIG. 1. Figure showing the Hanford original whitened strain +˜hwhitened(f) = ˜h(f)/ +� +Sn(f), the whitened glitch model and +the whitened clean data after subtracting the glitch. Times +are shown relative to the trigger time. +with templates starting at 20 Hz, which are roughly 100 +s long for the component masses discussed. In this sit- +uation the glitch must be removed. +Using BayesWave +[42, 43] we fit to the data a combined glitch, signal and +Gaussian noise model. We then subtract the glitch part +of the model from the original data and obtain the clean +data to be used for PE. The same procedure was used by +the LVK collaboration for GW170817 to clean the data +from a large glitch [46]. +We infer the CBC parameters of the signal using a +Bayesian analysis of the data from LIGO Livingston +and LIGO Hanford, following the methodology outlined +in Appendix B of [3]. +In analysing the data, we fit +two different waveform models: IMRPhenomPv2 [47] and +IMRPhenomXPHM [48], the latter including higher order +modes. +We then compare the posterior samples from +each of these, and find consistency between the two mod- +els, noting that both of them take into account precessing +spins. The TaylorF2 [49] waveform model has also been +tested and, despite it providing compatible results, it fails +to reach the same level of significance. +We use a low-frequency cutoff of 20 Hz in both detec- +tors for the likelihood evaluation and choose uninforma- +tive and wide priors (see Supplemental Material). The +primary tool used for sampling the posterior distribution +is the LALInference Markov Chain Monte Carlo imple- +mentation as described in [55]. The power spectral den- +sity used in the calculations of the likelihood is estimated +using BayesWave [42, 43]. The study uses the O2 open +access data [56] with a sampling frequency of 4096 Hz; +however the likelihood is integrated up to 1600 Hz. +The signal is present in the detector for ∼ 3000 cycles, +allowing us to constrain the source properties. The esti- +mated parameters are reported in Table I. The marginal- +ized posterior for the absolute value of the matched filter +SNR is 7.98+0.62 +−1.03 +for IMRPhenomPv2 and 7.94+0.70 +−1.05 +for + +Hanford +original data +15 +clean data +glitch +10 +Whitened Strain +5 +-5 +14.19-14.16-14.13 -14.1 -14.07-14.04-14.01-13.98-13.95-13.92 +Time [seco0nds1 from 2017-04-01 01:43:34.677 UTC (1175046232.677)3 +Parameter +IMRPhenomPv2 IMRPhenomXPHM +Signal to Noise Ratio +7.98+0.62 +−1.03 +7.94+0.70 +−1.05 +Primary mass (M⊙) +4.65+1.21 +−2.15 +4.71+1.57 +−2.18 +Secondary mass (M⊙) +0.77+0.50 +−0.12 +0.76+0.50 +−0.14 +Primary spin magnitude +0.32+0.47 +−0.26 +0.36+0.46 +−0.30 +Secondary spin magnitude +0.48+0.46 +−0.43 +0.47+0.46 +−0.42 +Total mass (M⊙) +5.42+1.10 +−1.65 +5.47+1.43 +−1.68 +Mass ratio (m2/m1 ≤ 1) +0.17+0.34 +−0.05 +0.16+0.34 +−0.06 +χeff [50, 51] +−0.06+0.17 +−0.32 +−0.05+0.22 +−0.35 +χp [52] +0.28+0.34 +−0.21 +0.33+0.33 +−0.26 +Luminosity Distance (Mpc) +119+82 +−48 +124+82 +−48 +Redshift +0.028+0.018 +−0.010 +0.028+0.017 +−0.011 +Ra (◦) +−2+34 +−35 +−1+34 +−37 +Dec (◦) +47+14 +−26 +46+14 +−29 +Final mass (M⊙) +5.34+1.11 +−1.70 +5.40+1.45 +−1.73 +Final spin +0.39+0.24 +−0.07 +0.42+0.22 +−0.10 +P(m2 < 1 M⊙) +85.5% +83.8% +P(m2 < 1.2 M⊙) +92.7% +92.7% +TABLE I. Parameters of SSM170401. All masses are in the +source frame. +We assume Planck15 Cosmology [53]. +The +statistical uncertainty of all the parameters is quantified by +the equal-tailed 90% credible intervals about the median of +the marginalized one-dimensional posteriors. Right ascension +(Ra) and declination (Dec) are measured in the International +Celestial Reference System (ICRS) [54]. +IMRPhenomXPHM. The median value of the SNR is lower +than that found by the search, which was 8.67. How- +ever, these two quantities are not directly comparable. +The SNR from the search is obtained by maximizing the +ranking statistic over a discrete template bank [41, 57– +59], while the quoted SNR from the PE is the median +value over the samples. Since the ranking statistic and +the SNR are closely related, the SNR that is more compa- +rable to that of the search would be the maximum SNR +as found by the PE. The values of this maximum PE SNR +are 9.09 for IMRPhenomPv2 and 9.18 for IMRPhenomXPHM. +These values are slightly larger than that of the search, +which is consistent with what would happen if the signal +was astrophysical. However, this is also expected in the +noise case due to the larger parameter space that allows +more flexibility for the PE analysis to fit the data. We +also notice the maximum value of the SNR to be larger +for IMRPhenomXPHM than for IMRPhenomPv2. In a simi- +lar way, this is expected for an astrophysical signal but +also for noise, since the waveform includes Higher Order +Modes and thus has more flexibility to fit the data. +The signal is then compatible with a compact binary +system having an unequal mass ratio q =0.17+0.34 +−0.05 +(all +uncertainties are quoted at 90% C.L.), a source frame +1 +2 +3 +4 +5 +6 +7 +msource +1 +[M ] +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +msource +2 +[M ] +IMRPhenomPv2 +IMRPhenomXPHM +FIG. 2. Posterior distributions for the primary and secondary +mass in the source frame. The 90% credible regions are in- +dicated by the solid contour in the joint distribution, and by +the dashed vertical and horizontal lines in the marginalized +distributions. +primary mass m1 = 4.65+1.21 +−2.15M⊙ and a source frame +secondary mass m2 = 0.77+0.50 +−0.12M⊙ as shown in Fig. 2. +The marginalised posterior distribution for the secondary +mass favors a mass lower than 1M⊙ (85.5% C.L.) and +provides strong support for a mass lower than 1.2M⊙ +(92.7% C.L.). Using the IMRPhenomXPHM waveform, we +find almost identical results, with a mass lower than 1M⊙ +at 83.8%C.L. +The left panel of Fig. 3 shows the posterior distribu- +tions for the magnitude and tilt angle of the individual +spins, measured at a reference frequency of 20 Hz. All +pixels in this plot have an equal prior probability. The +spin of the secondary BH is largely unconstrained, as ex- +pected for very unequal masses. +The primary spin, if +present, is likely to be misaligned with the orbital an- +gular momentum with a preference for small spin mag- +nitude (a1 =0.32+0.47 +−0.26). +As can be seen in the right +panel of Fig. 3, this leads to a χeff compatible with 0 +(χeff =−0.05+0.22 +−0.35) and an uninformative posterior in χp +(χp =0.33+0.33 +−0.26). +The luminosity distance and inclination angle θJN pos- +terior distributions are shown together in the left panel of +Fig. 4, since these two quantities are correlated. We find +a luminosity distance of dL =119+82 +−48Mpc. We identify +a bimodal distribution for θJN due to the fact that we +can not distinguish whether the system is being observed +face-on (θJN ∼ 0) or face-away (θJN ∼ π), but it being +edge-on (θJN ∼ π/2) is disfavoured. In the face-on(off) +configuration, the effects of precession [60] and higher or- +der modes in the signal are suppressed [61], as is the case + +4 +0.0 +0.2 +0.4 +0.6 +0.8 +/( +) +/( +) +× +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +posterior probability per pixel +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +p +1.00 +0.75 +0.50 +0.25 +0.00 +0.25 +0.50 +0.75 +1.00 +eff +IMRPhenomPv2 +IMRPhenomXPHM +Prior +FIG. 3. Left: posterior distribution for the individual spins of SSM170401 according to the IMRPhenomXPHM waveform model. +The radial coordinate in the plot denotes the dimensionless spin magnitude, while the angle denotes the spin tilt, defined as +the angle between the spin and the orbital angular momentum of the binary at a reference frequency of 20 Hz. A tilt of 0° +indicates that the spin is aligned with the orbital angular momentum. A nonzero magnitude and a tilt away from 0° and 180° +imply a precessing orbital plane. All bins have an equal prior probability. Right: posterior distributions for the effective spin +and effective in-plane spin parameters. The black lines in the right panel show the prior distributions for the effective spin +parameters. The 90% credible regions are indicated by the solid contour in the joint distribution, and by dashed vertical and +horizontal lines in the marginalized distributions. The large density for tilts close to 90° leads to non-zero values for χp and +low values for χeff. +here. +From the right panel of Fig. 4, it can be seen that +the signal came from a position in the sky for which the +LIGO network has good sensitivity to the two GW polar- +izations [62]. This, however, does not represent the area +with the best sensitivity, which would be located on top +of the continental US and its antipodes. +In summary, the PE shows that the chirp and compo- +nent masses, the effective spin, the luminosity distance +and the sky location can be reconstructed, and are con- +sistent with that of a BBH merger event. However, it +is known that Gaussian noise can also mimic such a sig- +nal [63], and given the low values of the SNR and iFAR, +it is not possible to ascertain the origin of the signal, as +it could very well have been generated by detector noise. +III. +DISCUSSION +Even if the significance of the trigger did not improve +with the present analysis and PE since it remains at the +threshold limit, it is interesting to speculate on the possi- +ble origin of the secondary component (with a preferred +mass below 1 M⊙ for such a compact object, if inter- +preted as a GW event). +The neutron star nature of the light compact object +seems disfavored. Indeed, neutron stars have relatively +well-determined masses from observations of binary sys- +tems, including pulsars or X-ray binaries involving an +accreting neutron star from a companion. Their masses +are contained within a narrow range 1.25-1.45 M⊙ [64], +further confirmed by the observation of GW170817 [46]. +Even though there is a recent claim [65] for a neutron star +of mass around 0.7 M⊙, modern core-collapse supernova +simulations [66, 67] indicate it is difficult to form neutron +stars with masses below one solar mass. Such a small +mass for a neutron star probably requires a QCD equa- +tion of state that is beyond theoretical predictions [68]. +Although the neutron star interpretation of the hypo- +thetical trigger SSM170401 is observationally disfavored, +given our current limited knowledge of the equation of +state we cannot exclude a neutron star origin. +On the other hand, PBHs [69–72], formed by the grav- +itational collapse of large inhomogeneities in the early +Universe are already considered as a possible explana- +tion of LVK GW detections, see e.g. [73–85]. Depending + +5 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +JN [radians] +50 +100 +150 +200 +250 +300 +350 +400 +DL [Mpc] +Prior +IMRPhenomPv2 +IMRPhenomXPHM +-135° +-90° +-45° +0° +45° +90° +135° +0° +30° +60° +60° +30° +0° +-30° +-60° +-60° +-30° +50% area: 711 deg² +90% area: 2,658 deg² +FIG. 4. Left: posterior distributions for the luminosity distance and the inclination angle of SSM170401, according to the +IMRPhenomXPHM and IMRPhenomPv2 waveform models. The inclination angle indicates the angle between the line of sight and +the total angular momentum of the binary. For nonprecessing binaries, this is equal to the angle between the orbital angular +momentum and the line of sight. The solid lines and the central contour denote 90% credible regions. Right: sky position of +the event as evaluated from the Greenwich meridian according to the IMRPhenomXPHM waveform model. +on the model, they may explain anything from a tiny +fraction of Dark Matter to its entirety. PBHs have been +the main motivation to conduct searches of SSM black +holes in the LVK data [38, 39, 41, 45, 86–88], in partic- +ular, the extended subsolar search with low-mass ratios +in O2 which reported SSM170401 as a possible candi- +date [41]. If some of the observed binary coalescences +are indeed due to PBHs, they must have a relatively ex- +tended mass distribution that would have been imprinted +by the thermal history of the Universe [78, 89]. +This +would lead to a peak in the mass distribution around a +solar mass which is naturally produced at the QCD tran- +sition [78, 89–94], and SSM170401 could be an example +of a subsolar PBH around the QCD-induced peak. The +spin posterior is quite broad and the spin is compatible +with zero, although a slight preference for a primary spin +around 0.3 is observed. In this case, the non-zero but +relatively low spin of the primary component may have +been acquired by matter accretion, previous mergers or +hyperbolic encounters [84, 95, 96]. +Dark Matter with very special particle composition +could also be at the origin of solar-mass black holes if +it can accumulate inside neutron stars and lead to their +collapse into a black hole. Several scenarios have been +proposed [22, 26, 97–100], but they all require very par- +ticular conditions, in order not to change the correspond- +ing Chandrasekhar limit. Such transmutations, if leading +to SSM black holes, would be accompanied by violent ex- +plosions that would have been observed. It is therefore +still unclear if such scenarios are realistic and compati- +ble with observations. In an alternative scenario, with +complex and dissipative Dark Matter composition, SSM +black holes could form through the cooling and gravita- +tional collapse of Dark Matter halos [23]. This model was +constrained by the LVK data in [39, 101, 102]. +Another possibility, if considered as a real GW event, +could be that the secondary component of SSM170401 is +a boson star, a hypothetical horizonless compact object +formed by an ultralight bosonic field. If the mass of the +bosonic particle is larger than 10−10eV/c2, the boson star +can have subsolar mass [103]. Note that due to Beken- +stein’s bound, any object of a given mass which is as +compact as a black hole can only be a black hole. Boson +stars necessarily must be larger, implying a lower ISCO +frequency in the middle or lower than the LIGO sensitiv- +ity band. So the viable parameter space for a boson star +is probably very limited. +Finally, we comment on the mass of the primary com- +ponent, which would preferably lie in the low mass gap +between 2.5 and 5M⊙ (61%C.L.). Assuming a real GW +merger event, it would most probably be a black hole. A +neutron star origin with mass above 2.5M⊙ is strongly +disfavoured. Other black hole candidates in the low mass +gap were observed in the GWTC-3 catalog, which may +bring additional support for a primordial origin since +PBHs should not have a mass gap. +IV. +CONCLUSIONS +In this work, we investigate the most significant can- +didate reported in [41], removing a prominent blip glitch + +6 +in the data and estimating the CBC parameters with +the state-of-the-art waveform families IMRPhenomPv2 and +IMRPhenomXPHM, taking into account contributions from +higher order modes and extending the frequency range +down to 20 Hz. However, with this improved modelling +with respect to the search, we find a 90% confidence level +network SNR of 7.94+0.70 +−1.05, which is lower than the SNR +of 8.6 obtained in the template-bank-based search. +Nevertheless, if one would assume that it is coming +from a real GW event, the trigger observed on the 1st of +April, 2017, is identified consistently in both LIGO de- +tectors with a light mass component, m2 = 0.76+0.50 +−0.14M⊙ +(90% credible interval). Such low mass is below one solar +mass and below 1.2 solar masses at 83.8% and 92.7% con- +fidence level, respectively. The compact binary coales- +cence presents a total mass of 5.47+1.43 +−1.68M⊙, correspond- +ing to a mass ratio of q =0.16+0.34 +−0.06, and a luminosity +distance of 124+82 +−48Mpc. +The values quoted here are a +result of using IMRPhenomXPHM but consistent results are +obtained with different analysis pipelines and waveform +families. At this point, the observational data and the +search from [41] do not show enough significance to claim +a firm GW observation. Nevertheless, our analysis shows +that the signal, if coming from a GW event, is consistent +with an SSM black hole. We discuss several scenarios for +the production of such a possible SSM black hole candi- +date and conclude that a neutron star origin is disfavored +or at least requires a non-standard matter equation of +state. Other possibilities could include primordial black +holes, black holes formed from the accretion of hypothet- +ical Dark Matter particles onto neutron stars, or boson +stars. Given that PBHs can also explain some intriguing +properties of other compact binary coalescences without +being restricted by the Chandrashekar mass, they can be +considered as our preferred hypothesis. +The data from the third observing run O3, as well as +data from the future planned runs with improved sensi- +tivity, O4 and O5, offer a great opportunity for discover- +ing additional SSM candidate events, and could increase +the statistical significance for the existence of a new class +of SSM compact objects. +ACKNOWLEDGMENTS +S.C. acknowledges support from the Francqui Foun- +dation +through +a +Starting +Grant. +K.M. +is +sup- +ported by King’s College London through a Postgrad- +uate International Scholarship. +M.S. is supported in +part by the Science and Technology Facility Council +(STFC), United Kingdom, under the research grant +ST/P000258/1. +This work is partially supported by +the Spanish grants PID2020-113701GB-I00, PID2021- +123012NB-C43 [MICINN-FEDER], and the Centro de +Excelencia Severo Ochoa Program CEX2020-001007-S +through IFT, some of which include ERDF funds from +the European Union. +IFAE is partially funded by the +CERCA program of the Generalitat de Catalunya. We +acknowledge the use of IUCAA LDG cluster Sarathi +for the computational/numerical work. This material is +based upon work supported by NSF’s LIGO Laboratory +which is a major facility fully funded by the National +Science Foundation. +[1] B. P. Abbott et al. (LIGO Scientific, Virgo), Phys. Rev. +Lett. 116, 061102 (2016), arXiv:1602.03837 [gr-qc]. +[2] B. P. Abbott et al. (LIGO Scientific, Virgo), Phys. Rev. +X 6, 041015 (2016), [Erratum: Phys.Rev.X 8, 039903 +(2018)], arXiv:1606.04856 [gr-qc]. +[3] B. P. Abbott et al. (LIGO Scientific, Virgo), Phys. Rev. +X 9, 031040 (2019), arXiv:1811.12907 [astro-ph.HE]. +[4] R. Abbott et al. (LIGO Scientific, Virgo), Phys. Rev. X +11, 021053 (2021), arXiv:2010.14527 [gr-qc]. +[5] R. Abbott et al. (LIGO Scientific, VIRGO), +(2021), +arXiv:2108.01045 [gr-qc]. +[6] R. Abbott et al. (LIGO Scientific, Virgo, KAGRA), +(2021), arXiv:2111.03606 [gr-qc]. +[7] B. Abbott, R. Abbott, T. Abbott, S. Abraham, F. Ac- +ernese, K. Ackley, C. Adams, V. Adya, C. Affeldt, +M. Agathos, K. Agatsuma, N. Aggarwal, O. Aguiar, +L. Aiello, A. Ain, P. Ajith, T. Akutsu, G. Allen, +and +A. Allocca, Living Reviews in Relativity 23 (2020), +10.1007/s41114-020-00026-9. +[8] R. Abbott et al. (LIGO Scientific, Virgo), Phys. Rev. +Lett. 125, 101102 (2020), arXiv:2009.01075 [gr-qc]. +[9] R. Abbott et al. (LIGO Scientific, Virgo), Astrophys. J. +Lett. 900, L13 (2020), arXiv:2009.01190 [astro-ph.HE]. +[10] S. +E. +Woosley, +Astrophys. +J. +836, +244 +(2017), +arXiv:1608.08939 [astro-ph.HE]. +[11] R. Abbott et al. (LIGO Scientific, Virgo), Astrophys. J. +896, L44 (2020), arXiv:2006.12611 [astro-ph.HE]. +[12] R. Abbott et al. (LIGO Scientific, VIRGO, KAGRA), +(2021), arXiv:2111.03634 [astro-ph.HE]. +[13] T. L. S. Collaboration, J. Aasi, B. P. Abbott, R. Ab- +bott, et al., Classical and Quantum Gravity 32, 074001 +(2015). +[14] F. Acernese, M. Agathos, et al., Classical and Quantum +Gravity 32, 024001 (2014). +[15] J. Garc´ıa-Bellido, A. D. Linde, +and D. Wands, Phys. +Rev. D 54, 6040 (1996), arXiv:astro-ph/9605094. +[16] I. Goldman and S. Nussinov, Phys. Rev. D 40, 3221 +(1989). +[17] C. Kouvaris and P. Tinyakov, Phys. Rev. D 83, 083512 +(2011), arXiv:1012.2039 [astro-ph.HE]. +[18] A. de Lavallaz and M. Fairbairn, Phys. Rev. D 81, +123521 (2010), arXiv:1004.0629 [astro-ph.GA]. +[19] J. Bramante and T. Linden, Phys. Rev. Lett. 113, +191301 (2014), arXiv:1405.1031 [astro-ph.HE]. +[20] J. Bramante and F. Elahi, Phys. Rev. D 91, 115001 +(2015), arXiv:1504.04019 [hep-ph]. +[21] J. Bramante, T. Linden, and Y.-D. Tsai, Phys. Rev. D +97, 055016 (2018), arXiv:1706.00001 [hep-ph]. + +7 +[22] C. Kouvaris, P. Tinyakov, and M. H. G. Tytgat, Phys. +Rev. Lett. 121, 221102 (2018), arXiv:1804.06740 [astro- +ph.HE]. +[23] S. Shandera, D. Jeong, and H. S. G. Gebhardt, Phys. +Rev. Lett. 120, 241102 (2018), arXiv:1802.08206 [astro- +ph.CO]. +[24] J. H. Chang, D. Egana-Ugrinovic, R. Essig, +and +C. Kouvaris, JCAP 03, 036 (2019), arXiv:1812.07000 +[hep-ph]. +[25] M. A. Latif, A. Lupi, D. R. G. Schleicher, G. D’Amico, +P. Panci, and S. Bovino, Mon. Not. Roy. Astron. Soc. +485, 3352 (2019), arXiv:1812.03104 [astro-ph.CO]. +[26] B. Dasgupta, R. Laha, +and A. Ray, Phys. Rev. Lett. +126, 141105 (2021), arXiv:2009.01825 [astro-ph.HE]. +[27] J. Gurian, +M. Ryan, +S. Schon, +D. Jeong, +and +S. Shandera, Astrophys. J. Lett. 939, L12 (2022), +arXiv:2209.00064 [astro-ph.CO]. +[28] M. Ryan and D. Radice, Phys. Rev. D 105, 115034 +(2022), arXiv:2201.05626 [astro-ph.HE]. +[29] M. +Hippert, +E. +Dillingham, +H. +Tan, +D. +Curtin, +J. +Noronha-Hostler, +and +N. +Yunes, +(2022), +arXiv:2211.08590 [astro-ph.HE]. +[30] J. M. Corral-Santana, J. Casares, T. Munoz-Darias, +F. E. Bauer, I. G. Martinez-Pais, +and D. M. Russell, +Astron. Astrophys. 587, A61 (2016), arXiv:1510.08869 +[astro-ph.HE]. +[31] B. Paczynski, Astrophys. J. 304, 1 (1986). +[32] �L. Wyrzykowski and I. Mandel, Astron. Astrophys. 636, +A20 (2020), arXiv:1904.07789 [astro-ph.SR]. +[33] H. Niikura et al., Nature Astron. 3, 524 (2019), +arXiv:1701.02151 [astro-ph.CO]. +[34] M. R. S. Hawkins, Astron. Astrophys. 643, A10 (2020), +arXiv:2010.15007 [astro-ph.CO]. +[35] M. R. S. Hawkins, Mon. Not. Roy. Astron. Soc. 512, +5706 (2022), arXiv:2204.09143 [astro-ph.CO]. +[36] B. P. Abbott et al. (LIGO Scientific, Virgo), Phys. +Rev. Lett. 121, 231103 (2018), arXiv:1808.04771 [astro- +ph.CO]. +[37] B. P. Abbott et al. (LIGO Scientific, Virgo), Phys. +Rev. Lett. 123, 161102 (2019), arXiv:1904.08976 [astro- +ph.CO]. +[38] A. +H. +Nitz +and +Y.-F. +Wang, +arXiv +(2021), +arXiv:2102.00868 [astro-ph.HE]. +[39] R. Abbott et al. (LIGO Scientific, Virgo, KAGRA), +(2021), arXiv:2109.12197 [astro-ph.CO]. +[40] R. Abbott et al. (LIGO Scientific, VIRGO, KAGRA), +(2022), arXiv:2212.01477 [astro-ph.HE]. +[41] K. S. Phukon, G. Baltus, S. Caudill, S. Clesse, A. De- +passe, M. Fays, H. Fong, S. J. Kapadia, R. Magee, +and A. J. Tanasijczuk, (2021), arXiv:2105.11449 [astro- +ph.CO]. +[42] N. J. Cornish and T. B. Littenberg, Classical and Quan- +tum Gravity 32, 135012 (2015). +[43] T. B. Littenberg and N. J. Cornish, Phys. Rev. D 91, +084034 (2015). +[44] D. Davis, T. B. Littenberg, I. M. Romero-Shaw, M. Mill- +house, J. McIver, F. D. Renzo, and G. Ashton, Classical +and Quantum Gravity 39, 245013 (2022). +[45] B. P. Abbott, R. Abbott, T. Abbott, S. Abraham, +F. Acernese, K. Ackley, C. Adams, R. X. Adhikari, +V. Adya, C. Affeldt, et al., Physical review letters 123, +161102 (2019). +[46] B. P. Abbott et al. (LIGO Scientific, Virgo), Phys. Rev. +Lett. 119, 161101 (2017), arXiv:1710.05832 [gr-qc]. +[47] S. +Khan, +K. +Chatziioannou, +M. +Hannam, +and +F. Ohme, Physical Review D 100 (2019), 10.1103/phys- +revd.100.024059. +[48] G. Pratten, C. Garc´ıa-Quir´os, M. Colleoni, A. Ramos- +Buades, H. Estell´es, M. Mateu-Lucena, R. Jaume, +M. Haney, D. Keitel, J. E. Thompson, and et al., Physi- +cal Review D 103 (2021), 10.1103/physrevd.103.104056. +[49] A. Buonanno, B. Iyer, E. Ochsner, Y. Pan, +and +B. S. Sathyaprakash, Phys. Rev. D 80, 084043 (2009), +arXiv:0907.0700 [gr-qc]. +[50] P. Ajith, M. Hannam, S. Husa, Y. Chen, B. Br¨ugmann, +N. Dorband, D. M¨uller, F. Ohme, D. Pollney, C. Reiss- +wig, L. Santamar´ıa, and J. Seiler, Phys. Rev. Lett. 106, +241101 (2011). +[51] L. Santamar´ıa, F. Ohme, P. Ajith, B. Br¨ugmann, +N. Dorband, M. Hannam, S. Husa, P. M¨osta, D. Pollney, +C. Reisswig, E. L. Robinson, J. Seiler, and B. Krishnan, +Phys. Rev. D 82, 064016 (2010). +[52] P. Schmidt, F. Ohme, and M. Hannam, Phys. Rev. D +91, 024043 (2015). +[53] Planck Collaboration, A&A 594, A13 (2016). +[54] E. F. Arias, P. Charlot, M. Feissel, and J. F. Lestrade, +Astronomy and Astrophysics 303, 604 (1995). +[55] J. Veitch, V. Raymond, B. Farr, W. Farr, P. Graff, +S. Vitale, B. Aylott, K. Blackburn, N. Christensen, +M. Coughlin, W. Del Pozzo, F. Feroz, J. Gair, C.- +J. Haster, V. Kalogera, T. Littenberg, I. Mandel, +R. O’Shaughnessy, M. Pitkin, C. Rodriguez, C. R¨over, +T. Sidery, R. Smith, M. Van Der Sluys, A. Vecchio, +W. Vousden, +and L. Wade, Phys. Rev. D 91, 042003 +(2015). +[56] R. Abbott et al. (LIGO Scientific, Virgo), SoftwareX +13, 100658 (2021), arXiv:1912.11716 [gr-qc]. +[57] C. Messick, K. Blackburn, P. Brady, P. Brockill, K. Can- +non, R. Cariou, S. Caudill, S. J. Chamberlin, J. D. +Creighton, R. Everett, et al., Physical Review D 95, +042001 (2017). +[58] K. Cannon, R. Cariou, A. Chapman, M. Crispin- +Ortuzar, N. Fotopoulos, M. Frei, C. Hanna, E. Kara, +D. Keppel, L. Liao, et al., The Astrophysical Journal +748, 136 (2012). +[59] K. Cannon, S. Caudill, C. Chan, B. Cousins, J. D. +Creighton, B. Ewing, H. Fong, P. Godwin, C. Hanna, +S. Hooper, R. Huxford, R. Magee, D. Meacher, C. Mes- +sick, S. Morisaki, D. Mukherjee, H. Ohta, A. Pace, +S. Privitera, +I. de Ruiter, +S. Sachdev, +L. Singer, +D. Singh, R. Tapia, L. Tsukada, D. Tsuna, T. Tsutsui, +K. Ueno, A. Viets, L. Wade, and M. Wade, SoftwareX +14, 100680 (2021). +[60] J. Calder´on Bustillo, P. Laguna, +and D. Shoemaker, +Phys. Rev. D 95, 104038 (2017). +[61] C. Mills and S. Fairhurst, Phys. Rev. D 103, 024042 +(2021), arXiv:2007.04313 [gr-qc]. +[62] N. Andersson, J. Baker, K. Belczynski, S. Bernuzzi, +E. Berti, L. Cadonati, P. Cerd´a-Dur´an, J. Clark, M. Fa- +vata, L. S. Finn, +and et al., Classical and Quantum +Gravity 30, 193002 (2013). +[63] G. Morras, J. F. Nu˜no Siles, J. Garc´ıa-Bellido, +and +E. R. Morales, (2022), arXiv:2209.05475 [gr-qc]. +[64] J. M. Lattimer, Universe 5 (2019). +[65] V. Doroshenko, V. Suleimanov, G. P¨uhlhofer, +and +A. Santangelo, Nature Astronomy 6, 1444 (2022). +[66] B. M¨uller, +T. M. Tauris, +A. Heger, +P. Banerjee, +Y. Z. Qian, J. Powell, C. Chan, D. W. Gay, +and + +8 +N. Langer, Mon. Not. Roy. Astron. Soc. 484, 3307 +(2019), arXiv:1811.05483 [astro-ph.SR]. +[67] A. Burrows, D. Radice, D. Vartanyan, H. Nagakura, +M. A. Skinner, and J. Dolence, Mon. Not. Roy. Astron. +Soc. 491, 2715 (2020), arXiv:1909.04152 [astro-ph.HE]. +[68] H. Sotani, K. Iida, K. Oyamatsu, +and A. Ohnishi, +PTEP 2014, 051E01 (2014), arXiv:1401.0161 [astro- +ph.HE]. +[69] Y. B. Zel’dovich and I. D. Novikov, Soviet Astronomy +10, 602 (1967). +[70] S. Hawking, Mon. Not. Roy. Astron. Soc. 152, 75 +(1971). +[71] B. J. Carr and S. W. Hawking, Mon. Not. Roy. Astron. +Soc. 168, 399 (1974). +[72] G. F. Chapline, Nature 253, 251 (1975). +[73] S. Bird, I. Cholis, J. B. Mu˜noz, Y. Ali-Ha¨ımoud, +M. Kamionkowski, E. D. Kovetz, A. Raccanelli, +and +A. G. Riess, Phys. Rev. Lett. 116, 201301 (2016), +arXiv:1603.00464 [astro-ph.CO]. +[74] S. Clesse and J. Garc´ıa-Bellido, Phys. Dark Univ. 15, +142 (2017), arXiv:1603.05234 [astro-ph.CO]. +[75] M. Sasaki, T. Suyama, T. Tanaka, +and S. Yokoyama, +Phys. +Rev. +Lett. +117, +061101 +(2016), +[Erratum: +Phys.Rev.Lett. 121, 059901 (2018)], arXiv:1603.08338 +[astro-ph.CO]. +[76] S. Clesse and J. Garc´ıa-Bellido, Phys. Dark Univ. 22, +137 (2018), arXiv:1711.10458 [astro-ph.CO]. +[77] N. Fernandez and S. Profumo, JCAP 08, 022 (2019), +arXiv:1905.13019 [astro-ph.HE]. +[78] B. Carr, S. Clesse, J. Garc´ıa-Bellido, +and F. K¨uhnel, +Phys. Dark Univ. 31, 100755 (2021), arXiv:1906.08217 +[astro-ph.CO]. +[79] K. Jedamzik, JCAP 09, 022 (2020), arXiv:2006.11172 +[astro-ph.CO]. +[80] K. Jedamzik, Phys. Rev. Lett. 126, 051302 (2021), +arXiv:2007.03565 [astro-ph.CO]. +[81] S. Clesse and J. Garc´ıa-Bellido, Phys. Dark Univ. 38, +101111 (2022), arXiv:2007.06481 [astro-ph.CO]. +[82] G. H¨utsi, M. Raidal, V. Vaskonen, +and H. Veerm¨ae, +JCAP 03, 068 (2021), arXiv:2012.02786 [astro-ph.CO]. +[83] A. Hall, A. D. Gow, and C. T. Byrnes, Phys. Rev. D +102, 123524 (2020), arXiv:2008.13704 [astro-ph.CO]. +[84] J. +Garc´ıa-Bellido, +J. +F. +Nu˜no +Siles, +and +E. +Ruiz +Morales, +Phys. +Dark +Univ. +31, +100791 +(2021), arXiv:2010.13811 [astro-ph.CO]. +[85] G. Franciolini, V. Baibhav, V. De Luca, K. K. Y. Ng, +K. W. K. Wong, E. Berti, P. Pani, A. Riotto, +and +S. Vitale, (2021), arXiv:2105.03349 [gr-qc]. +[86] B. P. Abbott et al. (LIGO Scientific, Virgo), Phys. +Rev. Lett. 121, 231103 (2018), arXiv:1808.04771 [astro- +ph.CO]. +[87] A. H. Nitz and Y.-F. Wang, Phys. Rev. Lett. 126, +021103 (2021), arXiv:2007.03583 [astro-ph.HE]. +[88] A. H. Nitz and Y.-F. Wang, (2022), arXiv:2202.11024 +[astro-ph.HE]. +[89] C. T. Byrnes, M. Hindmarsh, S. Young, and M. R. S. +Hawkins, JCAP 1808, 041 (2018), arXiv:1801.06138 +[astro-ph.CO]. +[90] K. Jedamzik, Phys. Rev. D 55, 5871 (1997), arXiv:astro- +ph/9605152. +[91] J. C. Niemeyer and K. Jedamzik, Phys. Rev. Lett. 80, +5481 (1998), arXiv:astro-ph/9709072. +[92] J. Garcia-Bellido, H. Murayama, and G. White, JCAP +12, 023 (2021), arXiv:2104.04778 [hep-ph]. +[93] A. +Escriv`a, +E. +Bagui, +and +S. +Clesse, +(2022), +arXiv:2209.06196 [astro-ph.CO]. +[94] G. Franciolini, I. Musco, P. Pani, +and A. Urbano, +(2022), arXiv:2209.05959 [astro-ph.CO]. +[95] V. De Luca, G. Franciolini, P. Pani, +and A. Riotto, +JCAP 04, 052 (2020), arXiv:2003.02778 [astro-ph.CO]. +[96] S. Jaraba and J. Garc´ıa-Bellido, Phys. Dark Univ. 34, +100882 (2021), arXiv:2106.01436 [gr-qc]. +[97] J. Kopp, R. Laha, T. Opferkuch, +and W. Shepherd, +JHEP 11, 096 (2018), arXiv:1807.02527 [hep-ph]. +[98] M. I. Gresham and K. M. Zurek, Phys. Rev. D 99, +083008 (2019), arXiv:1809.08254 [astro-ph.CO]. +[99] V. Takhistov, G. M. Fuller, +and A. Kusenko, Phys. +Rev. Lett. 126, 071101 (2021), arXiv:2008.12780 [astro- +ph.HE]. +[100] R. Garani, D. Levkov, +and P. Tinyakov, +(2021), +arXiv:2112.09716 [hep-ph]. +[101] D. Singh, M. Ryan, R. Magee, T. Akhter, S. Shandera, +D. Jeong, +and C. Hanna, Phys. Rev. D 104, 044015 +(2021), arXiv:2009.05209 [astro-ph.CO]. +[102] R. Abbott et al. (LIGO Scientific, Virgo, KAGRA), +“Search for subsolar-mass black hole binaries in the sec- +ond part of advanced ligo’s and advanced virgo’s third +observing run,” (2022). +[103] S. L. Liebling and C. Palenzuela, Living Rev. 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' in 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' The origin of this trigger is unclear, because the reported signal-to-noise ratio (SNR) of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='6 and inverse false alarm rate of about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='5 yr are too low to claim a gravitational-wave origin, but large enough to be intriguing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' When using more precise waveforms, extending the frequency range down to 20 Hz, removing a prominent blip glitch and marginalizing over all the model parameters, we find that the network signal-to-noise ratio posterior distribution lies mostly below the search value, with the 90% confidence interval being 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='94+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='70 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' If one assumes that the signal comes from a real gravitational-wave merger event, we find a light component m2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='76+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='50 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='14M⊙, suggesting a compact object of mass below one solar mass at 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='8% confidence level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Such a low mass for a compact object would suggest an unexpectedly light neutron star or a black hole of primordial or exotic origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' The primary mass would be m1 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='71+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='57 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='18M⊙, likely in the lower mass gap, for a mass ratio of q =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='16+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='34 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='06, at a distance of DL =124+82 −48Mpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' The improved sensitivity of the next observing runs would make it possible to observe similar signals with a higher SNR and to distinguish a sub-solar mass component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' INTRODUCTION The development of gravitational wave (GW) astron- omy, with about 90 binary black hole (BBH) coalescence events detected so far [1–6] by the LIGO-Virgo-KAGRA (LVK) collaboration [7], is driving a true revolution in astrophysics and cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' As the number of detected events grows with successive observing catalogs, prop- erties of the progenitors seem to challenge prior expec- tations for a population of astrophysical objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Re- cent examples are BBH events like GW190521 [8, 9] with its most massive component in the pair-instability mass gap [10], as well as events like GW190814 which has a very low mass ratio and a secondary in the lower mass gap [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Evidence for misaligned spins in the black hole population has been found [12], suggesting a dynamical binary formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' The frequency range of the LIGO [13] and Virgo [14] detectors makes them sensitive to compact object bina- ries with masses below 1M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' There is no compelling stellar evolution model that can produce neutron stars or black holes below 1 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Therefore, the detection of a subsolar-mass (SSM) black hole directly points to a new black hole formation mechanism operating in the Universe, an alternative to the astrophysical evolution and collapse of ordinary matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Primordial black holes (PBHs) are natural candidates since they can be pro- duced with a wide mass spectrum in the early Universe through the collapse of highly overdense regions [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' An SSM compact object detection provides the cleanest sig- nature for a PBH, though there are some proposals of dark matter with exotic properties that could also pro- duce subsolar-mass objects [16–29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Before the advent of GW astronomy, the only way to detect SSM black holes was via X-ray binaries [30] or microlensing [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' At present, some hints of the existence of such light black holes come from microlensing events towards the bulge [32], from Andromeda [33] and lensed quasars [34, 35], although the mass, the nature and the abundance of the lenses remain uncertain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Complementary to these astrophysical searches, com- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='11619v1 [gr-qc] 27 Jan 2023 2 pact binary coalescences (CBCs) with at least one sub- solar component have been searched for in the first (O1), second (O2) and third (O3) observing runs of LVK, with- out convincing evidence [36–40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Nevertheless, a further search for SSM black holes with low mass ratio in the O2 data has recently revealed four potential candidate events1 (we refer the reader to Table I of [41]) with a false alarm rate smaller than 2 yr−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' In this paper, we follow up this search and perform pa- rameter inference of the four events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Our primary goal is to further investigate these SSM triggers using the stan- dard parameter estimation (PE) methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' These allow us to extend the frequency range of the search and use more accurate waveforms including spin precession, and higher order modes, as well as the merger and ringdown phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' We also can visually inspect the quality of the data and subtract non-gaussianities using standard tools such as BayesWave [42–44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' We focus on the third candidate event reported in Ta- ble I of [41], observed by both LIGO Hanford and LIGO Livingston interferometers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' It is the most significant two detector trigger of the search and the only one having significant support for an SSM component after further inspection with PE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' We analyse in detail the data and perform a careful PE around this trigger, observed on April 1st 2017 and referred here as SSM170401.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Fur- thermore, we discuss the impact of a prominent glitch removal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' As a by-product, the PE allows us to infer the component masses, spins, distance and sky locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' In particular, we infer the probability of an SSM compo- nent, if one assumes that the signal is coming from a BBH merger event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' PROPERTIES OF THE CANDIDATE To obtain the properties of the candidate we interpret the signal as coming from the coalescence of two com- pact objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' The signal was found in data taken on 2017 April 1, 01:43:34 UTC during the O2 LIGO-Virgo observ- ing run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' This candidate was not reported by any of the LVK searches, both generic [3] and SSM specific [45], but it was reported as part of a search for SSM candidates in asymmetric binaries using the GstLAL pipeline [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' The trigger had detector frame masses of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='897 M⊙ and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='7795 M⊙, with a false-alarm-rate (FAR) of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='4134 yr−1 and a combined network signal-to-noise ratio (SNR) of ∼ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' The strain in Hanford presents a glitch 14 s before co- alescence, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' The search presented in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [41] uses templates starting at f=45 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' The loud- est template, in this case, is only 10 s long and so should be unaffected by the glitch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' However, PE was performed 1 Three additional subsolar-mass triggers were recently reported in [40] which could be the object of future analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Figure showing the Hanford original whitened strain ˜hwhitened(f) = ˜h(f)/ � Sn(f), the whitened glitch model and the whitened clean data after subtracting the glitch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Times are shown relative to the trigger time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' with templates starting at 20 Hz, which are roughly 100 s long for the component masses discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' In this sit- uation the glitch must be removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Using BayesWave [42, 43] we fit to the data a combined glitch, signal and Gaussian noise model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' We then subtract the glitch part of the model from the original data and obtain the clean data to be used for PE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' The same procedure was used by the LVK collaboration for GW170817 to clean the data from a large glitch [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' We infer the CBC parameters of the signal using a Bayesian analysis of the data from LIGO Livingston and LIGO Hanford, following the methodology outlined in Appendix B of [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' In analysing the data, we fit two different waveform models: IMRPhenomPv2 [47] and IMRPhenomXPHM [48], the latter including higher order modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' We then compare the posterior samples from each of these, and find consistency between the two mod- els, noting that both of them take into account precessing spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' The TaylorF2 [49] waveform model has also been tested and, despite it providing compatible results, it fails to reach the same level of significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' We use a low-frequency cutoff of 20 Hz in both detec- tors for the likelihood evaluation and choose uninforma- tive and wide priors (see Supplemental Material).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' The primary tool used for sampling the posterior distribution is the LALInference Markov Chain Monte Carlo imple- mentation as described in [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' The power spectral den- sity used in the calculations of the likelihood is estimated using BayesWave [42, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' The study uses the O2 open access data [56] with a sampling frequency of 4096 Hz;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' however the likelihood is integrated up to 1600 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' The signal is present in the detector for ∼ 3000 cycles, allowing us to constrain the source properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' The esti- mated parameters are reported in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' The marginal- ized posterior for the absolute value of the matched filter SNR is 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='98+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='62 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='03 for IMRPhenomPv2 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='94+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='70 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='05 for Hanford original data 15 clean data glitch 10 Whitened Strain 5 5 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='19-14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='16-14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='13 -14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='1 -14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='07-14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='04-14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='01-13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='98-13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='95-13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='92 Time [seco0nds1 from 2017-04-01 01:43:34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='677 UTC (1175046232.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='677)3 Parameter IMRPhenomPv2 IMRPhenomXPHM Signal to Noise Ratio 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='98+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='62 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='03 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='94+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='70 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='05 Primary mass (M⊙) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='65+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='21 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='15 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='71+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='57 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='18 Secondary mass (M⊙) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='77+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='50 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='76+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='50 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='14 Primary spin magnitude 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='32+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='47 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='36+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='46 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='30 Secondary spin magnitude 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='48+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='46 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='43 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='47+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='46 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='42 Total mass (M⊙) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='42+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='10 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='65 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='47+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='43 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='68 Mass ratio (m2/m1 ≤ 1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='17+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='34 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='16+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='34 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='06 χeff [50, 51] −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='06+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='17 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='32 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='05+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='22 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='35 χp [52] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='28+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='34 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='33+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='33 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='26 Luminosity Distance (Mpc) 119+82 −48 124+82 −48 Redshift 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='028+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='018 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='028+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='017 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='011 Ra (◦) −2+34 −35 −1+34 −37 Dec (◦) 47+14 −26 46+14 −29 Final mass (M⊙) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='34+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='11 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='70 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='40+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='45 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='73 Final spin 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='39+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='24 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='42+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='22 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='10 P(m2 < 1 M⊙) 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='5% 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='8% P(m2 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='2 M⊙) 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='7% 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='7% TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Parameters of SSM170401.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' All masses are in the source frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' We assume Planck15 Cosmology [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' The statistical uncertainty of all the parameters is quantified by the equal-tailed 90% credible intervals about the median of the marginalized one-dimensional posteriors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Right ascension (Ra) and declination (Dec) are measured in the International Celestial Reference System (ICRS) [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' IMRPhenomXPHM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' The median value of the SNR is lower than that found by the search, which was 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' How- ever, these two quantities are not directly comparable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' The SNR from the search is obtained by maximizing the ranking statistic over a discrete template bank [41, 57– 59], while the quoted SNR from the PE is the median value over the samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Since the ranking statistic and the SNR are closely related, the SNR that is more compa- rable to that of the search would be the maximum SNR as found by the PE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' The values of this maximum PE SNR are 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='09 for IMRPhenomPv2 and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='18 for IMRPhenomXPHM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' These values are slightly larger than that of the search, which is consistent with what would happen if the signal was astrophysical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' However, this is also expected in the noise case due to the larger parameter space that allows more flexibility for the PE analysis to fit the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' We also notice the maximum value of the SNR to be larger for IMRPhenomXPHM than for IMRPhenomPv2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' In a simi- lar way, this is expected for an astrophysical signal but also for noise, since the waveform includes Higher Order Modes and thus has more flexibility to fit the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' The signal is then compatible with a compact binary system having an unequal mass ratio q =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='17+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='34 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='05 (all uncertainties are quoted at 90% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' ), a source frame 1 2 3 4 5 6 7 msource 1 [M ] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='4 msource 2 [M ] IMRPhenomPv2 IMRPhenomXPHM FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Posterior distributions for the primary and secondary mass in the source frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' The 90% credible regions are in- dicated by the solid contour in the joint distribution, and by the dashed vertical and horizontal lines in the marginalized distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' primary mass m1 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='65+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='21 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='15M⊙ and a source frame secondary mass m2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='77+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='50 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='12M⊙ as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' The marginalised posterior distribution for the secondary mass favors a mass lower than 1M⊙ (85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='5% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=') and provides strong support for a mass lower than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='2M⊙ (92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='7% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Using the IMRPhenomXPHM waveform, we find almost identical results, with a mass lower than 1M⊙ at 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='8%C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' The left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' 3 shows the posterior distribu- tions for the magnitude and tilt angle of the individual spins, measured at a reference frequency of 20 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' All pixels in this plot have an equal prior probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' The spin of the secondary BH is largely unconstrained, as ex- pected for very unequal masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' The primary spin, if present, is likely to be misaligned with the orbital an- gular momentum with a preference for small spin mag- nitude (a1 =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='32+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='47 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' As can be seen in the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' 3, this leads to a χeff compatible with 0 (χeff =−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='05+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='22 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='35) and an uninformative posterior in χp (χp =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='33+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='33 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' The luminosity distance and inclination angle θJN pos- terior distributions are shown together in the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' 4, since these two quantities are correlated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' We find a luminosity distance of dL =119+82 −48Mpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' We identify a bimodal distribution for θJN due to the fact that we can not distinguish whether the system is being observed face-on (θJN ∼ 0) or face-away (θJN ∼ π), but it being edge-on (θJN ∼ π/2) is disfavoured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' In the face-on(off) configuration, the effects of precession [60] and higher or- der modes in the signal are suppressed [61], as is the case 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='8 /( ) /( ) × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='5 posterior probability per pixel 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='0 p 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='00 eff IMRPhenomPv2 IMRPhenomXPHM Prior FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Left: posterior distribution for the individual spins of SSM170401 according to the IMRPhenomXPHM waveform model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' The radial coordinate in the plot denotes the dimensionless spin magnitude, while the angle denotes the spin tilt, defined as the angle between the spin and the orbital angular momentum of the binary at a reference frequency of 20 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' A tilt of 0° indicates that the spin is aligned with the orbital angular momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' A nonzero magnitude and a tilt away from 0° and 180° imply a precessing orbital plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' All bins have an equal prior probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Right: posterior distributions for the effective spin and effective in-plane spin parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' The black lines in the right panel show the prior distributions for the effective spin parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' The 90% credible regions are indicated by the solid contour in the joint distribution, and by dashed vertical and horizontal lines in the marginalized distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' The large density for tilts close to 90° leads to non-zero values for χp and low values for χeff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' From the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' 4, it can be seen that the signal came from a position in the sky for which the LIGO network has good sensitivity to the two GW polar- izations [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' This, however, does not represent the area with the best sensitivity, which would be located on top of the continental US and its antipodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' In summary, the PE shows that the chirp and compo- nent masses, the effective spin, the luminosity distance and the sky location can be reconstructed, and are con- sistent with that of a BBH merger event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' However, it is known that Gaussian noise can also mimic such a sig- nal [63], and given the low values of the SNR and iFAR, it is not possible to ascertain the origin of the signal, as it could very well have been generated by detector noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' DISCUSSION Even if the significance of the trigger did not improve with the present analysis and PE since it remains at the threshold limit, it is interesting to speculate on the possi- ble origin of the secondary component (with a preferred mass below 1 M⊙ for such a compact object, if inter- preted as a GW event).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' The neutron star nature of the light compact object seems disfavored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Indeed, neutron stars have relatively well-determined masses from observations of binary sys- tems, including pulsars or X-ray binaries involving an accreting neutron star from a companion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Their masses are contained within a narrow range 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='25-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='45 M⊙ [64], further confirmed by the observation of GW170817 [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Even though there is a recent claim [65] for a neutron star of mass around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='7 M⊙, modern core-collapse supernova simulations [66, 67] indicate it is difficult to form neutron stars with masses below one solar mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Such a small mass for a neutron star probably requires a QCD equa- tion of state that is beyond theoretical predictions [68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Although the neutron star interpretation of the hypo- thetical trigger SSM170401 is observationally disfavored, given our current limited knowledge of the equation of state we cannot exclude a neutron star origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' On the other hand, PBHs [69–72], formed by the grav- itational collapse of large inhomogeneities in the early Universe are already considered as a possible explana- tion of LVK GW detections, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [73–85].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Depending 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='0 JN [radians] 50 100 150 200 250 300 350 400 DL [Mpc] Prior IMRPhenomPv2 IMRPhenomXPHM 135° 90° 45° 0° 45° 90° 135° 0° 30° 60° 60° 30° 0° 30° 60° 60° 30° 50% area: 711 deg² 90% area: 2,658 deg² FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Left: posterior distributions for the luminosity distance and the inclination angle of SSM170401, according to the IMRPhenomXPHM and IMRPhenomPv2 waveform models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' The inclination angle indicates the angle between the line of sight and the total angular momentum of the binary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' For nonprecessing binaries, this is equal to the angle between the orbital angular momentum and the line of sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' The solid lines and the central contour denote 90% credible regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Right: sky position of the event as evaluated from the Greenwich meridian according to the IMRPhenomXPHM waveform model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' on the model, they may explain anything from a tiny fraction of Dark Matter to its entirety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' PBHs have been the main motivation to conduct searches of SSM black holes in the LVK data [38, 39, 41, 45, 86–88], in partic- ular, the extended subsolar search with low-mass ratios in O2 which reported SSM170401 as a possible candi- date [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' If some of the observed binary coalescences are indeed due to PBHs, they must have a relatively ex- tended mass distribution that would have been imprinted by the thermal history of the Universe [78, 89].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' This would lead to a peak in the mass distribution around a solar mass which is naturally produced at the QCD tran- sition [78, 89–94], and SSM170401 could be an example of a subsolar PBH around the QCD-induced peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' The spin posterior is quite broad and the spin is compatible with zero, although a slight preference for a primary spin around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='3 is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' In this case, the non-zero but relatively low spin of the primary component may have been acquired by matter accretion, previous mergers or hyperbolic encounters [84, 95, 96].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Dark Matter with very special particle composition could also be at the origin of solar-mass black holes if it can accumulate inside neutron stars and lead to their collapse into a black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Several scenarios have been proposed [22, 26, 97–100], but they all require very par- ticular conditions, in order not to change the correspond- ing Chandrasekhar limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Such transmutations, if leading to SSM black holes, would be accompanied by violent ex- plosions that would have been observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' It is therefore still unclear if such scenarios are realistic and compati- ble with observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' In an alternative scenario, with complex and dissipative Dark Matter composition, SSM black holes could form through the cooling and gravita- tional collapse of Dark Matter halos [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' This model was constrained by the LVK data in [39, 101, 102].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Another possibility, if considered as a real GW event, could be that the secondary component of SSM170401 is a boson star, a hypothetical horizonless compact object formed by an ultralight bosonic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' If the mass of the bosonic particle is larger than 10−10eV/c2, the boson star can have subsolar mass [103].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Note that due to Beken- stein’s bound, any object of a given mass which is as compact as a black hole can only be a black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Boson stars necessarily must be larger, implying a lower ISCO frequency in the middle or lower than the LIGO sensitiv- ity band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' So the viable parameter space for a boson star is probably very limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Finally, we comment on the mass of the primary com- ponent, which would preferably lie in the low mass gap between 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='5 and 5M⊙ (61%C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Assuming a real GW merger event, it would most probably be a black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' A neutron star origin with mass above 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='5M⊙ is strongly disfavoured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Other black hole candidates in the low mass gap were observed in the GWTC-3 catalog, which may bring additional support for a primordial origin since PBHs should not have a mass gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' CONCLUSIONS In this work, we investigate the most significant can- didate reported in [41], removing a prominent blip glitch 6 in the data and estimating the CBC parameters with the state-of-the-art waveform families IMRPhenomPv2 and IMRPhenomXPHM, taking into account contributions from higher order modes and extending the frequency range down to 20 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' However, with this improved modelling with respect to the search, we find a 90% confidence level network SNR of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='94+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='70 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='05, which is lower than the SNR of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='6 obtained in the template-bank-based search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Nevertheless, if one would assume that it is coming from a real GW event, the trigger observed on the 1st of April, 2017, is identified consistently in both LIGO de- tectors with a light mass component, m2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='76+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='50 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='14M⊙ (90% credible interval).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Such low mass is below one solar mass and below 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='2 solar masses at 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='8% and 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='7% con- fidence level, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' The compact binary coales- cence presents a total mass of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='47+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='43 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='68M⊙, correspond- ing to a mass ratio of q =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='16+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='34 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='06, and a luminosity distance of 124+82 −48Mpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' The values quoted here are a result of using IMRPhenomXPHM but consistent results are obtained with different analysis pipelines and waveform families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' At this point, the observational data and the search from [41] do not show enough significance to claim a firm GW observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Nevertheless, our analysis shows that the signal, if coming from a GW event, is consistent with an SSM black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' We discuss several scenarios for the production of such a possible SSM black hole candi- date and conclude that a neutron star origin is disfavored or at least requires a non-standard matter equation of state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Other possibilities could include primordial black holes, black holes formed from the accretion of hypothet- ical Dark Matter particles onto neutron stars, or boson stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Given that PBHs can also explain some intriguing properties of other compact binary coalescences without being restricted by the Chandrashekar mass, they can be considered as our preferred hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' The data from the third observing run O3, as well as data from the future planned runs with improved sensi- tivity, O4 and O5, offer a great opportunity for discover- ing additional SSM candidate events, and could increase the statistical significance for the existence of a new class of SSM compact objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' ACKNOWLEDGMENTS S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' acknowledges support from the Francqui Foun- dation through a Starting Grant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' is sup- ported by King’s College London through a Postgrad- uate International Scholarship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' is supported in part by the Science and Technology Facility Council (STFC), United Kingdom, under the research grant ST/P000258/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' This work is partially supported by the Spanish grants PID2020-113701GB-I00, PID2021- 123012NB-C43 [MICINN-FEDER], and the Centro de Excelencia Severo Ochoa Program CEX2020-001007-S through IFT, some of which include ERDF funds from the European Union.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' IFAE is partially funded by the CERCA program of the Generalitat de Catalunya.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' We acknowledge the use of IUCAA LDG cluster Sarathi for the computational/numerical work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' This material is based upon work supported by NSF’s LIGO Laboratory which is a major facility fully funded by the National Science Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [1] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' (LIGO Scientific, Virgo), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' 116, 061102 (2016), arXiv:1602.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='03837 [gr-qc].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [2] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' (LIGO Scientific, Virgo), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' X 6, 041015 (2016), [Erratum: Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='X 8, 039903 (2018)], arXiv:1606.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='04856 [gr-qc].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [3] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' (LIGO Scientific, Virgo), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' X 9, 031040 (2019), arXiv:1811.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='12907 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [4] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' (LIGO Scientific, Virgo), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' X 11, 021053 (2021), arXiv:2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='14527 [gr-qc].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [5] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' (LIGO Scientific, VIRGO), (2021), arXiv:2108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='01045 [gr-qc].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [6] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' (LIGO Scientific, Virgo, KAGRA), (2021), arXiv:2111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='03606 [gr-qc].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [7] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Abbott, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Abbott, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Abbott, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Abraham, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Ac- ernese, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Ackley, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Adams, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Adya, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Affeldt, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Agathos, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Agatsuma, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Aggarwal, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Aguiar, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Aiello, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Ain, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Ajith, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Akutsu, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Allen, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Allocca, Living Reviews in Relativity 23 (2020), 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='1007/s41114-020-00026-9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [8] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' (LIGO Scientific, Virgo), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' 125, 101102 (2020), arXiv:2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='01075 [gr-qc].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [9] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' (LIGO Scientific, Virgo), Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' 900, L13 (2020), arXiv:2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='01190 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [10] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Woosley, Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' 836, 244 (2017), arXiv:1608.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='08939 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [11] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' (LIGO Scientific, Virgo), Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' 896, L44 (2020), arXiv:2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='12611 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [12] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' (LIGO Scientific, VIRGO, KAGRA), (2021), arXiv:2111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='03634 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [13] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Collaboration, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Aasi, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Abbott, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Ab- bott, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=', Classical and Quantum Gravity 32, 074001 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [14] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Acernese, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Agathos, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=', Classical and Quantum Gravity 32, 024001 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [15] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Garc´ıa-Bellido, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Linde, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Wands, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' D 54, 6040 (1996), arXiv:astro-ph/9605094.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [16] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Goldman and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Nussinov, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' D 40, 3221 (1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [17] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Kouvaris and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Tinyakov, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' D 83, 083512 (2011), arXiv:1012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='2039 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [18] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' de Lavallaz and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Fairbairn, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' D 81, 123521 (2010), arXiv:1004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='0629 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='GA].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [19] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Bramante and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Linden, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' 113, 191301 (2014), arXiv:1405.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='1031 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [20] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Bramante and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Elahi, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' D 91, 115001 (2015), arXiv:1504.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='04019 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [21] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Bramante, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Linden, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Tsai, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' D 97, 055016 (2018), arXiv:1706.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='00001 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' 7 [22] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Kouvaris, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Tinyakov, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Tytgat, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' 121, 221102 (2018), arXiv:1804.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='06740 [astro- ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [23] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Shandera, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Jeong, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Gebhardt, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' 120, 241102 (2018), arXiv:1802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='08206 [astro- ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='CO].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [24] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Chang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Egana-Ugrinovic, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Essig, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Kouvaris, JCAP 03, 036 (2019), arXiv:1812.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='07000 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [25] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Latif, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Lupi, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Schleicher, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' D’Amico, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Panci, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Bovino, Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' 485, 3352 (2019), arXiv:1812.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='03104 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='CO].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [26] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Dasgupta, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Laha, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Ray, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' 126, 141105 (2021), arXiv:2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='01825 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [27] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Gurian, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Ryan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Schon, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Jeong, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Shandera, Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' 939, L12 (2022), arXiv:2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='00064 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='CO].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [28] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Ryan and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Radice, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' D 105, 115034 (2022), arXiv:2201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='05626 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [29] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Hippert, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Dillingham, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Tan, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Curtin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Noronha-Hostler, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Yunes, (2022), arXiv:2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='08590 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [30] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Corral-Santana, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Casares, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Munoz-Darias, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Bauer, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Martinez-Pais, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Russell, Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' 587, A61 (2016), arXiv:1510.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='08869 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [31] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Paczynski, Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' 304, 1 (1986).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [32] �L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Wyrzykowski and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Mandel, Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' 636, A20 (2020), arXiv:1904.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='07789 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='SR].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [33] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Niikura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=', Nature Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' 3, 524 (2019), arXiv:1701.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='02151 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='CO].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [34] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Hawkins, Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' 643, A10 (2020), arXiv:2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='15007 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='CO].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [35] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Hawkins, Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' 512, 5706 (2022), arXiv:2204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='09143 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='CO].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [36] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' (LIGO Scientific, Virgo), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' 121, 231103 (2018), arXiv:1808.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='04771 [astro- ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='CO].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [37] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' (LIGO Scientific, Virgo), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' 123, 161102 (2019), arXiv:1904.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='08976 [astro- ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='CO].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [38] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Nitz and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Wang, arXiv (2021), arXiv:2102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='00868 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [39] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' (LIGO Scientific, Virgo, KAGRA), (2021), arXiv:2109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='12197 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='CO].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [40] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' (LIGO Scientific, VIRGO, KAGRA), (2022), arXiv:2212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='01477 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [41] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Phukon, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Baltus, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Caudill, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Clesse, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' De- passe, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Fays, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Fong, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Kapadia, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Magee, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Tanasijczuk, (2021), arXiv:2105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='11449 [astro- ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='CO].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [42] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Cornish and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Littenberg, Classical and Quan- tum Gravity 32, 135012 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [43] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Littenberg and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Cornish, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' D 91, 084034 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [44] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Davis, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Littenberg, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Romero-Shaw, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Mill- house, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' McIver, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Renzo, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Ashton, Classical and Quantum Gravity 39, 245013 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [45] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Abbott, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Abbott, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Abbott, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Abraham, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Acernese, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Ackley, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Adams, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Adhikari, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Adya, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Affeldt, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=', Physical review letters 123, 161102 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [46] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' (LIGO Scientific, Virgo), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' 119, 161101 (2017), arXiv:1710.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='05832 [gr-qc].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [47] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Khan, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Chatziioannou, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Hannam, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Ohme, Physical Review D 100 (2019), 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='1103/phys- revd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='024059.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [48] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Pratten, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Garc´ıa-Quir´os, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Colleoni, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Ramos- Buades, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Estell´es, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Mateu-Lucena, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Jaume, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Haney, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Keitel, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Thompson, and et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=', Physi- cal Review D 103 (2021), 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='1103/physrevd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='104056.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [49] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Buonanno, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Iyer, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Ochsner, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Pan, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Sathyaprakash, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' D 80, 084043 (2009), arXiv:0907.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='0700 [gr-qc].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [50] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Ajith, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Hannam, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Husa, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Chen, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Br¨ugmann, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Dorband, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' M¨uller, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Ohme, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Pollney, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Reiss- wig, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Santamar´ıa, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Seiler, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' 106, 241101 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [51] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Santamar´ıa, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Ohme, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Ajith, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Br¨ugmann, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Dorband, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Hannam, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Husa, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' M¨osta, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Pollney, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Reisswig, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Robinson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Seiler, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Krishnan, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' D 82, 064016 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [52] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Schmidt, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Ohme, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Hannam, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' D 91, 024043 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [53] Planck Collaboration, A&A 594, A13 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [54] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Arias, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Charlot, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Feissel, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Lestrade, Astronomy and Astrophysics 303, 604 (1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [55] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Veitch, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Raymond, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Farr, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Farr, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Graff, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Vitale, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Aylott, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Blackburn, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Christensen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Coughlin, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Del Pozzo, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Feroz, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Gair, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='- J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Haster, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Kalogera, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Littenberg, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Mandel, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' O’Shaughnessy, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Pitkin, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Rodriguez, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' R¨over, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Sidery, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Smith, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Van Der Sluys, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Vecchio, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Vousden, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Wade, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' D 91, 042003 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [56] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' (LIGO Scientific, Virgo), SoftwareX 13, 100658 (2021), arXiv:1912.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='11716 [gr-qc].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [57] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Messick, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Blackburn, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Brady, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Brockill, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Can- non, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Cariou, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Caudill, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Chamberlin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Creighton, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Everett, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=', Physical Review D 95, 042001 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [58] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Cannon, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Cariou, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Chapman, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Crispin- Ortuzar, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Fotopoulos, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Frei, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Hanna, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Kara, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Keppel, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Liao, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=', The Astrophysical Journal 748, 136 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [59] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Cannon, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Caudill, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Chan, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Cousins, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Creighton, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Ewing, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Fong, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Godwin, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Hanna, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Hooper, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Huxford, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Magee, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Meacher, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Mes- sick, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Morisaki, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Mukherjee, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Ohta, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Pace, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Privitera, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' de Ruiter, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Sachdev, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Singer, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Singh, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Tapia, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Tsukada, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Tsuna, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Tsutsui, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Ueno, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Viets, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Wade, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Wade, SoftwareX 14, 100680 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [60] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Calder´on Bustillo, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Laguna, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Shoemaker, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' D 95, 104038 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [61] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Mills and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Fairhurst, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' D 103, 024042 (2021), arXiv:2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='04313 [gr-qc].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [62] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Andersson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Baker, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Belczynski, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Bernuzzi, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Berti, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Cadonati, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Cerd´a-Dur´an, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Clark, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Fa- vata, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Finn, and et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=', Classical and Quantum Gravity 30, 193002 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [63] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Morras, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Nu˜no Siles, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Garc´ıa-Bellido, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Morales, (2022), arXiv:2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='05475 [gr-qc].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [64] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Lattimer, Universe 5 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [65] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Doroshenko, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Suleimanov, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' P¨uhlhofer, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Santangelo, Nature Astronomy 6, 1444 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [66] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' M¨uller, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Tauris, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Heger, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Banerjee, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Qian, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Powell, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Chan, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Gay, and 8 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Langer, Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' 484, 3307 (2019), arXiv:1811.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='05483 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='SR].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [67] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Burrows, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Radice, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Vartanyan, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Nagakura, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Skinner, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Dolence, Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' 491, 2715 (2020), arXiv:1909.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='04152 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [68] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Sotani, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Iida, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Oyamatsu, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Ohnishi, PTEP 2014, 051E01 (2014), arXiv:1401.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='0161 [astro- ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [69] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Zel’dovich and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Novikov, Soviet Astronomy 10, 602 (1967).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [70] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Hawking, Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' 152, 75 (1971).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [71] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Carr and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Hawking, Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' 168, 399 (1974).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [72] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Chapline, Nature 253, 251 (1975).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [73] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Bird, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Cholis, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Mu˜noz, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Ali-Ha¨ımoud, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Kamionkowski, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Kovetz, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Raccanelli, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Riess, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' 116, 201301 (2016), arXiv:1603.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='00464 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='CO].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [74] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Clesse and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Garc´ıa-Bellido, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Dark Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' 15, 142 (2017), arXiv:1603.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='05234 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='CO].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [75] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Sasaki, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Suyama, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Tanaka, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Yokoyama, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' 117, 061101 (2016), [Erratum: Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' 121, 059901 (2018)], arXiv:1603.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='08338 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='CO].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [76] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Clesse and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Garc´ıa-Bellido, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Dark Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' 22, 137 (2018), arXiv:1711.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='10458 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='CO].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [77] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Fernandez and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Profumo, JCAP 08, 022 (2019), arXiv:1905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='13019 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [78] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Carr, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Clesse, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Garc´ıa-Bellido, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' K¨uhnel, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Dark Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' 31, 100755 (2021), arXiv:1906.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='08217 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='CO].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [79] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Jedamzik, JCAP 09, 022 (2020), arXiv:2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='11172 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='CO].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [80] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Jedamzik, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' 126, 051302 (2021), arXiv:2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='03565 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='CO].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [81] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Clesse and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Garc´ıa-Bellido, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Dark Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' 38, 101111 (2022), arXiv:2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='06481 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='CO].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [82] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' H¨utsi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Raidal, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Vaskonen, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Veerm¨ae, JCAP 03, 068 (2021), arXiv:2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='02786 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='CO].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [83] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Hall, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Gow, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Byrnes, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' D 102, 123524 (2020), arXiv:2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='13704 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='CO].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [84] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Garc´ıa-Bellido, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Nu˜no Siles, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Ruiz Morales, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Dark Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' 31, 100791 (2021), arXiv:2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='13811 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='CO].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [85] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Franciolini, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Baibhav, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' De Luca, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Ng, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Wong, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Berti, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Pani, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Riotto, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Vitale, (2021), arXiv:2105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='03349 [gr-qc].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [86] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' (LIGO Scientific, Virgo), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' 121, 231103 (2018), arXiv:1808.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='04771 [astro- ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='CO].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [87] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Nitz and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Wang, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' 126, 021103 (2021), arXiv:2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='03583 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [88] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Nitz and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Wang, (2022), arXiv:2202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='11024 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [89] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Byrnes, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Hindmarsh, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Young, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Hawkins, JCAP 1808, 041 (2018), arXiv:1801.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='06138 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='CO].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [90] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Jedamzik, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' D 55, 5871 (1997), arXiv:astro- ph/9605152.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [91] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Niemeyer and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Jedamzik, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' 80, 5481 (1998), arXiv:astro-ph/9709072.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [92] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Garcia-Bellido, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Murayama, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' White, JCAP 12, 023 (2021), arXiv:2104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='04778 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [93] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Escriv`a, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Bagui, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Clesse, (2022), arXiv:2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='06196 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='CO].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [94] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Franciolini, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Musco, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Pani, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Urbano, (2022), arXiv:2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='05959 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='CO].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [95] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' De Luca, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Franciolini, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Pani, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Riotto, JCAP 04, 052 (2020), arXiv:2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='02778 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='CO].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [96] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Jaraba and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Garc´ıa-Bellido, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Dark Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' 34, 100882 (2021), arXiv:2106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='01436 [gr-qc].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [97] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Kopp, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Laha, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Opferkuch, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Shepherd, JHEP 11, 096 (2018), arXiv:1807.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='02527 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [98] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Gresham and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Zurek, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' D 99, 083008 (2019), arXiv:1809.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='08254 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='CO].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [99] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Takhistov, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Fuller, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Kusenko, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' 126, 071101 (2021), arXiv:2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='12780 [astro- ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [100] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Garani, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Levkov, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Tinyakov, (2021), arXiv:2112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='09716 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [101] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Singh, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Ryan, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Magee, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Akhter, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Shandera, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Jeong, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Hanna, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' D 104, 044015 (2021), arXiv:2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='05209 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='CO].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [102] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' (LIGO Scientific, Virgo, KAGRA), “Search for subsolar-mass black hole binaries in the sec- ond part of advanced ligo’s and advanced virgo’s third observing run,” (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' [103] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Liebling and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Palenzuela, Living Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' Rel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content=' 15, 6 (2012), arXiv:1202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} +page_content='5809 [gr-qc].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FJT4oBgHgl3EQfty2e/content/2301.11619v1.pdf'} diff --git a/G9FLT4oBgHgl3EQfHS_T/content/tmp_files/2301.11996v1.pdf.txt b/G9FLT4oBgHgl3EQfHS_T/content/tmp_files/2301.11996v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..fb1ce1420a03c4f4811c16eddca361be4c6854b2 --- /dev/null +++ b/G9FLT4oBgHgl3EQfHS_T/content/tmp_files/2301.11996v1.pdf.txt @@ -0,0 +1,2048 @@ +arXiv:2301.11996v1 [math.AP] 27 Jan 2023 +L2 DIFFUSIVE EXPANSION FOR NEUTRON TRANSPORT EQUATION +YAN GUO AND LEI WU +Abstract. Grazing set singularity leads to a surprising counter-example and breakdown [24] of the classical +mathematical theory for L∞ diffusive expansion (1.9) of neutron transport equation with in-flow boundary +condition in term of the Knudsen number ε, one of the most classical problems in the kinetic theory. Even +though a satisfactory new theory has been established by constructing new boundary layers with favorable +ε-geometric correction for convex domains [24, 7, 8, 22, 23], the severe grazing singularity from non-convex +domains has prevented any positive mathematical progress. We develop a novel and optimal L2 expansion +theory for general domain (including non-convex domain) by discovering a surprising ε +1 +2 gain for the average +of remainder. +Contents +1. +Introduction +1 +2. +Asymptotic Analysis +5 +3. +Remainder Equation +7 +4. +Remainder Estimate +10 +5. +Proof of Main Theorem +13 +References +13 +1. Introduction +1.1. Problem Formulation. We consider the steady neutron transport equation in a three-dimensional +C3 bounded domain (convex or non-convex) with in-flow boundary condition. In the spatial domain Ω ∋ +x = (x1, x2, x3) and the velocity domain S2 ∋ w = (w1, w2, w3), the neutron density uε(x, w) satisfies + + + +w · ∇xuε + ε−1� +uε − uε +� += 0 in Ω × S2, +uε(x0, w) = g(x0, w) for w · n < 0 and x0 ∈ ∂Ω, +(1.1) +where g is a given function denoting the in-flow data, +uε(x) := 1 +4π +� +S2 uε(x, w)dw, +(1.2) +n is the outward unit normal vector, with the Knudsen number 0 < ε ≪ 1. We intend to study the asymptotic +behavior of uε as ε → 0. +Based on the flow direction, we can divide the boundary γ := +� +(x0, w) : +x0 ∈ ∂Ω, w ∈ S2� +into the +incoming boundary γ−, the outgoing boundary γ+, and the grazing set γ0 based on the sign of w · n(x0). In +particular, the boundary condition of (1.1) is only given on γ−. +2020 Mathematics Subject Classification. Primary 35Q49, 82D75; Secondary 35Q62, 35Q20. +Key words and phrases. non-convex domains, transport equation, diffusive limit. +Y. Guo was supported by NSF Grant DMS-2106650. +L. Wu was supported by NSF Grant DMS-2104775. +1 + +2 +Y. GUO, L. WU +1.2. Normal Chart near Boundary. We follow the approach in [8, 23] to define the geometric quantities, +and the details can be found in Section 2.2. For smooth manifold ∂Ω, there exists an orthogonal curvilinear +coordinates system (ι1, ι2) such that the coordinate lines coincide with the principal directions at any x0 ∈ +∂Ω. Assume ∂Ω is parameterized by r = r(ι1, ι2). Let the vector length be Li := |∂ιir| and unit vector +ςi := L−1 +i ∂ιir for i = 1, 2. +Consider the corresponding new coordinate system (µ, ι1, ι2), where µ denotes the normal distance to the +boundary surface ∂Ω, i.e. +x = r − µn. +(1.3) +Define the orthogonal velocity substitution for w := (ϕ, ψ) as +−w · n = sin ϕ, +w · ς1 = cos ϕ sin ψ, +w · ς2 = cos ϕ cos ψ. +(1.4) +Finally, we define the scaled normal variable η = µ +ε , which implies ∂ +∂µ = 1 +ε +∂ +∂η . +1.3. Asymptotic Expansion and Remainder Equation. We seek a solution to (1.1) in the form +uε =U + U B + R = +� +U0 + εU1 + ε2U2 +� ++ U B +0 + R, +(1.5) +where the interior solution is +U(x, w) := U0(x, w) + εU1(x, w) + ε2U2(x, w), +(1.6) +and the boundary layer is +U B(η, ι1, ι2, w) := U B +0 (η, ι1, ι2, w). +(1.7) +Here U0, U1, U2 and U B +0 are constructed in Section 2.1 and Section 2.2, and R(x, v) is the remainder. +1.4. Literature. The study of the neutron transport equation in bounded domains, has attracted a lot +of attention since the dawn of the atomic age. +Besides its significance in nuclear sciences and medical +imaging, neutron transport equation is usually regarded as a linear prototype of the more important yet more +complicated nonlinear Boltzmann equation, and thus, is an ideal starting point to develop new theories and +techniques. We refer to [10, 11, 12, 13, 14, 15, 16, 17, 18] for the formal expansion with respect to ε and explicit +solution. The discussion on bounded domain and half-space cases can be found in [5, 4, 3, 1, 2, 19, 20, 21]. +The classical boundary layer of neutron transport equation dictates that U B +0 (η, ι1, ι2, w) satisfies the Milne +problem +sin ϕ∂U B +0 +∂η ++ U B +0 − U B +0 = 0. +(1.8) +From the formal expansion in ε (see (2.6)), it is natural to expect the remainder estimate [5] +∥R∥L∞ ≲ ε. +(1.9) +Even though this is valid for domains with flat boundary, a counter-example is constructed [24] so that (1.9) +is invalid for a 2D disk. This is due to the grazing set singularity. +To be more specific, in order to show the remainder estimates (1.9), the higher-order boundary layer +expansion U B +1 ∈ L∞ is necessary, which further requires ∂ιiU B +0 ∈ L∞. Nevertheless, though U B +0 ∈ L∞, it is +shown that the normal derivative ∂ηU B +0 is singular at the grazing set ϕ = 0. Furthermore, this singularity +∂ηU B +0 +/∈ L∞ will be transferred to ∂ιiU B +0 +/∈ L∞. A careful construction of boundary data [24] justifies this +invalidity, i.e. both the method and result of the boundary layer (1.8) are problematic. +A new construction of boundary layer [24] based on the ε-Milne problem with geometric correction for +� +U B +0 (η, ι1, ι2, w) +sin ϕ∂� +U B +0 +∂η +− +ε +1 − εη cos ϕ∂� +U B +0 +∂ϕ + � +U B +0 − � +U B +0 = 0 +(1.10) +has been shown to provide the satisfactory characterization of the L∞ diffusive expansion in 2D disk domains. +With more detailed regularity analysis and boundary layer decomposition techniques for (1.10), such result +has been generalized to 2D/3D smooth convex domains [7, 8, 22, 23] and even 2D annulus domain [25]. + +DIFFUSIVE EXPANSION OF NEUTRON TRANSPORT EQUATION +3 +In non-convex domains, the boundary layer with geometric correction is essentially +sin ϕ∂� +U B +0 +∂η +− +ε +1 + εη cos ϕ∂� +U B +0 +∂ϕ + � +U B +0 − � +U B +0 = 0 +(1.11) +Compared to (1.10), this sign flipping dramatically changes the characteristics. +0 +1 +2 +3 +4 +5 +6 +7 +8 +−1.5 +−1 +−0.5 +0 +0.5 +1 +1.5 +η +φ +Figure 1. Characteristics in Convex Domains +0 +1 +2 +3 +4 +5 +6 +7 +8 +−1.5 +−1 +−0.5 +0 +0.5 +1 +1.5 +η +φ +Figure 2. Characteristics in Non- +Convex Domains +In Figure 1 and Figure 2 [25], the horizontal direction represents the scaled normal variable η and the +vertical direction represents the velocity ϕ. There exists a “hollow” region in Figure 2 that the characteristics +may never track back to the left boundary η = 0 and ϕ > 0, making the W 1,∞ estimates impossible and +thus preventing higher-order boundary layer expansion. +In this paper, we will employ a fresh approach to design a cutoff boundary layer without the geometric +correction and justify the L2 diffusive expansion in smooth non-convex domains. +1.5. Notation and Convention. Let ⟨ · , · ⟩w denote the inner product for w ∈ S2, ⟨ · , · ⟩x for x ∈ Ω, +and ⟨ · , · ⟩ for (x, w) ∈ Ω × S2. Also, let ⟨ · , · ⟩γ± denote the inner product on γ± with measure dγ := +|w · n| dwdSx = |sin ϕ| cos ϕdwdSx. Denote the bulk and boundary norms +∥f∥L2 := +��� +Ω×S2 |f(x, w)|2 dwdx +� 1 +2 +, +|f|L2 +γ± := +�� +γ± +|f(x, w)|2 dγ +� 1 +2 +. +(1.12) +Define the L∞ norms +∥f∥L∞ := +ess sup +(x,w)∈Ω×S2 +��f(x, w) +��, +|f|L∞ +γ± := ess sup +(x,w)∈γ± +��f(x, w) +��. +(1.13) +Let ∥·∥W k,p +x +denote the usual Sobolev norm for x ∈ Ω and |·|W k,p +x +for x ∈ ∂Ω, and ∥·∥W k,p +x +Lq +w denote W k,p +norm for x ∈ Ω and Lq norm for w ∈ S2. The similar notation also applies when we replace Lq by Lq +γ. When +there is no possibility of confusion, we will ignore the (x, w) variables in the norms. +Throughout this paper, C > 0 denotes a constant that only depends on the domain Ω, but does not +depend on the data or ε. It is referred as universal and can change from one inequality to another. We write +a ≲ b to denote a ≤ Cb and a ≳ b to denote a ≥ Cb. Also, we write a ≃ b if a ≲ b and a ≳ b. We will use +o(1) to denote a sufficiently small constant independent of the data. +1.6. Main Results. +Theorem 1.1. Under the assumption +|g|W 3,∞L∞ +γ− ≲ 1, +(1.14) +there exists a unique solution uε(x, w) ∈ L∞(Ω × S2) to (1.1). Moreover, the solution obeys the estimate +∥uε − U0∥L2 ≲ ε +1 +2 . +(1.15) + +4 +Y. GUO, L. WU +Here U0(x) satisfies the Laplace equation with Dirichlet boundary condition +� +∆xU0(x) = 0 in Ω, +U0(x0) = Φ∞(x0) on ∂Ω, +(1.16) +in which Φ∞(ι1, ι2) = Φ∞(x0) for x0 ∈ ∂Ω is given by solving the Milne problem for Φ(η, ι1, ι2, w) + + + + + + + + + + + +sin ϕ∂Φ +∂η + Φ − Φ = 0, +Φ(0, ι1, ι2, w) = g(ι1, ι2, w) for +sin ϕ > 0, +lim +η→∞ Φ(η, ι1, ι2, w) = Φ∞(ι1, ι2). +(1.17) +Remark 1.2. In [24, 22, 23] for 2D/3D convex domains, as well as [25] for 2D annulus domain, it is justified +that for any 0 < δ ≪ 1 +���uε − � +U0 − � +U B +0 +��� +L2 ≲ ε +5 +6 −δ, +(1.18) +where � +U B +0 (η, ι1, ι2, w) is the boundary layer with geometric correction defined in (1.10), and � +U0 is the cor- +responding interior solution. +[21, Theorem 2.1] reveals that the difference between two types of interior +solutions +���� +U0 − U0 +��� +L2 ≲ ε +2 +3 . +(1.19) +Due to the rescaling η = ε−1µ, for general in-flow boundary data g, the boundary layer � +U B +0 ̸= 0 satisfies +���� +U B +0 +��� +L2 ≃ ε +1 +2 . +(1.20) +Hence, we conclude that +∥uε − U0∥L2 ≃ ε +1 +2 . +(1.21) +Therefore, this indicates that (1.15) in Theorem 1.1 achieves the optimal L2 bound of the diffusive approxi- +mation. +1.7. Methodology. It is well-known that the key of the remainder estimate is to control R. In a series of +work [24, 25, 7, 8, 22, 23] based on a L2 → L∞ framework, it is shown that +��R +�� +L2 ≲ ε−1 ��R − R +�� +L2 ≲ 1 +(1.22) +combined from the expected energy (entropy production) bound for ε−1 ��R − R +�� +L2. This bound requires +the next-order ε expansion of boundary layer approximation, which is impossible for non-convex domains, +and barely possible by the new boundary layer theory with the ε-geometric correction. The key improvement +in our work is +��R +�� +L2 ≲ ε +1 +2 +(1.23) +which is a consequence of the following conservation law for test function ξ(x) satisfying −∆xξ = R and +ξ +�� +∂Ω = 0: +− +� +R, w · ∇xξ +� += − +� +R − R, w · ∇xξ +� += +� +S, ξ +� +, +(1.24) +where +� +R, w ·∇xξ +� += 0 thanks to the oddness. This conservation law exactly cancels the worst contribution +of ε−1 ��R − R +�� +L2 in +��R +�� +L2 estimate, which comes from taking test function w · ∇xξ +� +γ +R +� +w · ∇xξ +� +(w · n) − +� +R, w · ∇x +� +w · ∇xξ +�� ++ ε−1� +R − R, w · ∇xξ +� += +� +S, w · ∇xξ +� +. +(1.25) +Such a key cancellation produces an extra crucial gain of ε +1 +2 . We then conclude the remainder estimate +without any further expansion of the (singular) boundary layer approximation. + +DIFFUSIVE EXPANSION OF NEUTRON TRANSPORT EQUATION +5 +In addition, we construct a new cut-off boundary layer near ϕ = 0 to avoid the singularity, and are able +to perform delicate and precise estimates to control the resulting complex forcing term S (see (3.7)–(3.10)), +in terms of the desired order ε for closure. +2. Asymptotic Analysis +2.1. Interior Solution. Inserting (1.6) into (1.1) and comparing the order of ε, following the analysis in +[8, 23], we deduce that +U0 = U 0, +∆xU 0 = 0, +(2.1) +U1 = U 1 − w · ∇xU0, +∆xU 1 = 0, +(2.2) +U2 = U 2 − w · ∇xU1, +∆xU 2 = 0. +(2.3) +We need the boundary layer to determine the boundary conditions for U0, U 1 and U 2. +2.2. Boundary Layer. +2.2.1. Geometric Substitutions. The construction of boundary layer requires a local description in a neigh- +borhood of the physical boundary ∂Ω. We follow the procedure in [8, 23]: +Substitution 1: Spacial Substitution. Following the notation in Section 1.2, under the coordinate system +(µ, ι1, ι2), we have +w · ∇x = −(w · n) ∂ +∂µ − +w · ς1 +L1(κ1µ − 1) +∂ +∂ι1 +− +w · ς2 +L2(κ2µ − 1) +∂ +∂ι2 +, +(2.4) +where κi(ι1, ι2) for i = 1, 2 is the principal curvature. +Substitution 2: Velocity Substitution. Under the orthogonal velocity substitution (1.4) for ϕ ∈ +� +−π +2 , π +2 +� +and +ψ ∈ [−π, π], we have +w · ∇x = sin ϕ ∂ +∂µ − +� sin2 ψ +R1 − µ + cos2 ψ +R2 − µ +� +cos ϕ ∂ +∂ϕ + cos ϕ sin ψ +L1(1 − κ1µ) +∂ +∂ι1 ++ cos ϕ cos ψ +L2(1 − κ2µ) +∂ +∂ι2 +(2.5) ++ +sin ψ +R1 − µ +�R1 cos ϕ +L1L2 +� +ς1 · +� +ς2 × +� +∂ι1ι2r × ς2 +��� +− sin ϕ cos ψ +� ∂ +∂ψ +− cos ψ +R2 − µ +�R2 cos ϕ +L1L2 +� +ς2 · +� +ς1 × +� +∂ι1ι2r × ς1 +��� +− sin ϕ sin ψ +� ∂ +∂ψ, +where Ri = κ−1 +i +represents the radius of curvature. Note that the Jacobian dw = cos ϕdϕdψ will be present +when we perform integration. +Substitution 3: Scaling Substitution. Considering the scaled normal variable η = ε−1µ, we have +w · ∇x =ε−1 sin ϕ ∂ +∂η − +� sin2 ψ +R1 − εη + cos2 ψ +R2 − εη +� +cos ϕ ∂ +∂ϕ + R1 cos ϕ sin ψ +L1(R1 − εη) +∂ +∂ι1 ++ R2 cos ϕ cos ψ +L2(R2 − εη) +∂ +∂ι2 +(2.6) ++ +sin ψ +R1 − εη +�R1 cos ϕ +L1L2 +� +ς1 · +� +ς2 × +� +∂ι1ι2r × ς2 +��� +− sin ϕ cos ψ +� ∂ +∂ψ +− +cos ψ +R2 − εη +�R2 cos ϕ +L1L2 +� +ς2 · +� +ς1 × +� +∂ι1ι2r × ς1 +��� +− sin ϕ sin ψ +� ∂ +∂ψ . +2.2.2. Milne Problem. Let Φ(η, ι1, ι2, w) be the solution to the Milne problem +sin ϕ∂Φ +∂η + Φ − Φ =0, +Φ(η, ι1, ι2) = 1 +4π +� π +−π +� +π +2 +− π +2 +Φ(η, ι1, ι2, w) cos ϕdϕdψ, +(2.7) +with boundary condition +Φ(0, ι1, ι2, w) = g(ι1, ι2, w) for +sin ϕ > 0. +(2.8) + +6 +Y. GUO, L. WU +We are interested in the solution that satisfies +lim +η→∞ Φ(η, ι1, ι2, w) = Φ∞(ι1, ι2) +(2.9) +for some Φ∞(ι1, ι2). Based on [8, Section 4], we have the well-posedness and regularity of (2.7). +Proposition 2.1. Under the assumption (1.14), there exist Φ∞(ι1, ι2) and a unique solution Φ to (2.7)such +that Ψ := Φ − Φ∞ satisfies + + + + + + + + + + + +sin ϕ∂Ψ +∂η + Ψ − Ψ = 0, +Ψ(0, ι1, ι2, w) = g(ι1, ι2, w) − Φ∞(ι1, ι2), +lim +η→0 Ψ(η, ι1, ι2, w) = 0, +(2.10) +and for some constant K > 0 and any 0 < r ≤ 3 +|Φ∞|W 3,∞ +ι1,ι2 + +��eKηΨ +�� +L∞ ≲1, +(2.11) +����eKη sin ϕ∂Ψ +∂η +���� +L∞ + +����eKη sin ϕ∂Ψ +∂ϕ +���� +L∞ + +����eKη ∂Ψ +∂ψ +���� +L∞ ≲1, +(2.12) +����eKη ∂rΨ +∂ιr +1 +���� +L∞ ++ +����eKη ∂rΨ +∂ιr +2 +���� +L∞ +≲1. +(2.13) +Let χ(y) ∈ C∞(R) and �χ(y) = 1 − χ(y) be smooth cut-off functions satisfying χ(y) = 1 if |y| ≤ 1 and +χ(y) = 0 if |y| ≥ 2. We define the boundary layer +U B +0 (η, ι1, ι2, w) := �χ +� +ε−1ϕ +� +χ(εη)Ψ(η, ι1, ι2, w). +(2.14) +Remark 2.2. Due to the cutoff in (2.14), we have +U B +0 (0, ι1, ι2, w) = �χ +� +ε−1ϕ +�� +g(ι1, ι2, w) − Φ∞(ι1, ι2) +� += �χ +� +ε−1ϕ +� +Ψ(0, ι1, ι2, w), +(2.15) +and +sin ϕ∂U B +0 +∂η ++ U B +0 − U B +0 = −�χ +� +ε−1ϕ +� +χ(εη)Ψ + Ψ�χ(ε−1ϕ)χ(εη). +(2.16) +2.3. Matching Procedure. We plan to enforce the matching condition for x0 ∈ ∂Ω and w · n < 0 +U0(x0) + U B +0 (x0, w) =g(x0, w) + O(ε). +(2.17) +Considering (2.15), it suffices to require +U0(x0) = Φ∞(x0) := Φ∞(ι1, ι2), +(2.18) +which yields +U0(x0) + Ψ(x0, w) =g(x0, w). +(2.19) +Hence, we obtain +U0(x0, w) + U B +0 (x0, w) = g(x0, w) + χ +� +ε−1ϕ +� +Ψ(0, ι1, ι2, w). +(2.20) +Construction of U0. Based on (2.1) and (2.18), define U0(x) satisfying +U0 = U0, +∆xU 0 = 0, +U0(x0) = Φ∞(x0). +(2.21) +From standard elliptic estimates [9] and Proposition 2.1, we have for any s ∈ [2, ∞) +∥U0∥W 3+ 1 +s ,s + |U0|W 3,s ≲ 1. +(2.22) +Construction of U1. Based on (2.2), define U1(x, w) satisfying +U1 = U 1 − w · ∇xU0, +∆xU 1 = 0, +U 1(x0) = 0. +(2.23) +From (2.22), we have for any s ∈ [2, ∞) +∥U1∥W 2+ 1 +s ,sL∞ + |U1|W 2,sL∞ ≲ 1. +(2.24) + +DIFFUSIVE EXPANSION OF NEUTRON TRANSPORT EQUATION +7 +Construction of U2. Based on (2.2), define U2(x, w) satisfying +U2 = U 2 − w · ∇xU1, +∆xU 2 = 0, +U 2(x0) = 0. +(2.25) +From (2.24), we have for any s ∈ [2, ∞) +∥U2∥W 1+ 1 +s ,sL∞ + |U2|W 1,sL∞ ≲ 1. +(2.26) +Summarizing the above analysis, we have the well-posedness and regularity estimates of the interior +solution and boundary layer: +Proposition 2.3. Under the assumption (1.14), we can construct U0, U1, U2, U B +0 as in (2.21)(2.23)(2.25)(2.14) +satisfying for any s ∈ [2, ∞) +∥U0∥W 3+ 1 +s ,s + |U0|W 3,s ≲1, +(2.27) +∥U1∥W 2+ 1 +s ,sL∞ + |U1|W 2,sL∞ ≲1, +(2.28) +∥U2∥W 1+ 1 +s ,sL∞ + |U2|W 1,sL∞ ≲1, +(2.29) +and for some constant K > 0 and any 0 < r ≤ 3 +��eKηU B +0 +�� +L∞ + +����eKη ∂rU B +0 +∂ιr +1 +���� +L∞ ++ +����eKη ∂rU B +0 +∂ιr +2 +���� +L∞ +≲1. +(2.30) +3. Remainder Equation +Denote the approximate solution +ua := +� +U0 + εU1 + ε2U2 +� ++ U B +0 . +(3.1) +Inserting (1.5) into (1.1), we have +w · ∇x +� +ua + R +� ++ ε−1� +ua + R +� +− ε−1� +ua + R +� += 0, +� +ua + R +���� +γ− += g, +(3.2) +which yields +w · ∇xR + ε−1� +R − R +� += −w · ∇xua − ε−1� +ua − ua +� +, +R +��� +γ− = +� +g − ua +���� +γ−. +(3.3) +3.1. Formulation of Remainder Equation. We consider the remainder equation + + + +w · ∇xR + ε−1� +R − R +� += S in Ω × S2, +R(x0, w) = h(x0, w) for w · n < 0 and x0 ∈ ∂Ω, +(3.4) +where R(x) = 1 +4π +� +S2 R(x, w)dw. Here the boundary data h is given by +h := −εw · ∇xU0 − ε2w · ∇xU1 − χ +� +ε−1ϕ +� +Ψ(0), +(3.5) +and the source term S is given by +S := S0 + S1 + S2 + S3, +(3.6) + +8 +Y. GUO, L. WU +where +S0 := − ε2w · ∇xU2, +(3.7) +S1 := +� sin2 ψ +R1 − εη + cos2 ψ +R2 − εη +� +cos ϕ∂U B +0 +∂ϕ , +(3.8) +S2 :=ε−1 sin φ�χ +� +ε−1ϕ +�∂χ(εη) +∂η +Ψ + R1 cos ϕ sin ψ +L1(R1 − εη) +∂U B +0 +∂ι1 ++ R2 cos ϕ cos ψ +L2(R2 − εη) +∂U B +0 +∂ι2 +(3.9) ++ +sin ψ +R1 − εη +�R1 cos ϕ +L1L2 +� +ς1 · +� +ς2 × +� +∂ι1ι2r × ς2 +��� +− sin ϕ cos ψ +�∂U B +0 +∂ψ +− +cos ψ +R2 − εη +�R2 cos ϕ +L1L2 +� +ς2 · +� +ς1 × +� +∂ι1ι2r × ς1 +��� +− sin ϕ sin ψ +�∂U B +0 +∂ψ , +S3 :=ε−1 +� +�χ +� +ε−1ϕ +� +χ(εη)Ψ − Ψ�χ +� +ε−1ϕ +� +χ(εη) +� +. +(3.10) +3.2. Weak Formulation. +Lemma 3.1 (Green’s Identity, Lemma 2.2 of [6]). Assume f(x, w), g(x, w) ∈ L2(Ω × S2) and w · ∇xf, w · +∇xg ∈ L2(Ω × S2) with f, g ∈ L2 +γ. Then +�� +Ω×S2 +�� +w · ∇xf +� +g + +� +w · ∇xg +� +f +� +dxdw = +� +γ +fg(w · n) = +� +γ+ +fgdγ − +� +γ− +fgdγ. +(3.11) +Using Lemma 3.1, we can derive the weak formulation of (3.4). For any test function g(x, w) ∈ L2(Ω×S2) +with w · ∇xg ∈ L2(Ω × S2) with g ∈ L2 +γ, we have +� +γ +Rg(w · n) − +�� +Ω×S2 R +� +w · ∇xg +� ++ ε−1 +�� +Ω×S2 +� +R − R +� +g = +�� +Ω×S2 Sg. +(3.12) +3.3. Estimates of Boundary and Source Terms. +Lemma 3.2. Under the assumption (1.14), for h defined in (3.5), we have +|h|L2 +γ− ≲ ε. +(3.13) +Proof. Based on Proposition 2.3, we have +|εw · ∇xU0|L2 +γ− + +��ε2w · ∇xU1 +�� +L2γ− ≲ ε. +(3.14) +Noting the cutoff χ +� +ε−1ϕ +� +restricts the support to |ϕ| ≲ ε and dγ measure contributes an extra sin ϕ, we +have +��χ +� +ε−1ϕ +� +Ψ(0) +�� +L2γ− ≲ ε. +(3.15) +Hence, our result follows. +□ +Lemma 3.3. Under the assumption (1.14), for S0 defined in (3.7), we have +∥S0∥L2 ≲ ε2. +(3.16) +Proof. This follows from Proposition 2.3. +□ +Lemma 3.4. Under the assumption (1.14), for S1 defined in (3.8), we have +��� +1 + η +� +S1 +�� +L2 ≲ 1. +(3.17) +Also, for the boundary layer U B +0 defined in (2.14), we have +��� +1 + η +� +U B +0 +�� +L2 ≲ ε +1 +2 , +��� +1 + η +� +U B +0 +�� +L2xL1w ≲ ε +1 +2 , +(3.18) +and +��� +�� +1 + η +� +S1, g +���� ≲ +��� +� +1 + η +� +⟨v⟩2 U B +0 +��� +L2 ∥∇wg∥L2 ≲ ε +1 +2 ∥∇wg∥L2 . +(3.19) + +DIFFUSIVE EXPANSION OF NEUTRON TRANSPORT EQUATION +9 +Proof. We split +S1 = S11 + S12 := +� sin2 ψ +R1 − εη + cos2 ψ +R2 − εη +� +cos ϕ∂Ψ +∂ϕ �χ +� +ε−1ϕ +� +χ(εη) +(3.20) ++ +� sin2 ψ +R1 − εη + cos2 ψ +R2 − εη +� +cos ϕ∂�χ +� +ε−1ϕ +� +∂ϕ +χ(εη)Ψ. +Note that S11 is nonzero only when |ϕ| ≥ ε and thus based on Proposition 2.1, we know +���� +∂Ψ +∂ϕ +���� ≤ |sin ϕ|−1 |Ψ| ≲ +ε−1. Hence, using dµ = εdη, we have +∥S11∥L2 ≲ +��� +|ϕ|≥ε +���� +∂Ψ +∂ϕ +���� +2 +dϕdµ +� 1 +2 +≲ +��� +|ϕ|≥ε +|sin ϕ|−2 |Ψ|2 dϕdµ +� 1 +2 +(3.21) +≲ +��� +|ϕ|≥ε +|sin ϕ|−2 e−2Kηdϕdµ +� 1 +2 +≲ +� +ε +�� +|ϕ|≥ε +|sin ϕ|−2 e−2Kηdϕdη +� 1 +2 +≲ +� +εε−1� 1 +2 = 1. +Noticing ∂�χ +� +ε−1ϕ +� +∂ϕ += ε−1�χ′� +ε−1ϕ +� +, and �χ′� +ε−1ϕ +� +is nonzero only when ε < |ϕ| < 2ε, based on Proposition +2.1, we have +∥S12∥L2 ≲ε−1 +��� +ε<|ϕ|<2ε +|Ψ|2 dϕdµ +� 1 +2 +≲ ε−1 +��� +ε<|ϕ|<2ε +e−2Kηdϕdµ +� 1 +2 +(3.22) +≲ε−1 +� +ε +�� +ε<|ϕ|<2ε +e−2Kηdϕdη +� 1 +2 +≲ ε−1 (εε) +1 +2 = 1. +Collecting (3.21) and (3.22), we have (3.17). Note that e−Kη will suppress the growth from the pre-factor +1 + η. +(3.18) comes from Proposition 2.1. +Then we turn to (3.19). +The most difficult term in +�� ⟨S1, g⟩ +�� is +essentially +���� +�∂U B +0 +∂ϕ , g +�����. Integration by parts with respect to ϕ implies +���� +�∂U B +0 +∂ϕ , g +����� ≲ +���� +� +U B +0 , ∂g +∂ϕ +����� ≲ +��U B +0 +�� +L2 +���� +∂g +∂ϕ +���� +L2 . +(3.23) +From (1.4) and ∂x +∂ϕ = 0, we know the substitution (µ, ι1, ι2, w) → (µ, ι1, ι2, w) implies +−∂w +∂ϕ · n = cos ϕ, +∂w +∂ϕ · ς1 = − sin ϕ sin ψ, +∂w +∂ϕ · ς2 = − sin ϕ cos ψ. +(3.24) +Hence, we know +���� +∂w +∂ϕ +���� ≲ 1, and thus +���� +∂g +∂ϕ +���� ≲ |∇wg| +���� +∂w +∂ϕ +���� ≲ |∇wg| . +(3.25) +Hence, we know that +���� +�∂U B +0 +∂ϕ , g +����� ≲ +��U B +0 +�� +L2 ∥∇wg∥L2 ≲ ε +1 +2 ∥∇wg∥L2 . +(3.26) +□ +Lemma 3.5. Under the assumption (1.14), for S2 defined in (3.9), we have +��� +1 + η +� +S2 +�� +L2 ≲ ε +1 +2 , +��� +1 + η +� +S2 +�� +L2xL1w ≲ ε +1 +2 . +(3.27) + +10 +Y. GUO, L. WU +Proof. Notice that +����ε−1 sin φ�χ +� +ε−1ϕ +�∂χ(εη) +∂η +���� ≲ 1. Based on Proposition 2.1 and Proposition 2.3, we directly +bound +∥S2∥L2 ≲ +��� � +|Φ|2 + +���� +∂Φ +∂ι1 +���� +2 ++ +���� +∂Φ +∂ι2 +���� +2 ++ +���� +∂Φ +∂ψ +���� +2 � +dϕdµ +� 1 +2 +(3.28) +≲ +��� +e−2Kηdϕdµ +� 1 +2 +≲ +� +ε +�� +e−2Kηdϕdη +� 1 +2 +≲ ε +1 +2 . +Then the L2 +xL1 +w estimate follows from a similar argument noting that there is no rescaling in w variables. +□ +Lemma 3.6. Under the assumption (1.14), for S3 defined in (3.10), we have +��� +1 + η +� +S3 +�� +L2 ≲ 1, +��� +1 + η +� +S3 +�� +L2xL1w ≲ ε +1 +2 . +(3.29) +Proof. Using χ = 1 − �χ, we split +S3 = S31 + S32 :=ε−1Ψχ +� +ε−1ϕ +� +χ(εη) − ε−1χ +� +ε−1ϕ +� +χ(εη)Ψ. +(3.30) +Noting that S31 is nonzero only when |ϕ| ≤ ε, based on Proposition 2.1, we have +∥S31∥L2 ≲ +��� +|ϕ|≤ε +��ε−1Ψ +��2 dϕdµ +� 1 +2 +≲ +� +ε−2 +�� +|ϕ|≤ε +e−2Kηdϕdµ +� 1 +2 +(3.31) +≲ +� +ε−1 +�� +|ϕ|≤ε +e−2Kηdϕdη +� 1 +2 +≲ +� +ε−1ε +� 1 +2 ≲ 1. +Analogously, noting that S32 contains w integral, we have +∥S32∥L2 ≲ +��� ���ε−1Ψχ(ε−1ϕ) +��� +2 +dϕdµ +� 1 +2 +≲ + +ε−2 +�� ����� +� +|ϕ|≤ε +Ψdϕ +����� +2 +dϕdµ + + +1 +2 +(3.32) +≲ + +ε−2 +�� ����� +� +|ϕ|≤ε +e−Kηdϕ +����� +2 +dϕdµ + + +1 +2 +≲ +� +ε−2 +�� +ε2e−2Kηdϕdµ +� 1 +2 +≲ +��� +e−2Kηdϕdµ +� 1 +2 +≲ +� +ε +�� +e−2Kηdϕdη +� 1 +2 +≲ ε +1 +2 . +Collecting (3.31) and (3.32), we have the L2 estimate. Similarly, we derive the L2 +xL1 +w bound: +∥S31∥L2xL1w ≲ +�� � � +|ϕ|≤ε +��ε−1Ψ +�� dϕ +�2 +dµ +� 1 +2 +≲ +�� +e−2Kηdµ +� 1 +2 +≲ +� +ε +� +e−2Kηdη +� 1 +2 +≲ ε +1 +2 , +(3.33) +∥S32∥L2xL1w ≲ +�� � � ���ε−1Ψχ(ε−1ϕ) +��� dϕ +�2 +dµ +� 1 +2 +≲ +� +ε−2 +� � � ����� +� +|ϕ|≤ε +Ψdϕ +����� dϕ +�2 +dµ +� 1 +2 +(3.34) +≲ +� +ε−2 +� � � +εe−Kηdϕ +�2 +dµ +� 1 +2 +≲ +�� +e−2Kηdµ +� 1 +2 +≲ +� +ε +� +e−2Kηdη +� 1 +2 +≲ ε +1 +2 . +□ +4. Remainder Estimate +4.1. Basic Energy Estimate. +Lemma 4.1. Under the assumption (1.14), we have +ε−1 |R|2 +L2γ+ + ε−2 ��R − R +��2 +L2 ≲ o(1)ε−1 ��R +��2 +L2 + 1. +(4.1) + +DIFFUSIVE EXPANSION OF NEUTRON TRANSPORT EQUATION +11 +Proof. Taking g = ε−1R in (3.12), we obtain +ε−1 +2 +� +γ +|R|2 (w · n) + ε−2� +R, R − R +� += ε−1� +R, S +� +. +(4.2) +Then using the orthogonality of R and R − R, we have +ε−1 +2 +|R|2 +L2γ+ + ε−2 ��R − R +��2 +L2 = ε−1� +R, S +� ++ ε−1 +2 +|h|2 +L2γ− . +(4.3) +Using Lemma 3.2, we know +ε−1 |R|2 +L2γ+ + ε−2 ��R − R +��2 +L2 ≲ ε + ε−1� +R, S0 + S1 + S2 + S3 +� +. +(4.4) +Using Lemma 3.3, we have +���ε−1� +R, S0 +���� ≲ ε−1 ∥R∥L2 ∥S0∥L2 ≲ ε ∥R∥L2 ≲ o(1) ∥R∥2 +L2 + ε2. +(4.5) +Using Lemma 3.4, Lemma 3.5 and Lemma 3.6, we have +���ε−1� +R − R, S1 + S2 + S3 +���� ≲ε−1 ��R − R +�� +L2 ∥S1 + S2 + S3∥L2 +(4.6) +≲ε−1 ��R − R +�� +L2 ≲ o(1)ε−2 ��R − R +��2 +L2 + 1. +Finally, we turn to ε−1� +R, S1 + S2 + S3 +� +. For S1, we integrate by parts with respect to ϕ and use Lemma +3.4 to obtain +���ε−1� +R, S1 +���� =ε−1 +���� +� +R, +� sin2 ψ +R1 − εη + cos2 ψ +R2 − εη +� +cos ϕ∂U B +0 +∂ϕ +����� +(4.7) +=ε−1 +���� +� +R, +� sin2 ψ +R1 − εη + cos2 ψ +R2 − εη +� +U B +0 sin ϕ +����� +≲ε−1 ��R +�� +L2 +��U B +1 +�� +L2xL1w ≲ ε− 1 +2 ��R +�� +L2 ≲ o(1)ε−1 ��R +��2 +L2 + 1. +Also, Lemma 3.5 and Lemma 3.6 yield +���ε−1� +R, S2 + S3 +���� ≲ε−1 ��R +�� +L2 +� +∥S2∥L2xL1w + ∥S3∥L2xL1w +� +≲ ε− 1 +2 ��R +�� +L2 ≲ o(1)ε−1 ��R +��2 +L2 + 1. +(4.8) +Collecting (4.5)(4.6)(4.7)(4.8), we obtain +���ε−1� +R, S0 + S1 + S2 + S3 +���� ≲ o(1)ε−2 ��R − R +��2 +L2 + o(1)ε−1 ∥R∥2 +L2 + 1. +(4.9) +Combining (4.9) and (4.4), we have (4.1). +□ +4.2. Kernel Estimate. +Lemma 4.2. Under the assumption (1.14), we have +��R +��2 +L2 ≲ +��R − R +��2 +L2 + |R|2 +L2γ+ + ε. +(4.10) +Proof. Denote ξ(x) satisfying +� +−∆xξ = R in Ω, +ξ(x0) = 0 on ∂Ω. +(4.11) +Based on standard elliptic estimates and trace estimates, we have +∥ξ∥H2 + |ξ|H +3 +2 ≲ +��R +�� +L2 . +(4.12) +Taking g = ξ in (3.12), we have +� +γ +Rξ(w · n) − +� +R, w · ∇xξ +� ++ ε−1� +R − R, ξ +� += +� +S, ξ +� +. +(4.13) +Using oddness, orthogonality and ξ +�� +∂Ω = 0, we obtain (1.24). +Then taking g = w · ∇xξ in (3.12), we obtain (1.25). + +12 +Y. GUO, L. WU +Adding ε−1×(1.24) and (1.25) to eliminate ε−1� +R − R, w · ∇xξ +� +, we obtain +� +γ +R +� +w · ∇xξ +� +(w · n) − +� +R, w · ∇x +� +w · ∇xξ +�� +=ε−1� +S, ξ +� ++ +� +S, w · ∇xξ +� +. +(4.14) +Notice that +− +� +R, w · ∇x +� +w · ∇xξ +�� += − +� +R, w · ∇x +� +w · ∇xξ +�� +− +� +R − R, w · ∇x +� +w · ∇xξ +�� +, +(4.15) +where (4.12) and Cauchy’s inequality yield +− +� +R, w · ∇x +� +w · ∇xξ +�� +≃ +��R +��2 +L2 , +(4.16) +��� +� +R − R, w · ∇x +� +w · ∇xξ +����� ≲ +��R − R +��2 +L2 + o(1) +��R +��2 +L2 . +(4.17) +Also, using (4.12) and Lemma 3.2, we have +���� +� +γ +R +� +w · ∇xξ +� +(w · n) +���� ≲ +� +|R|L2γ+ + |h|L2γ− +� +|∇xξ|L2 ≲ o(1) +��R +��2 +L2 + |R|2 +L2γ+ + ε2. +(4.18) +Inserting (4.15)–(4.18) into (4.14), we obtain +��R +��2 +L2 ≲ε2 + +��R − R +��2 +L2 + |R|2 +L2γ+ + +���ε−1� +S, ξ +���� + +��� +� +S, w · ∇xξ +���� . +(4.19) +Then we turn to the estimate of source terms in (4.19). Cauchy’s inequality and Lemma 3.3 yield +���ε−1� +S0, ξ +���� + +��� +� +S0, w · ∇xξ +���� ≲ ε−1 ∥S0∥L2 ∥ξ∥H1 ≲ ε +��R +�� +L2 ≲ o(1) +��R +��2 +L2 + ε2. +(4.20) +Similar to (4.7), we first integrate by parts with respect to ϕ in S1. Using ξ +�� +∂Ω = 0, (4.12), Hardy’s inequality +and Lemma 3.4, Lemma 3.5, Lemma 3.6, we have +���ε−1� +S1 + S2 + S3, ξ +���� ≲ +����ε−1� +U B +0 + S2 + S3, +� µ +0 +∂ξ +∂µ +����� = +���� +� +ηU B +0 + ηS2 + ηS3, 1 +µ +� µ +0 +∂ξ +∂µ +����� +(4.21) +≲ +��ηU B +0 + ηS2 + ηS3 +�� +L2xL1w +���� +1 +µ +� µ +0 +∂ξ +∂µ +���� +L2 +≲ +��ηU B +0 + ηS2 + ηS3 +�� +L2xL1w +���� +∂ξ +∂µ +���� +L2 +≲ ε +1 +2 ∥ξ∥H1 +≲ε +1 +2 ��R +�� +L2 ≲ o(1) +��R +��2 +L2 + ε. +Analogously, we integrate by parts with respect to ϕ in S1. Then using (4.12), fundamental theorem of +calculus, Hardy’s inequality and Lemma 3.4, Lemma 3.5, Lemma 3.6, we bound +��� +� +S1 + S2 + S3, w · ∇xξ +���� ≲ +����� +� +U B +0 + S2 + S3, ∇xξ +��� +µ=0 + +� µ +0 +∂ +� +∇xξ +� +∂µ +������ +(4.22) +≲ +���� +� +U B +0 + S2 + S3, ∇xξ +��� +µ=0 +����� + +�����ε +� +ηU B +0 + ηS2 + ηS3, 1 +µ +� µ +0 +∂ +� +∇xξ +� +∂µ +������ +≲ +��U B +0 + S2 + S3 +�� +L2xL1w |∇xξ|L2 + ε +��ηU B +0 + ηS2 + ηS3 +�� +L2 +����� +∂ +� +∇xξ +� +∂µ +����� +L2 +≲ε +1 +2 |∇xξ|L2 +∂Ω + ε ∥ξ∥H2 ≲ ε +1 +2 ��R +�� +L2 ≲ o(1) +��R +��2 +L2 + ε. +Hence, inserting (4.20), (4.21) and (4.22) into (4.19), we have shown (4.10). +□ +4.3. Synthesis. +Proposition 4.3. Under the assumption (1.14), we have +ε− 1 +2 |R|L2γ+ + ε− 1 +2 ��R +�� +L2 + ε−1 ��R − R +�� +L2 ≲ 1. +(4.23) + +DIFFUSIVE EXPANSION OF NEUTRON TRANSPORT EQUATION +13 +Proof. From (4.1), we have +ε−1 |R|2 +L2γ+ + ε−2 ��R − R +��2 +L2 ≲ o(1)ε−1 ��R +��2 +L2 + 1. +(4.24) +From (4.10), we have +��R +��2 +L2 ≲ +��R − R +��2 +L2 + |R|2 +L2γ+ + ε. +(4.25) +Inserting (4.25) into (4.24), we have +ε−1 |R|2 +L2γ+ + ε−2 ��R − R +��2 +L2 ≲ 1. +(4.26) +Inserting (4.26) into (4.25), we have +��R +��2 +L2 ≲ ε. +(4.27) +Hence, adding ε−1×(4.27) and (4.26), we have +ε−1 |R|2 +L2γ+ + ε−1 ��R +��2 +L2 + ε−2 ��R − R +��2 +L2 ≲ 1. +(4.28) +Then our result follows. +□ +5. Proof of Main Theorem +The well-posedness of (1.1) is well-known [5, 4, 24]. The construction of U0, Φ and Φ∞ follows from +Proposition 2.1 and Proposition 2.3, so we focus on the derivation of (1.15). +Based on Proposition 4.3 and (1.5), we have +��uε − U0 − εU1 − ε2U2 − U B +0 +�� +L2 ≲ ε +1 +2 . +(5.1) +Using Proposition 2.3, we have +��εU1 + ε2U2 +�� +L2 ≲ ε. +(5.2) +Using Proposition 2.3 and the rescaling η = ε−1µ, we have +��U B +0 +�� +L2 ≲ ε +1 +2 . +(5.3) +Then (1.15) follows from inserting (5.2)(5.3) into (5.1). +References +[1] C. Bardos, F. Golse, and B. Perthame, The Rosseland approximation for the radiative transfer equations, Comm. Pure +Appl. Math., 40 (1987), pp. 69–721. +[2] C. Bardos, F. Golse, B. Perthame, and R. Sentis, The nonaccretive radiative transfer equations: existence of solutions +and Rosseland approximation, J. Funct. Anal., 77 (1988), pp. 434–460. +[3] C. Bardos and K. D. Phung, Observation estimate for kinetic transport equations by diffusion approximation, C. R. +Math. Acad. Sci. Paris, 355 (2017), pp. 640–664. +[4] C. Bardos, R. Santos, and R. Sentis, Diffusion approximation and computation of the critical size, Trans. Amer. Math. +Soc., 284 (1984), pp. 617–649. +[5] A. Bensoussan, J.-L. Lions, and G. C. Papanicolaou, Boundary layers and homogenization of transport processes, +Publ. Res. Inst. Math. Sci., 15 (1979), pp. 53–157. +[6] R. Esposito, Y. Guo, C. Kim, and R. Marra, Non-isothermal boundary in the Boltzmann theory and Fourier law, +Comm. Math. Phys., 323 (2013), pp. 177–239. +[7] Y. Guo and L. Wu, Geometric correction in diffusive limit of neutron transport equation in 2D convex domains, Arch. +Rational Mech. Anal., 226 (2017), pp. 321–403. +[8] +, Regularity of Milne problem with geometric correction in 3D, Math. Models Methods Appl. Sci., 27 (2017), pp. 453– +524. +[9] N. V. Krylov, Lectures on elliptic and parabolic equations in Sobolev spaces., American Mathematical Society, Providence, +RI, 2008. +[10] E. W. Larsen, A functional-analytic approach to the steady, one-speed neutron transport equation with anisotropic scat- +tering, Comm. Pure Appl. Math., 27 (1974), pp. 523–545. +[11] +, Solutions of the steady, one-speed neutron transport equation for small mean free paths, J. Mathematical Phys., 15 +(1974), pp. 299–305. +[12] +, Neutron transport and diffusion in inhomogeneous media I, J. Mathematical Phys., 16 (1975), pp. 1421–1427. +[13] +, Asymptotic theory of the linear transport equation for small mean free paths II, SIAM J. Appl. Math., 33 (1977), +pp. 427–445. + +14 +Y. GUO, L. WU +[14] E. W. Larsen and J. D’Arruda, Asymptotic theory of the linear transport equation for small mean free paths I, Phys. +Rev., 13 (1976), pp. 1933–1939. +[15] E. W. Larsen and G. J. Habetler, A functional-analytic derivation of Case’s full and half-range formulas, Comm. Pure +Appl. Math., 26 (1973), pp. 525–537. +[16] E. W. Larsen and J. B. Keller, Asymptotic solution of neutron transport problems for small mean free paths, J. +Mathematical Phys., 15 (1974), pp. 75–81. +[17] E. W. Larsen and P. F. Zweifel, On the spectrum of the linear transport operator, J. Mathematical Phys., 15 (1974), +pp. 1987–1997. +[18] +, Steady, one-dimensional multigroup neutron transport with anisotropic scattering, J. Mathematical Phys., 17 +(1976), pp. 1812–1820. +[19] Q. Li, J. Lu, and W. Sun, Diffusion approximations and domain decomposition method of linear transport equations: +asymptotics and numerics, J. Comput. Phys., 292 (2015), pp. 141–167. +[20] +, Half-space kinetic equations with general boundary conditions, Math. Comp., 86 (2017), pp. 1269–1301. +[21] +, Validity and regularization of classical half-space equations, J. Stat. Phys., 166 (2017), pp. 398–433. +[22] L. Wu, Boundary layer of transport equation with in-flow boundary, Arch. Rational Mech. Anal., 235 (2020), pp. 2085– +2169. +[23] +, Diffusive limit of transport equation in 3D convex domains, Peking Math. J., 4 (2021), pp. 203–284. +[24] L. Wu and Y. Guo, Geometric correction for diffusive expansion of steady neutron transport equation, Comm. Math. +Phys., 336 (2015), pp. 1473–1553. +[25] L. Wu, X. Yang, and Y. Guo, Asymptotic analysis of transport equation in annulus, J. Stat. Phys., 165 (2016), pp. 585– +644. +(Y. Guo) +Division of Applied Mathematics, Brown University +Email address: yan guo@brown.edu +(L. Wu) +Department of Mathematics, Lehigh University +Email address: lew218@lehigh.edu + diff --git a/G9FLT4oBgHgl3EQfHS_T/content/tmp_files/load_file.txt b/G9FLT4oBgHgl3EQfHS_T/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b5b95044e70a5e5f0b76fbe7b740196cb0114862 --- /dev/null +++ b/G9FLT4oBgHgl3EQfHS_T/content/tmp_files/load_file.txt @@ -0,0 +1,751 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf,len=750 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='11996v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='AP] 27 Jan 2023 L2 DIFFUSIVE EXPANSION FOR NEUTRON TRANSPORT EQUATION YAN GUO AND LEI WU Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Grazing set singularity leads to a surprising counter-example and breakdown [24] of the classical mathematical theory for L∞ diffusive expansion (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='9) of neutron transport equation with in-flow boundary condition in term of the Knudsen number ε, one of the most classical problems in the kinetic theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Even though a satisfactory new theory has been established by constructing new boundary layers with favorable ε-geometric correction for convex domains [24, 7, 8, 22, 23], the severe grazing singularity from non-convex domains has prevented any positive mathematical progress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' We develop a novel and optimal L2 expansion theory for general domain (including non-convex domain) by discovering a surprising ε 1 2 gain for the average of remainder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Contents 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Introduction 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Asymptotic Analysis 5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Remainder Equation 7 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Remainder Estimate 10 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Proof of Main Theorem 13 References 13 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Introduction 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Problem Formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' We consider the steady neutron transport equation in a three-dimensional C3 bounded domain (convex or non-convex) with in-flow boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' In the spatial domain Ω ∋ x = (x1, x2, x3) and the velocity domain S2 ∋ w = (w1, w2, w3), the neutron density uε(x, w) satisfies \uf8f1 \uf8f2 \uf8f3 w · ∇xuε + ε−1� uε − uε � = 0 in Ω × S2, uε(x0, w) = g(x0, w) for w · n < 0 and x0 ∈ ∂Ω, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='1) where g is a given function denoting the in-flow data, uε(x) := 1 4π � S2 uε(x, w)dw, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='2) n is the outward unit normal vector, with the Knudsen number 0 < ε ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' We intend to study the asymptotic behavior of uε as ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Based on the flow direction, we can divide the boundary γ := � (x0, w) : x0 ∈ ∂Ω, w ∈ S2� into the incoming boundary γ−, the outgoing boundary γ+, and the grazing set γ0 based on the sign of w · n(x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' In particular, the boundary condition of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='1) is only given on γ−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Primary 35Q49, 82D75;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Secondary 35Q62, 35Q20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' non-convex domains, transport equation, diffusive limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Guo was supported by NSF Grant DMS-2106650.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Wu was supported by NSF Grant DMS-2104775.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' 1 2 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' GUO, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' WU 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Normal Chart near Boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' We follow the approach in [8, 23] to define the geometric quantities, and the details can be found in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' For smooth manifold ∂Ω, there exists an orthogonal curvilinear coordinates system (ι1, ι2) such that the coordinate lines coincide with the principal directions at any x0 ∈ ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Assume ∂Ω is parameterized by r = r(ι1, ι2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Let the vector length be Li := |∂ιir| and unit vector ςi := L−1 i ∂ιir for i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Consider the corresponding new coordinate system (µ, ι1, ι2), where µ denotes the normal distance to the boundary surface ∂Ω, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' x = r − µn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='3) Define the orthogonal velocity substitution for w := (ϕ, ψ) as −w · n = sin ϕ, w · ς1 = cos ϕ sin ψ, w · ς2 = cos ϕ cos ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='4) Finally, we define the scaled normal variable η = µ ε , which implies ∂ ∂µ = 1 ε ∂ ∂η .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Asymptotic Expansion and Remainder Equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' We seek a solution to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='1) in the form uε =U + U B + R = � U0 + εU1 + ε2U2 � + U B 0 + R, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='5) where the interior solution is U(x, w) := U0(x, w) + εU1(x, w) + ε2U2(x, w), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='6) and the boundary layer is U B(η, ι1, ι2, w) := U B 0 (η, ι1, ι2, w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='7) Here U0, U1, U2 and U B 0 are constructed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='1 and Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='2, and R(x, v) is the remainder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' The study of the neutron transport equation in bounded domains, has attracted a lot of attention since the dawn of the atomic age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Besides its significance in nuclear sciences and medical imaging, neutron transport equation is usually regarded as a linear prototype of the more important yet more complicated nonlinear Boltzmann equation, and thus, is an ideal starting point to develop new theories and techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' We refer to [10, 11, 12, 13, 14, 15, 16, 17, 18] for the formal expansion with respect to ε and explicit solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' The discussion on bounded domain and half-space cases can be found in [5, 4, 3, 1, 2, 19, 20, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' The classical boundary layer of neutron transport equation dictates that U B 0 (η, ι1, ι2, w) satisfies the Milne problem sin ϕ∂U B 0 ∂η + U B 0 − U B 0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='8) From the formal expansion in ε (see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='6)), it is natural to expect the remainder estimate [5] ∥R∥L∞ ≲ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='9) Even though this is valid for domains with flat boundary, a counter-example is constructed [24] so that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='9) is invalid for a 2D disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' This is due to the grazing set singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' To be more specific, in order to show the remainder estimates (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='9), the higher-order boundary layer expansion U B 1 ∈ L∞ is necessary, which further requires ∂ιiU B 0 ∈ L∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Nevertheless, though U B 0 ∈ L∞, it is shown that the normal derivative ∂ηU B 0 is singular at the grazing set ϕ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Furthermore, this singularity ∂ηU B 0 /∈ L∞ will be transferred to ∂ιiU B 0 /∈ L∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' A careful construction of boundary data [24] justifies this invalidity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' both the method and result of the boundary layer (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='8) are problematic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' A new construction of boundary layer [24] based on the ε-Milne problem with geometric correction for � U B 0 (η, ι1, ι2, w) sin ϕ∂� U B 0 ∂η − ε 1 − εη cos ϕ∂� U B 0 ∂ϕ + � U B 0 − � U B 0 = 0 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='10) has been shown to provide the satisfactory characterization of the L∞ diffusive expansion in 2D disk domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' With more detailed regularity analysis and boundary layer decomposition techniques for (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='10), such result has been generalized to 2D/3D smooth convex domains [7, 8, 22, 23] and even 2D annulus domain [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' DIFFUSIVE EXPANSION OF NEUTRON TRANSPORT EQUATION 3 In non-convex domains, the boundary layer with geometric correction is essentially sin ϕ∂� U B 0 ∂η − ε 1 + εη cos ϕ∂� U B 0 ∂ϕ + � U B 0 − � U B 0 = 0 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='11) Compared to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='10), this sign flipping dramatically changes the characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' 0 1 2 3 4 5 6 7 8 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='5 −1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='5 η φ Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Characteristics in Convex Domains 0 1 2 3 4 5 6 7 8 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='5 −1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='5 η φ Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Characteristics in Non- Convex Domains In Figure 1 and Figure 2 [25], the horizontal direction represents the scaled normal variable η and the vertical direction represents the velocity ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' There exists a “hollow” region in Figure 2 that the characteristics may never track back to the left boundary η = 0 and ϕ > 0, making the W 1,∞ estimates impossible and thus preventing higher-order boundary layer expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' In this paper, we will employ a fresh approach to design a cutoff boundary layer without the geometric correction and justify the L2 diffusive expansion in smooth non-convex domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Notation and Convention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Let ⟨ · , · ⟩w denote the inner product for w ∈ S2, ⟨ · , · ⟩x for x ∈ Ω, and ⟨ · , · ⟩ for (x, w) ∈ Ω × S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Also, let ⟨ · , · ⟩γ± denote the inner product on γ± with measure dγ := |w · n| dwdSx = |sin ϕ| cos ϕdwdSx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Denote the bulk and boundary norms ∥f∥L2 := ��� Ω×S2 |f(x, w)|2 dwdx � 1 2 , |f|L2 γ± := �� γ± |f(x, w)|2 dγ � 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='12) Define the L∞ norms ∥f∥L∞ := ess sup (x,w)∈Ω×S2 ��f(x, w) ��, |f|L∞ γ± := ess sup (x,w)∈γ± ��f(x, w) ��.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='13) Let ∥·∥W k,p x denote the usual Sobolev norm for x ∈ Ω and |·|W k,p x for x ∈ ∂Ω, and ∥·∥W k,p x Lq w denote W k,p norm for x ∈ Ω and Lq norm for w ∈ S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' The similar notation also applies when we replace Lq by Lq γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' When there is no possibility of confusion, we will ignore the (x, w) variables in the norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Throughout this paper, C > 0 denotes a constant that only depends on the domain Ω, but does not depend on the data or ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' It is referred as universal and can change from one inequality to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' We write a ≲ b to denote a ≤ Cb and a ≳ b to denote a ≥ Cb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Also, we write a ≃ b if a ≲ b and a ≳ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' We will use o(1) to denote a sufficiently small constant independent of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Main Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Under the assumption |g|W 3,∞L∞ γ− ≲ 1, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='14) there exists a unique solution uε(x, w) ∈ L∞(Ω × S2) to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Moreover, the solution obeys the estimate ∥uε − U0∥L2 ≲ ε 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='15) 4 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' GUO, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' WU Here U0(x) satisfies the Laplace equation with Dirichlet boundary condition � ∆xU0(x) = 0 in Ω, U0(x0) = Φ∞(x0) on ∂Ω, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='16) in which Φ∞(ι1, ι2) = Φ∞(x0) for x0 ∈ ∂Ω is given by solving the Milne problem for Φ(η, ι1, ι2, w) \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 sin ϕ∂Φ ∂η + Φ − Φ = 0, Φ(0, ι1, ι2, w) = g(ι1, ι2, w) for sin ϕ > 0, lim η→∞ Φ(η, ι1, ι2, w) = Φ∞(ι1, ι2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='17) Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' In [24, 22, 23] for 2D/3D convex domains, as well as [25] for 2D annulus domain, it is justified that for any 0 < δ ≪ 1 ���uε − � U0 − � U B 0 ��� L2 ≲ ε 5 6 −δ, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='18) where � U B 0 (η, ι1, ι2, w) is the boundary layer with geometric correction defined in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='10), and � U0 is the cor- responding interior solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' [21, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='1] reveals that the difference between two types of interior solutions ���� U0 − U0 ��� L2 ≲ ε 2 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='19) Due to the rescaling η = ε−1µ, for general in-flow boundary data g, the boundary layer � U B 0 ̸= 0 satisfies ���� U B 0 ��� L2 ≃ ε 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='20) Hence, we conclude that ∥uε − U0∥L2 ≃ ε 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='21) Therefore, this indicates that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='15) in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='1 achieves the optimal L2 bound of the diffusive approxi- mation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' It is well-known that the key of the remainder estimate is to control R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' In a series of work [24, 25, 7, 8, 22, 23] based on a L2 → L∞ framework, it is shown that ��R �� L2 ≲ ε−1 ��R − R �� L2 ≲ 1 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='22) combined from the expected energy (entropy production) bound for ε−1 ��R − R �� L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' This bound requires the next-order ε expansion of boundary layer approximation, which is impossible for non-convex domains, and barely possible by the new boundary layer theory with the ε-geometric correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' The key improvement in our work is ��R �� L2 ≲ ε 1 2 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='23) which is a consequence of the following conservation law for test function ξ(x) satisfying −∆xξ = R and ξ �� ∂Ω = 0: − � R, w · ∇xξ � = − � R − R, w · ∇xξ � = � S, ξ � , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='24) where � R, w ·∇xξ � = 0 thanks to the oddness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' This conservation law exactly cancels the worst contribution of ε−1 ��R − R �� L2 in ��R �� L2 estimate, which comes from taking test function w · ∇xξ � γ R � w · ∇xξ � (w · n) − � R, w · ∇x � w · ∇xξ �� + ε−1� R − R, w · ∇xξ � = � S, w · ∇xξ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='25) Such a key cancellation produces an extra crucial gain of ε 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' We then conclude the remainder estimate without any further expansion of the (singular) boundary layer approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' DIFFUSIVE EXPANSION OF NEUTRON TRANSPORT EQUATION 5 In addition, we construct a new cut-off boundary layer near ϕ = 0 to avoid the singularity, and are able to perform delicate and precise estimates to control the resulting complex forcing term S (see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='7)–(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='10)), in terms of the desired order ε for closure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Asymptotic Analysis 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Interior Solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Inserting (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='6) into (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='1) and comparing the order of ε, following the analysis in [8, 23], we deduce that U0 = U 0, ∆xU 0 = 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='1) U1 = U 1 − w · ∇xU0, ∆xU 1 = 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='2) U2 = U 2 − w · ∇xU1, ∆xU 2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='3) We need the boundary layer to determine the boundary conditions for U0, U 1 and U 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Boundary Layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Geometric Substitutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' The construction of boundary layer requires a local description in a neigh- borhood of the physical boundary ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' We follow the procedure in [8, 23]: Substitution 1: Spacial Substitution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Following the notation in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='2, under the coordinate system (µ, ι1, ι2), we have w · ∇x = −(w · n) ∂ ∂µ − w · ς1 L1(κ1µ − 1) ∂ ∂ι1 − w · ς2 L2(κ2µ − 1) ∂ ∂ι2 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='4) where κi(ι1, ι2) for i = 1, 2 is the principal curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Substitution 2: Velocity Substitution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Under the orthogonal velocity substitution (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='4) for ϕ ∈ � −π 2 , π 2 � and ψ ∈ [−π, π], we have w · ∇x = sin ϕ ∂ ∂µ − � sin2 ψ R1 − µ + cos2 ψ R2 − µ � cos ϕ ∂ ∂ϕ + cos ϕ sin ψ L1(1 − κ1µ) ∂ ∂ι1 + cos ϕ cos ψ L2(1 − κ2µ) ∂ ∂ι2 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='5) + sin ψ R1 − µ �R1 cos ϕ L1L2 � ς1 · � ς2 × � ∂ι1ι2r × ς2 ��� − sin ϕ cos ψ � ∂ ∂ψ − cos ψ R2 − µ �R2 cos ϕ L1L2 � ς2 · � ς1 × � ∂ι1ι2r × ς1 ��� − sin ϕ sin ψ � ∂ ∂ψ, where Ri = κ−1 i represents the radius of curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Note that the Jacobian dw = cos ϕdϕdψ will be present when we perform integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Substitution 3: Scaling Substitution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Considering the scaled normal variable η = ε−1µ, we have w · ∇x =ε−1 sin ϕ ∂ ∂η − � sin2 ψ R1 − εη + cos2 ψ R2 − εη � cos ϕ ∂ ∂ϕ + R1 cos ϕ sin ψ L1(R1 − εη) ∂ ∂ι1 + R2 cos ϕ cos ψ L2(R2 − εη) ∂ ∂ι2 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='6) + sin ψ R1 − εη �R1 cos ϕ L1L2 � ς1 · � ς2 × � ∂ι1ι2r × ς2 ��� − sin ϕ cos ψ � ∂ ∂ψ − cos ψ R2 − εη �R2 cos ϕ L1L2 � ς2 · � ς1 × � ∂ι1ι2r × ς1 ��� − sin ϕ sin ψ � ∂ ∂ψ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Milne Problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Let Φ(η, ι1, ι2, w) be the solution to the Milne problem sin ϕ∂Φ ∂η + Φ − Φ =0, Φ(η, ι1, ι2) = 1 4π � π −π � π 2 − π 2 Φ(η, ι1, ι2, w) cos ϕdϕdψ, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='7) with boundary condition Φ(0, ι1, ι2, w) = g(ι1, ι2, w) for sin ϕ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='8) 6 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' GUO, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' WU We are interested in the solution that satisfies lim η→∞ Φ(η, ι1, ι2, w) = Φ∞(ι1, ι2) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='9) for some Φ∞(ι1, ι2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Based on [8, Section 4], we have the well-posedness and regularity of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Under the assumption (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='14), there exist Φ∞(ι1, ι2) and a unique solution Φ to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='7)such that Ψ := Φ − Φ∞ satisfies \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 sin ϕ∂Ψ ∂η + Ψ − Ψ = 0, Ψ(0, ι1, ι2, w) = g(ι1, ι2, w) − Φ∞(ι1, ι2), lim η→0 Ψ(η, ι1, ι2, w) = 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='10) and for some constant K > 0 and any 0 < r ≤ 3 |Φ∞|W 3,∞ ι1,ι2 + ��eKηΨ �� L∞ ≲1, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='11) ����eKη sin ϕ∂Ψ ∂η ���� L∞ + ����eKη sin ϕ∂Ψ ∂ϕ ���� L∞ + ����eKη ∂Ψ ∂ψ ���� L∞ ≲1, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='12) ����eKη ∂rΨ ∂ιr 1 ���� L∞ + ����eKη ∂rΨ ∂ιr 2 ���� L∞ ≲1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='13) Let χ(y) ∈ C∞(R) and �χ(y) = 1 − χ(y) be smooth cut-off functions satisfying χ(y) = 1 if |y| ≤ 1 and χ(y) = 0 if |y| ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' We define the boundary layer U B 0 (η, ι1, ι2, w) := �χ � ε−1ϕ � χ(εη)Ψ(η, ι1, ι2, w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='14) Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Due to the cutoff in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='14), we have U B 0 (0, ι1, ι2, w) = �χ � ε−1ϕ �� g(ι1, ι2, w) − Φ∞(ι1, ι2) � = �χ � ε−1ϕ � Ψ(0, ι1, ι2, w), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='15) and sin ϕ∂U B 0 ∂η + U B 0 − U B 0 = −�χ � ε−1ϕ � χ(εη)Ψ + Ψ�χ(ε−1ϕ)χ(εη).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='16) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Matching Procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' We plan to enforce the matching condition for x0 ∈ ∂Ω and w · n < 0 U0(x0) + U B 0 (x0, w) =g(x0, w) + O(ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='17) Considering (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='15), it suffices to require U0(x0) = Φ∞(x0) := Φ∞(ι1, ι2), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='18) which yields U0(x0) + Ψ(x0, w) =g(x0, w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='19) Hence, we obtain U0(x0, w) + U B 0 (x0, w) = g(x0, w) + χ � ε−1ϕ � Ψ(0, ι1, ι2, w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='20) Construction of U0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Based on (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='1) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='18), define U0(x) satisfying U0 = U0, ∆xU 0 = 0, U0(x0) = Φ∞(x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='21) From standard elliptic estimates [9] and Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='1, we have for any s ∈ [2, ∞) ∥U0∥W 3+ 1 s ,s + |U0|W 3,s ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='22) Construction of U1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Based on (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='2), define U1(x, w) satisfying U1 = U 1 − w · ∇xU0, ∆xU 1 = 0, U 1(x0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='23) From (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='22), we have for any s ∈ [2, ∞) ∥U1∥W 2+ 1 s ,sL∞ + |U1|W 2,sL∞ ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='24) DIFFUSIVE EXPANSION OF NEUTRON TRANSPORT EQUATION 7 Construction of U2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Based on (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='2), define U2(x, w) satisfying U2 = U 2 − w · ∇xU1, ∆xU 2 = 0, U 2(x0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='25) From (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='24), we have for any s ∈ [2, ∞) ∥U2∥W 1+ 1 s ,sL∞ + |U2|W 1,sL∞ ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='26) Summarizing the above analysis, we have the well-posedness and regularity estimates of the interior solution and boundary layer: Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Under the assumption (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='14), we can construct U0, U1, U2, U B 0 as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='21)(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='23)(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='25)(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='14) satisfying for any s ∈ [2, ∞) ∥U0∥W 3+ 1 s ,s + |U0|W 3,s ≲1, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='27) ∥U1∥W 2+ 1 s ,sL∞ + |U1|W 2,sL∞ ≲1, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='28) ∥U2∥W 1+ 1 s ,sL∞ + |U2|W 1,sL∞ ≲1, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='29) and for some constant K > 0 and any 0 < r ≤ 3 ��eKηU B 0 �� L∞ + ����eKη ∂rU B 0 ∂ιr 1 ���� L∞ + ����eKη ∂rU B 0 ∂ιr 2 ���� L∞ ≲1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='30) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Remainder Equation Denote the approximate solution ua := � U0 + εU1 + ε2U2 � + U B 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='1) Inserting (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='5) into (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='1), we have w · ∇x � ua + R � + ε−1� ua + R � − ε−1� ua + R � = 0, � ua + R ���� γ− = g, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='2) which yields w · ∇xR + ε−1� R − R � = −w · ∇xua − ε−1� ua − ua � , R ��� γ− = � g − ua ���� γ−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='3) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Formulation of Remainder Equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' We consider the remainder equation \uf8f1 \uf8f2 \uf8f3 w · ∇xR + ε−1� R − R � = S in Ω × S2, R(x0, w) = h(x0, w) for w · n < 0 and x0 ∈ ∂Ω, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='4) where R(x) = 1 4π � S2 R(x, w)dw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Here the boundary data h is given by h := −εw · ∇xU0 − ε2w · ∇xU1 − χ � ε−1ϕ � Ψ(0), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='5) and the source term S is given by S := S0 + S1 + S2 + S3, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='6) 8 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' GUO, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' WU where S0 := − ε2w · ∇xU2, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='7) S1 := � sin2 ψ R1 − εη + cos2 ψ R2 − εη � cos ϕ∂U B 0 ∂ϕ , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='8) S2 :=ε−1 sin φ�χ � ε−1ϕ �∂χ(εη) ∂η Ψ + R1 cos ϕ sin ψ L1(R1 − εη) ∂U B 0 ∂ι1 + R2 cos ϕ cos ψ L2(R2 − εη) ∂U B 0 ∂ι2 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='9) + sin ψ R1 − εη �R1 cos ϕ L1L2 � ς1 · � ς2 × � ∂ι1ι2r × ς2 ��� − sin ϕ cos ψ �∂U B 0 ∂ψ − cos ψ R2 − εη �R2 cos ϕ L1L2 � ς2 · � ς1 × � ∂ι1ι2r × ς1 ��� − sin ϕ sin ψ �∂U B 0 ∂ψ , S3 :=ε−1 � �χ � ε−1ϕ � χ(εη)Ψ − Ψ�χ � ε−1ϕ � χ(εη) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='10) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Weak Formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='1 (Green’s Identity, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='2 of [6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Assume f(x, w), g(x, w) ∈ L2(Ω × S2) and w · ∇xf, w · ∇xg ∈ L2(Ω × S2) with f, g ∈ L2 γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Then �� Ω×S2 �� w · ∇xf � g + � w · ∇xg � f � dxdw = � γ fg(w · n) = � γ+ fgdγ − � γ− fgdγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='11) Using Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='1, we can derive the weak formulation of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' For any test function g(x, w) ∈ L2(Ω×S2) with w · ∇xg ∈ L2(Ω × S2) with g ∈ L2 γ, we have � γ Rg(w · n) − �� Ω×S2 R � w · ∇xg � + ε−1 �� Ω×S2 � R − R � g = �� Ω×S2 Sg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='12) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Estimates of Boundary and Source Terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Under the assumption (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='14), for h defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='5), we have |h|L2 γ− ≲ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='13) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Based on Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='3, we have |εw · ∇xU0|L2 γ− + ��ε2w · ∇xU1 �� L2γ− ≲ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='14) Noting the cutoff χ � ε−1ϕ � restricts the support to |ϕ| ≲ ε and dγ measure contributes an extra sin ϕ, we have ��χ � ε−1ϕ � Ψ(0) �� L2γ− ≲ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='15) Hence, our result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Under the assumption (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='14), for S0 defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='7), we have ∥S0∥L2 ≲ ε2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='16) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' This follows from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Under the assumption (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='14), for S1 defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='8), we have ��� 1 + η � S1 �� L2 ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='17) Also, for the boundary layer U B 0 defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='14), we have ��� 1 + η � U B 0 �� L2 ≲ ε 1 2 , ��� 1 + η � U B 0 �� L2xL1w ≲ ε 1 2 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='18) and ��� �� 1 + η � S1, g ���� ≲ ��� � 1 + η � ⟨v⟩2 U B 0 ��� L2 ∥∇wg∥L2 ≲ ε 1 2 ∥∇wg∥L2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='19) DIFFUSIVE EXPANSION OF NEUTRON TRANSPORT EQUATION 9 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' We split S1 = S11 + S12 := � sin2 ψ R1 − εη + cos2 ψ R2 − εη � cos ϕ∂Ψ ∂ϕ �χ � ε−1ϕ � χ(εη) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='20) + � sin2 ψ R1 − εη + cos2 ψ R2 − εη � cos ϕ∂�χ � ε−1ϕ � ∂ϕ χ(εη)Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Note that S11 is nonzero only when |ϕ| ≥ ε and thus based on Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='1, we know ���� ∂Ψ ∂ϕ ���� ≤ |sin ϕ|−1 |Ψ| ≲ ε−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Hence, using dµ = εdη, we have ∥S11∥L2 ≲ ��� |ϕ|≥ε ���� ∂Ψ ∂ϕ ���� 2 dϕdµ � 1 2 ≲ ��� |ϕ|≥ε |sin ϕ|−2 |Ψ|2 dϕdµ � 1 2 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='21) ≲ ��� |ϕ|≥ε |sin ϕ|−2 e−2Kηdϕdµ � 1 2 ≲ � ε �� |ϕ|≥ε |sin ϕ|−2 e−2Kηdϕdη � 1 2 ≲ � εε−1� 1 2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Noticing ∂�χ � ε−1ϕ � ∂ϕ = ε−1�χ′� ε−1ϕ � , and �χ′� ε−1ϕ � is nonzero only when ε < |ϕ| < 2ε, based on Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='1, we have ∥S12∥L2 ≲ε−1 ��� ε<|ϕ|<2ε |Ψ|2 dϕdµ � 1 2 ≲ ε−1 ��� ε<|ϕ|<2ε e−2Kηdϕdµ � 1 2 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='22) ≲ε−1 � ε �� ε<|ϕ|<2ε e−2Kηdϕdη � 1 2 ≲ ε−1 (εε) 1 2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Collecting (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='21) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='22), we have (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Note that e−Kη will suppress the growth from the pre-factor 1 + η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='18) comes from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Then we turn to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' The most difficult term in �� ⟨S1, g⟩ �� is essentially ���� �∂U B 0 ∂ϕ , g �����.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Integration by parts with respect to ϕ implies ���� �∂U B 0 ∂ϕ , g ����� ≲ ���� � U B 0 , ∂g ∂ϕ ����� ≲ ��U B 0 �� L2 ���� ∂g ∂ϕ ���� L2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='23) From (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='4) and ∂x ∂ϕ = 0, we know the substitution (µ, ι1, ι2, w) → (µ, ι1, ι2, w) implies −∂w ∂ϕ · n = cos ϕ, ∂w ∂ϕ · ς1 = − sin ϕ sin ψ, ∂w ∂ϕ · ς2 = − sin ϕ cos ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='24) Hence, we know ���� ∂w ∂ϕ ���� ≲ 1, and thus ���� ∂g ∂ϕ ���� ≲ |∇wg| ���� ∂w ∂ϕ ���� ≲ |∇wg| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='25) Hence, we know that ���� �∂U B 0 ∂ϕ , g ����� ≲ ��U B 0 �� L2 ∥∇wg∥L2 ≲ ε 1 2 ∥∇wg∥L2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='26) □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Under the assumption (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='14), for S2 defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='9), we have ��� 1 + η � S2 �� L2 ≲ ε 1 2 , ��� 1 + η � S2 �� L2xL1w ≲ ε 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='27) 10 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' GUO, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' WU Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Notice that ����ε−1 sin φ�χ � ε−1ϕ �∂χ(εη) ∂η ���� ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Based on Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='1 and Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='3, we directly bound ∥S2∥L2 ≲ ��� � |Φ|2 + ���� ∂Φ ∂ι1 ���� 2 + ���� ∂Φ ∂ι2 ���� 2 + ���� ∂Φ ∂ψ ���� 2 � dϕdµ � 1 2 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='28) ≲ ��� e−2Kηdϕdµ � 1 2 ≲ � ε �� e−2Kηdϕdη � 1 2 ≲ ε 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Then the L2 xL1 w estimate follows from a similar argument noting that there is no rescaling in w variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Under the assumption (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='14), for S3 defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='10), we have ��� 1 + η � S3 �� L2 ≲ 1, ��� 1 + η � S3 �� L2xL1w ≲ ε 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='29) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Using χ = 1 − �χ, we split S3 = S31 + S32 :=ε−1Ψχ � ε−1ϕ � χ(εη) − ε−1χ � ε−1ϕ � χ(εη)Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='30) Noting that S31 is nonzero only when |ϕ| ≤ ε, based on Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='1, we have ∥S31∥L2 ≲ ��� |ϕ|≤ε ��ε−1Ψ ��2 dϕdµ � 1 2 ≲ � ε−2 �� |ϕ|≤ε e−2Kηdϕdµ � 1 2 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='31) ≲ � ε−1 �� |ϕ|≤ε e−2Kηdϕdη � 1 2 ≲ � ε−1ε � 1 2 ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Analogously, noting that S32 contains w integral, we have ∥S32∥L2 ≲ ��� ���ε−1Ψχ(ε−1ϕ) ��� 2 dϕdµ � 1 2 ≲ \uf8eb \uf8edε−2 �� ����� � |ϕ|≤ε Ψdϕ ����� 2 dϕdµ \uf8f6 \uf8f8 1 2 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='32) ≲ \uf8eb \uf8edε−2 �� ����� � |ϕ|≤ε e−Kηdϕ ����� 2 dϕdµ \uf8f6 \uf8f8 1 2 ≲ � ε−2 �� ε2e−2Kηdϕdµ � 1 2 ≲ ��� e−2Kηdϕdµ � 1 2 ≲ � ε �� e−2Kηdϕdη � 1 2 ≲ ε 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Collecting (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='31) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='32), we have the L2 estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Similarly, we derive the L2 xL1 w bound: ∥S31∥L2xL1w ≲ �� � � |ϕ|≤ε ��ε−1Ψ �� dϕ �2 dµ � 1 2 ≲ �� e−2Kηdµ � 1 2 ≲ � ε � e−2Kηdη � 1 2 ≲ ε 1 2 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='33) ∥S32∥L2xL1w ≲ �� � � ���ε−1Ψχ(ε−1ϕ) ��� dϕ �2 dµ � 1 2 ≲ � ε−2 � � � ����� � |ϕ|≤ε Ψdϕ ����� dϕ �2 dµ � 1 2 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='34) ≲ � ε−2 � � � εe−Kηdϕ �2 dµ � 1 2 ≲ �� e−2Kηdµ � 1 2 ≲ � ε � e−2Kηdη � 1 2 ≲ ε 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Remainder Estimate 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Basic Energy Estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Under the assumption (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='14), we have ε−1 |R|2 L2γ+ + ε−2 ��R − R ��2 L2 ≲ o(1)ε−1 ��R ��2 L2 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='1) DIFFUSIVE EXPANSION OF NEUTRON TRANSPORT EQUATION 11 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Taking g = ε−1R in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='12), we obtain ε−1 2 � γ |R|2 (w · n) + ε−2� R, R − R � = ε−1� R, S � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='2) Then using the orthogonality of R and R − R, we have ε−1 2 |R|2 L2γ+ + ε−2 ��R − R ��2 L2 = ε−1� R, S � + ε−1 2 |h|2 L2γ− .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='3) Using Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='2, we know ε−1 |R|2 L2γ+ + ε−2 ��R − R ��2 L2 ≲ ε + ε−1� R, S0 + S1 + S2 + S3 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='4) Using Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='3, we have ���ε−1� R, S0 ���� ≲ ε−1 ∥R∥L2 ∥S0∥L2 ≲ ε ∥R∥L2 ≲ o(1) ∥R∥2 L2 + ε2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='5) Using Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='4, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='5 and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='6, we have ���ε−1� R − R, S1 + S2 + S3 ���� ≲ε−1 ��R − R �� L2 ∥S1 + S2 + S3∥L2 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='6) ≲ε−1 ��R − R �� L2 ≲ o(1)ε−2 ��R − R ��2 L2 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Finally, we turn to ε−1� R, S1 + S2 + S3 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' For S1, we integrate by parts with respect to ϕ and use Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='4 to obtain ���ε−1� R, S1 ���� =ε−1 ���� � R, � sin2 ψ R1 − εη + cos2 ψ R2 − εη � cos ϕ∂U B 0 ∂ϕ ����� (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='7) =ε−1 ���� � R, � sin2 ψ R1 − εη + cos2 ψ R2 − εη � U B 0 sin ϕ ����� ≲ε−1 ��R �� L2 ��U B 1 �� L2xL1w ≲ ε− 1 2 ��R �� L2 ≲ o(1)ε−1 ��R ��2 L2 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Also, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='5 and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='6 yield ���ε−1� R, S2 + S3 ���� ≲ε−1 ��R �� L2 � ∥S2∥L2xL1w + ∥S3∥L2xL1w � ≲ ε− 1 2 ��R �� L2 ≲ o(1)ε−1 ��R ��2 L2 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='8) Collecting (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='5)(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='6)(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='7)(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='8), we obtain ���ε−1� R, S0 + S1 + S2 + S3 ���� ≲ o(1)ε−2 ��R − R ��2 L2 + o(1)ε−1 ∥R∥2 L2 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='9) Combining (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='9) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='4), we have (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Kernel Estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Under the assumption (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='14), we have ��R ��2 L2 ≲ ��R − R ��2 L2 + |R|2 L2γ+ + ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='10) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Denote ξ(x) satisfying � −∆xξ = R in Ω, ξ(x0) = 0 on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='11) Based on standard elliptic estimates and trace estimates, we have ∥ξ∥H2 + |ξ|H 3 2 ≲ ��R �� L2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='12) Taking g = ξ in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='12), we have � γ Rξ(w · n) − � R, w · ∇xξ � + ε−1� R − R, ξ � = � S, ξ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='13) Using oddness, orthogonality and ξ �� ∂Ω = 0, we obtain (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Then taking g = w · ∇xξ in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='12), we obtain (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' 12 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' GUO, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' WU Adding ε−1×(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='24) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='25) to eliminate ε−1� R − R, w · ∇xξ � , we obtain � γ R � w · ∇xξ � (w · n) − � R, w · ∇x � w · ∇xξ �� =ε−1� S, ξ � + � S, w · ∇xξ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='14) Notice that − � R, w · ∇x � w · ∇xξ �� = − � R, w · ∇x � w · ∇xξ �� − � R − R, w · ∇x � w · ∇xξ �� , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='15) where (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='12) and Cauchy’s inequality yield − � R, w · ∇x � w · ∇xξ �� ≃ ��R ��2 L2 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='16) ��� � R − R, w · ∇x � w · ∇xξ ����� ≲ ��R − R ��2 L2 + o(1) ��R ��2 L2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='17) Also, using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='12) and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='2, we have ���� � γ R � w · ∇xξ � (w · n) ���� ≲ � |R|L2γ+ + |h|L2γ− � |∇xξ|L2 ≲ o(1) ��R ��2 L2 + |R|2 L2γ+ + ε2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='18) Inserting (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='15)–(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='18) into (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='14), we obtain ��R ��2 L2 ≲ε2 + ��R − R ��2 L2 + |R|2 L2γ+ + ���ε−1� S, ξ ���� + ��� � S, w · ∇xξ ���� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='19) Then we turn to the estimate of source terms in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Cauchy’s inequality and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='3 yield ���ε−1� S0, ξ ���� + ��� � S0, w · ∇xξ ���� ≲ ε−1 ∥S0∥L2 ∥ξ∥H1 ≲ ε ��R �� L2 ≲ o(1) ��R ��2 L2 + ε2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='20) Similar to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='7), we first integrate by parts with respect to ϕ in S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Using ξ �� ∂Ω = 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='12), Hardy’s inequality and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='4, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='5, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='6, we have ���ε−1� S1 + S2 + S3, ξ ���� ≲ ����ε−1� U B 0 + S2 + S3, � µ 0 ∂ξ ∂µ ����� = ���� � ηU B 0 + ηS2 + ηS3, 1 µ � µ 0 ∂ξ ∂µ ����� (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='21) ≲ ��ηU B 0 + ηS2 + ηS3 �� L2xL1w ���� 1 µ � µ 0 ∂ξ ∂µ ���� L2 ≲ ��ηU B 0 + ηS2 + ηS3 �� L2xL1w ���� ∂ξ ∂µ ���� L2 ≲ ε 1 2 ∥ξ∥H1 ≲ε 1 2 ��R �� L2 ≲ o(1) ��R ��2 L2 + ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Analogously, we integrate by parts with respect to ϕ in S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Then using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='12), fundamental theorem of calculus, Hardy’s inequality and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='4, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='5, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='6, we bound ��� � S1 + S2 + S3, w · ∇xξ ���� ≲ ����� � U B 0 + S2 + S3, ∇xξ ��� µ=0 + � µ 0 ∂ � ∇xξ � ∂µ ������ (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='22) ≲ ���� � U B 0 + S2 + S3, ∇xξ ��� µ=0 ����� + �����ε � ηU B 0 + ηS2 + ηS3, 1 µ � µ 0 ∂ � ∇xξ � ∂µ ������ ≲ ��U B 0 + S2 + S3 �� L2xL1w |∇xξ|L2 + ε ��ηU B 0 + ηS2 + ηS3 �� L2 ����� ∂ � ∇xξ � ∂µ ����� L2 ≲ε 1 2 |∇xξ|L2 ∂Ω + ε ∥ξ∥H2 ≲ ε 1 2 ��R �� L2 ≲ o(1) ��R ��2 L2 + ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Hence, inserting (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='20), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='21) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='22) into (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='19), we have shown (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Under the assumption (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='14), we have ε− 1 2 |R|L2γ+ + ε− 1 2 ��R �� L2 + ε−1 ��R − R �� L2 ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='23) DIFFUSIVE EXPANSION OF NEUTRON TRANSPORT EQUATION 13 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' From (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='1), we have ε−1 |R|2 L2γ+ + ε−2 ��R − R ��2 L2 ≲ o(1)ε−1 ��R ��2 L2 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='24) From (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='10), we have ��R ��2 L2 ≲ ��R − R ��2 L2 + |R|2 L2γ+ + ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='25) Inserting (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='25) into (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='24), we have ε−1 |R|2 L2γ+ + ε−2 ��R − R ��2 L2 ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='26) Inserting (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='26) into (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='25), we have ��R ��2 L2 ≲ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='27) Hence, adding ε−1×(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='27) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='26), we have ε−1 |R|2 L2γ+ + ε−1 ��R ��2 L2 + ε−2 ��R − R ��2 L2 ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='28) Then our result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Proof of Main Theorem The well-posedness of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='1) is well-known [5, 4, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' The construction of U0, Φ and Φ∞ follows from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='1 and Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='3, so we focus on the derivation of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Based on Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='3 and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='5), we have ��uε − U0 − εU1 − ε2U2 − U B 0 �� L2 ≲ ε 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='1) Using Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='3, we have ��εU1 + ε2U2 �� L2 ≲ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='2) Using Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='3 and the rescaling η = ε−1µ, we have ��U B 0 �� L2 ≲ ε 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='3) Then (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='15) follows from inserting (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='2)(5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='3) into (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' References [1] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Bardos, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Golse, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Perthame, The Rosseland approximation for the radiative transfer equations, Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Pure Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=', 40 (1987), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' 69–721.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' [2] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Bardos, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Golse, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Perthame, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Sentis, The nonaccretive radiative transfer equations: existence of solutions and Rosseland approximation, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Funct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=', 77 (1988), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' 434–460.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' [3] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Bardos and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Phung, Observation estimate for kinetic transport equations by diffusion approximation, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Paris, 355 (2017), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' 640–664.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' [4] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Bardos, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Santos, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Sentis, Diffusion approximation and computation of the critical size, Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=', 284 (1984), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' 617–649.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' [5] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Bensoussan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Lions, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Papanicolaou, Boundary layers and homogenization of transport processes, Publ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=', 15 (1979), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' 53–157.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' [6] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Esposito, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Guo, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Kim, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Marra, Non-isothermal boundary in the Boltzmann theory and Fourier law, Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=', 323 (2013), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' 177–239.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' [7] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Guo and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Wu, Geometric correction in diffusive limit of neutron transport equation in 2D convex domains, Arch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Rational Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=', 226 (2017), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' 321–403.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' [8] , Regularity of Milne problem with geometric correction in 3D, Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Models Methods Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=', 27 (2017), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' 453– 524.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' [9] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Krylov, Lectures on elliptic and parabolic equations in Sobolev spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=', American Mathematical Society, Providence, RI, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' [10] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Larsen, A functional-analytic approach to the steady, one-speed neutron transport equation with anisotropic scat- tering, Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Pure Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=', 27 (1974), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' 523–545.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' [11] , Solutions of the steady, one-speed neutron transport equation for small mean free paths, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Mathematical Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=', 15 (1974), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' 299–305.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' [12] , Neutron transport and diffusion in inhomogeneous media I, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Mathematical Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=', 16 (1975), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' 1421–1427.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' [13] , Asymptotic theory of the linear transport equation for small mean free paths II, SIAM J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=', 33 (1977), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' 427–445.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' 14 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' GUO, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' WU [14] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Larsen and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' D’Arruda, Asymptotic theory of the linear transport equation for small mean free paths I, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=', 13 (1976), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' 1933–1939.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' [15] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Larsen and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Habetler, A functional-analytic derivation of Case’s full and half-range formulas, Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Pure Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=', 26 (1973), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' 525–537.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' [16] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Larsen and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Keller, Asymptotic solution of neutron transport problems for small mean free paths, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Mathematical Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=', 15 (1974), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' 75–81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' [17] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Larsen and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Zweifel, On the spectrum of the linear transport operator, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Mathematical Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=', 15 (1974), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' 1987–1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' [18] , Steady, one-dimensional multigroup neutron transport with anisotropic scattering, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Mathematical Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=', 17 (1976), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' 1812–1820.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' [19] Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Li, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Lu, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Sun, Diffusion approximations and domain decomposition method of linear transport equations: asymptotics and numerics, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=', 292 (2015), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' 141–167.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' [20] , Half-space kinetic equations with general boundary conditions, Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=', 86 (2017), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' 1269–1301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' [21] , Validity and regularization of classical half-space equations, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=', 166 (2017), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' 398–433.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' [22] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Wu, Boundary layer of transport equation with in-flow boundary, Arch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Rational Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=', 235 (2020), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' 2085– 2169.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' [23] , Diffusive limit of transport equation in 3D convex domains, Peking Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=', 4 (2021), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' 203–284.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' [24] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Wu and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Guo, Geometric correction for diffusive expansion of steady neutron transport equation, Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=', 336 (2015), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' 1473–1553.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' [25] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Wu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Yang, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Guo, Asymptotic analysis of transport equation in annulus, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=', 165 (2016), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' 585– 644.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' (Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Guo) Division of Applied Mathematics, Brown University Email address: yan guo@brown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='edu (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content=' Wu) Department of Mathematics, Lehigh University Email address: lew218@lehigh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} +page_content='edu' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9FLT4oBgHgl3EQfHS_T/content/2301.11996v1.pdf'} diff --git a/GtFKT4oBgHgl3EQfcC5D/content/tmp_files/2301.11814v1.pdf.txt b/GtFKT4oBgHgl3EQfcC5D/content/tmp_files/2301.11814v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..a46ce7223671241adb6f3e129802cff5373579dd --- /dev/null +++ b/GtFKT4oBgHgl3EQfcC5D/content/tmp_files/2301.11814v1.pdf.txt @@ -0,0 +1,5902 @@ +Neutrinos from dense: flavor mechanisms, theoretical approaches, +observations, new directions +M. Cristina Volpe +CNRS, Universit´e Paris Cit´e, +Astroparticule et Cosmologie, F-75013 Paris, +France∗ +(Dated: January 30, 2023) +Neutrino masses and mixings produce vacuum oscillations, an established quantum me- +chanical phenomenon. In matter, the Mikheev-Smirnov-Wolfenstein effect, due to neu- +trino interactions with the background particles, triggers resonant flavor modification. +In dense environments, sizable neutrino-neutrino interactions, shock waves and turbu- +lence impact the neutrino flavor content under a variety of phenomena. +Theoretical +approaches of neutrino propagation range from the mean-field approximation to the +full quantum kinetic equations. Intriguing connections have been uncovered between +weakly interacting dense neutrino gases and other many-body systems and domains, +from condensed matter and nuclear physics to quantum computing. Besides the intrin- +sic theoretical interest, establishing how neutrinos change flavor contributes to answer +the longstanding open questions of how massive stars explode and of the r-process sites. +It is also important for future observations of core-collapse supernova neutrinos and of +the diffuse supernova neutrino background that should be discovered in the foreseeable +future. +CONTENTS +I. General and historical +1 +A. The birth of neutrino astronomy +1 +B. The oscillation discovery +2 +C. Unknown neutrino properties +3 +D. Future supernova neutrino observations +4 +E. The r-process and GW170817 +5 +F. Theoretical developments +6 +II. Neutrino flavor mechanisms in environments +8 +A. Mean-field equations +8 +B. The MSW effect +10 +C. The MSW effect in dense media +12 +D. Shock wave effects +13 +E. Turbulence effects +14 +F. MSW-like mechanisms +14 +1. Matter-neutrino resonance +15 +2. Spin or helicity coherence +16 +3. I-resonance +16 +G. Neutrino-neutrino interactions +17 +1. Slow modes +17 +2. Fast modes +19 +III. Flavor evolution: Theoretical frameworks +20 +A. The mean-field approximation +20 +B. The mean-field Hamiltonian: a derivation +21 +C. Beyond the usual mean-field +22 +D. Linearization +23 +1. The linearised equations +23 +2. A dispersion-relation approach +25 +E. Towards neutrino quantum kinetic equations +26 +F. Neutrinos in presence of strong gravitational fields +27 +G. Connections: from atomic nuclei to quantum devices +28 +IV. Past and future observations +29 +A. SN1987A +29 +B. From the next supernova +31 +∗ Electronic address: volpe@apc.in2p3.fr +C. The discovery of the diffuse supernova neutrino +background +33 +V. Conclusions and perspectives +35 +VI. Acknowledgments +37 +References +37 +I. GENERAL AND HISTORICAL +A. The birth of neutrino astronomy +In his famous letter to Lisa Meitner, Pauli (1930) hy- +pothesized the existence of a new fermion, the neutron. +He wanted to explain the observed continuous beta spec- +trum in the β-decay of atomic nuclei and to save the +laws of energy conservation and statistics. This particle +had to be as light as the electron with a mass not heav- +ier than 0.01 the one of the proton. Renamed neutrino +(”small neutral particle” in Italian), it remained elusive +until Cowan et al. (1956) detected electron anti-neutrinos +via inverse β-decay nearby reactors, the most powerful +man-made neutrino sources in terrestrial experiments. +The same year Lee and Yang (1956) examined the +question of parity conservation in weak interactions, +stimulated by the so-called θ-τ meson puzzle. +They +suggested, as a possible experimental test of the par- +ity non-conservation hypothesis, the measurement of a +pseudo-scalar observable, namely the angular distribu- +tion of electrons emitted in polarized 60Co decay. In a few +months Wu et al. (1957) successfully performed the ex- +periment, demonstrating weak interaction differentiates +the right from the left. In 1958 Goldhaber et al. (1958) +measured neutrinos from electron capture in 152Eu and +arXiv:2301.11814v1 [hep-ph] 27 Jan 2023 + +2 +found them to be left-handed. In the Glashow (1961), +Weinberg (1967), Salam (1957) (GWS) model, neutrinos +are described by Weyl spinors. +In his seminal work Bethe (1939) suggested that car- +bon and nitrogen act as catalysts in a chain reaction that +burns hydrogen into helium in luminous main sequence +stars (later known as the CNO cycle). Afterwards, solar +models predicted sizable νe fluxes from energy generation +due to hydrogen burning into helium in the proton-proton +(pp) reaction chain (Bahcall, 1964) and the CNO cycle +(Bahcall et al., 1968). It was Davis et al. (1968) who first +detected solar neutrinos with his pioneering radiochem- +ical experiment in the Homestake mine, using neutrino +capture on 37Cl (Davis, 1964). In a few months the mea- +surement revealed less neutrinos than expected (Bahcall +et al., 1968): the solar neutrino problem was born. +For over more than three decades, radiochemical, wa- +ter Cherenkov and scintillator experiments showed that, +depending on neutrino energy, one-third to one-half of +the predicted solar neutrino fluxes were actually reach- +ing the Earth (see for example (Giunti and Kim, 2007; +Haxton et al., 2013; Raffelt, 1996)). Both the Standard +Solar Model and neutrino properties were questioned. +Helioseismology +(Turck-Chieze +and +Lopes, +1993) +brought an important clue in favor of the Standard Solar +Model. In particular, the solar sound speed, measured +at a few % level, was agreeing with predictions. Among +the debated solutions was the possibility that neutrinos +could oscillate. Earlier Pontecorvo (1957, 1958) had sug- +gested that ν could transform into ¯ν, in analogy with +oscillations of neutral K0- ¯K0 mesons. +Wolfenstein (1978, 1979) pointed out that in matter +neutrinos can change flavor due to coherent forward scat- +tering and a flavor-dependent refraction index. Later on, +Mikheev and Smirnov (1986) realized that flavor conver- +sion in matter could be resonantly amplified: an adi- +abatic evolution at the resonance location could solve +the solar neutrino problem (Bethe, 1986; Bouchez et al., +1986; Haxton, 1986; Mikheev and Smirnov, 1986; Parke, +1986). +In 1987, the explosion of the blue supergiant Sk-69◦202 +brought evidence that core-collapse supernovae1 emit +neutrinos at the end of their life (Fig. +1). +SN1987A +was in the Large Magellanic Cloud (LMC), a satellite +galaxy of the Milky Way. Kamiokande-II (KII) (Hirata +et al., 1987), Irvine-Michigan-Brookhaven (IMB) (Bionta +et al., 1987) detectors and the Baksan Scintillator Tele- +scope (BST) (Alekseev et al., 1988) recorded a 10 seconds +burst of 24 events, with a few tens of MeV energy. Five +hours before, the Mont Blanc Liquid Scintillator Detec- +tor (LSD) (Aglietta et al., 1987) detected 5 events, which +1 SN II and Ib/c are massive stars that undergo gravitational core- +collapse. O-N-Mg supernovae have 8M⊙ < M < 10M⊙; whereas +iron core-collapse supernovae have M > 10M⊙. +SN type II, +unlike type Ib and Ic, have H lines. +FIG. 1 Hubble Space Telescope image of SN1987A, in the +Large Magellanic Cloud a neighboring galaxy of our Milky +way, 30 years after its explosion. +(Figure adapted from +(ESA/Hubble and NASA, 2011)). +remain controversial. +The SN1987A events confirmed that neutrinos take +away most of the gravitational energy, as Colgate and +White (1966) conjectured, and agreed overall with the +predicted neutrino fluxes and spectra. +Moreover, the +Bayesian analysis of the SN1987A time signal by Loredo +and Lamb (2002) corroborated a two-component (ac- +cretion+cooling) model at 2-3σ, which was confirmed +by the subsequent analysis by Pagliaroli et al. (2009b). +This supported the delayed neutrino-heating mechanism +of Bethe and Wilson (1985), thus rejecting the favored +prompt bounce-shock model by Colgate and White (1966). +On the particle physics side, the two dozens events +brought an impressive amount of constraints on unknown +neutrino properties (e.g. the neutrino magnetic moment, +charge radius or decay), on non-standard interactions +and particles such as axions (see for example (Raffelt, +1996; Zyla et al., 2020)). +The observation of neutrinos from the Sun and from +SN1987A pioneered neutrino astronomy2. The detection +of PeV neutrinos in the IceCube detector at the South +Pole (Aartsen et al., 2014) opened a new observational +window. One of the events detected so far is consistent +with blazar TXS 0506+056 (Aartsen et al., 2018). With +these observations, neutrino astronomy now covers from +MeVs to the highest neutrino energies. +B. The oscillation discovery +Primary cosmic rays interacting with the Earth’s at- +mosphere produce twice as many νµ as νe from π and +µ decay. Underground experiments searching for proton +instability, which was expected in some unified theories, +2 R. Davis and M. Koshiba (Kamiokande) were the recipients of +the 2002 Nobel Prize with R. Giacconi (X-ray astronomy). + +3 +reported a reduced νµ/νe ratio in the atmospheric back- +ground, with respect to Monte-Carlo simulations. This +was known as the atmospheric anomaly (see for example +Giunti and Kim, 2007). +In 1998 the Super-Kamiokande (Super-K) Collabora- +tion (Fukuda et al., 1998) discovered3 that atmospheric +νµ traversing the Earth (up-going) were less than ex- +pected, whereas up-going νe stayed unaffected. +The +zenith angle dependence of the µ-like and e-like events +gave unambiguous proof that νµ oscillated into ντ. +The Sudbury Neutrino Observatory (SNO) (Ahmad +et al., 2001a) and the Kamioka Liquid Scintillator An- +tineutrino Detector (KamLAND) (Eguchi et al., 2003) +experiments brought two further milestones in the clar- +ification of the solar neutrino problem. The first exper- +iment (Ahmad et al., 2001b), using heavy water, found +8B solar neutrinos to be in agreement with the Standard +Solar Model predictions. The different sensitivity of νe +and νµ, ντ to elastic scattering, combined with neutral- +and charged-current ν interactions on deuterium allowed +to identify the solar νµ, ντ fluxes at 5.3 σ (Ahmad et al., +2002). Moreover KamLAND measured ¯νe disappearance +at an average distance of 200 km from Japanese reac- +tors and unambiguously identified the large mixing angle +MSW solution. +These observations established that only half of low +energy (less than 2 MeV) solar νe reach the Earth because +of averaged vacuum oscillations; whereas high energy 8B +neutrinos are reduced to one-third due to the MSW effect. +The solar neutrino problem was finally solved. +The occurrence of vacuum oscillations implies that +neutrinos are elementary particles with non-zero masses +and mixings. Hence, the flavor and mass bases are re- +lated by the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) +unitary matrix4, analogous to the Cabibbo-Kobayashi- +Maskawa matrix in the quark sector (although with large +mixing angles). +Since 1998, atmospheric, solar, reactor and accelera- +tor experiments have determined most of the neutrino +oscillation parameters. +The mixing angles, θ23 ≈ 45◦, +θ12 ≈ 35◦ and θ13 ≈ 8.2◦, as the mass-squared differ- +ences ∆m2 +32 = m2 +3 − m2 +2 ≈ 8 × 10−5 eV2 (atmospheric) +and ∆m2 +21 = m2 +2 − m2 +1 ≈ 2 × 10−3 eV2 (solar) (Zyla +et al., 2020) are known with good accuracy (few percent +precision). +The discovery of neutrino vacuum oscillations was a +breakthrough: it opened a window beyond the Standard +model and had an impact in astrophysics and cosmology. +3 T. Kajita (Super-K) and A.B. McDonald (SNO) were recipients +of the Nobel Prize in 2015. +4 Note that, for an arbitrary number N of neutrino families, the +PMNS matrix depends on N(N − 1)/2 angles and N(N + 1)/2 +phases. For Majorana neutrinos, only N −1 phases remain, since +some phases can be reabsorbed by a redefinition of the charged +lepton fields. For Dirac neutrinos (N −1)(N −2)/2 are left, since +charged and neutrino lepton fields can be redefined. +C. Unknown neutrino properties +Key neutrino properties remain unknown and will be +the object of an intense experimental program. +Sixty +years after Christenson et al. (1964) discovered that weak +interaction breaks the CP symmetry in K0 decay, there +are indications that neutrinos do not oscillate as antineu- +trinos. Leptonic CP violation points to CP breaking val- +ues of the Dirac phase (see for example (Capozzi et al., +2021) for an analysis). +The ordering of the neutrino mass eigenstates needs +to be established, since the atmospheric mass-squared +difference sign has not been measured yet. The neutrino +mass ordering (or hierarchy) might be normal (∆m2 +32 > +0), or inverted (∆m2 +32 < 0). On the other hand, the sign +of the solar mass-squared difference is determined by the +presence of the MSW resonance in the Sun. Currently +data show a preference (at 2.5 σ) for normal ordering, +i.e. the third mass eigenstate is likely more massive than +the others (Capozzi et al., 2021). +The absolute neutrino mass scale is not identified +yet. The KATRIN experiment has obtained sub-eV up- +per limits (m < 0.8 eV at 90 % confidence level) on +the effective νe mass with tritium β-decay (Aker et al., +2022). Complementary information comes from cosmo- +logical observations which give (model dependent) infor- +mation on the sum of the neutrino masses (at sub-eV +level) (Zyla et al., 2020). +The origin of the neutrino masses remains an open is- +sue. See-saw mechanisms constitute a possibility to ex- +plain the smallness of neutrino masses and are investi- +gated in numerous theories beyond the Standard model +(see for example the reviews (Altarelli and Feruglio, 2010; +King, 2015)). Moreover, as Majorana pointed out long +ago (Majorana, 1937), neutrinos could well be their own +antiparticles. Searches for total lepton-number violating +0ν2β decay (Agostini et al., 2022; Giunti and Kim, 2007) +appear the most feasible path to uncover the neutrino na- +ture and give access to the Majorana CP violating phases. +As for neutrino electromagnetic properties, such as the +neutrino magnetic moment, only bounds exist (see for +example (Giunti et al., 2016)). Finally, the existence of +sterile neutrinos is actively debated. Sterile eV neutrinos +are discussed as solution5 of the reactor (Mention et al., +2011), the Gallium (Giunti and Laveder, 2011) and the +MiniBooNE anomalies (Aguilar-Arevalo et al., 2018) that +cannot be cast in the 3ν theoretical framework. On gen- +eral theoretical grounds, (heavy) sterile neutrinos could +5 Recent evaluations of the reactor neutrino fluxes have lowered +the statistical significance of the reactor anomaly (Giunti et al., +2022), whereas the Gallium one, confirmed by the counting +BEST experiment (Barinov et al., 2022), gives sterile mixing pa- +rameters in tension with the reactor anomaly. +Moreover, the +first results from the MicroBooNE experiment (Arg¨uelles et al., +2022) disfavor some explanations and part of the parameter space +identified by the MiniBooNE low energy excess. + +4 +explain the smallness of neutrino masses in see-saw mod- +els. +Neutrino properties are intertwined with neutrino fla- +vor evolution in dense sources and influence observations. +Therefore, as we shall discuss, neutrino signals from such +environments constitute a unique probe of astrophysical +media, or in cosmology, and to learn about unknown neu- +trino properties. +D. Future supernova neutrino observations +SN1987A remains to date the only core-collapse super- +nova observed through its neutrinos. Supernovae6 type +II and Ib/c are exciting and a rich laboratory for par- +ticle physics and astrophysics, requiring both multipur- +pose and dedicated neutrino observatories that can run +over long time periods. A network of neutrino detectors +around the world is awaiting for the next (extra)galactic +supernova explosion. Their majority is included in the +Supernova Early Warning System (SNEWS) (Al Kharusi +et al., 2021; Scholberg, 2000) which should alert as- +tronomers if such a lucky event takes place. +In the Milky Way the spatial probability distribution +of objects that are likely to become supernovae has its +maximum at the galaxy center at 8 kpc while its mean +is at 10 kpc. The latter is mostly adopted to make pre- +dictions. +Neutrino observatories will measure the time signal +and the spectra of νe, ¯νe, νx, ¯νx (x = µ, τ) with charged- +current νe scattering on nuclei, inverse β-decay, elastic +scattering on electrons and on protons (Beacom et al., +2002). If a supernova explodes in our galaxy (10 kpc), +detectors will measure7 about 40 (540) events in HALO +(HALO-2, (Vaananen and Volpe, 2011)), hundreds in +KamLAND, 300 in LVD (Vigorito et al., 2021) up to +3×103 in DUNE (40 ktons) (Abi et al., 2021), up to about +8 × 103 in JUNO (An et al., 2016), almost 104 in Super- +K (Beacom and Vogel, 1998), 105 in Hyper-Kamiokande +(Hyper-K, 248 ktons) (Abe et al., 2018) and 106 in Ice- +Cube8. From a supernova in Andromeda galaxy (M31, +773 kpc) which has a low supernova rate, 12 events are +expected in Hyper-K. +Supernovae are rare in our galaxy. +Typical quoted +number for the core-collapse supernova rate in our +Galaxy is 1-3/century. Rozwadowska et al. (2021), ob- +tained a more pessimistic mean time of occurrence of 61 +6 Note that SNe of type Ia undergo thermonuclear explosions. The +detection of the emitted neutrinos could help elucidating the pre- +cise mechanism for their explosion (Wright et al., 2017). +7 For +the +event +rates +see +also +https://github.com/SNOwGLoBES/snowglobes +(Scholberg, +2012) and SNEWPY (Baxter et al., 2022). +8 The rates correspond to a luminosity of 3 × 1053 ergs (or close +to it), average energies between 12 and 18 MeV (depending on +the neutrino species) with 100 % or more realistic efficiencies. ++24 +−14 based on neutrino and electromagnetic observations +of collapse events in the Milky Way and the Local Group. +Supernovae are frequent events in the universe. With a +1 Mt detector, about one supernova per year is expected +within 10 Mpc, due to nearby galaxies with higher rates +than the Milky Way. Within 4 Mpc, less than one event +per year would be detected (Ando et al., 2005). +Neutrinos from past supernova explosions form a relic, +or diffuse, supernova neutrino background (DSNB) (see +the reviews by Ando and Sato (2004); Beacom (2010); +Lunardini (2016); and Mathews et al. (2020)). Its flux, +integrated over cosmological redshift, depends on the red- +shifted supernova neutrino fluxes, on the core-collapse su- +pernova rate and on the fraction of failed supernovae that +turn into a black hole without an electromagnetic coun- +terpart (Keehn and Lunardini, 2012; Lunardini, 2009). +At present we only have upper limits. +Super-K (Malek et al., 2003) set the first upper limit +of 1.2 ¯νe cm−2s−1 (Eν > 19.3 MeV, 90 % CL) on the su- +pernova relic flux. The bound was improved with Super- +K IV data (Zhang et al., 2015) using ¯νe detection via +inverse β-decay and neutron tagging on protons. +The +DSNB search combining Super-K I to Super-K IV data +yields 2.7 ¯νe cm−2s−1 (Eν > 17.3 MeV, 90 % CL). The +KamLAND experiment obtained the upper value of 139 +¯νe cm−2s−1 (90 % CL) in the window [8.3, 31.8] MeV +(Gando et al., 2012). +The Borexino Collaboration ex- +tracted a model dependent limit of 112.3 ¯νe cm−2s−1 (90 +% CL) in the interval [7.8, 16.8] MeV (Agostini et al., +2021). +As for νe, the ensemble of SNO data provided the up- +per limit of 19 νe cm−2s−1 in the window [22.9, 36.9] MeV +(90 % CL) (Aharmim et al., 2006). The loosest limits are +φνx,¯νx < 1.3-1.8 103 cm−2s−1 (Eν > 19 MeV) (Lunardini +and Peres, 2008) (for x = µ, τ flavors). With neutrino- +nucleus coherent scattering in dark matter detectors, one +could improve this bound to 10 νx or ¯νx (Suliga et al., +2022). +Beacom and Vagins (2004) suggested to add gadolin- +ium (Gd) to water Cherenkov detectors9. Neutron cap- +ture by Gd improves10 inverse β-decay tagging through +the 8 MeV photons following the capture. The Super- +K+Gd experiment is currently taking data. +DSNB predictions (Ando and Sato, 2004; Chakraborty +et al., 2011a; De Gouvea et al., 2020; Ekanger et al., 2022; +Fukugita and Kawasaki, 2003; Galais et al., 2010; Ho- +riuchi et al., 2018; Ivanez-Ballesteros and Volpe, 2022; +Kresse et al., 2021; Lunardini, 2006; Moeller et al., 2018; +Priya and Lunardini, 2017; Tabrizi and Horiuchi, 2021; +Vissani and Pagliaroli, 2011; Yuksel and Beacom, 2007) +are close to current bounds. The analysis from the com- +bined SK-I to IV data (Abe et al., 2021) shows an excess +9 The idea was named GADZOOS for Gadolinium Antineutrino +Detectos Zealously Outperforming Old Kamiokande, Super! +10 An efficiency of 90 % is expected with 0.1 % Gd concentration. + +5 +at 1.5 σ over background prediction. The related sensi- +tivity analysis is on par with four of the most optimistic +predictions (Ando and Sato, 2004; Galais et al., 2010; +Horiuchi et al., 2021; Kresse et al., 2021) and a factor of +about 2-5 larger than the most conservative ones. +With Super-K+Gd, the upcoming JUNO, DUNE and +Hyper-K experiments, the DNSB should be discovered in +the forthcoming future. +E. The r-process and GW170817 +Neutrinos in dense environments are tightly connected +to two unresolved issues in astrophysics: the death of +massive stars and the origin of r-process11 elements. Cur- +rently, two- and three-dimensional simulations include re- +alistic neutrino transport, convection, turbulence and hy- +drodynamical instabilities such as SASI (see for example +Burrows et al. (2020); Foglizzo et al. (2015); Janka (2012, +2017); Kotake et al. (2006); Mezzacappa et al. (2020); +Radice et al. (2018a); and Takiwaki et al. (2021)). The +delayed neutrino-heating mechanism is believed to trig- +ger most of core-collapse supernova explosions12. +The +future observation of a galactic or extragalactic super- +nova could confirm or refute the current paradigm and +elucidate a six-decade quest. +As for r-process nucleosynthesis, Burbidge et al. (1957) +and Cameron (1958) first linked it to core-collapse su- +pernovae which have long been thought the main r- +process site13 (see for example Qian (2014), Kajino et al. +(2019))14. While simulations show that entropies are typ- +ically too low, the most energetic events seem to provide +the right conditions to attain a successful nucleosynthe- +sis (see Cowan et al. (2021) and Cˆot´e et al. (2019) for a +comprehensive review). +Another candidate site for the r-process are binary +neutron star mergers (BNS) as first suggested by Lat- +timer and Schramm (1974, 1976) (see for example (Cˆot´e +et al., 2019; Goriely et al., 2015; Kajino et al., 2019) for +reviews). As supernovae, BNS are powerful sources of +MeV neutrinos15. If the former are more frequent than +11 r-process stands for rapid neutron capture process. +12 Note that the most energetic events might require a magneto- +hydrodynamical mechanism (see for example Janka (2012)). +13 With a rule of thumb, if each supernova produces 10−4M⊙ el- +ements and there are 3 such events per century, in 1010 years +there are 3 104M⊙ r-process elements ejected in the Milky Way. +14 In the r-process, nuclei capture neutrons faster than β-decay. +The process, which makes about half of the heavy elements in our +galaxy, involves thousands of exotic nuclei far from the neutron +drip-line. The other half is produced in the s-process (s stands +for slow), whereas a small contribution comes from the p-process +(p stands for proton). +15 They emit 1051 to 1053 erg in νe, νµ, ντ and their antiparti- +cles with tens of MeV. Contrary to supernovae, binary neutron +star mergers are more neutron-rich and produce an excess of ¯νe +over νe (see e.g. Cusinato et al. (2021)). As for νµs and ντs, +their fluxes are predicted to be small compared to those of core- +the latter, simulations show that BSN offer more suitable +astrophysical conditions for a strong r-process16. More- +over studies have shown some r-process elements are syn- +thetised in accretion disks around black holes (Surman +et al., 2006) and black hole-neutron star mergers (Ca- +ballero et al., 2012; Surman et al., 2008). +The first detection of gravitational waves from the fu- +sion of two black-holes by the Virgo-LIGO Collaboration +(Abbott et al., 2016) has opened the era of gravitational +wave astronomy. +To date, GW170817 (Abbott et al., +2017a,b) is a unique multi-messenger event in which grav- +itational waves from a binary neutron star merger were +first detected, also concomitantly with a short-gamma +ray burst and a kilonova17 (Fig. 2). The ejecta opacities +show presence of actinides and lanthanides (see for exam- +ple the studies by (Cowan et al., 2021; Cˆot´e et al., 2019; +Tanaka et al., 2018)). This represents the first evidence +for r-process elements in binary neutron star mergers. +Before GW170817 dynamical ejecta were thought to +mainly contribute to a strong r-process (see for exam- +ple the discussion in Martin et al. (2015).) +But, the +kilonova observation has changed this paradigm. +In- +deed the comparison of the electromagnetic emission with +most models shows two components: the early and fast +pre-merger contribution from dynamical ejecta, the later +post-merger one due to viscosity and neutrino-driven +winds. The former is associated with red emission, the +latter with the blue one (Radice et al., 2018b). The role +of neutrinos on the pre- and post-merger ejecta and of +flavor evolution appears crucial and is currently object +of debate (see e.g. Nedora et al. (2021)). +Numerous r-process studies that include not only neu- +trinos but also neutrino flavor evolution find that mat- +ter becomes proton-rich and tends to harm the r-process +in core-collapse supernovae18. One should keep in mind +though, that studying flavor evolution in a consistent +manner is complex. Depending on the site and on the +assumptions made, one can find situations in which fla- +vor modification favors the r-process. In general, what +emerges from the investigations, is that neutrino flavor +evolution impacts the nucleosynthetic abundances when +collapse supernovae and have large theoretical uncertainties (see +for example Table VII of Frensel et al. (2017)). +16 A weak r-process produces elements in the first peak around +mass number A=80-90 and the second peak around A=130-138. +A strong r-process reaches the third peak at A=195-208. +17 The electromagnetic signal of a kilonova (Metzger et al., 2010), +or macronova (Kulkarni, 2005) is in-between the ones of novae +and supernovae. +Since the afterglow emission of the kilonova +associated with GW170817 extends over some days, it appears +that radioactivity injects some energy, powering the kilonova. +The optical/IR spectra in the IR emission peak are compatible +with elements heavier than iron to be responsible for absorption +and reemission of the radiation (Cowan et al., 2021). +18 Note that there are other nucleosynthesis processes where neu- +trinos influence element abundances, including neutrino nucle- +osynthesis (Langanke et al., 2019) and the νp process (Frohlich +et al., 2006). + +6 +FIG. 2 Hubble Space Telescope image of kilonova gradu- +ally fading, in the lenticular galaxy NGC 4993 (40 Mpc). +The event GW170817 was seen concomitantly in gravitational +waves, short gamma ray bursts and electromagnetic emission. +It represents the first observation from merging binary neu- +tron stars. (Figure adapted from (Telescope, 2017)). +one includes standard or non-standard properties and in- +teractions. +The quest for the identification of the r-process site(s) +and the supernova mechanism as well as the need of pre- +dictions for future observations motivated an in-depth +investigation of flavor evolution in dense environments, +as we shall now discuss. +F. Theoretical developments +Undoubtedly, understanding flavor evolution is an +intriguing theoretical problem, with many interesting +questions. +In dense environments, do new phenomena +emerge? What are the conditions to trigger them and +what is their impact? Do novel flavor mechanisms intro- +duce extra heating and help supernova explosions? How +do neutrinos behave in presence of shock waves and of +turbulence? Are there flavor mechanisms that favor the +r-process? Is the commonly employed mean-field approx- +imation sufficient? Do weakly interacting neutrino gases +behave like known many-body systems? What is the in- +terplay with unknown neutrino properties? What is the +role of strong gravitational fields? Are many-body cor- +relations important? The list encompasses many others. +Over thirty years theoretical studies have paved the way +to their answers. +First of all, investigations have shown that a variety +of novel conversion phenomena can occur due to matter, +shock waves, turbulence and νν interactions. Obviously, +there is the established MSW effect that takes place both +in astrophysical and cosmological environments. In par- +ticular, Dighe and Smirnov (2000) pointed out that, be- +cause of the large densities and of radiative corrections +(Botella et al., 1987), three MSW resonances occur in +core-collapse supernovae. While the MSW effect is cer- +tainly a reference in studies of flavor modification, the +novel phenomena uncovered go much beyond it. +Schirato and Fuller (2002) reported that shock waves +could modify the time signal of supernova neutrinos. +Tomas et al. (2004) showed the presence of front and re- +verse shocks in supernova simulations. Fogli et al. (2003) +and then Dasgupta and Dighe (2007) found that front +and reverse shocks produce multiple MSW resonances +and phase effects. Several authors (Choubey et al., 2007; +Fogli et al., 2005; Gava et al., 2009; Kneller et al., 2008; +Takahashi et al., 2003) studied their impact. +Concerning noisy media, Loreti and Balantekin (1994) +first studied the influence of matter fluctuations in re- +lation with solar neutrinos. +A few papers (Balantekin +et al., 1996; Loreti et al., 1995; Nunokawa et al., 1996) +showed that fluctuations of matter density profiles could +induce neutrino depolarization. +The impact of turbu- +lence on neutrinos was then explored in the context of +supernovae (Abbar, 2021; Fogli et al., 2006; Friedland +and Gruzinov, 2006; Kneller and Volpe, 2010; Lund and +Kneller, 2013), reaching similar, or opposite (Borriello +et al., 2014), conclusions. +Besides shock waves and turbulence, neutrino-neutrino +interactions have attracted a strong interest. In the early +nineties, Pantaleone (1992) pointed out that such interac- +tions become sizable when the neutrino number densities +are sufficiently large, and make neutrino propagation a +non-linear many-body problem. Samuel (1993) showed +that such interactions could trigger new flavor effects. +At first, studies for the early universe (Abazajian et al., +2002; Dolgov et al., 2002; Kostelecky et al., 1993; Kost- +elecky and Samuel, 1995; Mangano et al., 2005; Pastor +et al., 2002) implemented νν interactions (see also (Gava +and Volpe, 2010)). Moreover, Rudzsky (1990), Sigl and +Raffelt (1993) and McKellar and Thomson (1994) derived +neutrino quantum kinetic equations including neutrino +interactions with matter and neutrinos. +It was the work by Duan et al. (2006a) that trig- +gered the attention on νν interactions in core-collapse +supernovae. Balantekin and Yuksel (2005) also showed +that such interactions could produce significant effects +on the r-process. These works stimulated an intense ac- +tivity on νν interactions (see the reviews by Duan and +Kneller (2009), Duan et al. (2010), Mirizzi et al. (2016), +Volpe (2015)). +The first numerical simulations, based +on the stationary bulb model, studied in great detail the +mechanisms under which neutrinos first synchronised, +then underwent bipolar oscillations and finally spectral +splits19. In the literature, authors commonly refer to fla- +vor modes due to νν interactions as collective neutrino +oscillations20. +19 These are often called slow modes, as we shall discuss (section +II.F). +20 Note however, that the effects of νν interactions are not neces- + +Aug 22, 2017 +Aug 26, 2017 +Aug 28, 20177 +Moreover, Malkus et al. (2012) showed that in black +hole accretion disks the interplay between νν and mat- +ter interactions produced a new mechanism later called +neutrino matter resonance. +This was studied in the +context of merging compact objects (black hole-neutron +star, neutron star-neutron star) (Frensel et al., 2017; +Malkus et al., 2014, 2016; Vaananen and McLaughlin, +2016; Vlasenko and McLaughlin, 2018; Wu et al., 2016) +and of core-collapse supernovae, with non-standard in- +teractions (Stapleford et al., 2016). +Earlier than the bulb model, Sawyer (2005) showed +that the neutrino-neutrino interaction could trigger sig- +nificant flavor conversion on very short scales (see also +Sawyer (2009)). +It was only ten years after (Sawyer, +2016), when he considered different neutrinospheres for +νe and ¯νe and found flavor modes with nm scale, that +his findings triggered excitement. Indeed, for long, theo- +rists had searched for mechanisms that could take place +behind the shock wave and impact the explosion dynam- +ics of core-collapse supernovae. These modes were called +fast, in contrast with the ones found in the bulb model. +Subsequent studies showed that a sufficient but not +necessary condition for the occurrence of fast modes is +that the neutrinospheres of νe and ¯νe cross each other. +The conditions to have fast modes and their impact is ac- +tively investigated (see for example (Abbar et al., 2019, +2020; Abbar and Volpe, 2019; Chakraborty et al., 2016b; +Dasgupta et al., 2017; George et al., 2020; Glas et al., +2020; Just et al., 2022; Nagakura et al., 2021; Wu and +Tamborra, 2017; Wu et al., 2017) and the review by Tam- +borra and Shalgar (2021)). +While most of the developments focussed on the novel +flavor mechanisms and their impact, numerous studies +concentrated on the neutrino evolution equations them- +selves. +Indeed the majority of the literature employs +mean-field equations, derived by several authors (Bal- +antekin and Pehlivan, 2007; Friedland and Lunardini, +2003b; Samuel, 1993; Sawyer, 2005; Serreau and Volpe, +2014; Sirera and Perez, 1999; Volpe et al., 2013; Ya- +mada, 2000). However, in very dense stellar regions, or +in the early universe, where collisions matter, neutrino +quantum kinetic equations are necessary. +Such equa- +tions were obtained using different approaches (Blaschke +and Cirigliano, 2016; Froustey et al., 2020; McKellar and +Thomson, 1994; Rudzsky, 1990; Sigl and Raffelt, 1993; +Vlasenko et al., 2014a). +In principle, the theoretical +framework should consistently evolve from the collision- +dominated to the mean-field regime. +The full solution of neutrino quantum kinetic equa- +tions is achievable, if the medium is homogeneous and +isotropic as in the early universe (Bennett et al., 2021; +Froustey et al., 2020). +Unfortunately, in dense stellar +environments this becomes a formidable numerical task. +Not only the number of degrees of freedom is 7, but also, +sarily collective, neither oscillatory. +when νν interactions are sizable, we face a non-linear +many-body problem! There are currently serious efforts +to explore the role of collisions and their interplay with +flavor mechanisms (see for example (Capozzi et al., 2019; +Hansen et al., 2022; Richers et al., 2019; Xiong et al., +2022)). +One should keep in mind that, even in models with +reduced degrees of freedom and approximations, the de- +scription of neutrino propagation requires the solution +of a large number of stiff and coupled non-linear equa- +tions (in presence of νν interactions). In order to avoid it, +Banerjee et al. (2011) proposed to linearize the equations +of motion and replace the solution of the full non-linear +problem by eigenvalue equations, as first pointed out by +Sawyer (2009). +Vaananen and Volpe (2013) provided +an alternative derivation inspired by the Random-Phase- +Approximation used, for example, in the study of collec- +tive vibrations in atomic nuclei. Izaguirre et al. (2017) +cast the linearized equations in a dispersion-relation ap- +proach, commonly used in the study of fast modes. +Moreover, new numerical methods based on deep +learning techniques (see e.g. +(Armstrong, 2022; Arm- +strong et al., 2022; Rrapaj et al., 2021)) have been re- +cently employed. They go beyond theoretical approaches +using forward integration. +But are mean-field equations enough when neutrinos +start free-streaming? +This aspect is actively debated. +Balantekin and Pehlivan (2007) discussed corrections to +the mean-field approximation using the coherent-state +path-integral approach. +Volpe et al. (2013) pointed +out that the most general mean-field equations include +both pairing correlations, analogous to those of Bardeen- +Cooper-Schrieffer (Bardeen et al., 1957), and wrong- +helicity contributions due to the absolute neutrino mass. +Vlasenko et al. (2014a) derived neutrino quantum kinetic +equations including wrong-helicity contributions, termed +spin coherence. +Serreau and Volpe (2014) referred to +them as helicity coherence when they derived the most +general mean-field equations for anisotropic and inhomo- +geneous media. +In addition, Cherry et al. (2012) showed that a small +fraction of backward neutrinos (neutrino halo) can influ- +ence the neutrino flavor content. This result questioned +the validity of descriptions of neutrino flavor evolution as +an initial value problem. Halo effects were further studied +in O-Ne-Mg (Cherry et al., 2013) and iron core-collapse +supernovae (see for example (Sarikas et al., 2012b)). +Pehlivan et al. (2011) showed that the use of algebraic +methods and of the Bethe ansatz opens the way to exact +solutions of the quantum many-body problem of neutrino +propagation in dense media (without the matter term +and collisions). Further investigated, this has uncovered +the importance of many-body correlations (Birol et al., +2018; Pehlivan et al., 2014) and brought exciting con- +nections to quantum information theory (Cervia et al., +2019; Lacroix et al., 2022; Patwardhan et al., 2021; Rog- +gero, 2021a,b; Roggero et al., 2022) and quantum devices + +8 +(Amitrano et al., 2023; Hall et al., 2021). +Other intriguing connections have been established be- +tween a weakly interacting neutrino gas in a dense envi- +ronment and other many-body systems. Pehlivan et al. +(2011) showed that the neutrino Hamiltonian can be re- +lated to the (reduced) Bardeen-Cooper-Schrieffer Hamil- +tonian of Cooper pairs in superconductivity. Volpe et al. +(2013) applied the Born-Bogoliubov-Green-Kirkwood- +Yvon (BBGKY) hierarchy to an interacting neutrino +gas in a medium and established a formal connection +with atomic nuclei. Also, a futher connection exist be- +tween flavor evolution of an interacting neutrino gas with +the transition from laminar to turbulent regimes in non- +linear fluids, as pointed out by Mirizzi et al. (2015). +Besides these aspects flavor evolution in dense objects +is also interesting because it is tightly connected to neu- +trino properties and non-standard physics or particles, +e.g. +sterile neutrinos (Fetter et al., 2003; McLaughlin +et al., 1999; Tamborra et al., 2012; Wu et al., 2014; +Xiong et al., 2019), non-standard interactions (Blennow +et al., 2008; Chatelain and Volpe, 2018; Das et al., +2017; Esteban-Pretel et al., 2007a; Stapleford et al., 2016; +Wolfenstein, 1978), the neutrino mass ordering (Barger +et al., 2005; Dighe et al., 2003; Engel et al., 2003; Gava +et al., 2009; Serpico et al., 2012) or CP violation (Akhme- +dov et al., 2002; Balantekin et al., 2008; Gava and Volpe, +2008; Pehlivan et al., 2014; Popov and Studenikin, 2021). +There are numerous reviews on core-collapse super- +nova neutrinos. They focus on oscillations in media (Kuo +and Pantaleone, 1989), on the diffuse supernova ν back- +ground (Ando and Sato, 2004; Beacom, 2010; Lunardini, +2016; Mathews et al., 2020), on pre-supernova neutri- +nos (Kato et al., 2020), on νν interactions (Duan et al., +2010; Tamborra and Shalgar, 2021), on νν interactions +and turbulence (Duan and Kneller, 2009), on supernova +ν detection (Scholberg, 2012), on observations (Horiuchi +and Kneller, 2018), on production, oscillations and detec- +tion (Mirizzi et al., 2016) and on the neutrino evolution +equations (Volpe, 2015). +The goal of this review is to highlight the richness and +complexity of neutrino flavor evolution in dense media. +The review encompasses in particular two aspects of this +fascinating problem, namely flavor mechanisms and the +theoretical frameworks, and discusses aspects of super- +nova neutrino observations. Since fifteen years this is a +developing field, where new approaches, exciting connec- +tions and interesting ideas keep being proposed which +makes the writing of this review challenging. +The structure of the manuscript is as follows. Section +II is focussed on flavor mechanisms in media. Section III +focusses on the theoretical description of neutrino prop- +agation. Section IV is on past and future observations of +supernova neutrinos. Section V presents conclusions and +perspectives. +II. NEUTRINO FLAVOR MECHANISMS IN +ENVIRONMENTS +Neutrino flavor mechanisms are quantum mechanical +phenomena. +Suggested by Pontecorvo (1957), vacuum +oscillations are analogous to Rabi oscillations in atomic +physics, or K0 − ¯K0 oscillations in meson systems. +Oscillations in vacuum arise because the flavor (or in- +teraction) and mass (or propagation) bases do not coin- +cide. This produces an interference phenomenon among +the mass eigenstates when neutrinos propagate. The two +bases are related by the PMNS matrix21 U, that is +|να⟩ = U ∗ +αk |νk⟩ . +(1) +with k = 1, 2, 3, ..., N and α = e, µ, τ, ..., N the mass +and flavor indices22 respectively, N being an arbitrary +number of neutrino families. The matrix is unitary (U = +U †). For antineutrinos the same relation holds, with Uαk +instead of U ∗ +αk. +The massive neutrino states are eigenstates of the free +Hamiltonian H0 = diag(Ek), with eigenenergies +Ek = +� +⃗p2 +k + m2 +k , +(2) +momentum ⃗pk and mass mk. In the flavor basis, the free +Hamiltonian reads +Hf +vac = UHvacU †. +(3) +Let us recall the neutrino equations of motion mostly +used to determine how flavor changes in media. Based on +such equations several mechanisms emerge, that I shall +highlight. +A. Mean-field equations +Neutrino +flavor +states +evolve +according +to +the +Schr¨odinger-like equation23 +i d +dt|να(t)⟩ = H |να(t)⟩ , +(4) +with the initial condition |να(0)⟩ = |να⟩. When neutrinos +traverse dense matter, the neutrino Hamiltonian receives +different contributions, namely +H = Hf +vac + Hmat + Hνν + HNSI , +(5) +where Hmat comes from neutrino interactions with mat- +ter and Hνν from νν interactions. The last term HNSI is +21 From now on I use ℏ = c = 1. +22 A sum on repeated indices is subtended. +23 Unless differently stated, I shall consider neutrinos as plane waves +and neglect space-time curvature due to the presence of strong +gravitational fields. + +9 +present, if non-standard interactions exist between neu- +trinos and neutrinos, or neutrinos and matter. +We now have a closer look at the different contributions +to the neutrino Hamiltonian Eq.(5). When neutrinos tra- +verse a dense astrophysical medium, they interact with +the background electrons, protons, neutrons and neutri- +nos. The mean-field is the simplest widely used approxi- +mation to implement such interactions. +A number of authors have derived mean-field evolu- +tion equations including neutrino interactions with mat- +ter and neutrinos (Balantekin and Pehlivan, 2007; Fried- +land and Lunardini, 2003b; Samuel, 1993; Sawyer, 2005; +Serreau and Volpe, 2014; Sigl and Raffelt, 1993; Sirera +and Perez, 1999; Vlasenko et al., 2014a; Volpe et al., 2013; +Yamada, 2000). In words, the mean-field approximation +consists in adding the amplitudes associated with neu- +trino scattering on a background particle, weighted by +the quantum expectation value of the particle number +operator over the background. Integrating such quantity, +over the degrees of freedom of the background particle, +generates a potential that acts on the neutrino propagat- +ing through the medium (Fig. 3). +Let us first consider neutrino-matter interactions +whose contribution can be derived from the charged- and +neutral-current interactions terms of the GWS model. +Associated with charged-current ν-e scattering is the well +known mean-field Hamiltonian +Hmat = +√ +2GF(ne − n¯e) , +(6) +GF being the Fermi coupling constant and ne (n¯e) the +electron (positron) number density. Neutral-current in- +teractions of νe, νµ and ντ on neutrons, which give equal +mean-field contributions, do not influence oscillations; +whereas charged-current interactions on electrons and +protons cancel each other, in a neutral medium. +Equation (6) holds for a homogenous, isotropic and +unpolarized medium. This is a good approximation for +example for the Sun. +If the assumptions are relaxed, +more mean-field terms appear due for example to polar- +isation, as discussed by e.g. (Nunokawa et al., 1997b). +In dense media, like supernovae or compact objects rem- +nants, interesting features arise due to anisotropy, as we +shall see. +For 2ν flavors, the evolution equations (4) with the +vacuum and matter terms read +i d +dt +� |να⟩ +|νβ⟩ +� += H +� |να⟩ +|νβ⟩ +� +, +(7) +where +H = Hc + +� - ∆m2 +4E cos2 2θ + +√ +2GFne +∆m2 +4E sin2 2θ +∆m2 +4E sin2 2θ +∆m2 +4E cos2 2θ +� +, +(8) +after subtracting a term common to all flavors +Hc = +� +E + (m2 +1 + m2 +2)/4E +� +1 , +(9) +FIG. 3 +Mean-field approximation: +tadpole diagrams for +neutrino-electron (upper) and neutrino-neutrino scattering +(lower diagrams) that contribute to the neutrino evolution +equations in media. +(Figure adapted from Volpe et al. +(2013)). +proportional to the identity matrix, with the neutrino +energy E = |p|. +The quantity ω = ∆m2/(2E) is the +vacuum oscillation frequency. +Let us remind that, to investigate flavor evolution, one +often evolves neutrino amplitudes, effective spins or den- +sity matrices, instead of neutrino states (see for example +(Giunti and Kim, 2007)). It was Harris and Stodolsky +(1982) who first discussed the density matrix (and po- +larization vector) approach in relation to neutrinos, to +describe the coherence properties of a two state system +undergoing random fluctuations in a medium. +For 3ν +flavors the neutrino density matrix reads +ϱ = +� +� +� +⟨a† +e,iae,i⟩ ⟨a† +µ,jae,i⟩ ⟨a† +τ,kae,i⟩ +⟨a† +e,iaµ,j⟩ ⟨a† +µ,jaµ,j⟩ ⟨a† +µ,jaτ,k⟩ +⟨a† +e,iaτ,k⟩ ⟨a† +τ,kaµ,j⟩ ⟨a† +τ,kaτ,k⟩ +� +� +� , +(10) +where the quantum expectation values24 is over the back- +ground that neutrinos are traversing. Here the indices +i, j, k indicate the quantum numbers (usually momen- +tum and helicity), which characterize neutrino states. +A similar expression holds for antineutrinos, but with +¯ϱij = ⟨b† +ibj⟩25 instead of ϱij = ⟨a† +jai⟩. The diagonal en- +tries of Eq.(10) are the quantum expectation values of +the occupation number operator. +Instead of evolving neutrino states (4), one can solve +the Liouville Von Neumann equation for the neutrinos or +24 The operators a† and a are the particle creation and an- +nihilation operators that satisfy the equal-time anticommuta- +tion rules {a(p, h), a†(p′, h′)} = (2π)3δ3(p − p′)2Epδh,h′ and +{a(p, h), a(p′, h′)} = {a†(p, h), a†(p′, h′)} = 0 (h, h′ are helici- +ties). Similar rules hold for the antiparticle creation and annihi- +lation operators, b† and b. +25 Note that with this convention ϱ and for ¯ϱ have the same equa- +tions formally; inversely to ¯ϱij = ⟨b† +jbi⟩ which introduces com- +plex conjugates of ¯ϱ in the equations of motion (see for example +(Sigl and Raffelt, 1993; Volpe et al., 2013)). + +(p) +V(R) +(p) +(d)_a +e-(p) +v(R) +(k)vB(p) +(Vg(k) +Ve(p) +Vα(p) + Ve(p) +Va(ki) +Vg(k)10 +the antineutrino density matrix26, i.e. +idϱ +dt = [H, ϱ] +id ˙¯ϱ +dt = [ ¯H, ¯ϱ]. +(11) +Dense media have the peculiarity that neutral-current +νν interactions are sizable and contribute in the mean- +field approximation through the Hamiltonian +Hνν = +√ +2GF +� +p +(1 − ˆp · ˆp′) [ϱ(p) − ¯ϱ(p)] , +(12) +where +� +p += +� +dp/(2π)3 +ˆp = p/|p|. +(13) +The term ˆp · ˆp′, coming from the V -A structure of the +weak interactions, contributes in anisotropic media. +More generally, the mean-field equations of motion +that describe neutrino (and similarly antineutrinos) +propagation in dense environments read +i(∂t + v · ∇x + F · ∇p)ϱ = [H, ϱ] +(14) +with v = p/E the neutrino velocity. The second term +is an advective term which contributes in presence of +spatial inhomogeneities. The third term depends on a +possible external force F, such as the gravitational one, +that acting on the neutrinos could change its momentum +or energy (because of trajectory bending for example). +Since the Liouville operator depends on time, coordinate +and momentum, the problem one faces in determining +neutrino flavor is 7-dimensional, and therefore extremely +challenging numerically. +The solution of Eq.(11) with the mean-field terms +Eqs.(6) and (12), the vacuum term Eq.(3) reveals flavor +mechanism that mostly arise from the interplay between +the different contributions, as we now describe. +B. The MSW effect +The MSW effect is a reference phenomenon for flavor +evolution studies. Several of the uncovered mechanisms +are either MSW-like or multiple MSW phenomena. To +clarify this link, we remind some basics. +The MSW effect arises when a resonance condition +is satisfied, the resonance width is large and evolution +through it is adiabatic. It is equivalent to the two-level +problem in quantum mechanics (Cohen-Tannoudji et al., +1998). +Let us introduce the matter basis which, by definition, +instantaneously diagonalizes the neutrino Hamiltonian. +26 Note that in the equation of motion for antineutrinos the vacuum +contribution to the Hamiltonian has a minus sign. +Here we consider only the vacuum and the matter con- +tributions27. +The flavor basis is related to the matter +basis through the relation +|να⟩ = ˜U ∗ +αk |˜νk⟩ , +(15) +with k = 1, 2, 3, ..., N. In the unitary matrix ˜U, effective +mixing parameters in matter replace the vacuum ones. +From Eqs. (4) and (15), one gets the following equation +of motion for the matter basis +i d +dt|˜ν(t)⟩ = ˜H |˜ν(t)⟩ = +� +K + i ˜U † d ˜U +dt +� +|˜ν(t)⟩ , +(16) +where K = diag(˜k1, ˜k2, . . . ˜kN) depends on the matter +eigenvalues ˜ki (i = 1, N) and the matter Hamiltonian +˜H now includes the derivatives of the effective mixing +parameters in matter. These depend on the specific en- +vironment neutrinos are traversing. +Let us consider the explicit expressions for 2ν flavors +for which equation (15) reads +� |νe⟩ +|νx⟩ +� += +� 1 0 +0 eı ˜β +�� cos ˜θ +sin ˜θ +− sin ˜θ cos ˜θ +� � |˜ν1⟩ +|˜ν2⟩ +� +, +(17) +with ˜θ and ˜β the effective mixing angle and phase re- +spectively. Neglecting the phase, the evolution equation +of the matter basis (16) reads +i d +dt +� |˜ν1⟩ +|˜ν2⟩ +� += +� ˜k1 i ˙˜θ +-i ˙˜θ ˜k2 +� � |˜ν1⟩ +|˜ν2⟩ +� +, +(18) +where ˜k1, ˜k2 are given by +˜k2 − ˜k1 = +� +(∆m2 cos 2θ − A)2 + ∆m2 sin2 2θ . +(19) +In matter neutrinos acquire an effective mass. Figure 4 +shows how ˜k1 and ˜k2 evolve as a function of the electron +number density in an environment. +The effective mixing angle diagonalizing the 2 × 2 ma- +trix given by equation (8) satisfies +sin2 2˜θ = +∆m2 sin2 2θ +(∆m2 cos2 2θ − A)2 + ∆m2 sin2 2θ , +(20) +with A = 2EHmat Eq.(6). +One can see that, when the following equality holds +√ +2GFne = ∆m2 +2E cos 2θ , +(21) +the matter mixing angle Eq. (20) is maximal, i.e. ˜θ = +π/4, and the distance between the matter eigenvalues +minimal (Fig. 4). +27 More generally, a ”matter” basis can be introduced whatever +terms are included in the Hamiltonian. + +11 +FIG. 4 The Mikheev-Smirnov-Wolfenstein effect: the figure +shows the matter eigenvalues as a function of matter number +density (solid lines). The dashed lines are the matter eigen- +values in absence of mixings. At the MSW resonance location, +the matter mixing angle is maximal and the eigenvalues ”re- +pel” each other. The MSW effect is a two-level problem in +quantum mechanics. (Figure from (Akhmedov, 1999)). +Relation (21) corresponds to the difference of the diag- +onal elements in the Hamiltonian Eq.(8) being equal to +zero (or to a minimal distance of the matter eigenvalues +in the matter basis). It is the MSW resonance condition. +Its fulfillment gives the sign of the mass-squared differ- +ences. In fact, if θ < π/4 (first octant), cos 2θ > 0 and +the condition holds for ∆m2 > 0. If θ > π/4 (second +octant), cos 2θ < 0 and Eq.(21) requires ∆m2 < 0. +When the resonance condition is satisfied and its width +ΓMSW = ∆m2 sin2 2θ , +(22) +large, the fate of neutrinos depends on the adiabaticity of +the neutrino evolution through the resonance. This can +be quantified by the adiabaticity parameter +γ−1 = +| ˙˜θ| +˜k2 − ˜k1 += ∆m2 +2E +sin2 2θ +cos 2θ +ne +|dne/dr| . +(23) +Quantitatively, if γ−1 +≫ 1, evolution is fully non- +adiabatic, the matter eigenstates strongly mix at the res- +onance and νe exits as ν1. On the contrary, if γ−1 ≪ 1, +the mixing of matter eigenstates at the resonance is sup- +pressed. Then, neutrinos evolve independently, just ac- +quiring a phase, from the inner regions to the surface of +the star. Hence, if νe ≈ ˜νB at π/2 (high density region) +and evolves through the MSW resonance adiabatically, +it exits the star as a ν2 and is mostly detected as νx on +Earth (Fig. 4). Since we now know that the MSW effect +suppresses 8B solar neutrinos to 1/3 the value predicted +by the Standard Solar Model, this gives us the sign of +the corresponding mass-squared difference: ∆m2 +12 > 0. +The concept of adiabaticity is general. The adiabatic- +ity parameter, defined as the ratio of the modulus of the +off-diagonal term of the neutrino Hamiltonian over the +difference of the diagonal ones, can be generalized in pres- +ence, for example, of νν interactions as done in (Galais +FIG. 5 Spin formalism: picture of the neutrino evolution in +flavor space for 2ν flavors. When electron neutrinos are pro- +duced in the Sun, the vector µ precess around the matter +vector n and, if evolution is adiabatic, it follows n until it be- +comes the vacuum vector n0. Note that the third component +of the neutrino spin vector corresponds either to νe (upward) +or to νµ (downward). (Figure from Kim et al. (1988).) +et al., 2012). +In this case it also involves the deriva- +tives of the mixing phase that arise because the neutrino +Hamiltonian Eq.(12) is complex. +If evolution through the resonance(s) is adiabatic, neu- +trinos adjust to a smooth density profile during propa- +gation. +This contrasts with what happens when steep +variations of the density profiles are present, as e.g. in +presence of shock waves. +With the spin formalism, we look at these phenomena +with different eyes, since we follow neutrinos through the +evolution of effective spin28, subject to effective magnetic +fields (Cohen-Tannoudji et al., 1998). +In this context, +vacuum oscillation is a precession of neutrino spins Pp +in flavor space, around the vacuum (effective) magnetic +field Bvac tilted by 2θ (Kim et al., 1988, 1987) (Fig. 5). +From the evolution of the Pz component, one recovers +the vacuum oscillation formula. +As for the MSW effect, it takes place in matter when +Pp goes through the x-y plane, since the MSW reso- +nance condition corresponds to Pz = 0. Adiabatic evolu- +tion occurs when the precession frequency of Pp around +B = Bvac+Bmat is fast compared to the rate with which +B changes, so that spins follow the magnetic field dur- +ing propagation. On the contrary, if evolution is non- +adiabatic, Pp ”lags behind”. +The description in terms of neutrino isospins has been +largely exploited to study neutrino flavor evolution in +dense environments and in particular when νν interac- +tions are sizable (see for example the review by Duan +et al. (2010)). +28 Also called neutrino ”isospins” or ”polarization” vectors. + +Ve +Ve +(α) +(b)12 +FIG. 6 MSW effect in core-collapse supernovae: effective neu- +trino masses as a function of the electron number density +in absence (dashed) or in presence (solid lines) of neutrino +mixings. The diagram corresponds to normal mass ordering. +(Figure adapted from (Dighe and Smirnov, 2000).) +C. The MSW effect in dense media +Neutrinos face more than one resonance if the density +of the astrophysical or cosmological environment is large. +Dighe and Smirnov (2000) pointed out that there are +three such resonances in supernovae: the high (H-), the +low (L-) and the Vµτ (Fig. 6). +As the MSW resonance condition shows, the resonance +location depends on the neutrino energy and mixing pa- +rameters. From Eq. (21) one finds that the H- and L- +resonances, associated with (θ31, ∆m2 +31) and (θ21, ∆m2 +21) +respectively, take place at +ρres ≃ 1.4 × 106 +g +cm3 +� ∆m2 +1 eV2 +��15 MeV +E +��0.5 +Ye +� +cos 2θ . +(24) +Therefore for a 10 MeV neutrino ρres = 2.7 × 103 g/cm3 +for the H–resonance and ρres = 10 g/cm3 for the L- +resonance. +Moreover radiative corrections that differ- +entiate νµ and ντ (Botella et al., 1987), introduce the +potential Vµτ = 10−5Ve which becomes relevant if the +medium is very dense. The Vµτ resonance is associated +with the atmospheric mixing parameters (θ32, ∆m2 +32). +A characteristic feature of flavor phenomena is that +they induce spectral modifications. Let us consider the +neutrino spectra at the neutrinosphere, φ0 +¯νe and φ0 +¯νx, and +assume ν evolution through H- and L-resonances only in +terms of probabilities29. For example, if ¯νe, produced in +29 This simple approach assumes that the evolution at each res- +onance is factorizable and neglects the role of phases from the +neutrino amplitudes and of the neutrino mixing matrix U. +the inner stellar regions, traverse the three resonances, +their spectra become +φ¯νe = ¯pφ0 +¯νe + (1 − ¯p)φ0 +¯νx , +(25) +where ¯p is the spectral swapping probability. In partic- +ular, ¯p = 0.68 and ¯p = 0 for normal and inverted mass +ordering respectively. +As supernova simulations show, the neutrino spectra at +the neutrinospheres, φ0 +¯νe and φ0 +¯νx, are well described by +pinched Fermi-Dirac distributions (Dighe and Smirnov, +2000) or by power laws (Keil et al., 2003). +Because +of their microscopic interactions, the neutrino average +energies (approximately) satisfy the hierarchy ⟨Eνe⟩ < +⟨E¯νe⟩ < ⟨Eνx⟩, with typical energies Eνe ∈ [8, 14] MeV, +E¯νe ∈ [14, 18] MeV and Eνx ∈ [16, 20] MeV. In fact, νx +undergo neutral current interactions and decouple from +deeper hotter regions. Unlike νx, νe and ¯νe interact via +charged- and neutral-current interactions and decouple +from colder outer shells. +From Eq.(25) one sees that, due to the MSW effect, +¯νe acquire hotter spectra in case of inverted mass or- +dering. +Similarly for νe, if the mass ordering is nor- +mal. These (and other) spectral modifications will im- +pact charged-current events associated with inverse-β +decay or neutrino-nucleus interactions, in a scintillator, +Cherenkov, lead, or liquid argon detector, if a new su- +pernova event occurs. On the contrary, neutral-current +events are ”flavor blind” and therefore are not sensitive +to spectral swapping. +It is important to keep in mind that if, under some con- +ditions, the supernova fluxes of the different neutrino fla- +vors become (practically) degenerate, then spectral dis- +tortions due to flavor mechanisms, according e.g. to Eq. +(25), impact neither directly nor indirectly observations. +For this reason, the possibility of flavor equilibration is +often discussed in the literature and theorists have been +actively looking for this simplifying possibility (particu- +larly when νν interactions are sizable). +A few more remarks. +First, in core-collapse super- +novae, for typical matter profiles (in absence of shock +waves), the evolution through the L-resonance is adia- +batic. Second, since φ0 +¯ντ = φ0 +¯νµ at three level, the Vµτ res- +onance, which mixes νµ and ντ, does not produce spectral +modifications and therefore observables effects30. Third, +the only unknown parameter that impacts the standard +MSW effect is the neutrino mass ordering, since the sign +of ∆m2 +23 ≈ ∆m2 +13 is not determined yet. Therefore, the +detection of the neutrino signal from a future galactic su- +pernova could inform us as on this key property31, as we +30 At least at the current status of our knowledge. Note that recent +calculations including muons in supernova simulations yield νµ +and ντ fluxes with small differences (Bollig et al., 2017). +31 Similarly, numerous studies investigated ways to identify θ13 with +a supernova neutrino signal, until it was measured by the Daya- +Bay (An et al., 2012), the RENO (Ahn et al., 2012) and the +Double Chooz (Abe et al., 2012) experiments. + +H +Ve +H +Vt' +3m +Vt' +L +V1m +e +e13 +FIG. 7 Shock waves: matter density profiles, at different post- +bounce times, as a function of distance in an exploding core- +collapse supernova. Front and reverse shocks are visible. The +yellow and blue bands correspond to the densities and neu- +trino energies that fulfill the high (H-) and low (L-) MSW +resonance conditions. (Figure from (Tomas et al., 2004).) +shall see. +Let us now discuss multiple MSW resonances and +MSW-like mechanisms that arise in dense and sometimes +explosive environments. +D. Shock wave effects +Shock-related structures in supernova neutrino obser- +vations could inform us on shock reheating and propa- +gation, a unique observation of the explosion mechanism +on its becoming. The availability of large scale observa- +tories and a close supernova would offer the possibility +to observe such structures and other deviations from the +expected exponential cooling of the newly formed neu- +tron star. +Even if there are variations among models, +some features appear as sufficiently generic to deserve +investigation. +In an exploding supernova, shock waves constitute a +major perturbation of the electron fraction, defined as +Ye = (ne− − ne+)/(nn + np) (the ni with i = n, p the +neutron and the proton number densities), and of the +pre-supernova matter density profiles. The shock wave +reaches the H-resonance region in about 2 s after core- +bounce. Tomas et al. (2004) showed the presence of both +a front and a reverse shock, due to the earlier slower +ejecta meeting a hot supersonically expanding neutrino- +driven wind (Fig. 7). +Schirato and Fuller (2002) first pointed out that shock +waves could ”shut off” flavor evolution when passing +through an MSW resonance region. Because of the steep- +ness of the density profile, νµ,τ ⇋ νe would be suppressed +due to non-adiabatic evolution. Hence the νe, ¯νe fluxes +would become colder producing dips in the supernova +neutrino time signal. +The passage of shock waves in MSW resonance regions +engenders two effects: it makes the resonance temporar- +ily non-adiabatic and induces multiple MSW-resonances. +The evolution through the different resonance locations +can be treated as incoherent or as coherent. In the for- +mer, the MSW resonances are independent, in the latter +coherent evolution produces interference effects among +the matter eigenstates called phase effects. +Since the +MSW resonance condition is necessary for shock wave +effects, they occur either in the νe signal (for normal), or +in the ¯νe signal (for inverted mass ordering). +The change in the adiabaticity at the MSW resonance +locations impacts the evolution in the H-resonance re- +gion32 and modifies the neutrino average energies creat- +ing dips or bumps in the supernova time signal and the +corresponding rates. These features were investigated in +a series of works (Fogli et al., 2003, 2005; Kneller et al., +2008; Lunardini and Smirnov, 2003; Takahashi et al., +2003), see the review by Duan and Kneller (2009)). +Fogli et al. (2003) pointed out that multiple resonances +could produce phase effects that would average out for +large values of θ13. Dasgupta and Dighe (2007) investi- +gated them in detail. Phase effects require semi-adiabatic +and coherent evolution at the resonances33. They are dif- +ficult to be seen because even when the coherence con- +dition is met, the associated oscillations are smeared by +the energy resolution of detectors. +Let us consider the presence of a dip in a supernova +density profile as an example. A neutrino of energy E +encounters two resonances at locations x1 and x2. If |νh⟩ +and |νl⟩ are the heavier and lighter matter eigenstates +respectively, at x < x1, one has |νh⟩(x ≪ x1) ≈ |νe⟩. +While evolution before the resonance is adiabatic, the +resonance mixes the matter eigenstates just before the +crossing x < x1− yielding new matter eigenstates +� |νh(x1+)⟩ +|νl(x1+)⟩ +� += +� +cos χ1 +sin χ1eiφ +− sin χ1e−iφ +cos χ1 +� � |νh(x1−)⟩ +|νl(x1−)⟩ +� +, +(26) +where Pi = sin2 χ1 is the hopping probability for an iso- +lated resonance. The matter eigenstates acquire a rela- +tive phase up to the second resonance at x2. After the +latter the νe survival probability is (far from x2) +Pνe→νe = cos2(χ1−χ2)−sin 2χ1 sin 2χ2 sin2 � � x2 +x1 +∆ ˜m2 +4E dx +� +. +(27) +The last term, due to the interference between the mat- +ter eigenstates, oscillates with the neutrino energy and +32 Note that the shock wave also influences the neutrino evolution +through the L-resonance region. However its impact (at low en- +ergies and at late times) is negligible. +33 In a wave-packet description in flat spacetime, decoherence arises +at distances larger than the coherence length. For a typical wave- +packet width at production, i.e. +σ ≈ 10−11-10−12 cm−1 and +E ∈ [5, 80] MeV (average energy between two matter eigenstates) +one gets Lcoh ≈ 104 km. + +21012 +L +EL +0.1 sec +@1010 +1.0 sec +Q 109 +2.0 sec +108E +5.0 sec +107 +15.7 sec. +106 +105 +104 +103 +PH +102E +10 +PL +1 +10 +LLuL +Ll +107 +108 +109 +10 +.10 +R (cm)14 +with the resonance locations. +It produces fast oscilla- +tions (phase effects) as a function of energy or, for a +given energy, as a function of time because the shock +wave propagation slightly shifts such locations. In ab- +sence of coherence, the interference term averages out +and the two resonances at x1 and at x2 are independent. +Two studies implemented shock wave effects and νν in- +teractions (in the bulb model). Using a consistent treat- +ment that retains phase information, Gava et al. (2009) +showed that, depending on the neutrino energy, dips or +bumps are present in the positron time signal of scintil- +lators or Cherenkov detectors (inverted mass ordering). +Similar features are present in the electron time signal for +example of argon-based detectors such as DUNE for nor- +mal mass ordering. In contrast, the detailed analysis of +the time signal for the lead detector HALO-2 performed +by Ekinci et al. (2021) showed changes at the level of a +few percent for the one-neutron and two-neutron emis- +sion rates in neutrino-lead interactions, so too small to +be seen. +Besides shock waves, turbulence can play a significant +role in supernova explosions (see for example Radice et al. +(2018a)). +The influence of turbulence on the neutrino +flavor content has features in common with shock wave +effects, as we shall now discuss. +E. Turbulence effects +Noisy media, originating e.g. +from helioseismic g- +modes or temperature fluctuations, influence neutrino +flavor evolution, as pointed out in relation with the so- +lar neutrino problem (see for example Balantekin et al. +(1996); Nunokawa et al. (1996); and Sawyer (1990)). In +particular, Loreti and Balantekin (1994) showed that +randomly fluctuating matter density and magnetic fields +tend to depolarize neutrinos, i.e. the survival probability +averages to one-half. Neutrino propagation in stochastic +media was also discussed in Burgess and Michaud (1997) +and Torrente-Lujan (1999). +Interestingly, solar neutrino and KamLAND data con- +strain matter density fluctuations in our Sun at a few per- +cent level. This result holds for delta correlated (white) +noise, and correlation lengths of 10 km (see Balantekin +and Yuksel (2003b) and Guzzo et al. (2003)) to 100 km. +Hence, one can extract the solar oscillation parameters +independently from fluctuations (Burgess et al., 2004). +Simulations of exploding core-collapse supernovae +show that non-radial turbulent flows associated with +convection and SASI have explosion supportive effects +(Couch and Ott, 2015; Janka, 2012, 2017; Mezzacappa +et al., 2015; Radice et al., 2018a). +Hydrodynamic in- +stabilities generate large scale anisotropies between the +proto-neutron star and the supernova envelope. There- +fore, +supernova neutrinos reaching the Earth ”see” +stochastic matter density profiles. +Noisy media might influence the supernova neu- +trino flavor content significantly. +First investigations +evolved fluctuations-averaged density matrices, or prob- +abilities34, with delta-correlated fluctuations and static +(Loreti et al., 1995) or dynamic density profiles with front +and reverse shocks (Fogli et al., 2006). +Friedland and +Gruzinov (2006) argued for Kolmogorov correlated fluc- +tuations. +Kneller and Volpe (2010) evolved neutrino amplitudes +and built a statistical ensemble of instantiations for +the neutrino survival probabilities using one-dimensional +simulations and Kolmogorov fluctuations added. Retain- +ing the phase information, the approach revealed the +presence of multiple MSW resonances from turbulence +and a transition, when increasing the fluctuations ampli- +tude, from phase effects due to shock waves to a fluctua- +tions dominated regime. Lund and Kneller (2013) inves- +tigated the interplay between neutrino-neutrino interac- +tions, shock waves and turbulence using one-dimensional +dynamical simulations for three progenitors. These stud- +ies (Fogli et al., 2006; Friedland and Gruzinov, 2006; +Kneller and Volpe, 2010; Loreti et al., 1995) showed that +large amplitude fluctuations resulted into depolarization +of the neutrino probabilities. +Borriello et al. (2014) came to different conclusions. +The authors performed the first investigation exploiting +fluctuations from high resolution two-dimensional super- +nova simulations down to scales smaller than typical mat- +ter oscillation lengths35. These fluctuations followed bro- +ken power laws (with exponents 5/3 and 3)36 in agree- +ment with two-dimensional Kolgomorov-Kraichnan the- +ory of turbulence. Their analysis showed small damping +of the neutrino probabilities due to matter fluctuations +and absence of strong or full depolarization. +Clearly further work is needed to determine the impact +of turbulence on flavor evolution and to assess if matter +fluctuations introduce a loss of memory effects, or not. +F. MSW-like mechanisms +The MSW effect arises from the cancellation of the vac- +uum and the matter contributions. New resonance con- +ditions emerge from the interplay of the different terms +of the neutrino Hamiltonian Eq.(5) describing neutrino +propagation in a dense medium. Thus, various types of +MSW-like phenomena have been uncovered, in particu- +lar the matter-neutrino resonance, helicity coherence and +the I-resonance that I shall now discuss. +34 This gives a generalized Parke’s formula with a damping factor +(Burgess and Michaud, 1997). +35 Note that small scale fluctuations (less than 10 km) have smaller +scales than what can be resolved. +36 Three-dimensional simulations should bring turbulence spectra +with a Kolmogorov exponent of 5/3 at all scales. + +15 +FIG. 8 Matter-neutrino resonances: potentials of the neu- +trino Hamiltonian, as a function of distance, in an accretion +disk model of a black hole-neutron star merger. The νν inter- +action (Vν) and the matter (Ve) terms cross at the locations +of the symmetric and standard MNRs. The nutation region +is due to a cancellation between the νν interactions and the +vacuum (∆32) terms. The MSW region is also shown. (Figure +adapted from (Malkus et al., 2016).) +1. Matter-neutrino resonance +Accretion disks around compact objects – binary neu- +tron star merger remnants or black holes37 – produce +large amounts of neutrinos with luminosities and aver- +age energies similar to those of core-collapse supernovae. +An important difference is that, in these environments, +matter is neutron rich which produces an excess of the +¯νe flux over the νe one. Computationally, even the sim- +plest models require spherical symmetry breaking which +is numerically more involved. It is to be noted that, in +the context of core-collapse supernovae, spherical sym- +metry was assumed in numerous studies which yielded +interesting results. +In a collapsar type disk, Malkus et al. (2012) found a +novel conversion mechanism called the matter-neutrino +resonance (MNR). The MNR arises in regions above the +disk when the νν and ν-matter interactions cancel each +other Eqs.(6) and (12) (Figs. 8 and 9). Indeed, the excess +of the ¯νe flux over the νe one gives a different sign to the +two contributions. Moreover, because of the geometry of +the disks and the ¯νe decoupling deeper than νe, the Hνν +sign can flip (at some point). If the flip in sign is not +present the phenomenon is called standard MNR (Malkus +et al., 2014); whereas if it is present, the process is called +the symmetric MNR (Malkus et al., 2012, 2016) (Fig. 8). +Adiabatic evolution through the MNRs produces efficient +conversion of νe into νµ, ντ in the former; or of νe and ¯νe +in the latter. This can influence the electron fraction Ye +and favor disk wind nucleosynthesis of r-process elements +(Fig. 10). +37 From collapsing stars or from black hole-neutron star binaries +FIG. 9 Matter-neutrino resonances: the stars indicate the lo- +cations where the resonance condition is fulfilled along differ- +ent trajectories above a binary neutron star merger remnant. +(Figure from (Frensel et al., 2017).) +Zhu et al. (2016) and Frensel et al. (2017) showed +that patterns of flavor evolution depend on the neutrino +path38. Both studies were based on astrophysical inputs +from the detailed two-dimensional simulations of a binary +neutron star merger remnant by Perego et al. (2014). +Frensel et al. (2017) also found that the neutrino capture +rates on neutrons and protons for different initial condi- +tions and azimuthal angles showed variations by tens of +percent due to flavor mechanisms. +In these studies the flavor history of one neutrino is +taken as representative of all trajectories39. A consistent +treatment of the neutrino-neutrino interaction term of +the Hamiltonian Eq.(12) should implement the neutrino +evolution along different paths. Vlasenko and McLaugh- +lin (2018) showed that the MNRs take place even in a +more consistent treatment, leading to significant neutrino +conversion. +What is the underlying mechanism of the matter- +neutrino resonances? Wu et al. (2016), with a schematic +model, and Chatelain and Volpe (2017), with detailed +BNS simulations, showed the MNRs are multiple MSW +resonances. The crossing of the potentials shows the loca- +tion where the MNR starts (Fig. 8). Furthermore if one +looks at the full evolution of the Hamiltonian, the mat- +ter and the νν interaction terms cancel for tens of km, +conconmitantly with the process. Indeed, using a per- +turbative argument, Chatelain and Volpe (2017) showed +that the νν interactions adjust to the matter term over +long distances: the MNR condition is fulfilled multiple +times due to the non-linearity of the equations and non- +linear feedback. +38 First studies fixed the azimuthal angle θ to 45◦. +39 This is equivalent to the treatment, in the supernova context, of +νν interactions in the single-angle approximation (bulb model, +see section II.G). + +10-16 +symmetrie MNR +region +MNR +10-18 +region +I (erg) +Teglon +nutation +Potential +10-20 +MSW +10*22 +regicn +10-24 +105 +10° +10° +108 +10° +1010 +Position (cm)300 +1014 +250 +1013 +1012 +200 +1011 +109 +100 +108 +107 +50 +106 +105 +-150 +100 +50 +0 +50 +100 +150 +reyl [km]16 +FIG. 10 Nucleosynthetic abundances: the crosses show the +scaled solar abundances in comparison with predictions in a +black hole-accretion disk scenario. The lines correspond to +predictions in absence of neutrino oscillations (red), with os- +cillations (blue) or without the νν interaction (green). Visible +are the second and the third r-process peak as well as the rare +elements plateau in-between. The neutrino mass ordering is +normal. (Figure from (Malkus et al., 2012).) +2. Spin or helicity coherence +The derivation of extended mean-field equations be- +yond the ones usually employed in flavor studies Eq.(11) +uncovered new terms. Thanks to these, new resonances +become possible that can influence the neutrino content. +Volpe et al. (2013) derived mean-field equations in- +cluding pairing correlators and wrong-helicity contribu- +tions, due to the neutrino mass. +Afterwards Vlasenko +et al. (2014a) obtained quantum kinetic equations for +Majorana neutrinos, using the Closed-Time-Path formal- +ism, and pointed to wrong-helicity terms ∼ m/E naming +them spin coherence. Serreau and Volpe (2014) presented +the most general mean-field equations and called such +contributions helicity coherence. Present in anisotropic +media, they couple neutrinos with antineutrinos but are +suppressed, as expected, by the ratio m/E. +In a toy 2ν model Vlasenko et al. (2014b) first stud- +ied if helicity coherence modifies flavor. +The authors +found that it could trigger significant ν-¯ν transformation +through non-linear feedback. Motivated by such findings, +Chatelain and Volpe (2017) investigated these terms in +binary neutron star mergers with inputs from detailed +simulations. +In contrast with the previous findings, their results +showed that, while the resonance condition for helicity +coherence (similar to the MNR one) was fulfilled, adia- +batic evolution was absent for the ensemble of trajectories +considered. Indeed, the authors were able to show that +non-linear feedback could not induce multiple matching +of the resonance conditions40, contrarily to the MNR. +40 Unless peculiar matter density profiles are considered. Note that +The work of Tian et al. (2017) on the role of helicity +coherence in core-collapse supernovae reached the same +conclusion. +3. I-resonance +Non-standard interactions are present in theories be- +yond the Standard Model. +Limits on non-standard +neutrino-neutrino interactions are rather loose (Bilenky +and Santamaria, 1999); whereas oscillations and scat- +tering experiments provide tight constraints on non- +standard neutrino-matter interactions (NSI) (see for ex- +ample the reviews by Biggio et al. (2009); Davidson et al. +(2003); Farzan and Tortola (2018); and Ohlsson (2013)). +After decades of attempts, Akimov et al. (2017) (CO- +HERENT Collaboration) measured coherent neutrino- +nucleus scattering, giving, among others, new constraints +(Coloma et al., 2020; Giunti, 2020). +NSI are often evoked in the interpretation of neu- +trino oscillation experiments, as possible explanations +of anomalies. +If NSI exist, mixing angles and mass- +squared differences inferred by experiments are modified. +In dense astrophysical environments, NSI were studied +by several authors (Chatelain and Volpe, 2018; Esteban- +Pretel et al., 2007b, 2010; Fogli et al., 2002; Stapleford +et al., 2016): they significantly impact flavor evolution. +Moreover the role of non-standard neutrino-neutrino in- +teractions in supernovae was also studied by Blennow +et al. (2008). In the context of primordial nucleosynthe- +sis, NSI give a subleading contribution to the effective +number of degrees of freedom (Mangano et al., 2006). +Esteban-Pretel et al. (2007b, 2010) explored the com- +bined effect of νν interactions and ν-matter NSI in core- +collapse supernovae. For NSI couplings |ϵ| ≥ 10−2, the +I-resonance41, an MSW-like phenomenon, emerges. +It +takes place when the standard and non-standard matter +terms cancel each other, for ν and ¯ν simultaneously, and +independently from the neutrino energy. The I-resonance +triggers efficient conversions of νe → νµ,τ and ¯νe → ¯νµ,τ. +Stapleford et al. (2016) performed an extensive investi- +gation of NSI effects as a function of their couplings. The +authors found that, even for NSI strengths well below +bounds, NSIs produce (symmetric and standard) MNRs +in core-collapse supernovae and impact νν interactions +effects (in the bulb model) and the MSW-H resonance. +The first investigation of NSI effects in BNS was per- +formed by Chatelain and Volpe (2018) (Fig. 11). They +showed that neutrino-neutrino interactions play a role +on the I-resonance, contrarily to previous findings. In- +deed, when the νν interactions matter, the I-resonance +becomes a synchronized MSW effect. The investigation +the argument holds for supernovae as well. +41 I stands for ”internal”, since the phenomenon occurs close to the +neutrinosphere, in the most deleptonized inner layers. + +3 +4 +Y(A) +-6 +8 +100 +150 +200 +A17 +400 +200 +0 +200 +400 +x (km) +0 +100 +200 +300 +400 +500 +600 +z (km) +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +0.35 +0.40 +Ye +FIG. 11 I-resonances: locations where the I-resonance condi- +tion is fulfilled above a binary neutron star merger remnant. +Values of the electron fraction Ye, before flavor evolution, are +coded with colors. (Figure from (Chatelain and Volpe, 2018).) +of a large ensemble of trajectories, above a BNS remnant, +uncovered that, as in core-collapse supernovae, even very +small values of NSI parameters produce intricate patterns +of mechanisms, including MNRs, I- and synchronized I- +resonances. +G. Neutrino-neutrino interactions +Dense environments have sizable νν standard neutral- +current interactions because neutrinos are emitted in +large amounts. +In a seminal work, Pantaleone (1992) +pointed out that the νν interactions introduce off- +diagonal potentials42 and make ν evolution a non-linear +many-body problem. +In the last fifteen years theorists have tirelessly worked +to understand the neutrino-neutrino refraction effects, +the novel flavor mechanisms, how they arise and their +impact. They have established connections with other +many-body systems and figured out new approaches to +deal with such interactions. Several reviews dress a de- +tailed picture of these developments (Duan et al., 2010; +Duan and Kneller, 2009; Horiuchi and Kneller, 2018; Mi- +rizzi et al., 2016; Tamborra and Shalgar, 2021; Volpe, +2015). Here I shall highlight aspects emerged from the +efforts to solve this complex problem. +Flavor mechanisms due to νν interactions are cur- +rently classified as slow or as fast. +Slow modes occur +at typical distances of O(102-103) km from the neutri- +nosphere, whereas fast modes have scales of O(1) m or +much less and frequencies as large as µ ∼ +√ +2GFnν, nν +being the neutrino number density. Their rate can ex- +ceed the vacuum oscillation frequency by large factors, +e.g. µ/ω = 105. +42 The νν Hamiltonian has off-diagonal complex contributions, be- +cause of its dependence on the neutrino and antineutrino density +matrices Eq.(12) (Fig. 3). +FIG. 12 Neutrino-neutrino interactions in the bulb model: ge- +ometry of the neutrino emission from a newly formed proto- +neutron star (of radius R) in a core-collapse supernova ex- +plosion. The bulb model assumes both spherical symmetry +and azimuthal symmetry along the radial direction r. Each +neutrino state is characterized by the the momentum p and +the angle θR, the angle θ at the intersection point between a +neutrino emitted at θR (red line) and a radially propagating +neutrino (dashed line). The blue lines show the maximal angle +that contributes to the νν interaction Hamiltonian. (Figure +adapted from (Duan et al., 2010)). +1. Slow modes +Already twenty years ago Samuel (1993) showed that +νν interactions stimulated new flavor effects. Studies in +the cosmological context (see for example (Kostelecky +et al., 1993; Kostelecky and Samuel, 1995; Pastor et al., +2002)) uncovered a stunning mechanism where neutrino +spins ”sticked together”, precessing collectively around +an effective magnetic field. +Synchronized oscillations +were included for example in investigations of cosmologi- +cal neutrino-antineutrino asymmetries (Abazajian et al., +2002; Dolgov et al., 2002), also with CP violation (Gava +and Volpe, 2010). +Duan et al. (2006b) uncovered collective flavor modes43 +in supernovae, due to νν forward scattering. +Using +the bulb model (Fig. +12) Duan et al. (2006a) per- +formed ”single-angle” and demanding ”multi-angle” sim- +ulations44 which showed large-scale modes and spectral +splits45 (Duan et al., 2006a)46. Moreover, Balantekin and +43 The modes were named slow after the identification of fast modes +(see Section G.II). +44 Computationally, one can treat Hνν Eq.(12) in two ways. In the +simplified ”single-angle” approximation, the flavor history of a +neutrino, at a given angle with respect to the radial direction, +is representative of all angles. (Authors took π/4 and often 0◦, +which strictly speaking corresponds to ”non-interacting” neutri- +nos.). In contrast, multi-angle simulations include the full angu- +lar dependence of the νν potential. Concerning binary compact +objects, in which calculations are particularly challenging, it is +also common to treat Hνν as in ”single-angle” approximation +(see for example (Frensel et al., 2017; Malkus et al., 2014, 2012, +2016; Zhu et al., 2016)). +45 Splits are sharp boundary features at the edges of spectral swap +intervals. +46 Note that in these early works, the neutrino-neutrino and matter +number densities were such that self-induced flavor conversion +was intertwined with matter effects. + +PNS +R18 +FIG. 13 Bulb model: example of the neutrino spectral swap- +ping in supernovae. Due to νν interactions, the initial (quasi- +thermal) νe and νµ spectra undergo significant modification. +As the figure shows, there is a critical energy (Ecrit = 8 MeV) +for which, if E < Ecrit the νe flux is unchanged, whereas +for E > Ecrit the νx and the νe fluxes interchange. (Figure +adapted from (Duan et al., 2010)). +Yuksel (2005) found that the neutrino-neutrino refraction +impacted the equilibrium electron-fraction Ye, important +for r-process nucleosynthesis in neutrino-driven winds. +For several years the bulb model has received peculiar +attention. Simulations showed puzzling flavor behaviors +which triggered intense theoretical work. Named collec- +tive neutrino oscillations, these phenomena occur for siz- +able νν interactions and for non-zero mixings (even for +extremely small values of θ). +Three regimes emerge when neutrinos travel from the +neutrinospheres, where the neutrino number densities are +large, to regions where matter dominates. +They are +called the synchronization, the bipolar instability (Duan +et al., 2006a, 2007a, 2006b; Hannestad et al., 2006) and +the spectral splits (Dasgupta et al., 2009; Duan et al., +2006a; Fogli et al., 2007; Galais et al., 2012; Raffelt and +Smirnov, 2007) (Fig. 13). +The spin formalism in flavor space gives an image of +these three phases. First, as in the early universe at the +epoch of primordial nucleosynthesis, neutrino spins syn- +chronize in a stable collective mode. Second, neutrino +spins experience an instability where νe¯νe pairs convert +into νx¯νx ones, due to lepton-number conservation. They +perform precession and nutation around B and behave +like a pendulum (Duan et al., 2007a, 2006b, 2007b), or +a gyroscopic pendulum (Hannestad et al., 2006). Third, +they undergo either full or no conversion, depending on +the neutrino energy, generating spectral swapping and +splits. A pictorial image of these modes for 3ν flavors +was given in the e3-e8 triangle diagram by Dasgupta et al. +(2009) who employed Bloch vectors and the SU(3) alge- +bra. +Other approaches to these phenomena brought further +insight. Using the matter basis, Galais et al. (2012) found +that with the νν refraction the adiabaticity parameters +depend on the matter angle and phase derivative and +bipolar oscillations start when the latter diverges. More- +over, Galais and Volpe (2011) showed that spectral splits +arise from a magnetic resonance phenomenon: the swap- +ping emerges because the spins satisfy (or do not satisfy) +a magnetic resonance condition, depending on the neu- +trino energy. +Pehlivan et al. (2011) used a different angle of attack. +With an algebraic many-body approach and the Bethe +ansatz, they demonstrated that the splits emerged in the +transition from a quasi-particle to a particle description. +In a subsequent study Pehlivan et al. (2017) established +that the emergence of the splits, from the regions where +neutrinos strongly interact to those where they weakly +interact, is similar to the behavior of Cooper pairs in +the BEC-BCS crossover in experiments with ultra cold +atomic gases (Fig. 14). +While first studies revealed instabilities only in in- +verted mass ordering, Dasgupta et al. (2009) showed +that more plausible ratios of the νe, ¯νe, νx fluxes (i.e. +different from one) produced single and multiple spec- +tral splits in both hierarchies. Moreover, Esteban-Pretel +et al. (2008) argued that large matter densities intro- +duced decoherence in multi-angle calculations because +of the angular factor (1 − ˆp · ˆp′) in Eq.(12). Following +these findings, Chakraborty et al. (2011b,c) showed, us- +ing one-dimensional supernova simulations, that matter +suppressed collective effects when the matter exceeded +the νν number density. This finding should be confirmed +by multi-angle calculations using multi-dimensional su- +pernova simulations. +The studies assumed stationarity and homogeneity of +the medium where neutrinos propagate. Moreover, they +used the mean-field approximation and neglected col- +lisions. +Thanks to these approximations, the full 7- +dimensional problem of neutrino flavor evolution reduces +to a more tractable one, typically in two- or three- +dimensions, i.e. (E, r) or (E, r, θ). But even schematic +models with a reduced number of degrees of freedom are +often quite challenging to solve. +A step towards higher spatial dimensionality was pro- +vided by the so-called neutrino line model with two spa- +tial dimensions, either with only two neutrino beams +from each initial condition (Duan and Shalgar, 2015), +or with multi-angles at each point source (Abbar et al., +2015), showing inhomogeneous modes located at larger +neutrino densities than homogenous ones. +Abbar and +Duan (2015) and Dasgupta and Mirizzi (2015) identi- +fied temporal instabilities arising in non-stationary mod- +els since time can cancel a (albeit) constant matter term, +producing instabilities (quite) deep in the supernova. +Moreover, Cherry et al. (2012, 2013) uncovered that a +neutrino halo – a small amount of back-scattered neutri- +nos due to collisions – could completely reshape the fla- +vor patterns produced by forward scattering. This find- +ing cast doubts on the treatment of νν interactions as an +initial value problem and showed limitations of the mean- + +a +Y. initial +, initial +Y. final + final +20 +40 +60 +E (MeV)19 +FIG. 14 Neutrino-neutrino interactions in a supernova: cor- +respondence between neutrinos propagating from dense to di- +lute regions and the BEC to BCS limits in ultra cold atomic +gases. (Figure from (Pehlivan et al., 2017)). +field approximation. The halo effect was further studied +by Sarikas et al. (2012b) and Cherry et al. (2020). +Furthermore, Raffelt et al. (2013) found, in a linearized +study of the bulb model, azymuthal-angle-instabilities, +showing that solutions do not necessarily inherit the sym- +metries of the initial or boundary conditions (unless en- +forced). +This fact was dubbed spontaneous symmetry +breaking. Several works (Chakraborty et al., 2014; Mi- +rizzi, 2013) confirmed symmetry breaking solutions with +linearized analysis in supernovae. +Flavor evolution can even reveal chaotic behaviors. +This was mentioned by Raffelt and de Sousa Seixas +(2013) in a stationary model with two opposite neutrino +momenta. And, in fact, Hansen and Hannestad (2014) +clearly identified in the same model the exponential di- +vergence of Liapounov exponents of infinitely close initial +trajectories of the neutrino spin vectors. +So, after the discovery of novel phenomena in the bulb +model, models increased in complexity and included e.g. +non-stationarity, inhomogeneities, unconstrained sym- +metries and more. +These in-depth investigations kept +uncovering new features and the richness of flavor evo- +lution in dense media, due to the νν refraction. While +theorists thought they were shaping a solid understand- +ing, fast modes arrived ”on the stage”, triggering another +”runaway” of studies . . . +2. Fast modes +In a 3ν two-beam model Sawyer (2005, 2009) found +that νν interactions could ”speed-up” flavor transfor- +mation and produce counterintuitive modifications on a +short time scale of t = (2 +√ +2GFnν)−1 of order O(1) m. +Much later Sawyer (2016) considered non-trivial angu- +lar distributions at the neutrinospheres, with ¯νe emitted +deeper than νe. He identified modes with nm scale. +Sawyer’s findings triggered excitement again: +fast +modes took place close to the neutrinosphere. +They +could influence the supernova dynamics and nucleosyn- +thesis. Theorists had found the short scale modes they +were looking for, finally! +Contrarily to slow modes, fast modes have the pecu- +liarity that they are not triggered neither by the mix- +ings Eq.(3) nor by the matter contribution Eq.(6). Since +only the neutrino emission matters, Izaguirre et al. (2017) +introduced the angle distribution of the electron lepton +number (ELN) which for 2ν flavors reads +Gv = +√ +2GF +� ∞ +0 +dE E2 +2π2 [φνe(E, v) − φ¯νe(E, v)] . (28) +As indicated by the results of Sawyer (2016) and +pointed out by Dasgupta et al. (2017) and Izaguirre et al. +(2017), fast modes take place when the angular distribu- +tions of νe and ¯νe cross each other along a given direction +(i.e. Gv changes sign): this is an ELN crossing. Its pres- +ence is a necessary but not a sufficient condition for fast +modes to occur. +In the last years many studies of fast modes have been +realized based on the linearized approach (see for exam- +ple (Abbar and Duan, 2018; Abbar et al., 2019, 2020; +Capozzi et al., 2017; Chakraborty and Chakraborty, +2020; Chakraborty et al., 2016a,b; Delfan Azari et al., +2019; George et al., 2020; Padilla-Gay et al., 2021; Shal- +gar et al., 2020; Tamborra et al., 2014a; Wu and Tam- +borra, 2017; Xiong et al., 2020) and Tamborra and Shal- +gar (2021) for a review). +It was already advocated by Sawyer (2005) that fast +modes could bring flavor equilibration of the different +neutrino species. The ansatz has been admitted in the +literature for some time. +In fact it has the advantage +that it simplifies the problem since the neutrino spectra +emerge identical from a supernova. However, in a two- +beam model, Abbar and Volpe (2019) evolved for the first +time fast modes in the full non-linear regime, showing +they do not necessarily lead to flavor equilibration. +Fast modes behave differently in three-flavors with re- +spect to two-flavors. For the former the concept of ELN +crossing needs to be generalized to µLN and τLN cross- +ings. Chakraborty and Chakraborty (2020) first inves- +tigated such effects with the dispersion relation treating +both time and space. The authors pointed out the im- +portance of three-flavor effects. The further analysis of +Capozzi et al. (2020), that went up to the non-linear +regime, showed instabilities over tens of ns scale which +were either absent (in 2ν flavors), or got damped. Clearly +these findings emphasize the need for three-flavor analy- +sis of fast modes. +Since the scale for fast modes is so short, there can +be regions requiring the treatment of flavor mechanisms +and collisions when the medium is very dense. Indeed +the fast rate exceeds the collision rate even within a su- +pernova core. Using a one-dimensional model, with two +momentum modes, Capozzi et al. (2019) analyzed the +interplay between collisions and fast modes and showed +that collisions can trigger the conditions for fast conver- +sions. Significant efforts are made to study flavor con- +version modes in presence of collisions (see for example + +BCS +BEC +Weakly interacting regime +Strongly interacting regime +V +Proto-neutron +star +V.20 +FIG. 15 Fast modes in multidimensional supernova simula- +tions: Mollweide projection of the ¯νe-over-νe ratio at a dis- +tance r = 65.6 km in a snapshot at post-bounce time of 200 +ms of a 3D supernova model. The white crosses indicate the +location where fast modes occur, based on a linearized anal- +ysis. (Figure from (Abbar et al., 2020)). +Hansen et al. (2022); Richers et al. (2019); and Xiong +et al. (2022))). It is necessary to keep in mind, though, +that a consistent treatment in multidimensional simula- +tions is numerically very challenging and therefore far +ahead. +Fast modes are present in schematic models, but do +they occur in detailed supernova simulations? The first +investigation by Tamborra et al. (2017), based on one- +dimensional simulations, concluded for the absence of +fast modes. Searches in multidimensional simulations re- +vealed the presence of fast modes, in contrast with this +early finding. +Abbar et al. (2020) identified fast-growing modes +in two- and three-dimensional simulations when α = +nνe/n¯νe is of the order of 1. A linear stability analysis +confirmed the presence of fast modes in correspondence +with the angular crossings (Fig. 15), even deep in the +supernova core. +Their influence on the neutrino spec- +tra was found to be small, since the neutrino spectra are +already very similar at the location of the crossings. +Glas et al. (2020) and Delfan Azari et al. (2020) also +found ELN crossings nearby the neutrinosphere and con- +firmed the presence of fast modes in detailed supernova +simulations (with full Boltzmann transport) in three- and +two-dimensions respectively. It is to be noted that multi- +dimensional supernova simulations do not provide full in- +formation of the neutrino angular distributions as a func- +tion of time. Therefore methods have been developed, as +in Dasgupta et al. (2018), to identify fast modes using +the moments of the angular distributions. +Beyond the studies in the supernova context, Wu and +Tamborra (2017) performed the first analysis of fast +modes in accretion disks resulting from binary compact +objects mergers. +They found the conditions for fast +modes to be generically met because of the excess of ¯νes +over νes and of the geometry of such environments. +As for nucleosynthesis, Xiong et al. (2020) studied the +influence of fast oscillations in neutrino-driven winds in +a low and a high mass core-collapse supernova. +They +showed that partial (or total) flavor equilibration creates +more proton-rich conditions (Ye > 0.5) enhancing the νp +process and mass ejection. Wu et al. (2017) considered +the impact of fast modes on the r-process in a neutrino +driven wind nearby a black hole remnant from compact +binary mergers. Under the approximate assumption of +flavor equilibration, fast modes produced an increase of +lanthanides (more generally nuclei with A > 130) up to +a factor of 103, due to the decrease of Ye, showing a +potentially high impact on kilonova light-curves. +On the contrary, in an analysis for an hyper-massive +BNS merger remnant, George et al. (2020) finds lan- +thanides to be little affected by fast pairwise conversion +(under the same equilibration hypothesis). +The subject of fast modes and their impact undergoes +a fast development where interesting aspects keep being +uncovered. +Clearly, these short scale modes will keep +attracting attention in the coming years. +III. FLAVOR EVOLUTION: THEORETICAL +FRAMEWORKS +Neutrinos propagating in a dense environment consti- +tute a unique, weakly interacting, many-body system. Its +description benefits of a lucky situation in some respects, +since one does not have to deal with phenomenological +interactions as for atomic nuclei. It is a specific case also, +because theoretical approaches developed for many-body +systems need to be extended to particles with mixings. +The literature is rich with theoretical approaches for +this system (see the review by Volpe (2015)). These range +from the mean-field and the extended mean-field equa- +tions, the linearized equations and a dispersion relation +approach to the neutrino quantum kinetic equations, as +I shall now describe. +A. The mean-field approximation +The mean-field constitutes the simplest approximation +to describe neutrino propagation in astrophysical envi- +ronments. +Neutrino mean-field equations were derived +by several authors (Balantekin and Pehlivan, 2007; Fried- +land and Lunardini, 2003b; Samuel, 1993; Sawyer, 2005; +Serreau and Volpe, 2014; Sirera and Perez, 1999; Volpe +et al., 2013; Yamada, 2000). +It is common to determine neutrino propagation using +the formalisms of Green’s functions, of density matri- +ces, neutrino (iso)spins, or of neutrino amplitudes. The +density matrix formalism is widely used. The ν and ¯ν +one-body densities are defined as +ϱ1,ij(p, h, p′, h′) = ⟨a† +j(p′, h′)ai(p, h)⟩ , +¯ϱ1,ij(p, h, p′, h′) = ⟨b† +i(p, h)bj(p′, h′)⟩ , +(29) +where the quantum expectation value are over the astro- +physical or cosmological background; i, j ∈ [1, N] with +N the number of neutrino families. +The diagonal ele- +ments of the one-body density matrix correspond to the +expectation value of the number operator and are the + + 65.6 km +1.0 +0.821 +only contributions for particles without mixings. +The +off-diagonal elements (i ̸= j) implement coherence due +to the mixings. Sometimes it is said that neutrino evo- +lution requires a ”matrix of densities” quoting Sigl and +Raffelt (1993). +A simple way to derive mean-field evolution equations +is through the Ehrenfest theorem +i ˙ϱ1,ij = ⟨[a† +jai, H]⟩ , +(30) +where H is the neutrino Hamiltonian Eq.(5). +The mean-field approximation consists in neglecting +the correlated part of the two-body density47 +ϱ12 = ϱ1ϱ2 − c12 , +(31) +i.e. setting c12 = 0. Since only the uncorrelated part +is retained, the two particles propagate independently. +Thus the full many-body system evolves as made up of +independent particles. +Another way to define the mean-field approximation is +saying that one replaces two-body interaction terms by +one-body ones as +I1I2 → I1⟨I2⟩ + ⟨I1⟩I2 + ⟨I1⟩⟨I2⟩ . +(32) +If one takes the charged- or the neutral-current interac- +tion terms of the GWS model, the corresponding mean- +field Hamiltonian has the general bilinear form (Serreau +and Volpe, 2014) +HMF(t) = +� +dx ¯ψi(t, x)Γij(t, x)ψj(t, x), +(33) +where Γij(t, x) is the interaction kernel that depends on +the specific interaction terms that one considers, ψi de- +notes the neutrino field in the mass basis (for the ith +mass eigenstate)48 +ψj(t, x) = +� +h +� +p +[uj(p, h)aj(p, h)e−ip·x+vj(p, h)b† +je+ip·x] +(34) +with p · x = pµxµ and uj(p, h) and vj(p, h) the four- +components complex spinors, solution of the Dirac equa- +tion. +In the mean-field approximation, the interaction +kernel is quadratic (and not quartic) in the creation and +annihilation operators. +A theoretical framework to treat the evolution of +many-body systems is given by the Born and Green +(1946), +Bogoliubov (1946), +Kirkwood (1935), +Yvon +(1935) (BBGKY) hierarchy. The hierarchy, introduced +47 Note that the (reduced) density matrices are also referred to +as one-particle, two-particle, . . . , instead of one-body, two-body +and so on. +48 Here we write down expressions considering neutrino Dirac fields. +The generalization to Majorana fields is straightforward. +for a non-relativistic many-body system, replaces the ex- +act evolution of the quantum many-body system by a +hierarchy of integro-differential equations for n-body den- +sity matrices49 +ϱ1...n = ⟨a† +n . . . a† +1a1 . . . an⟩ +(35) +that can be truncated at different levels. The mean-field +approximation corresponds to truncating the hierarchy +at the lowest level, i.e. assuming c12 = 0 in Eq.(31). +While BBGKY was originally for a non-relativistic +many-body system, Calzetta and Hu (1988) generalized +it for relativistic many-body systems which involves, in +particular, an infinite hierarchy of equations. +B. The mean-field Hamiltonian: a derivation +To determine the contribution from a mean-field +Hamiltonian, one has to add the scattering amplitudes +for the corresponding scattering process as +Vkr(ρ) = +� +s,p +v(kp,rs)ρsp , +(36) +and sum (or integrate) over the background. The quan- +tity ρsp is +ρsp = ⟨a† +pas⟩ , +(37) +where k, p, r, s are, each, a set of single-particle in- +dices like (⃗p, h), characterizing the single-particle neu- +trino states of a Fock space. +Let us derive the mean-field term of the matter Hamil- +tonian, associated with a tadpole diagram (Fig. 3) as an +example. The charged-current interaction term associ- +ated with νe-e scatting reads +HCC =GF +√ +2[ ¯ψνe(t, x)γµ(1 − γ5)ψe(t, x)] +× [ ¯ψe(t, x)γµ(1 − γ5)ψνe(t, x)] . +(38) +The first step in determining Eq. (36) is to evaluate +the matrix elements +v(kp,rs) ≡ ⟨k, p|HCC|r, s⟩ . +(39) +More explicitly one needs to calculate +v(kp,rs) =GF +√ +2⟨νe, e| +� +dx [ ¯ψe(t, x)γµ(1 − γ5)ψe(t, x)] +× [ ¯ψνe(t, x)γµ(1 − γ5)ψνe(t, x)]|νe, e⟩ , +(40) +where the Fierz transformation has been applied to +Eq.(38). +49 Note that Wang and Cassing (1985) reformulated the hierarchy +as a set of equations for n-body correlation functions. + +22 +By introducing the Fourier expansions of the electron +and the neutrino quantum fields Eq.(34), one can simplify +the matrix element using the general relation (from the +equal-time anti-commutation rules) +⟨e(1)|a†(2)a(3)|e(4)⟩ =(2π)3δ3(p1 − p2)2Ep1δh1,h2 +(2π)3δ3(p3 − p4)2Ep3δh3,h4 , +(41) +where here the labels (1,2,3,4) stand for a set of single +particle quantum numbers (p, h) for the two incoming +and the two outgoing particles. Using this relation one +gets +v(kp,rs) =GF +√ +2 +� +dx [¯uνe(k′)γµ(1 − γ5)uνe(k)] +× [¯ue(p′)γµ(1 − γ5)ue(p)]ei(p+k−p′−k′)·x , +(42) +where the first two factors in the integral depend on +spinorial products, whereas the last one ensures momen- +tum conservation. +The second step to determine the mean-field term is +to evaluate +V (ρ) =GF +√ +2 +� +he,h′e +� +pe,p′e +� +dx [¯uνe(k′)γµ(1 − γ5)uνe(k)] +× [¯ue(p′)γµ(1 − γ5)ue(p)]ei(p+k−p′−k′)·x +× ρ(pe,he,p′e,h′e) , +(43) +and perform the integration over the degrees of freedom +of the electron background (at finite temperature T). +One very often makes the following ansatz +ρ(pe,he,p′e,h′e) = ⟨ψ(T)|a† +e(p′ +e, h′ +e)ae(pe, he)|ψ(T)⟩ += (2π)3δ3(pe − p′ +e)δhe,h′e2Epρp , +(44) +that is one assumes that the background particles are +uncorrelated (independent) and that the medium is ho- +mogeneous. This hypothesis corresponds to considering +forward-scattering only, where the electrons (and there- +fore neutrino) momenta are unchanged. +By plugging Eq.(44) into Eq. (43), the spinorial prod- +ucts in Eq. (43) can be evaluated thus giving +8pµ(kµ − msµ) = 8EpEk(1 − ˆp · ˆk)(1 − hν) , +(45) +where we have introduced the 4-vector +sµ = hν +�|k| +m , Ek +m|k| +� +, +(46) +and already imposed momentum conservation. Therefore +one gets +V (ρe) =GF +√ +2 (2π)3δ3(k − k′)8Ek(1 − hν) +� +p +(1 − ˆp · ˆk)ρp , +=2a +√ +2GF +� +dp +(2π)3 (1 − ˆp · ˆk)ρp , +(47) +with50 a = (2π)3δ3(k − k′)Ek(1 − hν). +If the medium is isotropic the angular dependence in +Eq.(47) averages out. Since the total number of electrons +in the medium is given by +Ne ≡ +� +he +� +p +⟨a† +e(p, he)ae(p, he)⟩ = 2V +� +dp +(2π)3 ρp , (48) +equation (47) becomes +V (ρe) = +√ +2GFne +(49) +where ne = N/V is the electron number density. This +is the well known mean-field Hamiltonian responsible for +the MSW effect in matter Eq. (6). +Following the procedure just outlined, one can derive +any mean-field contribution to the neutrino Hamiltonian, +such as the those coming from νν interactions, or from +NSI. +C. Beyond the usual mean-field +The mean-field equations Eq.(11) with vacuum mix- +ings, the standard charged- and neutral-current ν-matter +and νν interactions have been widely used in studies of +flavor evolution in dense astrophysical environments. A +few works explored extensions of such equations to estab- +lish the robustness of the mean-field approximation and +the possible necessity to go beyond. +It was pointed out by several authors that the Hamil- +tonian with νν interactions is analogous to an interacting +system of spins that have a spin-exchange interaction and +feel an external magnetic field. Balantekin and Pehlivan +(2007) provided a derivation of the mean field equations +using the algebraic formulation of the neutrino Hamilto- +nian51 +H = +� +ω +ωB0 · Jω + µ +� +p,q +(1 − cos θpq′)Jp · Jq , +(50) +where the last terms depends on the generators of SU(2) +algebra(s)52. +The authors used a coherent-state path- +integral approach and showed that the mean-field equa- +tions correspond to the saddle point approximation of the +path-integral for the full many-body system. Moreover, +they pointed out contributions beyond the mean-field as +corrections to the saddle-point solution. +50 Note that the factor a goes away when calculating the neutrino +mean-field evolution equations. +51 Here µ = +√ +2GF/V . +52 They depend on the operators J+(p) = a† +x(p)ae(p), J−(p) = +a† +e(p)ax(p), J0(p) = +1 +2 +� +a† +x(p)ax(p) − a† +e(p)ae(p) +� +, which sat- +isfy the commutation relations [J0(p), J±(p)] += +±δ3(p − +q)J±(p) and [J+(p), J−(q)] = 2δ3(p − p)J0(p). + +23 +Volpe et al. (2013) used the BBGKY hierarchy to de- +rive mean-field equations53for the ν and ¯ν one-body den- +sity matrices Eqs. (29) in a dense astrophysical environ- +ment. +Moreover, thanks to the hierarchy, the authors +pointed out that the neutrino evolution equations had +further terms at the mean-field level, namely two-point +correlators from wrong-helicity contributions due to neu- +trino masses and from pairing (or abnormal) densities54. +For Dirac neutrinos the latter read +κij(t, q, h, q′, h′) = ⟨bj(t, q, h′)ai(t, q, h)⟩ , +(51) +and their hermitian conjugates. +Volpe et al. (2013) +showed that in presence of pairing correlators one can +cast the extended mean-field evolution equations, simi- +larly to Eq. (11) as +i ˙R = [H, R] . +(52) +The quantities H and R are the generalized Hamilto- +nian and density that includes both the ν and ¯ν density +matrices. By introducing a Bogoliubov transformation, +such a system of ν and ¯ν with pairing correlators can be +described in terms of independent quasi-particles (Vaana- +nen and Volpe, 2013). +Furthermore, in their derivation of neutrino quantum +kinetic equations, Vlasenko et al. (2014a) pointed out +contributions from the correlators, named spin coherence +ζij(t, q) = ⟨a† +j(t, q, +)ai(t, q, −)⟩ . +(53) +They are due to the neutrino mass and suppressed by the +factor m/E. +Serreau and Volpe (2014) derived the most general +mean-field equations for inhomogeneous and anisotropic +media considering Dirac as well as Majorana neutrinos. +Such equations include contributions either from pair- +ing or from wrong-helicity correlators – helicity coher- +ence. Using the approach of Serreau and Volpe (2014), +Kartavtsev et al. (2015) also included contributions from +neutrino electromagnetic properties. +Spin or helicity coherence requires anisotropy of the +medium to be non-zero. The corresponding generalized +Hamiltonian can again be cast in the form Eq. (52) but +this time it has both flavor and helicity structure (Serreau +and Volpe, 2014; Vlasenko et al., 2014a). Helicity coher- +ence couples ν with ¯ν, i.e. active and sterile neutrinos if +ν are Dirac particles, or neutrinos and antineutrinos if ν +are Majorana particles. +53 In the context of atomic nuclei, the neutrino mean-field equa- +tions correspond to the so-called Time Dependent Hartree-Fock +approximation. If the initial state for the many-body system is a +Slater determinant, it remains a Slater determinant at all times +(Ring and Schuck, 2004). +54 Sigl and Raffelt (1993) mentioned such correlations but dis- +carded them. Note that neutrino-antineutrino correlations were +included in the neutrino evolution equations in the context of +baryogenesis via leptogenesis by Fidler et al. (2012). +The impact on flavor evolution of the supplementary +terms from the correlators (51) and (53) was investigated +as well. Kartavtsev et al. (2015) pointed out that the +pairing correlators do not influence flavor because the +large kinetic contributions cannot be removed55. +This +fact makes the influence of the off-diagonal contributions +from pairing correlators tiny. +Concerning helicity coherence (see section F.II), simu- +lations in 3ν flavors with detailed astrophysical inputs +from binary neutron star merger remnants (Chatelain +and Volpe, 2017) or supernovae (Tian et al., 2017) showed +that non-linear feedback does not operate in detailed set- +tings. As a consequence, helicity coherence does not seem +to influence the neutrino flavor in media as for now. +D. Linearization +Linearization is a widespread approach. It is used in +many domains of physics, e.g. in nuclear physics, con- +densed matter or in hydrodynamics. The linearization +procedure transforms the solution of the equations of +motion into eigenvalue equations, making the numerical +problem more tractable. +The first application of linearization to the neutrino +mean-field equations in a supernova was done by Sawyer +(2009). Afterwards, Banerjee et al. (2011) derived a lin- +earized version of the equations of motion in the bulb +model. Since then, the procedure has been widely em- +ployed in the study of both slow and fast modes. Vaana- +nen and Volpe (2013) provided an alternative derivation +of the linearized equations, by generalizing the random- +phase-approximation (RPA) commonly used in the study +of atomic nuclei. Subsequently Izaguirre et al. (2017) re- +cast the linearized equations in a dispersion relation ap- +proach. +1. The linearised equations +Let us have a closer look at the linearized version of +the equations of motion for supernova neutrinos. Here we +follow Banerjee et al. (2011) and consider the bulb model, +which includes νν neutral-current interactions (section +II.G.1). For 2ν flavors, the neutrino flux matrices can be +rewritten as +ρℓ,r = φνe,ℓ,r + φνx,ℓ,r +2 ++ gℓ +� sℓ,r +Sℓ,r +S∗ +ℓ,r - sℓ,r +� +, +(54) +where ℓ = (ω, u), Sℓ,r is an Hermitian matrix and +gℓ,r = (φνe,ℓ,r − φνx,ℓ,r) +2 +. +(55) +55 Usually, the diagonal contributions, proportional to the identity +matrix, are substracted. They do not impact neutrino flavor. + +24 +FIG. 16 Linearisation: example of the solution of the eigen- +value equations for neutrino Fermi-Dirac distributions at the +neutrinosphere (upper figure). The positive frequencies corre- +spond to neutrinos, whereas the negative ones to antineutri- +nos. The quantity κ1, as a function of the νν interaction cou- +pling constant µ, is the imaginary part of one of the two un- +stable solutions (lower figure). (Figure adapted from Banerjee +et al. (2011).) +The elements of Sℓ satisfy the normalization condition +s2 +ℓ,r + |Sℓ,r|2 = 1. The quantities (1 + s)/2 are the sur- +vival probabilities. The indices (ω, u, r) are the vacuum +oscillation frequency ω = ∆m2/2E, u = sin2 θR with +u ∈ [0, 1] characterizes the neutrino emission at the neu- +trinosphere (at r = R), r is the distance defining the +νν intersection point along the symmetry direction (Fig. +12). +In the linearization procedure one considers the ini- +tial state as ”quasi-static” and performs small variations +around it. In our case this amounts to considering the +initial states at the neutrinosphere Sℓ,R = diag(1, −1) +and ρℓ,R = diag(φ0 +νe, φ0 +νx) in Eq. +(54) and consider a +small amplitude approximation, i.e. +sℓ,r ≃ 1 +|Sℓ,r| ≪ 1. +(56) +Under this hypothesis, the neutrino mean-field equations +Eq.(11) with the contributions from vacuum Eq. +(3), +matter Eq. (6), and νν interaction Eq. (12) become +i∂rSℓ,r = [ω + u(λ + ϵ)] Sℓ,r +− µr +� 1 +0 +du′ +� +∞ +−∞ +dω′(u + u′)gℓ′Sℓ′,r , (57) +when r ≫ R. In Eq.(57) the third term on the r.h.s. is +the total lepton number ϵ = +� +du dω gℓ,r (normalized +to φ0 +¯νe); whereas the second and the last terms on the +r.h.s. are the matter and the νν terms, with the following +coupling constants +λ = +√ +2GF(ne − n¯e) +µr = +√ +2GFφ0 +¯νeR2 +8πr4 +. +(58) +One seeks for solutions of Eq. (57) of the type +Sℓ,r = Qℓ,re−iΩr , +(59) +which leads to the eigenvalue equations +[ω + (λ + ϵ)u − Ω]Qℓ,r = +−µr +� 1 +0 +du′ +� +∞ +−∞ +dω′(u + u′)gℓ′Qℓ′,r . +(60) +If the eigenvalue Ω ∈ ℜ, the initial condition is stable +and the system performs small oscillations around it. If +Ω ∈ C, one faces an instability in flavor space56: the +system deviates exponentially from the initial state. This +is often called a runaway solution. Figure 16 gives an +example of the application of linearized equations in the +supernova context. +Therefore, a complex eigenvalue indicates the start of +flavor modification when neutrinos depart from the neu- +trinosphere. However, it is worth emphasizing that lin- +earization does not provide any information on the full +non-linear regime. +Indeed the linearized equations are +inherently based on the small amplitude approximation +Eq.(56). Only the full numerical solution of the equa- +tions of motion tells us how significant flavor conversion +is, at large scales. +In the study of atomic nuclei or metallic clusters, lin- +earized equations are obtained with RPA. With this ap- +proach one determines small variations of the matter den- +sity around the initial state. If the eigenvalues are real +that indicates that the initial state is a true ground state; +if they are complex, then the initial state is not a ground +state of the system. The latter situation is in fact what +one looks for in the neutrino case. Also, in RPA, one +can face ”spurious” solutions that are numerical artifacts +(see for example (Ring and Schuck, 2004)). These were +also found in the neutrino context e.g. when multi-angle +calculations of the νν interaction do not include a suffi- +ciently large number of angle bins (Sarikas et al., 2012a). +Here is how Vaananen and Volpe (2013) generalized +RPA to neutrinos. At initial time, the density matrices +ϱ and ¯ϱ correspond to a stationary state +[h0, ϱ0] = 0 +[¯h0, ¯ϱ0] = 0 . +(61) +Since RPA is a small amplitude approximation, one per- +forms small variations δϱ(t) of the density57 around ϱ0: +δϱ = ρ0 + δϱ(t) = ϱ0 + ϱ′e−iΩt + ϱ′†eiΩ∗t. +(62) +56 Note that the linearized equations admit the pair (Ω, Ω∗) as +solutions. +57 Similarly for δ¯ρ. + +3 F +2 [ +1 +-1[ +-2E +-3 E +-2 +-1 +0 +1 +2 +Frequencyo0.12 +0.10 +V800 +K1 +0.06 +00 +0.02 +0.00 +0 +50 +100 +150 +200 +μ25 +where ϱ′ here stands for the off-diagonal terms of the +density matrix. The mean-field Hamiltonian around this +solution changes accordingly +h(ϱ) = h0 + δh +δϱ +��� +ϱ0δϱ + . . . +¯h(¯ϱ) = ¯h0 + δ¯h +δ¯ϱ +��� +¯ϱ0δ¯ϱ + . . . . +(63) +By implementing Eq. (62) and retaining terms at lowest +order, one obtains linearized equations of motion58 which +can be cast in the following matrix form59 +� A B +¯B +¯A +� � ϱ′ +¯ϱ′ +� += Ω +� ϱ′ +¯ϱ′ +� +. +(64) +The condition for their applicability, that the system is +initially in a ”quasi-static” state Eqs. (29), is satisfied +in the matter basis. Therefore, the linearized equations +of motion Eqs. (64) are applicable at any time of the ν +evolution. +2. A dispersion-relation approach +A third formulation of the linearized equations was +suggested by Izaguirre et al. (2017), for fast modes. +The authors cast them in a dispersion-relation approach +where the neutrino modes are neutrino flavor (iso)spin +waves, described by a four vector c and a ”polarization” +vector, in matter and neutrino backgrounds. Instead of +Eq.(60) one seeks for plane waves (in an homogeneous +and stationary background) +Sv(t, r) = Qv(Ω, K)e−i(Ωt−K·r) . +(65) +After linearizing the mean-field equations (11), one gets +the following equation +vµkµQv = − +� dv +4π vµv′ +µGvQv′ +(66) +where vµ = (1, v), and (Ω, K) is replaced60 by (ω, k). By +considering +Qv = aµkµ/vµkµ , +(67) +58 The detailed derivation and the explicit expressions for the +A, B, ¯ +A, ¯B matrices can be found in Vaananen and Volpe (2013). +59 In the application of the RPA approach to atomic nuclei, the ini- +tial state is the nucleus ground state, while the variations around +it determine the excited states. +The quantities ρ′ and ¯ρ′ are +called the forward and backward amplitudes and correspond to +particle-hole and hole-particle excitations. The RPA and its nu- +merous variants (e.g. QRPA, CRPA, SRPA) are used to study +the excited states of atomic nuclei, e.g. the Giant Resonances, +or to calculate the transition matrix elements of single β, 2β(2ν) +and 2β(0ν) decay. +60 Going to a comoving frame, to get rid of the background contri- +bution. +FIG. 17 Dispersion-relation approach: results for a two-beam +model with angular modes G1 and G2. The red region cor- +responds to complex values of ω or k for real k or ω with +k = (0, 0, kz) for which fast modes are grow either in time +(temporal) or in space (spatial instability). (Figure adapted +from Izaguirre et al. (2017).) +one can recast Equation (66) as a dispersion relation ω = +ω(k) as +Πµνaν = 0 +(68) +seeking for non-trivial solutions, such that det[Πµν] = 0. +The ”polarisation tensor” in Eq. (68) reads +Πµν = ηµν + +� dv +4π Gv +vµvν +ω − v · k , +(69) +with ηµν = diag(+, −, −, −). Equation (68) is a quartic +equation in ω whose roots give four possible dispersion- +relation branches: a) (ω, k) ∈ ℜ for stable solutions; b) +ω or k ∈ C for an unstable solution that grows in time, +or in space (Fig. 17); c) ω, k ∈ C, for a mode growing in +space and time. +A classification of instabilities was suggested by +Capozzi et al. (2017) following the one for plasmas by +Sturrock (1958) and Briggs (1964). +According to this +classification, the linear instabilities can be of four cate- +gories: the completely stable or stable with damping cor- +respond to spatially stable modes; the absolute and con- +vective grow either spatially or temporally. +Note that +most of flavor evolution studies (for slow or fast modes) +were performed by evolving either space or time. +The work was further deepened by Yi et al. (2019) +who pointed out the complex dispersion relation branches +to be bounded by its critical points and their useful- +ness in identifying fast modes. Among the applications +is the study by Martin et al. (2020) in a dynamic one- +dimensional model. The authors showed that fast modes +evolve coherently in the non-linear regime in space and +in time, when the corresponding ELN crossings undergo +absolute or convective instabilities in the linear regime. + +4F +cos(8.) = 0.9 +Gi = -0.5 +cos(62) = 0.3 +G2 = +0.5 +2 +Complex w for real kz +E +Complexk, for real w +Frequency +0 +-4 +-2 +0 +2 +4 +Wave number k26 +E. Towards neutrino quantum kinetic equations +The study of the interplay between collisions and fla- +vor modes is numerically challenging. Thus its investi- +gation is still at its premises. +First it requires a con- +sistent theoretical framework where one goes from the +dense collision-dominated regime to the dilute regions in +which mean-field equations are sufficient. Second, the di- +mensionality remains high in astrophysical environments; +whereas in the early universe isotropy and homogeneity +reduces the dimensionality of the problem making it nu- +merically tractable. +Numerous authors derived neutrino quantum kinetic +equations (QKEs) for the early universe (Blaschke and +Cirigliano, 2016; Froustey et al., 2020; McKellar and +Thomson, 1994; Rudzsky, 1990; Sigl and Raffelt, 1993; +Stodolsky, 1987; Vlasenko et al., 2014a). Such QKEs are +being used for the study of neutrino flavor evolution in +dense astrophysical environments. The general form of +QKEs, including collision terms (proportional to G2 +F) and +mean-field contributions (linear in GF) read +i(∂t + v · ∇)ϱp(t) = [h, ϱ] + C[ϱ, ¯ϱ] , +(70) +and similarly for ¯ϱ, with C the collision term. While the +mean-field Hamiltonian introduces coherence, the colli- +sion term is responsible for production and absorption of +neutrinos and of kinematical decoherence among neutri- +nos with different momenta. +Rudzsky (1990) gave an early derivation of the Boltz- +mann equation for relativistic distribution functions with +mixings. Sigl and Raffelt (1993) derived quantum kinetic +equations for a matrix of densities for the early universe, +implementing antineutrinos for the first time. They in- +cluded (anti)neutrino scattering on neutrons, protons, +electrons, positrons, ν and pair annihilation in a per- +turbative approach using the assumption of molecular +chaos. In this approximation, the collision timescale is +short enough that correlations do not develop between +collisions: the incoming and outgoing particles in the col- +lision integrals are free single-particle states. +Vlasenko et al. (2014a) gave an alternative derivation +of the flavored quantum Boltzmann equations for Majo- +rana neutrinos using the Closed Time Path (or ”in-in”) +formalism and the 2 Particle-Irreducible (2PI) effective +action. Blaschke and Cirigliano (2016) extended their re- +sults and obtained the full collision term for neutrinos for +anisotropic media. Extending the work by Volpe et al. +(2013) for supernova neutrinos, Froustey et al. (2020) +rederived the neutrino quantum kinetic equations with +the BBGKY hierarchy. +A separation of scales? +In dense matter, an important length-scale is the neu- +trino mean-free path, i.e. λ = (σρ)−1, with ρ the matter +number density and σ the interaction cross section of a +neutrino with a particle of the medium. Close to matter +saturation density, at ρ = 3 × 1014 g/cm3, and for a typ- +ical cross section σ = 6 × 10−41 cm2, a 10 MeV neutrino +has a mean free path of about a meter. At densities of +ρ = 1010 g/cm3, the idealized location where neutrinos +start free streaming, λ is of tens of kilometers. +The flavor length-scale is another important quantity. +For a long time, the MSW resonance(s) have provided +the only flavor length-scale in flavor studies. As previ- +ously discussed, considering for example typical super- +nova matter profiles, the H-resonance is approximately +located at ρ = 103 g/cm3; whereas the L-resonance is +at about ρ = 1 g/cm3. If the MSW effect was the only +flavor phenomenon, the mean-free path and the flavor +length-scales would be well separated. +For many years, this argument has supported the use +of two distinct theoretical treatments, one for the dense +region, where the particles of the medium acts as random +scatterer, and one for the dilute region, where neutrinos +free stream. This translates, formally, in the use of rel- +ativistic Boltzmann transport equations (Bruenn, 1985; +Cardall et al., 2013; Lindquist, 1966) for the former, and +of mean-field equations for the latter. +It is common practice numerically to simplify the +seven-dimensional transport neglecting, in particular, the +mixing and the mean-field terms. On the other hand, +flavor studies usually separate the two regions of ap- +plicability treating the neutrinosphere as an idealized, +sharp, surface; although the neutrino decoupling region61 +is build up by collisions, and is energy and flavor depen- +dent. +Since the work of Duan, Fuller, Carlson and Qian +(Duan et al., 2006a), fifteen years of investigations have +introduced new scales in the problem showing that the +separation of the collision and of the flavor scales does +not necessarily hold in presence of neutrino-neutrino in- +teractions. This is particularly true in presence of fast +modes, that take place very close to the neutrinosphere +and have wavelengths shorter than the collision one. +In the supernova context, studies have started of the +inclusion of collisions as well as the contributions from the +mixings and the mean-field. In particular, the influence +of collisions on fast modes is being investigated (Capozzi +et al., 2019; Hansen et al., 2022; Martin et al., 2021; Shal- +gar and Tamborra, 2021). +Capozzi et al. (2019) high- +lighted collisions can trigger fast modes. Martin et al. +(2021) studied, in the linear and non-linear regimes, a +homogeneous gas model with mono-energetic neutrino +undergoing direction changing ν-nucleon elastic scatter- +ing. They showed that collisions can suppress fast modes +and the ELN distributions tend asymptotically to be- +come isotropic62 as a function of time. +Hansen et al. +(2022) found that collisions can enhance or suppress fast +61 This is conventionally defined as the region where the opacity is +2/3. +62 For an isotropic ELN, there are no fast modes. + +27 +flavor conversions depending on the hypothesis that neu- +trino emission is close to isotropic, or forward peaked. +(Richers et al., 2019) studied the influence of collisions +on slow modes. +It is interesting to note that steps towards a consis- +tent solution of the full QKEs were also done in the early +universe. Indeed, for the off-diagonal contributions of the +collision term, a damping approximations was extensively +used by authors (see for example (Dolgov et al., 2002; +Gava and Volpe, 2010; Mangano et al., 2005). Froustey +et al. (2020) and Bennett et al. (2021) recently performed +the first calculations with the full collision term, includ- +ing the mixings and mean-field terms. +With radiative +corrections to the plasma equation of state, this has given +a very precise value of the effective number of degrees of +freedom Neff = 3.0440 at the epoch of primordial nucle- +osynthesis. +Moreover Cirigliano et al. (2017) investigated the role +of anisotropies in a dense neutrino gas with two spatial +dimensions, in presence of νν interactions and collisions, +and showed instabilities are not necessarily suppressed by +kinematical decoherence. +Hansen et al. (2021) showed +that ν-¯ν asymmetry can significantly grow due to the +non-linearity of the evolution and influence Neff in pres- +ence of small anisotropies. +Much more work is needed to achieve definite under- +standing of the impact of collisions on flavor evolution in +dense astrophysical environments. A unified and consis- +tent solution of the full neutrino QKEs clearly represents +a longterm goal. +F. Neutrinos in presence of strong gravitational fields +The theoretical description of neutrino propagation +discussed so far are in flat spacetime. In core-collapse +supernovae, accretion disks around black holes or com- +pact binary mergers, there is a compact central object +producing a strong gravitational field. Gravity modifies +neutrino propagation and impacts flavor evolution. To +investigate its role, the neutrino equations of motion need +to be extended to curved spacetime, as done by Cardall +and Fuller (1997); Chatelain and Volpe (2020); Deaton +et al. (2018); Dvornikov (2013); Dvornikov et al. (2005a); +and Piriz et al. (1996). +So far, the influence of gravity has received limited at- +tention, although the first works exploring its role date +back to the eighties. It was Stodolsky (Stodolsky, 1979) +who first considered the problem of finding the quan- +tum mechanical phase acquired by a particle propagating +along a classical trajectory in presence of gravitational +fields. In order to discuss and compare matter and light +interferometry, he introduced the quantum mechanical +phase along a path, from the spacetime point A to the +spacetime point B, +Φ = +� B +A +mds, +(71) +FIG. 18 Decoherence in curved spacetime: drawing of neu- +trino wave-packet propagation from a production point P to a +”detection” point D in presence of strong gravitational fields. +The widths correspond to the trajectories distributions due +to the finite wave-packet width. +Each wave-packet associ- +ated with a mass eigenstate follows a trajectory close to null +geodesics. (Figure from Chatelain and Volpe (2017).) +where m is the particle mass. +The quantity ds is the +infinitesimal line element along the particle worldline +ds2 = gµνdxµdxν, +(72) +with gµν the metric tensor and xµ a coordinate system. +The covariant phase Eq. (71) can be rewritten as +Φ = +� B +A +pµdxµ, +(73) +pµ = mgµν dxν +ds being the particle canonical momentum. +Afterwards, the impact of gravitational fields on the +vacuum oscillation phase was investigated by many au- +thors63. +In particular Ahluwalia and Burgard (1996, +1998); and Bhattacharya et al. (1999), Cardall and Fuller +(1997); Chatelain and Volpe (2020); Fornengo et al. +(1997); and Godunov and Pastukhov (2011) considered +the case of a Schwatzschild metric of a static and spher- +ically symmetric gravitational field, and Lambiase et al. +(2005); Mosquera Cuesta et al. (2017); Visinelli (2015); +and Wudka (2001) focussed on the Kerr-Newman met- +ric. Dvornikov et al. (2005b) pointed out a new mecha- +nism called ”spin light” that neutrinos emit in presence +of gravitational fields. +In core-collapse supernovae the role of trajectory bend- +ing and energy redshift was studied by few authors. For +example, Fuller and Qian (1996) argued that the differ- +ence in the gravitational redshift between ¯νe and νe can +increase the electron fraction and impact r-process nucle- +osysnthesis above the nascent proto-neutron star. Yang +and Kneller (2017) found that trajectory bending nearby +a very compact source in a supernovae produces a neu- +trino ”halo” similar to the one identified by Cherry et al. +63 It is to be noted that the discussions, in some of the early works, +about the possibility to separate the two contributions to the +oscillation phase, from the mixings and the gravitational field, +are of academic interest. + +28 +(2012) due to νν interactions. Caballero et al. (2012) pro- +vided the full nucleosynthetic outcomes of r-process ele- +ments in black hole accretion disks models. Their results +clearly show the importance of the inclusion of trajectory +bending and neutrino energy redshift when determining +element abundances. +Decoherence is also an important aspect of neutrino +flavor evolution. +Indeed, in a wave-packet descrip- +tion of neutrino propagation the wave-packets associ- +ated with neutrino mass eigenstates can decohere sup- +pressing flavor oscillations (see for example (Giunti and +Kim, 2007). +In vacuum one quantifies decoherence by +wave-packet separation through the ”coherence length”, +Lcoh ≃ (4 +√ +2E2/|∆m2|)σx for Gaussian wave-packets, +with σx the intrinsic wave-packet dispersion. Note that +the Daya-Bay experiment investigated the effects of a +wave-packet description of vacuum oscillations and set +the first limit on its width, finding it not significant (An +et al., 2017). +For supernova neutrinos, since neutrinos travel over +large distances, decoherence effects by wave-packet sep- +aration can be sizable as discussed e.g. +in Kersten +and Smirnov (2016). +Akhmedov et al. (2017) investi- +gated such effects in the density matrix formalism and +showed that, in vacuum, they induce a damping of the +off-diagonal terms. Moreover they studied decoherence +effects in presence of dense matter and neutrino back- +grounds for the cases of adiabatic and non-adiabatic evo- +lution. +Extending the formalism to curved-spacetime, Chate- +lain and Volpe (2017) investigated the impact of wave- +packet decoherence in a Schwarzschild metric (Fig. 18). +They pointed out that, in curved space-time, instead of +the coherence length a coherence proper time τcoh quan- +tifies decoherence. This is defined as the time at which +the difference between the proper times at a ”detection” +point D satisfies τ = σt +� +B(rD)64. Neglecting matter +and νν interactions, decoherence was found to produce +modifications of the proper time by several tens of a per- +cent. +G. Connections: from atomic nuclei to quantum devices +Several authors have unravelled exciting connections +between a weakly interacting neutrino gas and other +many body systems (Mirizzi et al., 2015; Pehlivan et al., +2011; Vaananen and Volpe, 2013; Volpe et al., 2013), or +investigated the role of many-body correlations and of +entanglement (Amitrano et al., 2023; Bell et al., 2003, +2002a,b; Birol et al., 2018; Cervia et al., 2019; Fried- +land and Lunardini, 2003a,b; Lacroix et al., 2022; Mar- +tin et al., 2023,?, 2022; Patwardhan et al., 2021; Pehli- +64 Here B(r) = 1 − rs/r and rs = 2M are the Schwarzschild radius +and M the mass of the central object. +FIG. 19 Probability that each neutrino is found in the 11 +state, [Pν]11 when µ ≤ ω0 for a number of neutrinos N = 16. +Shown are the initial configuration (a νe at ωi with i = 1, 8 +and νx at ωi with i = 8, 16). The many-body results (purple) +are compared with the mean-field ones (green). The entan- +glement entropy, that encodes information on the deviation +from the mean-field limit, peaks at the spectral split frequen- +cies ω/ω0 = 2 and 7 (ω0 is the vacuum oscillation frequency). +(Figure adapted from (Patwardhan et al., 2021)). +van et al., 2014; Roggero, 2021a,b; Roggero et al., 2022). +Besides being interesting from the theoretical point of +view, these studies brought novel ways to approach the +problem of neutrino propagation in dense environments +and opened new numerical treatments, in particular us- +ing quantum devices. +Pehlivan et al. (2011) showed that the neutrino Hamil- +tonian in dense environments Eq. (50) (without matter +and for constant µ) is formally equivalent to the reduced +Bardeen-Cooper-Schieffer (BCS) Hamiltonian in the the- +ory of superconductivity (Bardeen et al., 1957) +HBCS = +� +k +ϵktz +k + GT +T − , +(74) +with the quasi-spin operators +t+ +k = c† +k,↓c† +k,↑ t− +k = ck,↑ck,↓ tz +k = (c† +k,↓ck,↓ −c† +k,↑ck,↑ −1) , +(75) +which describe Cooper pairs of of valence electrons in +a lattice. +This result highlighted that the neutrino +Hamiltonian is exactly solvable since, as pointed out by +Richardson (1966), the BCS Hamiltonian has analytical +solutions thanks to the algebraic Bethe ansatz method; +whereas Gaudin (1976) showed the exact solvability of +the model because of the number of quantum invariants. +The work of Pehlivan et al. (2011) was further elaborated +in Pehlivan et al. (2014) in presence of neutrino magnetic +moments coupling to magnetic fields. +These developments have brought the first compar- +isons of mean-field and exact results (for a small num- +ber of particles), showing in some cases significant differ- +ences. Cervia et al. (2019) and Patwardhan et al. (2021) +have employed concepts from quantum information the- +ory, in particular entanglement entropy, to quantify the + +Menn-field +Inny-body +log(2) +0.8 +.0.5 +0.6 +0.25 +0.2 +7 +8 +10 +12 +14 +16 +m/m29 +FIG. 20 Analogy between non-linear fluids and supernova +neutrinos with νν interactions: as the translation symme- +try is broken in a two-dimensional model, the streamlines +of the νe flux along the vertical direction become irregular, +showing large variations and converging towards preferred di- +rections. This behavior is analogous to the transition from +laminar to turbulent regime. (Figure adapted from (Mirizzi +et al., 2015)). +entanglement between neutrino states. +The entanglement entropy of a neutrino with frequency +ω with the rest of the neutrinos is defined as +S(ω) = −Tr[ϱ(red) +ω +log(ϱ(red) +ω +)] = − +� +s=± +λs,ωlog(λs,ω) +(76) +with the reduced density matrix obtained by tracing over +all other neutrinos +ϱ(red) +ω += Trω′̸=ωϱ +(77) +and the eigenvalues given by65 +λ±,ω = 1 +2(1 ± |Pω|) . +(78) +with P the polarization vector. If the neutrino mode is +maximally entangled with its environment, then |Pω| = +0, and the entanglement entropy S(ω) = log(2). In the +mean-field approximation, the many-body wave function +is the factorized product of single-particle wave functions, +giving |Pω| = 1 and S(ω) = 0. +Therefore the entan- +glement entropy provides information on the deviations +from the mean-field limit due to many-body effects. +Figure 19 shows the entanglement entropy for a sys- +tem of the order of 10 particles, the results showing it +is the highest for the neutrinos whose energies are the +closest to the spectral split (see section II.G). Roggero +(2021a,b) and Martin et al. (2022) and Roggero et al. +(2022) performed further investigations of the entangle- +ment entropy for neutrino systems, following the real- +time dynamics of systems of larger size, up to 102 and +103 particles. +65 We remind that ϱ(red) +ω += 1 +2 (1 + σ · Pω). +Furthermore, as previously mentioned, Volpe et al. +(2013) established a formal connection between neutri- +nos propagating in dense media and atomic nuclei, or +metallic clusters, through the BBGKY hierarchy. More- +over it is to be noted that the pairing correlators Eq. +(51) are formally analogous to pairing correlations in the +BCS theory for superconductivity in condensed matter +or pairing in nuclei. +An analogy with fluids was pointed out by Mirizzi et al. +(2015) in a two-dimensional model of supernova neutri- +nos. The authors discussed that the instability produced +by νν interactions, breaking the spatial symmetry, has +a nice analogy with nonlinear fluid instability. In par- +ticular, the transition from a coherent to an incoher- +ent regime in flavor behaves like a streaming flow that +changes from laminar to turbulent regime (Fig. 20). +New developments also concern numerical methods +that are at variance with forward integration techniques. +An example are the recent efforts to employ an inference +procedure, as in the statistical data assimilation explored +by Armstrong (2022); Armstrong et al. (2020); and Rra- +paj et al. (2021), that looks for the optimization of a +cost function. The method does not require knowledge +of the initial conditions, but rather constraints at some +locations of the coordinate axis (not necessarily at the +bounds) that parametrize the model equations of motion. +Finally, recent studies are opening the exciting pos- +sibility to investigate correlations and entanglement of +strongly correlated neutrinos on quantum computers. +Hall et al. (2021) studied the evolution, the entanglement +entropy (of a pair) and concurrence of a 4 particle neu- +trino system using, for the first time, a quantum device. +Amitrano et al. (2023) considered trapped-ion qubits. +Having highlighted aspects of our understanding of +neutrino flavor evolution at dense, we now turn to ob- +servations. +IV. PAST AND FUTURE OBSERVATIONS +A. SN1987A +On the 23rd of February 1987 Sk-69◦202 exploded in +the Large Magellanic Cloud, a satellite galaxy of the +Milky Way, producing SN1987A (Arnett et al., 1989; +Bethe, 1990; Raffelt, 1996). It was the first naked-eye +since Kepler’s supernova (Ia) in 1604. +The supernova +was at 50 ± 5 kpc from the Earth. Measurements based +on the expanding photosphere method agreed within 10 +% (Schmidt et al., 1992). SN1987A was unique in many +respects. Thirty years after, there are finally indications +for a compact object, likely a neutron star, at its location +(Alp et al., 2018; Cigan et al., 2019; Page et al., 2020). +The earlier SN1054, seen by Chinese astronomers, left +a pulsar in the Crab nebula. SN1987A progenitor, the +first known, was a blue supergiant, whereas supernovae +progenitors were thought to be red supergiants (the blue + +12 +10 +10 +0 +0.2 +0.4 +0.8 +[2元/k] +x30 +FIG. 21 SN1987A events at Kamiokande, IMB and Baksan: +energies correspond to secondary positrons produced in in- +verse β-decay. Events have been shifted at same t=0 (clock +relative offsets are unknown). +(Figure from Fiorillo et al. +(2022).) +problem) (Arnett et al., 1989; Bethe, 1990). The inner +ring, large mixing and asymmetrical ejecta of SN1987A +(Arnett et al., 1989; Janka, 2012; Podsiadlowski, 1992) +indicated strong asphericity in the explosion. +SN1987A, the closest supernova in the last hundreds +of years, was observed in all wavelengths from gamma +rays to radio, and for the first time, neutrinos from the +collapse of the stellar core were detected. Suzuki (2008), +at its 20th anniversary, gave a lively description of this +pioneering observation. The water C¸erenkov detector KII +(Hirata et al., 1987) observed a neutrino burst66 of 11 +events of energy 7.5 to 36 MeV in 13 s; while IMB (Bionta +et al., 1987) measured 8 events in 6 s with 20 MeV to 40 +MeV. BST (Alekseev et al., 1988) detected a burst of +5 events in 9 s and, about 5 h before, the Mont Blanc +Liquid Scintillator Detector (LSD) (Aglietta et al., 1987) +recorded 5 events during 7 s with energy ≥ 7 MeV (Fig. +21). Since no signal has been found correspondingly in +the other detectors, the LSD events remain controversial. +66 The probability that the observed burst is a random fluctuation +over a constant background is of about 6 × 10−7 (Raffelt, 1996). +Baade (1934) first suggested that in supernovae the +tremendous energy comes from the gravitational collapse +of the inner core into a neutron star. Hoyle and Fowler +(1960) proposed stars could dye due to thermonuclear +runaway (SN Ia) of degenerate material or implosion of +the stellar core (SN II and Ib/c). Colgate and Johnson +(1960) pointed out that the collapse could be followed by +core bounce and shock formation. The shock would ex- +pel most of the star mass by propagating into the man- +tle. +To support the prompt-shock model, Colgate and +White (1966) hypothesized that most of the gravitational +binding energy of the imploding core, Eb ∼ GM 2 +NS/RNS, +namely [1.5, 4.5]×10−53 erg, would be emitted with neu- +trinos. A few percent of this energy deposited by neutri- +nos back into matter energy, could drive the supernova +explosion. Bethe and Wilson (1985) shaped it into the +delayed neutrino-heating mechanism. +Is the delayed neutrino-heating mechanism that drives +the explosion of most (type II and Ib/c) supernovae? +Early one-dimensional67 calculations faced shock stag- +nation. SN1987A observations of asymmetries and the +presence of strong hydrodynamic mixing processes dur- +ing the explosion gave momentum to the development +of multi-dimensional simulations. +Nowadays, although +some explosions are overemphasized by the imposed sym- +metry, two-dimensional simulations of different progeni- +tors successfully expel the supernova mantle, aided by re- +alistic neutrino transport, convection, turbulence and hy- +drodynamic instabilities in particular the SASI (Blondin +et al., 2003) and LESA (Tamborra et al., 2014a)(see +the reviews (Burrows, 2013; Foglizzo et al., 2015; Janka, +2012; Mezzacappa et al., 2015; Radice et al., 2018a; Taki- +waki et al., 2021)). Weaker than two-dimensional, three- +dimensional calculations show indications for successful +explosions. The answer to this longstanding unresolved +question seem to lie in a foreseeable future. +The observation of the 24 events from SN1987A is im- +portant for fundamental physics. +The neutrino signal +gave the start of the explosion, the energy released in +the gravitational collapse, the temperature at the neu- +trinosphere, information on the explosion mechanism on +the one hand and limits on neutrino properties and non- +standard physics on the other. +Assuming energy equipartition among the neutrino +species, analysis show the total energy associated with +SN987A events to be 5 × 10−52 erg at best-fit value +(Vissani, 2015) and (Loredo and Lamb, 2002; Pagliaroli +et al., 2009b; Sato and Suzuki, 1987), confirming Col- +gate and White’s hypothesis. The spectra agree reason- +ably well with thermal expectations with ¯νe tempera- +tures T = 4 MeV (Arnett et al., 1989; Loredo and Lamb, +2002; Pagliaroli et al., 2009b; Schramm and Truran, 1990; +67 A comparison of one-dimensional simulations show an encourag- +ing agreement among groups, in spite of the differences in the +numerical methods and approximations (O’Connor et al., 2018). + +50 +Kamiokande +10 +n +50 +40 +IMB +++ +50 +18 +Baksan + 20 +0 +2 +4 +6 +8 +10 +Time after first event []31 +Vissani, 2015), the neutrino time signal confirms the ex- +pected supernova pulse duration and neutron star cool- +ing. +Therefore, SN1987A events confirmed the global +picture68 of the neutrino emission during a gravitational +core-collapse supernova explosion. +Moreover, the Bayesian analysis (Loredo and Lamb, +2002) and the one of Pagliaroli et al. (2009b) of the +time signal supported the delayed-shock over the favored +prompt-shock model by unambiguously showing the pres- +ence of an accretion phase. From nuclear matter equa- +tions of state, the formation of a neutron star (instead of +a black hole) was favored, with mass (Sato and Suzuki, +1987) and radius (Loredo and Lamb, 2002; Pagliaroli +et al., 2009b) compatible with expectations. The radius +of the neutrino emitting surface was found to be 18 km +(Raffelt, 1996). +Analyses of the SN1987A events have produced a +wealth of information on non-standard neutrino prop- +erties, interactions or particles (Raffelt, 1996; Tanabashi +et al., 2018). For example, neutrinos flew through space +for 1.6 ×105 years, yielding a bound on the neutrino life- +time of τνe/m > 5×105 s/eV (rest frame) (Bethe, 1990). +The non-observation of a γ-ray signal over background, +in correspondence with the neutrino time signal, gave +stringent bounds on the neutrino lifetime from radiative +decays (Mohapatra and Pal, 1998; Payez et al., 2015; +Raffelt, 1996; Tanabashi et al., 2018). Optical brighten- +ing followed neutrino emission by a few hours. Neutrino +propagation through space, at nearly the same speed as +photons, gave a tight constraint on the neutrino speed +cν, i.e. |(c − cν)/c| < 2 × 10−9 (Longo, 1987). Moreover, +the absence of a dispersion of the neutrino pulse gave +upper limits on the neutrino charge and on the νe mass +(about 20 eV). Energy-loss arguments on the shortening +of ν time signal, associated with the neutron star cooling, +gave limits on axions (Payez et al., 2015), on right-handed +neutrinos or currents, and on the neutrino magnetic mo- +ment (Barbieri and Mohapatra, 1988). Concerning flavor +modification, Jegerlehner et al. (1996) studied the sensi- +tivity of SN1987A events to the MSW effect and found +that the ¯νe getting hotter they would be marginally com- +patible with observations69. +Using modern supernova +fluxes, likelihood analysis by Vissani (2015) show that +the MSW impact on the two dozen events is small and +comparable to variations due to other parameters. +Clearly, the historical SN1987A is representative of +how much, patience for such rare events, can be rewarded +in knowledge and progress. +68 Note that the KII and IMB events showed a forward-peaked +angular distribution, instead of isotropic, which is likely to be +due to a statistical fluctuation. +69 Indeed the ⟨E¯νe⟩ (and ⟨E¯νµ,τ ⟩ with MSW) were marginally com- +patible with SN1987A events because of incomplete microphysics +in the supernova simulations of that epoch. +B. From the next supernova +Neutrinos emitted in the first instants of the core- +collapse will be detected several hours before optical +emission and guide optical instruments (Abe et al., 2016). +If the supernova is nearby, pre-supernova neutrinos from +thermal (Odrzywolek et al., 2004) and weak processes +(Patton et al., 2017) in the late stages of the stellar evo- +lution could be observed preceding core-collapse and give +advanced warning (Yoshida et al., 2016) as well as in- +formation on the supernova progenitor (see the review +by Kato et al. (2020)). SK could detect about 200 pre- +supernova neutrinos 12 hours before collapse of a 15-25 +M star at 0.2 kpc (such as Betelgeuse); whereas SK+Gd +and KamLAND could reach 0.6 kpc (Abe et al., 2016) +and 0.69 kpc (Asakura et al., 2016) respectively. More- +over the measurement of the full neutrino lightcurve, up +to 100 s, if the supernova is close enough would yield +interesting information on the late cooling phases of the +proto-neutron star formed. +The observation of the next supernova will benefit of +the SNEWS network (Al Kharusi et al., 2021). From the +complementarity of the technologies available, we shall +measure the time and energy of neutrino flavors through +inverse β-decay, neutrino-nucleus scattering, neutral cur- +rent scattering on electrons as well as on protons (Beacom +et al., 2002). +For inverse β-decay the cross sections are precisely +known (Ricciardi et al., 2022; Strumia and Vissani, +2003). On the contrary, the cross sections associated with +charged-current ν-nucleus interactions are still affected +by theoretical uncertainties, with the exception of heavy +water whose cross sections are known with a few percent +precision (Balantekin and Yuksel, 2003a) and 12C (Hayes +and Towner, 2000; Volpe et al., 2000). Volpe (2004, 2007) +suggested to use a novel technique, i.e. the low energy +beta-beam, to perform measurements. These will finally +be performed at the Spallation Neutron Source (Avi- +gnone et al., 2001; Barbeau et al., 2021). Besides pro- +viding a better knowledge of the spin and spin-isospin +weak nuclear response to neutrinos, in the energy range +of interest for the detection of supernova neutrinos, such +measurements could shed further light on the issue of the +quenching of the axial-vector coupling constant, also for +forbidden states, as pointed out by Volpe (2005). +The observation of the supernova time signal will be +rich of fundamental lessons. The νe flux from the 50 ms +neutronization-burst represent less than 1% of the to- +tal neutrino luminosity. The accretion phase that lasts +about 500 ms and the neutron star cooling about 10 sec- +onds take away most of the gravitational binding energy. +If close enough, the neutrino time signal from a future su- +pernova will confirm the delayed-neutrino heating mecha- +nism which is the current paradigm for most of supernova +explosions. +If the supernova is close enough, the precise measure- +ment of the time signal will be crucial to definitely assess + +32 +the explosion mechanism through the identification of os- +cillations with high frequencies, correlated with SASI, +whose measurement requires a very precise time resolu- +tion. In this respect, modifications from flavor evolution +should not swamp the signature (see for example (M¨uller +and Janka, 2014; Tamborra et al., 2014b; Walk et al., +2020)). +The measurement of early (< 20 ms) stages of neu- +trino emission would give information on the bounce time +(Halzen and Raffelt, 2009). Note that this is key to esti- +mate the burst time of the gravitational waves (Pagliaroli +et al., 2009a) which are mainly produced by the oscilla- +tions of the newly formed proto-neutron star (Abdika- +malov et al., 2020). +The concomitant multimessenger +event of neutrino and gravitational waves from a core- +collapse supernova was discussed also for example by +Halim et al. (2021). +From the point of view of flavor evolution, +the +neutronization-burst represents a unique phase. Only the +MSW effect appears to influence the neutrino spectra. +Neither fast nor slow modes are at work, as we under- +stand them now. The former requires crossings in the +neutrinos and antineutrinos angular distributions, the +latter ν¯ν pairs (in the bulb model70). +Moreover there +are neither effects from shock waves, since shock waves +reach the MSW region after 1-2 s, nor from turbulence. +Therefore, the neutronization-burst appears to be a good +laboratory to explore non-standard properties. +These +include for example non-standard νν interactions (Das +et al., 2017) or neutrino non-radiative decay (Ando, 2004; +de Gouvˆea et al., 2020). +Since flavor mechanisms produce neutrino spectral +modifications (see Section II.B), an important question +to ask is with which precision we will be able to recon- +struct the supernova neutrino fluxes when the next su- +pernova blows off (see for example (Gallo Rosso, 2021; Lu +et al., 2016; Lujan-Peschard et al., 2014; Vaananen and +Volpe, 2011)). The answer obviously depends on which +observatories will be operating at that time and on the +supernova distance. +It is interesting to note that a precise determination +and reconstruction of the supernova neutrino spectra +might not be trivial in likelihood analysis where the en- +semble of the parameters are left free to vary, even in +the simplest case with the MSW effect. Indeed Minakata +et al. (2008) pointed out the presence of parameter de- +generacies that can in principle be broken by combining +detection channels (Gallo Rosso et al., 2017). However, +while most of the neutrino parameters appear to be pre- +cisely measurable (for a supernova at 10 kpc) identifying +the neutrino pinching for some of the flavors might be +more tricky (Gallo Rosso et al., 2018). +70 In more complex model as well, the νν interaction did not appear +to influence this early phase. +As for unknown neutrino properties, the neutrino sig- +nal from the next supernova could be a good laboratory +to determine the neutrino mass ordering for which there +are currently hints with a low statistical significance. The +passage of the shock wave can be pictured as it goes +through the MSW region (see section II.D). Shock waves +effects can be important and produce distortions of the +positron (or electron) time signals, depending on the neu- +trino energy and mass ordering (see for example (Fogli +et al., 2003, 2005; Kneller et al., 2008; Lunardini and +Smirnov, 2003; Takahashi et al., 2003) and the reviews +by Duan and Kneller (2009) and Horiuchi and Kneller +(2018)). Also, the rise time of the neutronization-burst +can be used to determine the neutrino mass ordering in +a detector like IceCube (Serpico et al., 2012). Although +these signatures are interesting, it is likely that Earth- +based experiments like JUNO, DUNE or Hyper-K will +measure the neutrino mass ordering before the next su- +pernova blows off. In particular the latter should measure +it at about 3 σ after 6 (An et al., 2016) or 10 years (Abe +et al., 2011) respectively. +As for CP violation in the lepton sector, hints for +sin δ < 0 (90 % CL) indicates that the CP violating +phase should be discovered soon from the DUNE and +Hyper-K experiments. The effects of the Dirac CP vio- +lating phase was studied in the context of core-collapse +supernovae. Akhmedov et al. (2002) concluded that there +should be no impact of the Dirac phase on the νe fluxes +in a supernova, even if the νµ and ντ fluxes are unequal. +In contrast with such findings Balantekin et al. (2008) +demonstrated that the Dirac phase can impact the elec- +tron neutrino fluxes if the muon and tau neutrino fluxes +differ, because of e.g. +radiative corrections or of non- +standard interactions, such as flavor-changing neutral +currents. The result relies on a factorization condition +of the neutrino Hamiltonian, i.e. H(δ) = S†H(δ = 0)S +with S†(δ) = diag(1, 1, eiδ). +These findings were generalized in presence of νν inter- +actions by Gava and Volpe (2008), beyond the mean-field +to the full many-body problem (Pehlivan et al., 2014) and +on the neutrino degeneracy parameter in the early uni- +verse (Gava and Volpe, 2010). +Numerical calculations +showed the impact of the phase to be small (Balantekin +et al., 2008). However, the combined effect of the Ma- +jorana CP violating phase(s) and the neutrino magnetic +moment could trigger sizable effects, opening the possi- +bility for new resonances, as pointed out by (Popov and +Studenikin, 2021). +Obviously, even when the mass ordering and CP vi- +olation will be precisely measured, supernova neutrinos +will remain interesting probes for non-standard physics. +Indeed there are numerous flavor mechanisms related to +other key unknown neutrino properties71 that have been +71 These are not the main focus of this review. + +33 +extensively discussed in the literature, such as sterile neu- +trinos, non-standard interactions or the neutrino mag- +netic moment (see for example Giunti and Studenikin +(2015); Nunokawa et al. (1997a); Pehlivan et al. (2014); +and Sasaki and Takiwaki (2021)) which can give sizable +modification of the neutrino spectra in presence of strong +magnetic fields, as in core-collapse supernovae or nearby +compact objects. +C. The discovery of the diffuse supernova neutrino +background +Complementary to the observations from one super- +nova is the DSNB, made of neutrinos emitted by past +core-collapse supernovae, and which is nearly isotropic +and constant in time (see the reviews by Ando and Sato +(2004); Beacom (2010); Lunardini (2016); and Mathews +et al. (2020)). The DSNB depends on cosmological, as- +trophysical and particle physics aspects. +The DSNB flux, including a progenitor dependence, +reads +φνα(Eν) = c +� � +dM dz +���dtc +dz +��� RSN(z, M) φνα(E′ +ν, M) , +(79) +where z ∈ [0, zmax] is the cosmological redshift, c is +the speed of light, E′ +ν is the neutrino energy at the +star location at redshift z, related to the energy Eν on +Earth through E′ +ν = Eν(1 + z), φνα(E′ +ν, M) is the time- +integrated neutrino flux (fluence) for a progenitor of mass +M. In Eq. (79) usually z ∈ [0, 5] and M ∈ [8, 125] M⊙. +However only the lowest redshifts, i.e. z ∈ [0, 2] give the +most important contribution to the DSNB flux. More- +over considering 100 M⊙ instead of 125 M⊙ does not in- +troduce any significant difference. +The progenitor mass dependence of the DSNB flux +was pointed out by Lunardini and Tamborra (2012). +It is to be noted that the most general expression for +the DSNB flux should also have an explicit dependence +on the galaxy metallicity as considered for example by +Nakazato et al. (2015). +The first factor in Eq.(79) is the cosmological time that +depends on the cosmological model. Usually the ΛCDM +model is assumed. The expansion history of the universe +is then +��� dz +dtc +��� = H0(1 + z) +� +ΩΛ + (1 + z)3Ωm , +(80) +with Ωm and ΩΛ the matter and the dark energy cosmic +energy densities, H0 = 70 km s−1 Mpc−1 is the Hub- +ble constant. DSNB predictions show that he DSNB is +not sensitive to variations compatible with the Hubble +tension72. Barranco et al. (2018) investigated the influ- +ence of cosmological models other than the ΛCDM on +the DSNB. +72 There is currently a tension between the Hubble constant value +The second important input in Eq.(79) is the evolving +core-collapse supernova rate73 RSN(z, M) that is related +to the star-formation rate history ˙ρ∗(z) according to +RSN(z, M) = ˙ρ∗(z) +φ(M)dM +� 125 M⊙ +0.5 M⊙ φ(M)MdM +, +(81) +where φ(M) is the initial mass function. +With his seminal work Salpeter (1955) introduced the +power law initial mass function +φ(M) ∼ Mχ , +(82) +for M ∈ [0.5, 1] M⊙. The quantity φ(M)d(M) gives the +number of stars in the mass interval [M, M + dM]. Since, +the Salpeter IMF has been employed, χ being determined +with an uncertainty of about 10%. Baldry and Glaze- +brook (2003) introduced a modified broken power law +for the IMF with χ = −1.5 at 0.1 M⊙ ≤ M ≤ 0.5 M⊙ +and χ = −2.12 for M > 0.5 M⊙. Note that such a mod- +ified IMF gives a similar result for RSN(z, M) (Horiuchi +et al., 2009). The universality of the IMF at high masses +can be questioned, with respect to the local environment +and the cosmic time, as discussed for example by Ziegler +et al. (2022). +The cosmic star-formation history can be deduced +from observations (see e.g. (Hopkins and Beacom, 2006; +Reddy et al., 2008; Rujopakarn et al., 2010)) and is de- +scribed by a piecewise continuous form of a broken power +law (Yuksel et al., 2008) (see also (Madau and Dickinson, +2014; Singh and Rentala, 2021)) +˙ρ∗(z) = ˙ρ0 +� +(1 + z)αη + +�1 + z +B +�βη ++ +�1 + z +C +�γη�−1/η +, +(83) +with α = 3.4, β = −0.3, γ = −3.5 the logarithmic slopes +at low, intermediate and high redshift. The quantity η = +−10 is the smoothing function and the constants defining +the redshift breaks are B = 5000, C = 9. +Currently, the local core-collapse supernova rate is +known with the following precision +RSN(0) = +� 125 M⊙ +8 M⊙ +RSN(0, M)dM += (1.25 ± 0.5) × 10−4yr−1Mpc−3 . +(84) +It constitutes one of the largest uncertainties in the +DSNB predictions. +Indeed there is a disagreement by +a factor 2 at 0 ≤ z ≤ 1 between the core-collapse super- +nova rate deduced from the star-formation rate history +extracted with the ”distance ladder method”, H0 = 74.03 ± +1.42 km s−1Mpc−1, and the one obtained from the Cosmo- +logical Microwave Background (CMB), i.e. +H0 += +67.4 ± +0.5 km s−1Mpc−1 (Di Valentino et al., 2021). +73 Number per unit time per unit comoving volume + +34 +and the one from direct core-collapse supernova obser- +vations (Horiuchi et al., 2011)). +This is known as the +”supernova rate problem”. +Several parametrization of the cosmic star-formation +rate history are available in the literature. The one given +by equation (84) from Horiuchi et al. (2009) and Yuksel +et al. (2008) includes GRB data z > 4 and is commonly +employed. +The parametrization in (Fogli et al., 2004) +is outdated, whereas the one in (Priya and Lunardini, +2017) present kinks. (Mathews et al., 2014) suggested an +alternative parametrization by including only the subset +of the star-formation rate data corrected for extinction +by dust74. +The third and last important factor is the neutrino flux +from one single-supernova with progenitor mass M. Lu- +nardini (Lunardini, 2009) pointed out the relic supernova +background can receive a significant contribution from +failed supernovae (directly collapsing into a black hole). +Indeed, due to the compression of baryonic matter during +black hole formation, the supernova generates large neu- +trino fluxes with higher average energies and larger differ- +ences among flavors than optical supernovae, depending +on the (soft or stiff) equation of state, as pointed out +by Sumiyoshi et al. (2007). Note that Schilbach et al. +(2019) investigated the DSNB only coming form black +hole accretion disks. +Although the fraction of supernovae that turn into a +black hole is sub-leading, this contribution influence the +tail of the DSNB spectrum and contributes substantially +to the DSNB rates. If one includes the neutrino spectra +from core-collapse supernovae that leave a neutron star +or a black hole, Eq.(79) becomes +φνα(Eν) =c +� +dz (1 + z) +���dtc +dz +��� +× +�� +Ω +dM RSN(z, M) φNS +να (E′ +ν, M) ++ +� +Σ +dM RSN(z, M) φBH +να (E′ +ν, M) +� +, +where Ω and Σ correspond to the range of masses for +which the collapse gives a NS or a BH. Thus the BH +fraction is given by +fBH = +� +Σ dMφ(M) +� 125M⊙ +8M⊙ +dMφ(M) +. +(85) +The fraction of failed supernovae is currently debated. +It constitutes another important factor of uncertainty +in the DSNB predictions. It has been argued by Hori- +uchi et al. (2014, 2018); O’Connor and Ott (2011); and +74 Moreover the authors argued that the ”supernova rate problem” +could be solved by the inclusion of contributions from binaries, +from failed supernovae and from electron-capture ONeMg super- +novae. +Ugliano et al. (2016) that the star compactness could be +a good indicator of the fraction of supernovae leaving +black holes. This is in contrast with Ertl et al. (2016) +who suggested as indicators two parameters, M4 and µ4, +giving the enclosed mass and its derivative (s = 4, dimen- +sionless entropy per nucleon), to better predict successful +explosions in the neutrino driven wind mechanism. +Predictions of the DSNB flux and rates have different +levels of sophistication either with respect to astrophys- +ical inputs, neutrino flavor mechanism, neutrino proper- +ties and new physics. Concerning the astrophysical de- +pendence, Ivanez-Ballesteros and Volpe (2022); Moeller +et al. (2018); and Tabrizi and Horiuchi (2021) included +a progenitor mass dependence and the fraction of failed +supernovae based on one-dimensional supernova simula- +tions. Note that according to the detailed simulations +by Kresse et al. (2021) the BH fraction75 ranges from 17 +% to 41 %. Horiuchi et al. (2021, 2018); Kresse et al. +(2021); and Moeller et al. (2018) performed extensive su- +pernova simulations to include a detailed progenitor mass +dependence, and also the contribution from binary sys- +tems (which is very uncertain). It is to be noted that +Fukugita and Kawasaki (2003); Lunardini (2006); Vissani +and Pagliaroli (2011); and Yuksel and Beacom (2007) +furnished DSNB predictions based on SN1987A observa- +tions for the relic neutrino spectra. +Apart from the progenitor dependence, predictions on +the DSNB neutrino spectra and rates are influenced by +flavor conversion mechanisms. The established MSW ef- +fect is routinely implemented in predictions (Ando and +Sato, 2004; Chakraborty et al., 2011a; De Gouvea et al., +2020; Ekanger et al., 2022; Galais et al., 2010; Horiuchi +et al., 2018; Ivanez-Ballesteros and Volpe, 2022; Kresse +et al., 2021; Moeller et al., 2018; Priya and Lunardini, +2017; Tabrizi and Horiuchi, 2021). So far only a few stud- +ies implemented flavor effects beyond the MSW mecha- +nism. Galais et al. (2010) investigated shock waves and +νν interactions effects in the bulb model and found vari- +ations up to 10%-20% due to the shock waves. Nakazato +(2013) found that the DSNB rates also depend on the +shock wave revival time. +Interestingly, the DSNB is also sensitive to non- +standard physics (Farzan and Palomares-Ruiz, 2014; +Goldberg et al., 2006; de Gouvˆea et al., 2022; Reno +et al., 2021). +In particular the DSNB is sensitive to +non-radiative two-body decay in the window τ/m ∈ +[109, 1011) s/eV (see Ando (2003); De Gouvea et al. +(2020); Fogli et al. (2004); Ivanez-Ballesteros and Volpe +(2022); and Tabrizi and Horiuchi (2021)). This window +is unique compared to terrestrial experiments, astrophys- +ical sources like the Sun or a supernova, and cosmological +probes such as BBN or the CMB (for the latter see e.g. +(Chen et al., 2022)). +75 Priya and Lunardini (2017) also considered a more conservative +value of fBH = 0.09. + +35 +Clearly, if the current hint is borne out, the DSNB +will become a unique laboratory for astrophysics, particle +physics and for the search of new physics. +V. CONCLUSIONS AND PERSPECTIVES +In our journey from the beginnings of neutrino physics +and astronomy we went through some of the discover- +ies that paved our knowledge of these elusive particles +and opened new horizons. +After the breakthroughs of +the evidence for neutrino oscillations and of the solution +of the solar neutrino problem, experiments and theory +put milestones in our understanding of neutrino masses +and mixings, of neutrinos from stellar and cosmological +environments and set important limits on new physics. +Still, neutrino physics and astrophysics remain nowadays +a very active domain of research. +Among the most challenging unsolved issues is the evo- +lution and flavor modification of neutrinos from dense +compact objects. What makes this problem so intrigu- +ing and challenging is that, besides shock waves and +turbulence inherent to exploding environments, nearby +compact objects one has sizable neutrino-neutrino inter- +actions that render neutrino flavor evolution a complex +non-linear many-body problem. Efforts to solve it are +motivated not only by theoretical interest but also, obvi- +ously, by observations. +So far, +investigations of shock wave effects have +mostly used parametric matter density profiles of one- +dimensional supernova simulations. Dips, or bumps, are +characteristic features of the neutrino time signals due to +the shock wave passage in MSW regions and, in partic- +ular, in the H-resonance one. The identification of such +structures offers a mean to identify the neutrino mass or- +dering; normal if the shock wave passage is ”seen” in the +νe time signals, inverted, if ”seen” in the ¯νe one. Since +experiments like JUNO, DUNE or Hyper-K are likely to +unambiguously measure the neutrino mass ordering be- +fore next supernova, the imprint of the shock waves in the +time signals will give a picture of the explosion dynamics. +It is useful to keep in mind that multi-dimensional +supernova simulations present strong anisotropies which +can produce large angular variations of the front and the +reverse shocks. Moreover, down-flows colliding with hot +matter that expands due to convection can induce multi- +ple shocks. As a consequence, the exact structures might +be direction dependent and possibly evolve chaotically. +Therefore, further investigations are necessary to assess +if the generic features of the shock wave passage, iden- +tified in one-dimensional studies, remain, when one im- +plements information from multi-dimensional supernova +simulations. +Turbulence also contributes to the not-yet understood +core-collapse supernova explosion mechanism. It is an- +other important aspect that impacts flavor evolution +since it introduces matter density fluctuations which +might produce neutrino depolarization, as early pointed +out. Their characteristics – amplitude, scale and power +spectrum – should be extracted from multi-dimensional +supernova simulations. Since this is a difficult numerical +task, most of the available studies have used parametric +matter profiles where fluctuations are superimposed. So +far, only one investigation has exploited information from +a two-dimensional supernova simulation, finding weak in- +dications that depolarization takes place, in contrast with +all previous findings. Clearly, new studies are called for, +with inputs from two- and three-dimensional simulations, +to establish if neutrino probabilities do have a loss of +memory effect, or not, due to turbulence, in an explod- +ing supernova. +Interestingly, studies have uncovered that, besides the +established MSW effect present in compact objects, mul- +tiple MSW resonances are produced by shock waves +or turbulence in supernovae, or more generally because +of non-linear feedback. +In particular the MSW-like +phenomena that were pointed out include the matter- +neutrino resonances, the resonance due to helicity coher- +ence, or the I- and synchronized I-resonances triggered +by non-standard neutrino-matter interactions. +Another feature that impacts the neutrino flavor in +dense media are neutrino-neutrino interactions which +were first studied in the nineties in the context of the +early universe. +Their investigation in core-collapse su- +pernovae and compact binary mergers has triggered an +intense theoretical activity in the last fifteen years. In +fact, novel unexpected flavor phenomena, located much +deeper than the MSW region, have attracted enthusiasm +because of the potential impact on the supernova explo- +sion mechanism and on nucleosynthesis, besides the one +on future observations of supernova neutrinos. +Neutrino evolution in presence of neutrino-neutrino in- +teractions is still an unsolved problem. From the studies +performed so far we have learnt that, relaxing an approx- +imation, or going beyond approaches, unforeseen aspects +emerge that can overturn how we represent the picture of +neutrino flavor evolution. The first, widely investigated, +bulb model revealed collective slow modes that are trig- +gered by mixings. With frequencies √µω at typical dis- +tances of O(102-103) km from the neutrinosphere, such +modes occur in regions where they cannot induce extra +heating to help explosions, whereas they can influence +the r-process, as shown by numerous studies. +Moreover the interplay of νν interactions with other +contributions, such as the standard and non-standard +neutrino-matter interactions, opens the way to new +MSW-like phenomena (e.g. the matter-neutrino and the +I-resonances). +With time we have learnt that the in- +clusion of new degrees of freedom, as in non-stationary +models, or in models with two-dimensional spatial de- +grees of freedom like the line model, opens up new re- +gions for flavor instabilities. +There are also situations +where small initial perturbations, that do not pertain the +same symmetries as the initial neutrino emission, give + +36 +solutions that spontaneously break symmetries (e.g. the +azymuthal one). And in some cases, even chaotic flavor +evolution can emerge. +If, instead of only forward-scattering neutrinos, one +includes a small amount of back-scattered neutrinos or +a better description of the ν angular emission, then the +whole picture can be overturned as it came unexpectedly. +The first option cast doubts on the treatment of neutrino +evolution as an initial value problem. +For the second, +crossings of the νe and ¯νe angular distributions turned +out to trigger short scale flavor modes, i.e. O(1) m or +much less, very close to the neutrinosphere. These fast +modes are currently very actively investigated. +It is now established that they occur in two- and three- +dimensional supernova simulations, nearby the neutri- +nosphere and even inside the proton-neutron star. If the +neutrino spectra are similar, at the fast mode location, +as it appears, their influence on the spectra is small. So, +far, only a couple of studies have evolved fast modes to +the full non-linear regime. +There are indications that +fast modes can influence the r-process in binary neutron +star mergers and the νp-process in core-collapse super- +novae. Also three-flavor effects were shown to be impor- +tant to determine when flavor evolution is modified on +large scales. The study of fast modes, the conditions for +their occurrence and impact is at present a fast develop- +ing field. Their understanding will require more work. +With a few exceptions, all the findings concerning fla- +vor evolution in dense environments available in the lit- +erature use the mean-field approximation. +Linearized +mean-field equations and a dispersion relation approach +for fast modes are commonly used to study when neutrino +flavor modification is triggered. This has the advantage +of solving an eigenvalue equation close to the initial quasi- +static condition but looses the long-term evolution of the +full non-linear problem. +A special effort was devoted to check the validity of the +mean-field equations. This lead to new evolution equa- +tions, to the re-derivation of quantum kinetic equations +and to the first attempts to solve kinetic equations with +the inclusion of mixings in schematic models. +Neutrinos in the early universe, where neutrino kinetic +equations are needed, represent a different case in many +respects. The homogeneity and isotropy of the medium +made possible the first consistent calculations of neutrino +evolution with the full collision term, the mixings and +mean-field term. +In the supernova context, extended mean-field evo- +lution equations were derived using in particular the +coherent-state path integral, the closed time path inte- +gral and the BBGKY hierarchy. Such equations included, +in particular, contributions by supplementary two-point +correlators, i.e. helicity coherence and pairing correla- +tors. For the former calculations based on detailed sim- +ulations of binary compact mergers and core-collapse su- +pernovae showed that they do not trigger significant fla- +vor evolution due to non-linear feedback, as perturbative +arguments also show. For the latter, no flavor modifica- +tion appears because the kinetic terms dominate. +The impact of collisions on flavor evolution is currently +an open problem which is numerically very challenging +because of its high dimensionality. For a long time, the +argument of the separation of scales between flavor mech- +anisms (the MSW effect) and the collision-dominated re- +gion justified the use of mean-field equations. With the +advent of νν interaction studies for dense astrophysical +environments, the identification of slow and then fast +modes has deeply changed our vision. +The interplay between the collisions and fast modes +is receiving particular attention. Studies of models with +lower dimensionality and approximate treatment of colli- +sions (e.g. direction changing and neutrino-nucleon only) +uncovered the possibility that collisions can trigger fast +modes, or suppress them, or enhance them, depending for +example on the angular distribution of the neutrino emis- +sion at the neutrinosphere. +While the models studied +so far necessarily have many approximations and limita- +tions, there are clear hints that we need to go towards fur- +ther complexity since even if the collision rate is smaller +than flavor scale, collisions are likely to be important. +And, in fact, it goes without saying that even the cross- +ings between the neutrino and antineutrino angular dis- +tributions, associated with the occurrence of fast modes, +should emerge from collisions in a fully consistent treat- +ment. +All these developments are based on theoretical ap- +proaches in flat spacetime. However, strong gravitational +fields are present nearby compact objects. Their impact +on flavor evolution is still in an exploratory phase. An +extension of the equations of motion in curved spacetime +has been discussed by a few authors. +Several studies investigated the impact on the vacuum +oscillation phase for different metrics and recently on +the decoherence by wave packet separation, in a wave- +packet treatment of neutrino evolution in curved space- +time. A ”halo” effect was found in an exploding super- +nova, whereas it was clearly shown that the inclusion of +gravity effects (trajectory bending, energy redshift) influ- +ence r-process nucleosynthesis in accretion disks around +black holes. +Clearly, gravitational effects on neutrino +propagation and flavor evolution should deserve more at- +tention in the coming years. +Intriguing connections between a system of weakly in- +teracting neutrinos and other domains have been uncov- +ered, often opening new unforeseen possibilities. In par- +ticular, an algebraic formulation and the Bethe ansatz +showed the νν many-body Hamiltonian to be solvable +(under some conditions). This and further works have +yielded the first comparison between mean-field and +many-body results highlighting the role of many-body +correlations in particular through the entanglement en- +tropy. Moreover first calculations based on an inference +procedure and on quantum devices are appearing. The +latter open exciting new possibilities that are and will + +37 +certainly attract a lot of interest in the coming years. +After the 24 ¯νe events of SN1987A, we are eagerly +awaiting for the next supernova to precisely measure the +neutrino light-curves hopefully this time, if the supernova +is close and we are patient enough. It goes without say- +ing that this observation is crucial both for astrophysics +and for particle physics. We will learn a lot on the super- +nova explosion, possibly having definite evidence for the +explosion mechanism and the favored neutrino heating +mechanism, the onset time of the explosion, important +for gravitational wave detection. We will get a picture +of the shock wave passage in the MSW region from the +time and energy signals, and eventually signatures of the +SASI instability. With the advent of SNEWS 2.0 we will +be able lo locate the exploding star through its neutrinos. +Besides, the upcoming discovery of the diffuse super- +nova neutrino background will be crucial. It will be the +second time ever we observe neutrinos from core-collapse +supernovae, with a unique sensitivity to the evolving +core-collapse supernova rate, the fraction of failed super- +novae and binaries, flavor mechanisms and non-standard +neutrino properties such as neutrino decay. The diffuse +supernova neutrino background promise to remain for +many years an incredible laboratory for astrophysics and +particle physics. +In this journey, I highlighted aspects of our current un- +derstanding of flavor evolution in dense media, setting it +in the context of the historical developments in neutrino +physics, of what we now know and of what we would +like to discover in the coming years. With a bit of an +historical perspective also on the theoretical progress in +this field, I have discussed numerous aspects that appear +now as clear (although with its limitations and approx- +imations) and the numerous theoretical challenges still +ahead. +Serious progress has been done. But one lesson we have +learnt, that new possibilities can always come up and +completely change the way we look at this complex prob- +lem. Clearly, the novel developments recently emerged +and future ones might give, once more, a completely new +insight on this fascinating subject. +VI. ACKNOWLEDGMENTS +Along the years I had interesting discussions with nu- +merous researchers in the field. In particular, I wish to +thank Baha Balantekin, Carlo Giunti, Thomas Janka, +Gail McLaughlin, George Fuller, Manfred Lindner, Kate +Scholberg, Francesco Vissani, and also Evgeny Akhme- +dov, Alessandro Mirizzi, Georg Raffelt, Alexey Smirnov, +Rebecca Surman, Irene Tamborra as well as many others +with whom I discussed at workshops and conferences, or +I exchanged correspondence. +Moreover I would like to thank very warmly to two of +my former PhD students, Am´elie Chatelain and J´erˆome +Gava, for the enthusiasm we shared working together on +this intriguing domain. +REFERENCES +Aartsen, M. G., et al. (IceCube) (2014), Phys. Rev. Lett. 113, +101101, arXiv:1405.5303 [astro-ph.HE]. +Aartsen, M. G., et al. (IceCube) (2018), Science 361 (6398), +147, arXiv:1807.08794 [astro-ph.HE]. +Abazajian, K. N., J. F. Beacom, and N. F. Bell (2002), Phys. +Rev. D 66, 013008, arXiv:astro-ph/0203442. +Abbar, +S. +(2021), +Phys. +Rev. +D +103 +(4), +045014, +arXiv:2007.13655 [astro-ph.HE]. +Abbar, S., +and H. Duan (2015), Phys. Lett. B 751, 43, +arXiv:1509.01538 [astro-ph.HE]. +Abbar, S., and H. Duan (2018), Phys. Rev. D 98 (4), 043014, +arXiv:1712.07013 [hep-ph]. +Abbar, S., H. Duan, +and S. Shalgar (2015), Phys. Rev. D +92 (6), 065019, arXiv:1507.08992 [hep-ph]. +Abbar, S., H. Duan, K. Sumiyoshi, T. Takiwaki, +and +M. C. Volpe (2019), Phys. Rev. D 100 (4), 043004, +arXiv:1812.06883 [astro-ph.HE]. +Abbar, S., H. Duan, K. Sumiyoshi, T. Takiwaki, +and +M. C. Volpe (2020), Phys. Rev. D 101 (4), 043016, +arXiv:1911.01983 [astro-ph.HE]. +Abbar, S., and M. C. Volpe (2019), Phys. Lett. B 790, 545, +arXiv:1811.04215 [astro-ph.HE]. +Abbott, B., et al. (LIGO Scientific, Virgo) (2016), Phys. Rev. +Lett. 116 (6), 061102, arXiv:1602.03837 [gr-qc]. +Abbott, B., et al. (2017a), Astrophys. J. Lett. 848 (2), L12, +arXiv:1710.05833 [astro-ph.HE]. +Abbott, B. P., et al. (LIGO Scientific, Virgo) (2017b), Phys. +Rev. Lett. 119 (16), 161101, arXiv:1710.05832 [gr-qc]. +Abdikamalov, E., G. Pagliaroli, +and D. Radice (2020), +arXiv:2010.04356 [astro-ph.SR]. +Abe, K., et al. (2011), arXiv:1109.3262 [hep-ex]. +Abe, K., et al. (Super-Kamiokande) (2016), Astropart. Phys. +81, 39, arXiv:1601.04778 [astro-ph.HE]. +Abe, +K., +et +al. +(Hyper-Kamiokande) +(2018), +arXiv:1805.04163 [physics.ins-det]. +Abe, K., et al. (Super-Kamiokande) (2021), Phys. Rev. D +104 (12), 122002, arXiv:2109.11174 [astro-ph.HE]. +Abe, Y., et al. (Double Chooz) (2012), Phys. Rev. Lett. 108, +131801, arXiv:1112.6353 [hep-ex]. +Abi, B., et al. (DUNE) (2021), Eur. Phys. J. C 81 (5), 423, +arXiv:2008.06647 [hep-ex]. +Aglietta, M., et al. (1987), Europhys. Lett. 3, 1315. +Agostini, M., G. Benato, J. A. Detwiler, J. Men´endez, +and +F. Vissani (2022), arXiv:2202.01787 [hep-ex]. +Agostini, M., et al. (Borexino) (2021), Astropart. Phys. 125, +102509, arXiv:1909.02422 [hep-ex]. +Aguilar-Arevalo, A., et al. (MiniBooNE) (2018), Phys. Rev. +Lett. 121 (22), 221801, arXiv:1805.12028 [hep-ex]. +Aharmim, B., et al. (SNO) (2006), Astrophys. J. 653, 1545, +arXiv:hep-ex/0607010. +Ahluwalia, D. V., +and C. Burgard (1996), Gen. Rel. Grav. +28, 1161, arXiv:gr-qc/9603008 [gr-qc]. +Ahluwalia, D. V., and C. Burgard (1998), Phys. Rev. D57, +4724, arXiv:gr-qc/9803013 [gr-qc]. +Ahmad, Q. R., et al. (SNO) (2001a), Phys. Rev. Lett. 87, +071301, arXiv:nucl-ex/0106015 [nucl-ex]. +Ahmad, Q. R., et al. (SNO) (2001b), Phys. Rev. Lett. 87, +071301, arXiv:nucl-ex/0106015. +Ahmad, Q. R., et al. (SNO) (2002), Phys. Rev. Lett. 89, +011301, arXiv:nucl-ex/0204008 [nucl-ex]. + +38 +Ahn, J. K., et al. (RENO) (2012), Phys. Rev. Lett. 108, +191802, arXiv:1204.0626 [hep-ex]. +Aker, M., et al. (KATRIN) (2022), Nature Phys. 18 (2), 160, +arXiv:2105.08533 [hep-ex]. +Akhmedov, E., J. Kopp, and M. Lindner (2017), JCAP 09, +017, arXiv:1702.08338 [hep-ph]. +Akhmedov, E. K. (1999), in ICTP Summer School in Particle +Physics, pp. 103–164, arXiv:hep-ph/0001264. +Akhmedov, E. K., C. Lunardini, and A. Y. Smirnov (2002), +Nucl. Phys. B 643, 339, arXiv:hep-ph/0204091. +Akimov, D., et al. (COHERENT) (2017), Science 357 (6356), +1123, arXiv:1708.01294 [nucl-ex]. +Al Kharusi, S., et al. (SNEWS) (2021), New J. Phys. 23 (3), +031201, arXiv:2011.00035 [astro-ph.HE]. +Alekseev, E. N., L. N. Alekseeva, I. V. Krivosheina, and V. I. +Volchenko (1988), Phys. Lett. B205, 209. +Alp, +D., +et +al. +(2018), +Astrophys. +J. +864 +(2), +174, +arXiv:1805.04526 [astro-ph.HE]. +Altarelli, G., +and F. Feruglio (2010), Rev. Mod. Phys. 82, +2701, arXiv:1002.0211 [hep-ph]. +Amitrano, V., A. Roggero, P. Luchi, F. Turro, L. Vespucci, +and F. Pederiva (2023), Phys. Rev. D 107 (2), 023007, +arXiv:2207.03189 [quant-ph]. +An, F., et al. (JUNO) (2016), J. Phys. G 43 (3), 030401, +arXiv:1507.05613 [physics.ins-det]. +An, F. P., et al. (Daya Bay) (2012), Phys. Rev. Lett. 108, +171803, arXiv:1203.1669 [hep-ex]. +An, F. P., et al. (Daya Bay) (2017), Eur. Phys. J. C 77 (9), +606, arXiv:1608.01661 [hep-ex]. +Ando, +S. +(2003), +Phys. +Lett. +B +570, +11, +arXiv:hep- +ph/0307169. +Ando, S. (2004), Phys. Rev. D 70, 033004, arXiv:hep- +ph/0405200. +Ando, S., J. F. Beacom, +and H. Yuksel (2005), Phys. Rev. +Lett. 95, 171101, arXiv:astro-ph/0503321. +Ando, S., +and K. Sato (2004), New J. Phys. 6, 170, +arXiv:astro-ph/0410061. +Arg¨uelles, C. A., I. Esteban, M. Hostert, K. J. Kelly, +J. Kopp, P. A. N. Machado, I. Martinez-Soler, and Y. F. +Perez-Gonzalez (2022), Phys. Rev. Lett. 128 (24), 241802, +arXiv:2111.10359 [hep-ph]. +Armstrong, +E. (2022), Phys. Rev. D 105 (8), 083012, +arXiv:2111.07412 [physics.comp-ph]. +Armstrong, E., A. V. Patwardhan, A. A. Ahmetaj, M. M. +Sanchez, S. Miskiewicz, M. Ibrahim, and I. Singh (2022), +Phys. Rev. D 105 (10), 103003, arXiv:2201.08505 [astro- +ph.HE]. +Armstrong, E., A. V. Patwardhan, E. Rrapaj, S. F. Ardizi, +and G. M. Fuller (2020), Phys. Rev. D 102 (4), 043013, +arXiv:2006.07725 [astro-ph.HE]. +Arnett, W. D., J. N. Bahcall, R. P. Kirshner, +and S. E. +Woosley (1989), Ann. Rev. Astron. Astrophys. 27, 629. +Asakura, +K., +et al. (KamLAND) (2016), Astrophys. J. +818 (1), 91, arXiv:1506.01175 [astro-ph.HE]. +Avignone, F. T., V. P. Gudkov, L. Chatterjee, Y. Efremenko, +and M. Strayer (2001), Nucl. Phys. B Proc. Suppl. 91, 113. +Baade, Z. F., W. (1934), On super-novae, Vol. 20. +Bahcall, J. N. (1964), Phys. Rev. Lett. 12, 300. +Bahcall, J. N., N. A. Bahcall, +and G. Shaviv (1968), Phys. +Rev. Lett. 20, 1209, [,45(1968)]. +Balantekin, A., J. Fetter, +and F. Loreti (1996), Phys. Rev. +D 54, 3941, arXiv:astro-ph/9604061. +Balantekin, A., and Y. Pehlivan (2007), J. Phys. G 34, 47, +arXiv:astro-ph/0607527. +Balantekin, A., +and H. Yuksel (2005), New J. Phys. 7, 51, +arXiv:astro-ph/0411159. +Balantekin, A. B., J. Gava, and C. Volpe (2008), Phys. Lett. +B 662, 396, arXiv:0710.3112 [astro-ph]. +Balantekin, A. B., and H. Yuksel (2003a), Phys. Rev. C 68, +055801, arXiv:hep-ph/0307227. +Balantekin, A. B., and H. Yuksel (2003b), Phys. Rev. D 68, +013006, arXiv:hep-ph/0303169. +Baldry, I. K., and K. Glazebrook (2003), Astrophys. J. 593, +258, arXiv:astro-ph/0304423. +Banerjee, A., A. Dighe, and G. Raffelt (2011), Phys. Rev. D +84, 053013, arXiv:1107.2308 [hep-ph]. +Barbeau, P. S., Y. Efremenko, +and K. Scholberg (2021), +arXiv:2111.07033 [hep-ex]. +Barbieri, R., and R. N. Mohapatra (1988), Phys. Rev. Lett. +61, 27. +Bardeen, J., L. Cooper, and J. Schrieffer (1957), Phys. Rev. +108, 1175. +Barger, V., P. Huber, and D. Marfatia (2005), Phys. Lett. B +617, 167, arXiv:hep-ph/0501184. +Barinov, V. V., et al. (2022), Phys. Rev. Lett. 128 (23), +232501, arXiv:2109.11482 [nucl-ex]. +Barranco, J., A. Bernal, and D. Delepine (2018), J. Phys. G +45 (5), 055201, arXiv:1706.03834 [astro-ph.CO]. +Baxter, A. L., et al. (SNEWS) (2022), Astrophys. J. 925 (2), +107, arXiv:2109.08188 [astro-ph.IM]. +Beacom, J. F. (2010), Ann. Rev. Nucl. Part. Sci. 60, 439, +arXiv:1004.3311 [astro-ph.HE]. +Beacom, J. F., W. M. Farr, and P. Vogel (2002), Phys. Rev. +D 66, 033001, arXiv:hep-ph/0205220. +Beacom, J. F., +and M. R. Vagins (2004), Phys. Rev. Lett. +93, 171101, arXiv:hep-ph/0309300 [hep-ph]. +Beacom, J. F., and P. Vogel (1998), Phys. Rev. D 58, 053010, +arXiv:hep-ph/9802424. +Bell, N. F., A. A. Rawlinson, and R. F. Sawyer (2003), Phys. +Lett. B 573, 86, arXiv:hep-ph/0304082. +Bell, N. F., R. F. Sawyer, +and R. R. Volkas (2002a), Phys. +Rev. A 65, 052105, arXiv:quant-ph/0106082. +Bell, N. F., R. F. Sawyer, R. R. Volkas, +and Y. Y. Y. +Wong (2002b), Phys. Rev. A 65, 042328, arXiv:quant- +ph/0109014. +Bennett, J. J., G. Buldgen, P. F. De Salas, M. Drewes, S. Gari- +azzo, S. Pastor, and Y. Y. Y. Wong (2021), JCAP 04, 073, +arXiv:2012.02726 [hep-ph]. +Bethe, H. (1986), Phys. Rev. Lett. 56, 1305. +Bethe, H. A. (1939), Phys. Rev. 55, 434. +Bethe, H. A. (1990), Rev. Mod. Phys. 62, 801. +Bethe, H. A., +and J. R. Wilson (1985), Astrophys. J. 295, +14. +Bhattacharya, T., S. Habib, +and E. Mottola (1999), Phys. +Rev. D 59, 067301. +Biggio, C., M. Blennow, and E. Fernandez-Martinez (2009), +JHEP 08, 090, arXiv:0907.0097 [hep-ph]. +Bilenky, M. S., and A. Santamaria (1999), in Neutrino Mix- +ing: Meeting in Honor of Samoil Bilenky’s 70th Birthday, +arXiv:hep-ph/9908272. +Bionta, R. M., et al. (1987), Phys. Rev. Lett. 58, 1494. +Birol, S., Y. Pehlivan, A. Balantekin, and T. Kajino (2018), +Phys. Rev. D 98 (8), 083002, arXiv:1805.11767 [astro- +ph.HE]. +Blaschke, D. N., +and V. Cirigliano (2016), Phys. Rev. D +94 (3), 033009, arXiv:1605.09383 [hep-ph]. +Blennow, M., A. Mirizzi, +and P. D. Serpico (2008), Phys. +Rev. D 78, 113004, arXiv:0810.2297 [hep-ph]. +Blondin, J. M., A. Mezzacappa, +and C. DeMarino (2003), +Astrophys. J. 584, 971, arXiv:astro-ph/0210634. + +39 +Bogoliubov, N. (1946), Journal of Physics USSR 10, 1175. +Bollig, R., H. T. Janka, A. Lohs, G. Martinez-Pinedo, +C. J. Horowitz, +and T. Melson (2017), Phys. Rev. Lett. +119 (24), 242702, arXiv:1706.04630 [astro-ph.HE]. +Born, M., and H. Green (1946), Proc. Roy. Soc. Lond. A188 +10, 1175. +Borriello, E., S. Chakraborty, H.-T. Janka, E. Lisi, and A. Mi- +rizzi (2014), JCAP 11, 030, arXiv:1310.7488 [astro-ph.SR]. +Botella, F. J., C. S. Lim, and W. J. Marciano (1987), Phys. +Rev. D 35, 896. +Bouchez, J., M. Cribier, J. Rich, M. Spiro, D. Vignaud, and +W. Hampel (1986), Z. Phys. C 32, 499. +Briggs, (1964), (M.I.T Press, Cambridge). +Bruenn, S. W. (1985), Astrophys. J. Suppl. 58, 771. +Burbidge, M. E., G. R. Burbidge, W. A. Fowler, and F. Hoyle +(1957), Rev. Mod. Phys. 29, 547. +Burgess, C. P., N. S. Dzhalilov, M. Maltoni, T. I. Rashba, +V. B. Semikoz, M. A. Tortola, and J. W. F. Valle (2004), +JCAP 01, 007, arXiv:hep-ph/0310366. +Burgess, C. P., and D. Michaud (1997), Annals Phys. 256, +1, arXiv:hep-ph/9606295. +Burrows, +A. +(2013), +Rev. +Mod. +Phys. +85, +245, +arXiv:1210.4921 [astro-ph.SR]. +Burrows, A., D. Radice, D. Vartanyan, H. Nagakura, M. A. +Skinner, +and J. Dolence (2020), Mon. Not. Roy. Astron. +Soc. 491 (2), 2715, arXiv:1909.04152 [astro-ph.HE]. +Caballero, O. L., G. C. McLaughlin, and R. Surman (2012), +Astrophys. J. 745, 170, arXiv:1105.6371 [astro-ph.HE]. +Calzetta, E., and B. L. Hu (1988), Phys. Rev. D 37, 2878. +Cameron, A. G. W. (1958), Ann. Rev. Nucl. Part. Sci. 8, 299. +Capozzi, F., M. Chakraborty, S. Chakraborty, +and M. Sen +(2020), Phys. Rev. Lett. 125, 251801, arXiv:2005.14204 +[hep-ph]. +Capozzi, F., B. Dasgupta, E. Lisi, A. Marrone, and A. Mi- +rizzi (2017), Phys. Rev. D 96 (4), 043016, arXiv:1706.03360 +[hep-ph]. +Capozzi, F., B. Dasgupta, A. Mirizzi, M. Sen, +and G. Sigl +(2019), Phys. Rev. Lett. 122 (9), 091101, arXiv:1808.06618 +[hep-ph]. +Capozzi, F., E. Di Valentino, E. Lisi, A. Marrone, A. Mel- +chiorri, +and A. Palazzo (2021), Phys. Rev. D 104 (8), +083031, arXiv:2107.00532 [hep-ph]. +Cardall, C. Y., E. Endeve, and A. Mezzacappa (2013), Phys. +Rev. D 88, 023011, arXiv:1305.0037 [astro-ph.HE]. +Cardall, C. Y., +and G. M. Fuller (1997), Phys. Rev. D55, +7960, arXiv:hep-ph/9610494 [hep-ph]. +Cervia, M. J., A. V. Patwardhan, A. Balantekin, d. S. Copper- +smith, and C. W. Johnson (2019), Phys. Rev. D 100 (8), +083001, arXiv:1908.03511 [hep-ph]. +Chakraborty, M., and S. Chakraborty (2020), JCAP 01, 005, +arXiv:1909.10420 [hep-ph]. +Chakraborty, S., S. Choubey, and K. Kar (2011a), Phys. Lett. +B 702, 209, arXiv:1006.3756 [hep-ph]. +Chakraborty, +S., +T. +Fischer, +A. +Mirizzi, +N. +Saviano, +and +R. +Tomas +(2011b), +Phys. +Rev. +D +84, +025002, +arXiv:1105.1130 [hep-ph]. +Chakraborty, +S., +T. +Fischer, +A. +Mirizzi, +N. +Saviano, +and R. Tomas (2011c), Phys. Rev. Lett. 107, 151101, +arXiv:1104.4031 [hep-ph]. +Chakraborty, S., R. Hansen, I. Izaguirre, +and G. Raffelt +(2016a), Nucl. Phys. B 908, 366, arXiv:1602.02766 [hep- +ph]. +Chakraborty, S., R. S. Hansen, I. Izaguirre, +and G. Raffelt +(2016b), JCAP 03, 042, arXiv:1602.00698 [hep-ph]. +Chakraborty, S., A. Mirizzi, N. Saviano, and D. d. S. Seixas +(2014), Phys. Rev. D 89 (9), 093001, arXiv:1402.1767 [hep- +ph]. +Chatelain, A., +and C. Volpe (2017), Phys. Rev. D 95 (4), +043005, arXiv:1611.01862 [hep-ph]. +Chatelain, A., and M. C. Volpe (2018), Phys. Rev. D 97 (2), +023014, arXiv:1710.11518 [hep-ph]. +Chatelain, A., +and M. C. Volpe (2020), Phys. Lett. B801, +135150, arXiv:1906.12152 [hep-ph]. +Chen, J. Z., I. M. Oldengott, G. Pierobon, and Y. Y. Y. Wong +(2022), Eur. Phys. J. C 82 (7), 640, arXiv:2203.09075 [hep- +ph]. +Cherry, J. F., J. Carlson, A. Friedland, G. M. Fuller, +and A. Vlasenko (2012), Phys. Rev. Lett. 108, 261104, +arXiv:1203.1607 [hep-ph]. +Cherry, J. F., J. Carlson, A. Friedland, G. M. Fuller, +and A. Vlasenko (2013), Phys. Rev. D 87, 085037, +arXiv:1302.1159 [astro-ph.HE]. +Cherry, J. F., G. M. Fuller, S. Horiuchi, K. Kotake, T. Taki- +waki, and T. Fischer (2020), Phys. Rev. D 102 (2), 023022, +arXiv:1912.11489 [astro-ph.HE]. +Choubey, S., N. Harries, and G. Ross (2007), Phys. Rev. D +76, 073013, arXiv:hep-ph/0703092. +Christenson, J. H., J. W. Cronin, V. L. Fitch, and R. Turlay +(1964), Phys. Rev. Lett. 13, 138. +Cigan, +P., +et +al. +(2019), +Astrophys. +J. +886, +51, +arXiv:1910.02960 [astro-ph.HE]. +Cirigliano, V., M. W. Paris, +and S. Shalgar (2017), Phys. +Lett. B 774, 258, arXiv:1706.07052 [hep-ph]. +Cohen-Tannoudji, C., B. Diu, and F. Lal¨oe (1998), Hermann, +Editeurs des sciences et des arts 1-2. +Colgate, S. A., and M. H. Johnson (1960), Phys. Rev. Lett. +5, 235... . +Colgate, S. A., and R. H. White (1966), Astrophys. J. 143, +626. +Coloma, P., I. Esteban, M. C. Gonzalez-Garcia, and M. Mal- +toni (2020), JHEP 02, 023, [Addendum: JHEP 12, 071 +(2020)], arXiv:1911.09109 [hep-ph]. +Couch, S. M., and C. D. Ott (2015), Astrophys. J. 799 (1), +5, arXiv:1408.1399 [astro-ph.HE]. +Cowan, C. L., F. Reines, F. B. Harrison, H. W. Kruse, and +A. D. McGuire (1956), Science 124, 103. +Cowan, J. J., C. Sneden, J. E. Lawler, A. Aprahamian, +M. Wiescher, K. Langanke, G. Mart´ınez-Pinedo, and F.- +K. Thielemann (2021), Rev. Mod. Phys. 93 (1), 15002, +arXiv:1901.01410 [astro-ph.HE]. +Cusinato, +M., +F. +M. +Guercilena, +A. +Perego, +D. +Lo- +goteta, D. Radice, S. Bernuzzi, +and S. Ansoldi (2021), +10.1140/epja/s10050-022-00743-5, +arXiv:2111.13005 +[astro-ph.HE]. +Cˆot´e, +B., +et +al. +(2019), +Astrophys. +J. +875 +(2), +106, +arXiv:1809.03525 [astro-ph.HE]. +Das, A., A. Dighe, +and M. Sen (2017), JCAP 05, 051, +arXiv:1705.00468 [hep-ph]. +Dasgupta, B., and A. Dighe (2007), Phys. Rev. D 75, 093002, +arXiv:hep-ph/0510219. +Dasgupta, B., A. Dighe, G. G. Raffelt, +and A. Y. Smirnov +(2009), Phys. Rev. Lett. 103, 051105, arXiv:0904.3542 +[hep-ph]. +Dasgupta, B., and A. Mirizzi (2015), Phys. Rev. D 92 (12), +125030, arXiv:1509.03171 [hep-ph]. +Dasgupta, B., A. Mirizzi, and M. Sen (2017), JCAP 02, 019, +arXiv:1609.00528 [hep-ph]. +Dasgupta, B., A. Mirizzi, and M. Sen (2018), Phys. Rev. D +98 (10), 103001, arXiv:1807.03322 [hep-ph]. +Davidson, S., C. Pena-Garay, N. Rius, +and A. Santamaria + +40 +(2003), JHEP 03, 011, arXiv:hep-ph/0302093. +Davis, J., Raymond, D. S. Harmer, +and K. C. Hoffman +(1968), Phys. Rev. Lett. 20, 1205. +Davis, R. (1964), Phys. Rev. Lett. 12, 303. +De Gouvea, A., I. Martinez-Soler, Y. F. Perez-Gonzalez, +and +M. +Sen +(2020), +Phys. +Rev. +D +102, +123012, +arXiv:2007.13748 [hep-ph]. +Deaton, M. B., E. O’Connor, Y. L. Zhu, A. Bohn, J. Jesse, +F. Foucart, M. D. Duez, +and G. C. McLaughlin (2018), +Phys. Rev. D 98 (10), 103014, arXiv:1806.10255 [astro- +ph.HE]. +Delfan Azari, M., S. Yamada, T. Morinaga, W. Iwakami, +H. Okawa, H. Nagakura, and K. Sumiyoshi (2019), Phys. +Rev. D 99 (10), 103011, arXiv:1902.07467 [astro-ph.HE]. +Delfan Azari, M., S. Yamada, T. Morinaga, H. Nagakura, +S. Furusawa, A. Harada, H. Okawa, W. Iwakami, +and +K. Sumiyoshi (2020), Phys. Rev. D 101 (2), 023018, +arXiv:1910.06176 [astro-ph.HE]. +Di Valentino, E., O. Mena, S. Pan, L. Visinelli, W. Yang, +A. Melchiorri, D. F. Mota, A. G. Riess, and J. Silk (2021), +Class. Quant. Grav. 38 (15), 153001, arXiv:2103.01183 +[astro-ph.CO]. +Dighe, A. S., M. T. Keil, +and G. G. Raffelt (2003), JCAP +06, 005, arXiv:hep-ph/0303210. +Dighe, A. S., +and A. Y. Smirnov (2000), Phys. Rev. D 62, +033007, arXiv:hep-ph/9907423. +Dolgov, A. D., S. H. Hansen, S. Pastor, S. T. Petcov, G. G. +Raffelt, and D. V. Semikoz (2002), Nucl. Phys. B 632, 363, +arXiv:hep-ph/0201287. +Duan, H., G. M. Fuller, J. Carlson, and Y.-Z. Qian (2006a), +Phys. Rev. D 74, 105014, arXiv:astro-ph/0606616. +Duan, H., G. M. Fuller, J. Carlson, and Y.-Z. Qian (2007a), +Phys. Rev. D 75, 125005, arXiv:astro-ph/0703776. +Duan, H., G. M. Fuller, and Y.-Z. Qian (2006b), Phys. Rev. +D 74, 123004, arXiv:astro-ph/0511275. +Duan, H., G. M. Fuller, and Y.-Z. Qian (2007b), Phys. Rev. +D 76, 085013, arXiv:0706.4293 [astro-ph]. +Duan, H., G. M. Fuller, +and Y.-Z. Qian (2010), Ann. Rev. +Nucl. Part. Sci. 60, 569, arXiv:1001.2799 [hep-ph]. +Duan, H., and J. P. Kneller (2009), J. Phys. G 36, 113201, +arXiv:0904.0974 [astro-ph.HE]. +Duan, H., +and S. Shalgar (2015), Phys. Lett. B 747, 139, +arXiv:1412.7097 [hep-ph]. +Dvornikov, M. (2013), JCAP 06, 015, arXiv:1306.2659 [hep- +ph]. +Dvornikov, M., A. Grigoriev, and A. Studenikin (2005a), Int. +J. Mod. Phys. D 14, 309, arXiv:hep-ph/0406114. +Dvornikov, M., A. Grigoriev, and A. Studenikin (2005b), Int. +J. Mod. Phys. D 14, 309, arXiv:hep-ph/0406114. +Eguchi, K., et al. (KamLAND) (2003), Phys. Rev. Lett. 90, +021802, arXiv:hep-ex/0212021 [hep-ex]. +Ekanger, N., S. Horiuchi, K. Kotake, +and K. Sumiyoshi +(2022), Phys. Rev. D 106 (4), 043026, arXiv:2206.05299 +[astro-ph.HE]. +Ekinci, B., Y. Pehlivan, and A. V. Patwardhan (2021), Phys. +Rev. D 103 (4), 043016, arXiv:2101.01797 [astro-ph.HE]. +Engel, J., G. McLaughlin, and C. Volpe (2003), Phys. Rev. +D 67, 013005, arXiv:hep-ph/0209267. +Ertl, +T., +H. +T. +Janka, +S. +Woosley, +T. +Sukhbold, +and M. Ugliano (2016), Astrophys. J. 818 (2), 124, +arXiv:1503.07522 [astro-ph.SR]. +ESA/Hubble,, +and +NASA +(2011), +https://esahubble.org/images/potw1142a/. +Esteban-Pretel, A., A. Mirizzi, S. Pastor, R. Tomas, G. G. +Raffelt, P. D. Serpico, +and G. Sigl (2008), Phys. Rev. D +78, 085012, arXiv:0807.0659 [astro-ph]. +Esteban-Pretel, A., R. Tomas, +and J. Valle (2007a), Phys. +Rev. D 76, 053001, arXiv:0704.0032 [hep-ph]. +Esteban-Pretel, A., R. Tomas, +and J. W. F. Valle (2007b), +Phys. Rev. D 76, 053001, arXiv:0704.0032 [hep-ph]. +Esteban-Pretel, A., R. Tomas, +and J. W. F. Valle (2010), +Phys. Rev. D 81, 063003, arXiv:0909.2196 [hep-ph]. +Farzan, Y., +and S. Palomares-Ruiz (2014), JCAP 06, 014, +arXiv:1401.7019 [hep-ph]. +Farzan, Y., +and M. Tortola (2018), Front. in Phys. 6, 10, +arXiv:1710.09360 [hep-ph]. +Fetter, J., G. McLaughlin, A. Balantekin, +and G. Fuller +(2003), Astropart. Phys. 18, 433, arXiv:hep-ph/0205029. +Fidler, C., M. Herranen, K. Kainulainen, and P. M. Rahkila +(2012), JHEP 02, 065, arXiv:1108.2309 [hep-ph]. +Fiorillo, D. F. G., G. G. Raffelt, +and E. Vitagliano (2022), +arXiv:2209.11773 [hep-ph]. +Fogli, G., E. Lisi, D. Montanino, and A. Mirizzi (2003), Phys. +Rev. D 68, 033005, arXiv:hep-ph/0304056. +Fogli, G. L., E. Lisi, A. Marrone, +and A. Mirizzi (2007), +JCAP 12, 010, arXiv:0707.1998 [hep-ph]. +Fogli, G. L., E. Lisi, A. Mirizzi, +and D. Montanino (2002), +Phys. Rev. D 66, 013009, arXiv:hep-ph/0202269. +Fogli, G. L., E. Lisi, A. Mirizzi, +and D. Montanino (2004), +Phys. Rev. D 70, 013001, arXiv:hep-ph/0401227. +Fogli, G. L., E. Lisi, A. Mirizzi, +and D. Montanino (2005), +JCAP 04, 002, arXiv:hep-ph/0412046. +Fogli, G. L., E. Lisi, A. Mirizzi, +and D. Montanino (2006), +JCAP 06, 012, arXiv:hep-ph/0603033. +Foglizzo, T., et al. (2015), Publ. Astron. Soc. Austral. 32, +e009, arXiv:1501.01334 [astro-ph.HE]. +Fornengo, N., C. Giunti, C. W. Kim, +and J. Song (1997), +Phys. Rev. D56, 1895, arXiv:hep-ph/9611231 [hep-ph]. +Frensel, M., M.-R. Wu, C. Volpe, and A. Perego (2017), Phys. +Rev. D 95 (2), 023011, arXiv:1607.05938 [astro-ph.HE]. +Friedland, +A., +and A. Gruzinov (2006), +arXiv:astro- +ph/0607244. +Friedland, A., +and C. Lunardini (2003a), JHEP 10, 043, +arXiv:hep-ph/0307140. +Friedland, A., and C. Lunardini (2003b), Phys. Rev. D 68, +013007, arXiv:hep-ph/0304055. +Frohlich, C., G. Martinez-Pinedo, M. Liebendorfer, F. K. +Thielemann, E. Bravo, W. R. Hix, K. Langanke, and N. T. +Zinner (2006), Phys. Rev. Lett. 96, 142502, arXiv:astro- +ph/0511376. +Froustey, J., C. Pitrou, and M. C. Volpe (2020), JCAP 12, +015, arXiv:2008.01074 [hep-ph]. +Fukuda, Y., et al. (Super-Kamiokande) (1998), Phys. Rev. +Lett. 81, 1562, arXiv:hep-ex/9807003 [hep-ex]. +Fukugita, M., and M. Kawasaki (2003), Mon. Not. Roy. As- +tron. Soc. 340, L7, arXiv:astro-ph/0204376. +Fuller, G. M., +and Y.-Z. Qian (1996), Nucl. Phys. A 606, +167, arXiv:astro-ph/9611126. +Galais, S., J. Kneller, +and C. Volpe (2012), J. Phys. G 39, +035201, arXiv:1102.1471 [astro-ph.SR]. +Galais, S., J. Kneller, C. Volpe, +and J. Gava (2010), Phys. +Rev. D 81, 053002, arXiv:0906.5294 [hep-ph]. +Galais, S., +and C. Volpe (2011), Phys. Rev. D 84, 085005, +arXiv:1103.5302 [astro-ph.SR]. +Gallo Rosso, A. (2021), JCAP 06, 046, arXiv:2012.12579 [hep- +ph]. +Gallo Rosso, A., F. Vissani, and M. C. Volpe (2017), JCAP +11, 036, arXiv:1708.00760 [hep-ph]. +Gallo Rosso, A., F. Vissani, and M. C. Volpe (2018), JCAP +04, 040, arXiv:1712.05584 [hep-ph]. + +41 +Gando, A., et al. (KamLAND) (2012), Astrophys. J. 745, +193, arXiv:1105.3516 [astro-ph.HE]. +Gaudin, M. (1976), . +Gava, J., J. Kneller, C. Volpe, +and G. McLaughlin (2009), +Phys. Rev. Lett. 103, 071101, arXiv:0902.0317 [hep-ph]. +Gava, J., +and C. Volpe (2008), Phys. Rev. D 78, 083007, +arXiv:0807.3418 [astro-ph]. +Gava, J., and C. Volpe (2010), Nucl. Phys. B 837, 50, [Er- +ratum: Nucl.Phys.B 957, 115035 (2020)], arXiv:1002.0981 +[hep-ph]. +George, M., M.-R. Wu, I. Tamborra, R. Ardevol-Pulpillo, +and H.-T. Janka (2020), Phys. Rev. D 102 (10), 103015, +arXiv:2009.04046 [astro-ph.HE]. +Giunti, +C. +(2020), +Phys. +Rev. +D +101 +(3), +035039, +arXiv:1909.00466 [hep-ph]. +Giunti, C., and C. W. Kim (2007), Fundamentals of Neutrino +Physics and Astrophysics. +Giunti, C., K. A. Kouzakov, Y.-F. Li, A. V. Lokhov, A. I. +Studenikin, and S. Zhou (2016), Annalen Phys. 528, 198, +arXiv:1506.05387 [hep-ph]. +Giunti, C., and M. Laveder (2011), Phys. Rev. C 83, 065504, +arXiv:1006.3244 [hep-ph]. +Giunti, C., Y. F. Li, C. A. Ternes, and Z. Xin (2022), Phys. +Lett. B 829, 137054, arXiv:2110.06820 [hep-ph]. +Giunti, C., and A. Studenikin (2015), Rev. Mod. Phys. 87, +531, arXiv:1403.6344 [hep-ph]. +Glas, R., H. T. Janka, F. Capozzi, M. Sen, B. Dasgupta, +A. Mirizzi, +and G. Sigl (2020), Phys. Rev. D 101 (6), +063001, arXiv:1912.00274 [astro-ph.HE]. +Glashow, S. L. (1961), Nucl. Phys. 22, 579. +Godunov, S., +and G. Pastukhov (2011), Phys. Atom. Nucl. +74, 302, arXiv:0906.5556 [hep-ph]. +Goldberg, H., G. Perez, +and I. Sarcevic (2006), JHEP 11, +023, arXiv:hep-ph/0505221. +Goldhaber, M., L. Grodzins, and A. W. Sunyar (1958), Phys. +Rev. 109, 1015. +Goriely, S., A. Bauswein, O. Just, E. Pllumbi, +and H.-T. +Janka (2015), Mon. Not. Roy. Astron. Soc. 452 (4), 3894, +arXiv:1504.04377 [astro-ph.SR]. +de Gouvˆea, A., I. Martinez-Soler, Y. F. Perez-Gonzalez, +and M. Sen (2022), Phys. Rev. D 106 (10), 103026, +arXiv:2205.01102 [hep-ph]. +de Gouvˆea, A., I. Martinez-Soler, and M. Sen (2020), Phys. +Rev. D 101 (4), 043013, arXiv:1910.01127 [hep-ph]. +Guzzo, M. M., P. C. de Holanda, +and N. Reggiani (2003), +Phys. Lett. B 569, 45, arXiv:hep-ph/0303203. +Halim, O., C. Casentini, M. Drago, V. Fafone, K. Scholberg, +C. F. Vigorito, +and G. Pagliaroli (2021), JCAP 11, 021, +arXiv:2107.02050 [astro-ph.HE]. +Hall, B., A. Roggero, A. Baroni, and J. Carlson (2021), Phys. +Rev. D 104 (6), 063009, arXiv:2102.12556 [quant-ph]. +Halzen, F., +and G. G. Raffelt (2009), Phys. Rev. D 80, +087301, arXiv:0908.2317 [astro-ph.HE]. +Hannestad, S., G. G. Raffelt, G. Sigl, +and Y. Y. Y. Wong +(2006), Phys. Rev. D 74, 105010, [Erratum: Phys.Rev.D +76, 029901 (2007)], arXiv:astro-ph/0608695. +Hansen, R. S., and S. Hannestad (2014), Phys. Rev. D 90 (2), +025009, arXiv:1404.3833 [hep-ph]. +Hansen, R. S. L., S. Shalgar, and I. Tamborra (2021), JCAP +07, 017, arXiv:2012.03948 [astro-ph.CO]. +Hansen, R. S. L., S. Shalgar, and I. Tamborra (2022), Phys. +Rev. D 105 (12), 123003, arXiv:2204.11873 [astro-ph.HE]. +Harris, R. A., and L. Stodolsky (1982), Phys. Lett. B 116, +464. +Haxton, W. (1986), Phys. Rev. Lett. 57, 1271. +Haxton, W. C., R. G. Hamish Robertson, +and A. M. +Serenelli (2013), Ann. Rev. Astron. Astrophys. 51, 21, +arXiv:1208.5723 [astro-ph.SR]. +Hayes, A. C., +and I. S. Towner (2000), Phys. Rev. C 61, +044603, arXiv:nucl-th/9907049. +Hirata, K., et al. (Kamiokande-II) (1987), Phys. Rev. Lett. +58, 1490. +Hopkins, A. M., and J. F. Beacom (2006), Astrophys. J. 651, +142, arXiv:astro-ph/0601463. +Horiuchi, S., J. F. Beacom, and E. Dwek (2009), Phys. Rev. +D 79, 083013, arXiv:0812.3157 [astro-ph]. +Horiuchi, S., J. F. Beacom, C. S. Kochanek, J. L. Prieto, K. Z. +Stanek, and T. A. Thompson (2011), Astrophys. J. 738, +154, arXiv:1102.1977 [astro-ph.CO]. +Horiuchi, S., T. Kinugawa, T. Takiwaki, K. Takahashi, +and K. Kotake (2021), Phys. Rev. D 103 (4), 043003, +arXiv:2012.08524 [astro-ph.HE]. +Horiuchi, S., +and J. P. Kneller (2018), J. Phys. G 45 (4), +043002, arXiv:1709.01515 [astro-ph.HE]. +Horiuchi, S., K. Nakamura, T. Takiwaki, K. Kotake, +and +M. Tanaka (2014), Mon. Not. Roy. Astron. Soc. 445, L99, +arXiv:1409.0006 [astro-ph.HE]. +Horiuchi, +S., +K. Sumiyoshi, +K. Nakamura, +T. Fischer, +A. Summa, T. Takiwaki, H.-T. Janka, +and K. Ko- +take (2018), Mon. Not. Roy. Astron. Soc. 475 (1), 1363, +arXiv:1709.06567 [astro-ph.HE]. +Hoyle, F., and W. A. Fowler (1960), Astrophys. J. 132, 565, +[Erratum: Astrophys.J. 134, 1028 (1961)]. +Ivanez-Ballesteros, +P., +and +M. +C. +Volpe +(2022), +arXiv:2209.12465 [hep-ph]. +Izaguirre, I., G. Raffelt, and I. Tamborra (2017), Phys. Rev. +Lett. 118 (2), 021101, arXiv:1610.01612 [hep-ph]. +Janka, H.-T. (2012), Ann. Rev. Nucl. Part. Sci. 62, 407, +arXiv:1206.2503 [astro-ph.SR]. +Janka, +H. +T. +(2017), +10.1007/978-3-319-21846-5 109, +arXiv:1702.08825 [astro-ph.HE]. +Jegerlehner, B., F. Neubig, and G. Raffelt (1996), Phys. Rev. +D 54, 1194, arXiv:astro-ph/9601111. +Just, O., S. Abbar, M.-R. Wu, I. Tamborra, H.-T. Janka, +and F. Capozzi (2022), Phys. Rev. D 105 (8), 083024, +arXiv:2203.16559 [astro-ph.HE]. +Kajino, T., W. Aoki, A. Balantekin, R. Diehl, M. Famiano, +and G. Mathews (2019), Prog. Part. Nucl. Phys. 107, 109, +arXiv:1906.05002 [astro-ph.HE]. +Kartavtsev, A., G. Raffelt, and H. Vogel (2015), Phys. Rev. +D 91 (12), 125020, arXiv:1504.03230 [hep-ph]. +Kato, C., K. Ishidoshiro, and T. Yoshida (2020), Ann. Rev. +Nucl. Part. Sci. 70, 121, arXiv:2006.02519 [astro-ph.HE]. +Keehn, J. G., +and C. Lunardini (2012), Phys. Rev. D 85, +043011, arXiv:1012.1274 [astro-ph.CO]. +Keil, M. T., G. G. Raffelt, +and H.-T. Janka (2003), Astro- +phys. J. 590, 971, arXiv:astro-ph/0208035. +Kersten, J., and A. Y. Smirnov (2016), Eur. Phys. J. C 76 (6), +339, arXiv:1512.09068 [hep-ph]. +Kim, C. W., J. Kim, +and W. K. Sze (1988), Phys. Rev. D +37, 1072. +Kim, C. W., W. K. Sze, and S. Nussinov (1987), Phys. Rev. +D 35, 4014. +King, S. F. (2015), J. Phys. G 42, 123001, arXiv:1510.02091 +[hep-ph]. +Kirkwood, J. (1935), J.Chem.Phys. 3 300, 1175. +Kneller, J. P., G. C. McLaughlin, and J. Brockman (2008), +Phys. Rev. D 77, 045023, arXiv:0705.3835 [astro-ph]. +Kneller, J. P., and C. Volpe (2010), Phys. Rev. D 82, 123004, +arXiv:1006.0913 [hep-ph]. + +42 +Kostelecky, V. A., J. T. Pantaleone, +and S. Samuel (1993), +Phys. Lett. B 315, 46. +Kostelecky, V. A., and S. Samuel (1995), Phys. Rev. D 52, +621, arXiv:hep-ph/9506262. +Kotake, K., K. Sato, and K. Takahashi (2006), Rept. Prog. +Phys. 69, 971, arXiv:astro-ph/0509456. +Kresse, D., T. Ertl, +and H.-T. Janka (2021), Astrophys. J. +909 (2), 169, arXiv:2010.04728 [astro-ph.HE]. +Kulkarni, S. R. (2005), arXiv:astro-ph/0510256 [astro-ph]. +Kuo, T.-K., +and J. T. Pantaleone (1989), Rev. Mod. Phys. +61, 937. +Lacroix, D., A. B. Balantekin, M. J. Cervia, A. V. Patward- +han, and P. Siwach (2022), Phys. Rev. D 106 (12), 123006, +arXiv:2205.09384 [nucl-th]. +Lambiase, G., G. Papini, R. Punzi, and G. Scarpetta (2005), +Phys. Rev. D 71, 073011, arXiv:gr-qc/0503027. +Langanke, K., G. Martinez-Pinedo, and A. Sieverding (2019), +10.22661/AAPPSBL.2018.28.6.41, +arXiv:1901.03741 +[astro-ph.HE]. +Lattimer, J. M., +and D. N. Schramm (1974), Astrophys. J. +192, L145. +Lattimer, J. M., +and D. N. Schramm (1976), Astrophys. J. +210, 549. +Lee, T. D., and C.-N. Yang (1956), Phys. Rev. 104, 254. +Lindquist, R. W. (1966), Ann. of Physics 37, 487. +Longo, M. J. (1987), Phys. Rev. D36, 3276. +Loredo, T. J., +and D. Q. Lamb (2002), Phys. Rev. D65, +063002, arXiv:astro-ph/0107260 [astro-ph]. +Loreti, F., and A. Balantekin (1994), Phys. Rev. D 50, 4762, +arXiv:nucl-th/9406003. +Loreti, F., Y. Qian, G. Fuller, +and A. Balantekin (1995), +Phys. Rev. D 52, 6664, arXiv:astro-ph/9508106. +Lu, J.-S., Y.-F. Li, and S. Zhou (2016), Phys. Rev. D 94 (2), +023006, arXiv:1605.07803 [hep-ph]. +Lujan-Peschard, C., G. Pagliaroli, +and F. Vissani (2014), +JCAP 07, 051, arXiv:1402.6953 [astro-ph.SR]. +Lunardini, C. (2006), Astropart. Phys. 26, 190, arXiv:astro- +ph/0509233. +Lunardini, +C. +(2009), +Phys. +Rev. +Lett. +102, +231101, +arXiv:0901.0568 [astro-ph.SR]. +Lunardini, +C. +(2016), +Astropart. +Phys. +79, +49, +arXiv:1007.3252 [astro-ph.CO]. +Lunardini, C., +and O. L. Peres (2008), JCAP 08, 033, +arXiv:0805.4225 [astro-ph]. +Lunardini, C., +and A. Y. Smirnov (2003), JCAP 06, 009, +arXiv:hep-ph/0302033. +Lunardini, C., +and I. Tamborra (2012), JCAP 07, 012, +arXiv:1205.6292 [astro-ph.SR]. +Lund, T., +and J. P. Kneller (2013), Phys. Rev. D 88 (2), +023008, arXiv:1304.6372 [astro-ph.HE]. +Madau, P., and M. Dickinson (2014), Ann. Rev. Astron. As- +trophys. 52, 415, arXiv:1403.0007 [astro-ph.CO]. +Majorana, E. (1937), Nuovo Cim. 14, 171. +Malek, M., et al. (Super-Kamiokande) (2003), Phys. Rev. +Lett. 90, 061101, arXiv:hep-ex/0209028 [hep-ex]. +Malkus, A., A. Friedland, +and G. McLaughlin (2014), +arXiv:1403.5797 [hep-ph]. +Malkus, A., J. Kneller, G. McLaughlin, +and R. Surman +(2012), Phys. Rev. D 86, 085015, arXiv:1207.6648 [hep-ph]. +Malkus, A., G. McLaughlin, +and R. Surman (2016), Phys. +Rev. D 93 (4), 045021, arXiv:1507.00946 [hep-ph]. +Mangano, G., G. Miele, S. Pastor, T. Pinto, O. Pisanti, and +P. D. Serpico (2005), Nucl. Phys. B 729, 221, arXiv:hep- +ph/0506164. +Mangano, G., G. Miele, S. Pastor, T. Pinto, O. Pisanti, and +P. D. Serpico (2006), Nucl. Phys. B 756, 100, arXiv:hep- +ph/0607267. +Martin, D., A. Perego, A. Arcones, F.-K. Thielemann, O. Ko- +robkin, and S. Rosswog (2015), Astrophys. J. 813 (1), 2, +arXiv:1506.05048 [astro-ph.SR]. +Martin, J. D., J. Carlson, V. Cirigliano, and H. Duan (2021), +Phys. Rev. D 103, 063001, arXiv:2101.01278 [hep-ph]. +Martin, J. D., A. Roggero, H. Duan, and J. Carlson (2023), +arXiv:2301.07049 [hep-ph]. +Martin, J. D., A. Roggero, H. Duan, J. Carlson, +and +V. Cirigliano (2022), Phys. Rev. D 105 (8), 083020, +arXiv:2112.12686 [hep-ph]. +Martin, J. D., C. Yi, and H. Duan (2020), Phys. Lett. B 800, +135088, arXiv:1909.05225 [hep-ph]. +Mathews, G. J., L. Boccioli, J. Hidaka, and T. Kajino (2020), +Mod. Phys. Lett. A 35 (25), 2030011, arXiv:1907.10088 +[astro-ph.HE]. +Mathews, G. J., J. Hidaka, T. Kajino, and J. Suzuki (2014), +Astrophys. J. 790, 115, arXiv:1405.0458 [astro-ph.CO]. +McKellar, B. H., and M. J. Thomson (1994), Phys. Rev. D +49, 2710. +McLaughlin, G., J. Fetter, A. Balantekin, +and G. Fuller +(1999), Phys. Rev. C 59, 2873, arXiv:astro-ph/9902106. +Mention, G., M. Fechner, T. Lasserre, T. A. Mueller, D. Lhuil- +lier, M. Cribier, +and A. Letourneau (2011), Phys. Rev. +D83, 073006, arXiv:1101.2755 [hep-ex]. +Metzger, B. D., G. Martinez-Pinedo, S. Darbha, E. Quataert, +A. Arcones, D. Kasen, R. Thomas, P. Nugent, I. V. Panov, +and N. T. Zinner (2010), Mon. Not. Roy. Astron. Soc. 406, +2650, arXiv:1001.5029 [astro-ph.HE]. +Mezzacappa, A., E. Endeve, O. E. Bronson Messer, +and +S. W. Bruenn (2020), Liv. Rev. Comput. Astrophys. 6, 4, +arXiv:2010.09013 [astro-ph.HE]. +Mezzacappa, A., et al. (2015), +arXiv:1501.01688 [astro- +ph.SR]. +Mikheev, S. P., and A. Yu. Smirnov (1986), Nuovo Cim. C9, +17. +Minakata, H., H. Nunokawa, R. Tomas, and J. W. F. Valle +(2008), JCAP 12, 006, arXiv:0802.1489 [hep-ph]. +Mirizzi, +A. +(2013), +Phys. +Rev. +D +88 +(7), +073004, +arXiv:1308.1402 [hep-ph]. +Mirizzi, A., G. Mangano, and N. Saviano (2015), Phys. Rev. +D 92 (2), 021702, arXiv:1503.03485 [hep-ph]. +Mirizzi, A., I. Tamborra, H.-T. Janka, N. Saviano, K. Schol- +berg, R. Bollig, L. Hudepohl, and S. Chakraborty (2016), +Riv. Nuovo Cim. 39 (1-2), 1, arXiv:1508.00785 [astro- +ph.HE]. +Moeller, K., A. M. Suliga, I. Tamborra, +and P. B. Denton +(2018), JCAP 05, 066, arXiv:1804.03157 [astro-ph.HE]. +Mohapatra, R. N., +and P. B. Pal (1998), World Sci. +Lect. +Notes +Phys. +60, +1, +[World +Sci. +Lect. +Notes +Phys.72,1(2004)]. +Mosquera Cuesta, H. J., G. Lambiase, +and J. P. Pereira +(2017), Phys. Rev. D 95 (2), 025011, arXiv:1701.00431 [gr- +qc]. +M¨uller, B., and H.-T. Janka (2014), Astrophys. J. 788, 82, +arXiv:1402.3415 [astro-ph.SR]. +Nagakura, H., L. Johns, A. Burrows, and G. M. Fuller (2021), +Phys. Rev. D 104 (8), 083025, arXiv:2108.07281 [astro- +ph.HE]. +Nakazato, +K. +(2013), +Phys. +Rev. +D +88 +(8), +083012, +arXiv:1306.4526 [astro-ph.HE]. +Nakazato, K., E. Mochida, Y. Niino, and H. Suzuki (2015), +Astrophys. J. 804 (1), 75, arXiv:1503.01236 [astro-ph.HE]. +Nedora, V., S. Bernuzzi, D. Radice, B. Daszuta, A. En- + +43 +drizzi, A. Perego, A. Prakash, M. Safarzadeh, F. Schi- +anchi, and D. Logoteta (2021), Astrophys. J. 906 (2), 98, +arXiv:2008.04333 [astro-ph.HE]. +Nunokawa, H., Y. Z. Qian, and G. M. Fuller (1997a), Phys. +Rev. D 55, 3265, arXiv:astro-ph/9610209. +Nunokawa, H., A. Rossi, V. B. Semikoz, and J. W. F. Valle +(1996), Nucl. Phys. B 472, 495, arXiv:hep-ph/9602307. +Nunokawa, +H., +V. B. Semikoz, +A. Y. Smirnov, +and +J. W. F. Valle (1997b), Nucl. Phys. B 501, 17, arXiv:hep- +ph/9701420. +O’Connor, E., and C. D. Ott (2011), Astrophys. J. 730, 70, +arXiv:1010.5550 [astro-ph.HE]. +O’Connor, E., et al. (2018), J. Phys. G 45 (10), 104001, +arXiv:1806.04175 [astro-ph.HE]. +Odrzywolek, A., M. Misiaszek, +and M. Kutschera (2004), +Astropart. Phys. 21, 303, arXiv:astro-ph/0311012. +Ohlsson, +T. +(2013), +Rept. +Prog. +Phys. +76, +044201, +arXiv:1209.2710 [hep-ph]. +Padilla-Gay, I., S. Shalgar, +and I. Tamborra (2021), JCAP +01, 017, arXiv:2009.01843 [astro-ph.HE]. +Page, D., M. V. Beznogov, I. Garibay, J. M. Lattimer, +M. Prakash, +and H.-T. Janka (2020), Astrophys. J. +898 (2), 125, arXiv:2004.06078 [astro-ph.HE]. +Pagliaroli, G., F. Vissani, E. Coccia, +and W. Fulgione +(2009a), Phys. Rev. Lett. 103, 031102, arXiv:0903.1191 +[hep-ph]. +Pagliaroli, G., F. Vissani, M. L. Costantini, +and A. Ianni +(2009b), Astropart. Phys. 31, 163, arXiv:0810.0466 [astro- +ph]. +Pantaleone, J. T. (1992), Phys. Lett. B 287, 128. +Parke, S. J. (1986), Phys. Rev. Lett. 57, 1275. +Pastor, S., G. G. Raffelt, +and D. V. Semikoz (2002), Phys. +Rev. D 65, 053011, arXiv:hep-ph/0109035. +Patton, K. M., C. Lunardini, R. J. Farmer, and F. X. Timmes +(2017), Astrophys. J. 851 (1), 6, arXiv:1709.01877 [astro- +ph.HE]. +Patwardhan, A. V., M. J. Cervia, +and A. B. Balantekin +(2021), Phys. Rev. D 104 (12), 123035, arXiv:2109.08995 +[hep-ph]. +Pauli, W. (1930), Wissenschaftlicher Briefwechsel mit Bohr, +Einstein, Heisenberg,, Letter 259. +Payez, A., C. Evoli, T. Fischer, M. Giannotti, A. Mirizzi, +and A. Ringwald (2015), JCAP 02, 006, arXiv:1410.3747 +[astro-ph.HE]. +Pehlivan, Y., A. Balantekin, +and T. Kajino (2014), Phys. +Rev. D 90 (6), 065011, arXiv:1406.5489 [hep-ph]. +Pehlivan, Y., A. Balantekin, T. Kajino, +and T. Yoshida +(2011), Phys. Rev. D 84, 065008, arXiv:1105.1182 [astro- +ph.CO]. +Pehlivan, +Y., +A. +L. +Suba¸sı, +N. +Ghazanfari, +S. +Birol, +and H. Y¨uksel (2017), Phys. Rev. D 95 (6), 063022, +arXiv:1603.06360 [astro-ph.HE]. +Perego, A., S. Rosswog, R. M. Cabez´on, O. Korobkin, +R. K¨appeli, A. Arcones, +and M. Liebend¨orfer (2014), +Mon. Not. Roy. Astron. Soc. 443 (4), 3134, arXiv:1405.6730 +[astro-ph.HE]. +Piriz, D., M. Roy, +and J. Wudka (1996), Phys. Rev. D 54, +1587, arXiv:hep-ph/9604403. +Podsiadlowski, P. (1992), Astronomical Society of the Pacific +104, 717. +Pontecorvo, B. (1957), Sov. Phys. JETP 6, 429, [Zh. Eksp. +Teor. Fiz.33,549(1957)]. +Pontecorvo, B. (1958), Sov. Phys. JETP 7, 172, [Zh. Eksp. +Teor. Fiz.34,247(1957)]. +Popov, A., and A. Studenikin (2021), Phys. Rev. D 103 (11), +115027, arXiv:2102.07991 [hep-ph]. +Priya, +A., +and C. Lunardini (2017), JCAP 11, 031, +arXiv:1705.02122 [astro-ph.HE]. +Qian, Y.-Z. (2014), J. Phys. G 41, 044002, arXiv:1310.4462 +[astro-ph.SR]. +Radice, D., E. Abdikamalov, C. D. Ott, P. M¨osta, S. M. +Couch, +and L. F. Roberts (2018a), J. Phys. G 45 (5), +053003, arXiv:1710.01282 [astro-ph.HE]. +Radice, +D., +A. Perego, +K. Hotokezaka, +S. A. Fromm, +S. Bernuzzi, +and L. F. Roberts (2018b), Astrophys. J. +869 (2), 130, arXiv:1809.11161 [astro-ph.HE]. +Raffelt, G., S. Sarikas, and D. de Sousa Seixas (2013), Phys. +Rev. Lett. 111 (9), 091101, [Erratum: Phys.Rev.Lett. 113, +239903 (2014)], arXiv:1305.7140 [hep-ph]. +Raffelt, G., and D. de Sousa Seixas (2013), Phys. Rev. D 88, +045031, arXiv:1307.7625 [hep-ph]. +Raffelt, G. G. (1996), Stars as laboratories for fundamental +physics. +Raffelt, G. G., +and A. Y. Smirnov (2007), Phys. Rev. D +76, 081301, [Erratum: +Phys.Rev.D 77, 029903 (2008)], +arXiv:0705.1830 [hep-ph]. +Reddy, N. A., C. C. Steidel, M. Pettini, K. L. Adelberger, +A. E. Shapley, D. K. Erb, +and M. Dickinson (2008), As- +trophys. J. Suppl. 175, 48, arXiv:0706.4091 [astro-ph]. +Reno, M. H., Y. S. Jeong, S. Palomares-Ruiz, and I. Sarcevic +(2021), PoS ICHEP2020, 603, arXiv:2012.05380 [hep-ph]. +Ricciardi, G., N. Vignaroli, and F. Vissani (2022), JHEP 08, +212, arXiv:2206.05567 [hep-ph]. +Richardson, R. W. (1966), Phys. Rev. 141, 949. +Richers, S. A., G. C. McLaughlin, J. P. Kneller, +and +A. Vlasenko (2019), Phys. Rev. D 99 (12), 123014, +arXiv:1903.00022 [astro-ph.HE]. +Ring, P., and P. Schuck (2004), The nuclear many body prob- +lem. +Roggero, +A. (2021a), Phys. Rev. D 104 (12), 123023, +arXiv:2103.11497 [hep-ph]. +Roggero, +A. (2021b), Phys. Rev. D 104 (10), 103016, +arXiv:2102.10188 [hep-ph]. +Roggero, A., E. Rrapaj, and Z. Xiong (2022), Phys. Rev. D +106 (4), 043022, arXiv:2203.02783 [astro-ph.HE]. +Rozwadowska, K., F. Vissani, and E. Cappellaro (2021), New +Astron. 83, 101498, arXiv:2009.03438 [astro-ph.HE]. +Rrapaj, +E., +A. +V. +Patwardhan, +E. +Armstrong, +and +G. M. Fuller (2021), Phys. Rev. D 103 (4), 043006, +arXiv:2010.00695 [astro-ph.HE]. +Rudzsky, M. A. (1990), Astrophysics and Space Science 165, +65. +Rujopakarn, W., et al. (2010), Astrophys. J. 718, 1171, +arXiv:1006.4359 [astro-ph.CO]. +Salam, A. (1957), Nuovo Cim. 5, 299. +Salpeter, E. E. (1955), Astrophys. J. 121, 161. +Samuel, S. (1993), Phys. Rev. D 48, 1462. +Sarikas, S., D. de Sousa Seixas, and G. Raffelt (2012a), Phys. +Rev. D 86, 125020, arXiv:1210.4557 [hep-ph]. +Sarikas, S., I. Tamborra, G. Raffelt, L. Hudepohl, and H.-T. +Janka (2012b), Phys. Rev. D 85, 113007, arXiv:1204.0971 +[hep-ph]. +Sasaki, H., and T. Takiwaki (2021), Phys. Rev. D 104 (2), +023018, arXiv:2106.02181 [hep-ph]. +Sato, K., and H. Suzuki (1987), Phys. Lett. B196, 267. +Sawyer, R. F. (1990), Phys. Rev. D 42, 3908. +Sawyer, R. F. (2005), Phys. Rev. D 72, 045003, arXiv:hep- +ph/0503013. +Sawyer, +R. +F. +(2009), +Phys. +Rev. +D +79, +105003, +arXiv:0803.4319 [astro-ph]. + +44 +Sawyer, R. F. (2016), Phys. Rev. Lett. 116 (8), 081101, +arXiv:1509.03323 [astro-ph.HE]. +Schilbach, T. S. H., O. L. Caballero, and G. C. McLaughlin +(2019), Phys. Rev. D 100 (4), 043008, arXiv:1808.03627 +[astro-ph.HE]. +Schirato, R. C., +and G. M. Fuller (2002), +arXiv:astro- +ph/0205390. +Schmidt, B. P., R. P. Kirshnerm, and R. G. Eastman (1992), +Astrophys. J. 395, 366, arXiv:astro-ph/9204004 [astro-ph]. +Scholberg, +K. +(2000), +AIP +Conf. +Proc. +523 +(1), +355, +arXiv:astro-ph/9911359. +Scholberg, K. (2012), Ann. Rev. Nucl. Part. Sci. 62, 81, +arXiv:1205.6003 [astro-ph.IM]. +Schramm, D. N., and J. W. Truran (1990), Phys. Rept. 189, +89. +Serpico, P. D., S. Chakraborty, T. Fischer, L. Hudepohl, H.- +T. Janka, and A. Mirizzi (2012), Phys. Rev. D 85, 085031, +arXiv:1111.4483 [astro-ph.SR]. +Serreau, J., +and C. Volpe (2014), Phys. Rev. D 90 (12), +125040, arXiv:1409.3591 [hep-ph]. +Shalgar, S., I. Padilla-Gay, +and I. Tamborra (2020), JCAP +06, 048, arXiv:1911.09110 [astro-ph.HE]. +Shalgar, S., +and I. Tamborra (2021), Phys. Rev. D 103, +063002, arXiv:2011.00004 [astro-ph.HE]. +Sigl, G., and G. Raffelt (1993), Nucl. Phys. B 406, 423. +Singh, +R., +and +V. +Rentala +(2021), +JCAP +08, +019, +arXiv:2007.02951 [astro-ph.HE]. +Sirera, M., +and A. Perez (1999), Phys. Rev. D 59, 125011, +arXiv:hep-ph/9810347. +Stapleford, C. J., D. J. V¨a¨an¨anen, J. P. Kneller, G. C. +McLaughlin, +and B. T. Shapiro (2016), Phys. Rev. D +94 (9), 093007, arXiv:1605.04903 [hep-ph]. +Stodolsky, L. (1979), Gen. Rel. Grav. 11, 391. +Stodolsky, L. (1987), Phys. Rev. D 36, 2273. +Strumia, A., +and F. Vissani (2003), Phys. Lett. B 564, 42, +arXiv:astro-ph/0302055. +Sturrock, P. A. (1958), Phys. Rev. 112, 1488. +Suliga, A. M., J. F. Beacom, and I. Tamborra (2022), Phys. +Rev. D 105 (4), 043008, arXiv:2112.09168 [astro-ph.HE]. +Sumiyoshi, K., S. Yamada, and H. Suzuki (2007), Astrophys. +J. 667, 382, arXiv:0706.3762 [astro-ph]. +Surman, R., G. C. McLaughlin, and W. R. Hix (2006), As- +trophys. J. 643, 1057, arXiv:astro-ph/0509365. +Surman, R., G. C. McLaughlin, M. Ruffert, H. T. Janka, +and W. R. Hix (2008), Astrophys. J. Lett. 679, L117, +arXiv:0803.1785 [astro-ph]. +Suzuki, A. (2008), Topics in astroparticle and underground +physics. Proceedings, 10th International Conference, TAUP +2007, Sendai, Japan, September 11-15, 2007, J. Phys. Conf. +Ser. 120, 072001. +Tabrizi, +Z., +and S. Horiuchi (2021), JCAP 05, 011, +arXiv:2011.10933 [hep-ph]. +Takahashi, K., K. Sato, H. Dalhed, +and J. Wilson (2003), +Astropart. Phys. 20, 189, arXiv:astro-ph/0212195. +Takiwaki, T., K. Kotake, and T. Foglizzo (2021), Mon. Not. +Roy. Astron. Soc. 508 (1), 966, arXiv:2107.02933 [astro- +ph.HE]. +Tamborra, I., F. Hanke, H.-T. Janka, B. M¨uller, G. G. Raf- +felt, +and A. Marek (2014a), Astrophys. J. 792 (2), 96, +arXiv:1402.5418 [astro-ph.SR]. +Tamborra, I., L. Huedepohl, G. Raffelt, +and H.-T. Janka +(2017), Astrophys. J. 839, 132, arXiv:1702.00060 [astro- +ph.HE]. +Tamborra, I., G. Raffelt, F. Hanke, H.-T. Janka, +and +B. +Mueller +(2014b), +Phys. +Rev. +D +90 +(4), +045032, +arXiv:1406.0006 [astro-ph.SR]. +Tamborra, I., G. G. Raffelt, L. Hudepohl, and H.-T. Janka +(2012), JCAP 01, 013, arXiv:1110.2104 [astro-ph.SR]. +Tamborra, I., and S. Shalgar (2021), Ann. Rev. Nucl. Part. +Sci. 71, 165, arXiv:2011.01948 [astro-ph.HE]. +Tanabashi, M., et al. (Particle Data Group) (2018), Phys. +Rev. D 98 (3), 030001. +Tanaka, M., et al. (2018), Astrophys. J. 852 (2), 109, +arXiv:1708.09101 [astro-ph.HE]. +Telescope, H. S. (2017), https://www.nasa.gov. +Tian, J. Y., A. V. Patwardhan, +and G. M. Fuller (2017), +Phys. Rev. D 95 (6), 063004, arXiv:1610.08586 [astro- +ph.HE]. +Tomas, R., M. Kachelriess, G. Raffelt, A. Dighe, H.-T. +Janka, and L. Scheck (2004), JCAP 09, 015, arXiv:astro- +ph/0407132. +Torrente-Lujan, +E. +(1999), +Phys. +Rev. +D +59, +073001, +arXiv:hep-ph/9807361. +Turck-Chieze, S., +and I. Lopes (1993), Astrophys. J. 408, +347. +Ugliano, M., H. Janka, A. Marek, +and A. Arcones (2016), +Astrophys. J. 821, 69, arXiv:1602.06327 [astro-ph.HE]. +Vaananen, D., +and G. McLaughlin (2016), Phys. Rev. D +93 (10), 105044, arXiv:1510.00751 [hep-ph]. +Vaananen, +D., +and C. Volpe (2011), JCAP 10, 019, +arXiv:1105.6225 [astro-ph.SR]. +Vaananen, D., and C. Volpe (2013), Phys. Rev. D 88, 065003, +arXiv:1306.6372 [hep-ph]. +Vigorito, C. F., G. Bruno, W. Fulgione, +and A. Molinario +(LVD) (2021), PoS ICRC2021, 1111. +Visinelli, +L. +(2015), +Gen. +Rel. +Grav. +47 +(5), +62, +arXiv:1410.1523 [gr-qc]. +Vissani, F. (2015), J. Phys. G42, 013001, arXiv:1409.4710 +[astro-ph.HE]. +Vissani, F., and G. Pagliaroli (2011), Astron. Astrophys. 528, +L1, arXiv:1102.0447 [astro-ph.HE]. +Vlasenko, A., G. M. Fuller, and V. Cirigliano (2014a), Phys. +Rev. D 89 (10), 105004, arXiv:1309.2628 [hep-ph]. +Vlasenko, A., G. M. Fuller, +and V. Cirigliano (2014b), +arXiv:1406.6724 [astro-ph.HE]. +Vlasenko, A., +and G. C. McLaughlin (2018), Phys. Rev. D +97 (8), 083011, arXiv:1801.07813 [astro-ph.HE]. +Volpe, C. (2004), J. Phys. G 30, L1, arXiv:hep-ph/0303222. +Volpe, C. (2005), J. Phys. G 31, 903, arXiv:hep-ph/0501233. +Volpe, C. (2007), J. Phys. G34, R1, arXiv:hep-ph/0605033 +[hep-ph]. +Volpe, C. (2015), Int. J. Mod. Phys. E 24 (09), 1541009, +arXiv:1506.06222 [astro-ph.SR]. +Volpe, C., N. Auerbach, G. Colo, T. Suzuki, and N. Van Giai +(2000), Phys. Rev. C 62, 015501, arXiv:nucl-th/0001050. +Volpe, C., D. Vaananen, and C. Espinoza (2013), Phys. Rev. +D 87 (11), 113010, arXiv:1302.2374 [hep-ph]. +Walk, L., I. Tamborra, H.-T. Janka, A. Summa, +and +D. +Kresse +(2020), +Phys. +Rev. +D +101 +(12), +123013, +arXiv:1910.12971 [astro-ph.HE]. +Wang, S. J., and W. Cassing (1985), Annals Phys. 159, 328. +Weinberg, S. (1967), Phys. Rev. Lett. 19, 1264. +Wolfenstein, L. (1978), Phys. Rev. D17, 2369, [,294(1977)]. +Wolfenstein, L. (1979), Phys. Rev. D 20, 2634. +Wright, W. P., J. P. Kneller, S. T. Ohlmann, F. K. Roepke, +K. Scholberg, +and I. R. Seitenzahl (2017), Phys. Rev. D +95 (4), 043006, arXiv:1609.07403 [astro-ph.HE]. +Wu, C. S., E. Ambler, R. W. Hayward, D. D. Hoppes, and +R. P. Hudson (1957), Phys. Rev. 105, 1413. +Wu, M.-R., H. Duan, and Y.-Z. Qian (2016), Phys. Lett. B + +45 +752, 89, arXiv:1509.08975 [hep-ph]. +Wu, M.-R., T. Fischer, L. Huther, G. Mart´ınez-Pinedo, +and Y.-Z. Qian (2014), Phys. Rev. D 89 (6), 061303, +arXiv:1305.2382 [astro-ph.HE]. +Wu, M.-R., and I. Tamborra (2017), Phys. Rev. D 95 (10), +103007, arXiv:1701.06580 [astro-ph.HE]. +Wu, M.-R., I. Tamborra, O. Just, +and H.-T. Janka (2017), +Phys. Rev. D 96 (12), 123015, arXiv:1711.00477 [astro- +ph.HE]. +Wudka, +J. (2001), Phys. Rev. D 64, 065009, arXiv:gr- +qc/0010077. +Xiong, Z., A. Sieverding, M. Sen, and Y.-Z. Qian (2020), As- +trophys. J. 900 (2), 144, arXiv:2006.11414 [astro-ph.HE]. +Xiong, +Z., +M.-R. +Wu, +G. +Mart´ınez-Pinedo, +T. +Fis- +cher, +M. George, +C.-Y. Lin, +and L. Johns (2022), +arXiv:2210.08254 [astro-ph.HE]. +Xiong, Z., M.-R. Wu, and Y.-Z. Qian (2019), 10.3847/1538- +4357/ab2870, arXiv:1904.09371 [astro-ph.HE]. +Yamada, S. (2000), Phys. Rev. D 62, 093026, arXiv:astro- +ph/0002502. +Yang, Y., +and J. P. Kneller (2017), Phys. Rev. D 96 (2), +023009, arXiv:1705.09723 [astro-ph.HE]. +Yi, C., L. Ma, J. D. Martin, and H. Duan (2019), Phys. Rev. +D 99 (6), 063005, arXiv:1901.01546 [hep-ph]. +Yoshida, T., K. Takahashi, H. Umeda, +and K. Ishidoshiro +(2016), Phys. Rev. D 93 (12), 123012, arXiv:1606.04915 +[astro-ph.HE]. +Yuksel, H., +and J. F. Beacom (2007), Phys. Rev. D 76, +083007, arXiv:astro-ph/0702613. +Yuksel, H., M. D. Kistler, J. F. Beacom, and A. M. Hopkins +(2008), Astrophys. J. Lett. 683, L5, arXiv:0804.4008 [astro- +ph]. +Yvon, J. (1935), Act. Sci. et Ind. 203. +Zhang, H., et al. (Super-Kamiokande) (2015), Astropart. +Phys. 60, 41, arXiv:1311.3738 [hep-ex]. +Zhu, Y.-L., A. Perego, and G. C. McLaughlin (2016), Phys. +Rev. D 94 (10), 105006, arXiv:1607.04671 [hep-ph]. +Ziegler, J. J., T. D. P. Edwards, A. M. Suliga, I. Tam- +borra, S. Horiuchi, S. Ando, +and K. Freese (2022), +arXiv:2205.07845 [astro-ph.GA]. +Zyla, P. A., et al. (Particle Data Group) (2020), PTEP +2020 (8), 083C01. + diff --git a/GtFKT4oBgHgl3EQfcC5D/content/tmp_files/load_file.txt b/GtFKT4oBgHgl3EQfcC5D/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e81feb50d3eacff96de9b90a11bfd4d63146bdd5 --- /dev/null +++ b/GtFKT4oBgHgl3EQfcC5D/content/tmp_files/load_file.txt @@ -0,0 +1,6067 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf,len=6066 +page_content='Neutrinos from dense: flavor mechanisms, theoretical approaches, observations, new directions M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Cristina Volpe CNRS, Universit´e Paris Cit´e, Astroparticule et Cosmologie, F-75013 Paris, France∗ (Dated: January 30, 2023) Neutrino masses and mixings produce vacuum oscillations, an established quantum me- chanical phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' In matter, the Mikheev-Smirnov-Wolfenstein effect, due to neu- trino interactions with the background particles, triggers resonant flavor modification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' In dense environments, sizable neutrino-neutrino interactions, shock waves and turbu- lence impact the neutrino flavor content under a variety of phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Theoretical approaches of neutrino propagation range from the mean-field approximation to the full quantum kinetic equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Intriguing connections have been uncovered between weakly interacting dense neutrino gases and other many-body systems and domains, from condensed matter and nuclear physics to quantum computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Besides the intrin- sic theoretical interest, establishing how neutrinos change flavor contributes to answer the longstanding open questions of how massive stars explode and of the r-process sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' It is also important for future observations of core-collapse supernova neutrinos and of the diffuse supernova neutrino background that should be discovered in the foreseeable future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' CONTENTS I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' General and historical 1 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The birth of neutrino astronomy 1 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The oscillation discovery 2 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Unknown neutrino properties 3 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Future supernova neutrino observations 4 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The r-process and GW170817 5 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Theoretical developments 6 II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Neutrino flavor mechanisms in environments 8 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Mean-field equations 8 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The MSW effect 10 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The MSW effect in dense media 12 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Shock wave effects 13 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Turbulence effects 14 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' MSW-like mechanisms 14 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Matter-neutrino resonance 15 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Spin or helicity coherence 16 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' I-resonance 16 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Neutrino-neutrino interactions 17 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Slow modes 17 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Fast modes 19 III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Flavor evolution: Theoretical frameworks 20 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The mean-field approximation 20 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The mean-field Hamiltonian: a derivation 21 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Beyond the usual mean-field 22 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Linearization 23 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The linearised equations 23 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' A dispersion-relation approach 25 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Towards neutrino quantum kinetic equations 26 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Neutrinos in presence of strong gravitational fields 27 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Connections: from atomic nuclei to quantum devices 28 IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Past and future observations 29 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' SN1987A 29 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' From the next supernova 31 ∗ Electronic address: volpe@apc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='in2p3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='fr C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The discovery of the diffuse supernova neutrino background 33 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Conclusions and perspectives 35 VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Acknowledgments 37 References 37 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' GENERAL AND HISTORICAL A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The birth of neutrino astronomy In his famous letter to Lisa Meitner, Pauli (1930) hy- pothesized the existence of a new fermion, the neutron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' He wanted to explain the observed continuous beta spec- trum in the β-decay of atomic nuclei and to save the laws of energy conservation and statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' This particle had to be as light as the electron with a mass not heav- ier than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='01 the one of the proton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Renamed neutrino (”small neutral particle” in Italian), it remained elusive until Cowan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (1956) detected electron anti-neutrinos via inverse β-decay nearby reactors, the most powerful man-made neutrino sources in terrestrial experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The same year Lee and Yang (1956) examined the question of parity conservation in weak interactions, stimulated by the so-called θ-τ meson puzzle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' They suggested, as a possible experimental test of the par- ity non-conservation hypothesis, the measurement of a pseudo-scalar observable, namely the angular distribu- tion of electrons emitted in polarized 60Co decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' In a few months Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (1957) successfully performed the ex- periment, demonstrating weak interaction differentiates the right from the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' In 1958 Goldhaber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (1958) measured neutrinos from electron capture in 152Eu and arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='11814v1 [hep-ph] 27 Jan 2023 2 found them to be left-handed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' In the Glashow (1961), Weinberg (1967), Salam (1957) (GWS) model, neutrinos are described by Weyl spinors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' In his seminal work Bethe (1939) suggested that car- bon and nitrogen act as catalysts in a chain reaction that burns hydrogen into helium in luminous main sequence stars (later known as the CNO cycle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Afterwards, solar models predicted sizable νe fluxes from energy generation due to hydrogen burning into helium in the proton-proton (pp) reaction chain (Bahcall, 1964) and the CNO cycle (Bahcall et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 1968).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' It was Davis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (1968) who first detected solar neutrinos with his pioneering radiochem- ical experiment in the Homestake mine, using neutrino capture on 37Cl (Davis, 1964).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' In a few months the mea- surement revealed less neutrinos than expected (Bahcall et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 1968): the solar neutrino problem was born.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' For over more than three decades, radiochemical, wa- ter Cherenkov and scintillator experiments showed that, depending on neutrino energy, one-third to one-half of the predicted solar neutrino fluxes were actually reach- ing the Earth (see for example (Giunti and Kim, 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Haxton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Raffelt, 1996)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Both the Standard Solar Model and neutrino properties were questioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Helioseismology (Turck-Chieze and Lopes, 1993) brought an important clue in favor of the Standard Solar Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' In particular, the solar sound speed, measured at a few % level, was agreeing with predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Among the debated solutions was the possibility that neutrinos could oscillate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Earlier Pontecorvo (1957, 1958) had sug- gested that ν could transform into ¯ν, in analogy with oscillations of neutral K0- ¯K0 mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Wolfenstein (1978, 1979) pointed out that in matter neutrinos can change flavor due to coherent forward scat- tering and a flavor-dependent refraction index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Later on, Mikheev and Smirnov (1986) realized that flavor conver- sion in matter could be resonantly amplified: an adi- abatic evolution at the resonance location could solve the solar neutrino problem (Bethe, 1986;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Bouchez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 1986;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Haxton, 1986;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Mikheev and Smirnov, 1986;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Parke, 1986).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' In 1987, the explosion of the blue supergiant Sk-69◦202 brought evidence that core-collapse supernovae1 emit neutrinos at the end of their life (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' SN1987A was in the Large Magellanic Cloud (LMC), a satellite galaxy of the Milky Way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Kamiokande-II (KII) (Hirata et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 1987), Irvine-Michigan-Brookhaven (IMB) (Bionta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 1987) detectors and the Baksan Scintillator Tele- scope (BST) (Alekseev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 1988) recorded a 10 seconds burst of 24 events, with a few tens of MeV energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Five hours before, the Mont Blanc Liquid Scintillator Detec- tor (LSD) (Aglietta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 1987) detected 5 events, which 1 SN II and Ib/c are massive stars that undergo gravitational core- collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' O-N-Mg supernovae have 8M⊙ < M < 10M⊙;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' whereas iron core-collapse supernovae have M > 10M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' SN type II, unlike type Ib and Ic, have H lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 1 Hubble Space Telescope image of SN1987A, in the Large Magellanic Cloud a neighboring galaxy of our Milky way, 30 years after its explosion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (Figure adapted from (ESA/Hubble and NASA, 2011)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' remain controversial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The SN1987A events confirmed that neutrinos take away most of the gravitational energy, as Colgate and White (1966) conjectured, and agreed overall with the predicted neutrino fluxes and spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Moreover, the Bayesian analysis of the SN1987A time signal by Loredo and Lamb (2002) corroborated a two-component (ac- cretion+cooling) model at 2-3σ, which was confirmed by the subsequent analysis by Pagliaroli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2009b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' This supported the delayed neutrino-heating mechanism of Bethe and Wilson (1985), thus rejecting the favored prompt bounce-shock model by Colgate and White (1966).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' On the particle physics side, the two dozens events brought an impressive amount of constraints on unknown neutrino properties (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' the neutrino magnetic moment, charge radius or decay), on non-standard interactions and particles such as axions (see for example (Raffelt, 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Zyla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2020)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The observation of neutrinos from the Sun and from SN1987A pioneered neutrino astronomy2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The detection of PeV neutrinos in the IceCube detector at the South Pole (Aartsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2014) opened a new observational window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' One of the events detected so far is consistent with blazar TXS 0506+056 (Aartsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' With these observations, neutrino astronomy now covers from MeVs to the highest neutrino energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The oscillation discovery Primary cosmic rays interacting with the Earth’s at- mosphere produce twice as many νµ as νe from π and µ decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Underground experiments searching for proton instability, which was expected in some unified theories, 2 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Davis and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Koshiba (Kamiokande) were the recipients of the 2002 Nobel Prize with R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Giacconi (X-ray astronomy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 3 reported a reduced νµ/νe ratio in the atmospheric back- ground, with respect to Monte-Carlo simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' This was known as the atmospheric anomaly (see for example Giunti and Kim, 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' In 1998 the Super-Kamiokande (Super-K) Collabora- tion (Fukuda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 1998) discovered3 that atmospheric νµ traversing the Earth (up-going) were less than ex- pected, whereas up-going νe stayed unaffected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The zenith angle dependence of the µ-like and e-like events gave unambiguous proof that νµ oscillated into ντ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The Sudbury Neutrino Observatory (SNO) (Ahmad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2001a) and the Kamioka Liquid Scintillator An- tineutrino Detector (KamLAND) (Eguchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2003) experiments brought two further milestones in the clar- ification of the solar neutrino problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The first exper- iment (Ahmad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2001b), using heavy water, found 8B solar neutrinos to be in agreement with the Standard Solar Model predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The different sensitivity of νe and νµ, ντ to elastic scattering, combined with neutral- and charged-current ν interactions on deuterium allowed to identify the solar νµ, ντ fluxes at 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='3 σ (Ahmad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Moreover KamLAND measured ¯νe disappearance at an average distance of 200 km from Japanese reac- tors and unambiguously identified the large mixing angle MSW solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' These observations established that only half of low energy (less than 2 MeV) solar νe reach the Earth because of averaged vacuum oscillations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' whereas high energy 8B neutrinos are reduced to one-third due to the MSW effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The solar neutrino problem was finally solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The occurrence of vacuum oscillations implies that neutrinos are elementary particles with non-zero masses and mixings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Hence, the flavor and mass bases are re- lated by the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) unitary matrix4, analogous to the Cabibbo-Kobayashi- Maskawa matrix in the quark sector (although with large mixing angles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Since 1998, atmospheric, solar, reactor and accelera- tor experiments have determined most of the neutrino oscillation parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The mixing angles, θ23 ≈ 45◦, θ12 ≈ 35◦ and θ13 ≈ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='2◦, as the mass-squared differ- ences ∆m2 32 = m2 3 − m2 2 ≈ 8 × 10−5 eV2 (atmospheric) and ∆m2 21 = m2 2 − m2 1 ≈ 2 × 10−3 eV2 (solar) (Zyla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2020) are known with good accuracy (few percent precision).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The discovery of neutrino vacuum oscillations was a breakthrough: it opened a window beyond the Standard model and had an impact in astrophysics and cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 3 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Kajita (Super-K) and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' McDonald (SNO) were recipients of the Nobel Prize in 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 4 Note that, for an arbitrary number N of neutrino families, the PMNS matrix depends on N(N − 1)/2 angles and N(N + 1)/2 phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' For Majorana neutrinos, only N −1 phases remain, since some phases can be reabsorbed by a redefinition of the charged lepton fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' For Dirac neutrinos (N −1)(N −2)/2 are left, since charged and neutrino lepton fields can be redefined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Unknown neutrino properties Key neutrino properties remain unknown and will be the object of an intense experimental program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Sixty years after Christenson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (1964) discovered that weak interaction breaks the CP symmetry in K0 decay, there are indications that neutrinos do not oscillate as antineu- trinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Leptonic CP violation points to CP breaking val- ues of the Dirac phase (see for example (Capozzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2021) for an analysis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The ordering of the neutrino mass eigenstates needs to be established, since the atmospheric mass-squared difference sign has not been measured yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The neutrino mass ordering (or hierarchy) might be normal (∆m2 32 > 0), or inverted (∆m2 32 < 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' On the other hand, the sign of the solar mass-squared difference is determined by the presence of the MSW resonance in the Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Currently data show a preference (at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='5 σ) for normal ordering, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' the third mass eigenstate is likely more massive than the others (Capozzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The absolute neutrino mass scale is not identified yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The KATRIN experiment has obtained sub-eV up- per limits (m < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='8 eV at 90 % confidence level) on the effective νe mass with tritium β-decay (Aker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Complementary information comes from cosmo- logical observations which give (model dependent) infor- mation on the sum of the neutrino masses (at sub-eV level) (Zyla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The origin of the neutrino masses remains an open is- sue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' See-saw mechanisms constitute a possibility to ex- plain the smallness of neutrino masses and are investi- gated in numerous theories beyond the Standard model (see for example the reviews (Altarelli and Feruglio, 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' King, 2015)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Moreover, as Majorana pointed out long ago (Majorana, 1937), neutrinos could well be their own antiparticles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Searches for total lepton-number violating 0ν2β decay (Agostini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Giunti and Kim, 2007) appear the most feasible path to uncover the neutrino na- ture and give access to the Majorana CP violating phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' As for neutrino electromagnetic properties, such as the neutrino magnetic moment, only bounds exist (see for example (Giunti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2016)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Finally, the existence of sterile neutrinos is actively debated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Sterile eV neutrinos are discussed as solution5 of the reactor (Mention et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2011), the Gallium (Giunti and Laveder, 2011) and the MiniBooNE anomalies (Aguilar-Arevalo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2018) that cannot be cast in the 3ν theoretical framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' On gen- eral theoretical grounds, (heavy) sterile neutrinos could 5 Recent evaluations of the reactor neutrino fluxes have lowered the statistical significance of the reactor anomaly (Giunti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2022), whereas the Gallium one, confirmed by the counting BEST experiment (Barinov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2022), gives sterile mixing pa- rameters in tension with the reactor anomaly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Moreover, the first results from the MicroBooNE experiment (Arg¨uelles et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2022) disfavor some explanations and part of the parameter space identified by the MiniBooNE low energy excess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 4 explain the smallness of neutrino masses in see-saw mod- els.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Neutrino properties are intertwined with neutrino fla- vor evolution in dense sources and influence observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Therefore, as we shall discuss, neutrino signals from such environments constitute a unique probe of astrophysical media, or in cosmology, and to learn about unknown neu- trino properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Future supernova neutrino observations SN1987A remains to date the only core-collapse super- nova observed through its neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Supernovae6 type II and Ib/c are exciting and a rich laboratory for par- ticle physics and astrophysics, requiring both multipur- pose and dedicated neutrino observatories that can run over long time periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' A network of neutrino detectors around the world is awaiting for the next (extra)galactic supernova explosion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Their majority is included in the Supernova Early Warning System (SNEWS) (Al Kharusi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Scholberg, 2000) which should alert as- tronomers if such a lucky event takes place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' In the Milky Way the spatial probability distribution of objects that are likely to become supernovae has its maximum at the galaxy center at 8 kpc while its mean is at 10 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The latter is mostly adopted to make pre- dictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Neutrino observatories will measure the time signal and the spectra of νe, ¯νe, νx, ¯νx (x = µ, τ) with charged- current νe scattering on nuclei, inverse β-decay, elastic scattering on electrons and on protons (Beacom et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' If a supernova explodes in our galaxy (10 kpc), detectors will measure7 about 40 (540) events in HALO (HALO-2, (Vaananen and Volpe, 2011)), hundreds in KamLAND, 300 in LVD (Vigorito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2021) up to 3×103 in DUNE (40 ktons) (Abi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2021), up to about 8 × 103 in JUNO (An et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2016), almost 104 in Super- K (Beacom and Vogel, 1998), 105 in Hyper-Kamiokande (Hyper-K, 248 ktons) (Abe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2018) and 106 in Ice- Cube8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' From a supernova in Andromeda galaxy (M31, 773 kpc) which has a low supernova rate, 12 events are expected in Hyper-K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Supernovae are rare in our galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Typical quoted number for the core-collapse supernova rate in our Galaxy is 1-3/century.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rozwadowska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2021), ob- tained a more pessimistic mean time of occurrence of 61 6 Note that SNe of type Ia undergo thermonuclear explosions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The detection of the emitted neutrinos could help elucidating the pre- cise mechanism for their explosion (Wright et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 7 For the event rates see also https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='com/SNOwGLoBES/snowglobes (Scholberg, 2012) and SNEWPY (Baxter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 8 The rates correspond to a luminosity of 3 × 1053 ergs (or close to it), average energies between 12 and 18 MeV (depending on the neutrino species) with 100 % or more realistic efficiencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' +24 −14 based on neutrino and electromagnetic observations of collapse events in the Milky Way and the Local Group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Supernovae are frequent events in the universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' With a 1 Mt detector, about one supernova per year is expected within 10 Mpc, due to nearby galaxies with higher rates than the Milky Way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Within 4 Mpc, less than one event per year would be detected (Ando et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Neutrinos from past supernova explosions form a relic, or diffuse, supernova neutrino background (DSNB) (see the reviews by Ando and Sato (2004);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Beacom (2010);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lunardini (2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' and Mathews et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2020)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Its flux, integrated over cosmological redshift, depends on the red- shifted supernova neutrino fluxes, on the core-collapse su- pernova rate and on the fraction of failed supernovae that turn into a black hole without an electromagnetic coun- terpart (Keehn and Lunardini, 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lunardini, 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' At present we only have upper limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Super-K (Malek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2003) set the first upper limit of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='2 ¯νe cm−2s−1 (Eν > 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='3 MeV, 90 % CL) on the su- pernova relic flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The bound was improved with Super- K IV data (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2015) using ¯νe detection via inverse β-decay and neutron tagging on protons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The DSNB search combining Super-K I to Super-K IV data yields 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='7 ¯νe cm−2s−1 (Eν > 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='3 MeV, 90 % CL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The KamLAND experiment obtained the upper value of 139 ¯νe cm−2s−1 (90 % CL) in the window [8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='3, 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='8] MeV (Gando et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The Borexino Collaboration ex- tracted a model dependent limit of 112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='3 ¯νe cm−2s−1 (90 % CL) in the interval [7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='8, 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='8] MeV (Agostini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' As for νe, the ensemble of SNO data provided the up- per limit of 19 νe cm−2s−1 in the window [22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='9, 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='9] MeV (90 % CL) (Aharmim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The loosest limits are φνx,¯νx < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='3-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='8 103 cm−2s−1 (Eν > 19 MeV) (Lunardini and Peres, 2008) (for x = µ, τ flavors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' With neutrino- nucleus coherent scattering in dark matter detectors, one could improve this bound to 10 νx or ¯νx (Suliga et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Beacom and Vagins (2004) suggested to add gadolin- ium (Gd) to water Cherenkov detectors9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Neutron cap- ture by Gd improves10 inverse β-decay tagging through the 8 MeV photons following the capture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The Super- K+Gd experiment is currently taking data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' DSNB predictions (Ando and Sato, 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Chakraborty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2011a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' De Gouvea et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Ekanger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Fukugita and Kawasaki, 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Galais et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Ho- riuchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Ivanez-Ballesteros and Volpe, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Kresse et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lunardini, 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Moeller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Priya and Lunardini, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Tabrizi and Horiuchi, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Vissani and Pagliaroli, 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Yuksel and Beacom, 2007) are close to current bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The analysis from the com- bined SK-I to IV data (Abe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2021) shows an excess 9 The idea was named GADZOOS for Gadolinium Antineutrino Detectos Zealously Outperforming Old Kamiokande, Super!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 10 An efficiency of 90 % is expected with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='1 % Gd concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 5 at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='5 σ over background prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The related sensi- tivity analysis is on par with four of the most optimistic predictions (Ando and Sato, 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Galais et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Horiuchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Kresse et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2021) and a factor of about 2-5 larger than the most conservative ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' With Super-K+Gd, the upcoming JUNO, DUNE and Hyper-K experiments, the DNSB should be discovered in the forthcoming future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The r-process and GW170817 Neutrinos in dense environments are tightly connected to two unresolved issues in astrophysics: the death of massive stars and the origin of r-process11 elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Cur- rently, two- and three-dimensional simulations include re- alistic neutrino transport, convection, turbulence and hy- drodynamical instabilities such as SASI (see for example Burrows et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Foglizzo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2015);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Janka (2012, 2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Kotake et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2006);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Mezzacappa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Radice et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2018a);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' and Takiwaki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2021)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The delayed neutrino-heating mechanism is believed to trig- ger most of core-collapse supernova explosions12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The future observation of a galactic or extragalactic super- nova could confirm or refute the current paradigm and elucidate a six-decade quest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' As for r-process nucleosynthesis, Burbidge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (1957) and Cameron (1958) first linked it to core-collapse su- pernovae which have long been thought the main r- process site13 (see for example Qian (2014), Kajino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2019))14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' While simulations show that entropies are typ- ically too low, the most energetic events seem to provide the right conditions to attain a successful nucleosynthe- sis (see Cowan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2021) and Cˆot´e et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2019) for a comprehensive review).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Another candidate site for the r-process are binary neutron star mergers (BNS) as first suggested by Lat- timer and Schramm (1974, 1976) (see for example (Cˆot´e et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Goriely et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Kajino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2019) for reviews).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' As supernovae, BNS are powerful sources of MeV neutrinos15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' If the former are more frequent than 11 r-process stands for rapid neutron capture process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 12 Note that the most energetic events might require a magneto- hydrodynamical mechanism (see for example Janka (2012)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 13 With a rule of thumb, if each supernova produces 10−4M⊙ el- ements and there are 3 such events per century, in 1010 years there are 3 104M⊙ r-process elements ejected in the Milky Way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 14 In the r-process, nuclei capture neutrons faster than β-decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The process, which makes about half of the heavy elements in our galaxy, involves thousands of exotic nuclei far from the neutron drip-line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The other half is produced in the s-process (s stands for slow), whereas a small contribution comes from the p-process (p stands for proton).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 15 They emit 1051 to 1053 erg in νe, νµ, ντ and their antiparti- cles with tens of MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Contrary to supernovae, binary neutron star mergers are more neutron-rich and produce an excess of ¯νe over νe (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Cusinato et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2021)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' As for νµs and ντs, their fluxes are predicted to be small compared to those of core- the latter, simulations show that BSN offer more suitable astrophysical conditions for a strong r-process16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' More- over studies have shown some r-process elements are syn- thetised in accretion disks around black holes (Surman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2006) and black hole-neutron star mergers (Ca- ballero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Surman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The first detection of gravitational waves from the fu- sion of two black-holes by the Virgo-LIGO Collaboration (Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2016) has opened the era of gravitational wave astronomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' To date, GW170817 (Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2017a,b) is a unique multi-messenger event in which grav- itational waves from a binary neutron star merger were first detected, also concomitantly with a short-gamma ray burst and a kilonova17 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The ejecta opacities show presence of actinides and lanthanides (see for exam- ple the studies by (Cowan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Cˆot´e et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Tanaka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2018)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' This represents the first evidence for r-process elements in binary neutron star mergers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Before GW170817 dynamical ejecta were thought to mainly contribute to a strong r-process (see for exam- ple the discussion in Martin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=') But, the kilonova observation has changed this paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' In- deed the comparison of the electromagnetic emission with most models shows two components: the early and fast pre-merger contribution from dynamical ejecta, the later post-merger one due to viscosity and neutrino-driven winds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The former is associated with red emission, the latter with the blue one (Radice et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2018b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The role of neutrinos on the pre- and post-merger ejecta and of flavor evolution appears crucial and is currently object of debate (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Nedora et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2021)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Numerous r-process studies that include not only neu- trinos but also neutrino flavor evolution find that mat- ter becomes proton-rich and tends to harm the r-process in core-collapse supernovae18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' One should keep in mind though, that studying flavor evolution in a consistent manner is complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Depending on the site and on the assumptions made, one can find situations in which fla- vor modification favors the r-process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' In general, what emerges from the investigations, is that neutrino flavor evolution impacts the nucleosynthetic abundances when collapse supernovae and have large theoretical uncertainties (see for example Table VII of Frensel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2017)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 16 A weak r-process produces elements in the first peak around mass number A=80-90 and the second peak around A=130-138.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' A strong r-process reaches the third peak at A=195-208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 17 The electromagnetic signal of a kilonova (Metzger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2010), or macronova (Kulkarni, 2005) is in-between the ones of novae and supernovae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Since the afterglow emission of the kilonova associated with GW170817 extends over some days, it appears that radioactivity injects some energy, powering the kilonova.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The optical/IR spectra in the IR emission peak are compatible with elements heavier than iron to be responsible for absorption and reemission of the radiation (Cowan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 18 Note that there are other nucleosynthesis processes where neu- trinos influence element abundances, including neutrino nucle- osynthesis (Langanke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2019) and the νp process (Frohlich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 6 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 2 Hubble Space Telescope image of kilonova gradu- ally fading, in the lenticular galaxy NGC 4993 (40 Mpc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The event GW170817 was seen concomitantly in gravitational waves, short gamma ray bursts and electromagnetic emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' It represents the first observation from merging binary neu- tron stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (Figure adapted from (Telescope, 2017)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' one includes standard or non-standard properties and in- teractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The quest for the identification of the r-process site(s) and the supernova mechanism as well as the need of pre- dictions for future observations motivated an in-depth investigation of flavor evolution in dense environments, as we shall now discuss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Theoretical developments Undoubtedly, understanding flavor evolution is an intriguing theoretical problem, with many interesting questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' In dense environments, do new phenomena emerge?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' What are the conditions to trigger them and what is their impact?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Do novel flavor mechanisms intro- duce extra heating and help supernova explosions?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' How do neutrinos behave in presence of shock waves and of turbulence?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Are there flavor mechanisms that favor the r-process?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Is the commonly employed mean-field approx- imation sufficient?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Do weakly interacting neutrino gases behave like known many-body systems?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' What is the in- terplay with unknown neutrino properties?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' What is the role of strong gravitational fields?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Are many-body cor- relations important?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The list encompasses many others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Over thirty years theoretical studies have paved the way to their answers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' First of all, investigations have shown that a variety of novel conversion phenomena can occur due to matter, shock waves, turbulence and νν interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Obviously, there is the established MSW effect that takes place both in astrophysical and cosmological environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' In par- ticular, Dighe and Smirnov (2000) pointed out that, be- cause of the large densities and of radiative corrections (Botella et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 1987), three MSW resonances occur in core-collapse supernovae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' While the MSW effect is cer- tainly a reference in studies of flavor modification, the novel phenomena uncovered go much beyond it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Schirato and Fuller (2002) reported that shock waves could modify the time signal of supernova neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Tomas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2004) showed the presence of front and re- verse shocks in supernova simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Fogli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2003) and then Dasgupta and Dighe (2007) found that front and reverse shocks produce multiple MSW resonances and phase effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Several authors (Choubey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Fogli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Gava et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Kneller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Takahashi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2003) studied their impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Concerning noisy media, Loreti and Balantekin (1994) first studied the influence of matter fluctuations in re- lation with solar neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' A few papers (Balantekin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Loreti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Nunokawa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 1996) showed that fluctuations of matter density profiles could induce neutrino depolarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The impact of turbu- lence on neutrinos was then explored in the context of supernovae (Abbar, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Fogli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Friedland and Gruzinov, 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Kneller and Volpe, 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lund and Kneller, 2013), reaching similar, or opposite (Borriello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2014), conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Besides shock waves and turbulence, neutrino-neutrino interactions have attracted a strong interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' In the early nineties, Pantaleone (1992) pointed out that such interac- tions become sizable when the neutrino number densities are sufficiently large, and make neutrino propagation a non-linear many-body problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Samuel (1993) showed that such interactions could trigger new flavor effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' At first, studies for the early universe (Abazajian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Dolgov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Kostelecky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Kost- elecky and Samuel, 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Mangano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Pastor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2002) implemented νν interactions (see also (Gava and Volpe, 2010)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Moreover, Rudzsky (1990), Sigl and Raffelt (1993) and McKellar and Thomson (1994) derived neutrino quantum kinetic equations including neutrino interactions with matter and neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' It was the work by Duan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2006a) that trig- gered the attention on νν interactions in core-collapse supernovae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Balantekin and Yuksel (2005) also showed that such interactions could produce significant effects on the r-process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' These works stimulated an intense ac- tivity on νν interactions (see the reviews by Duan and Kneller (2009), Duan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2010), Mirizzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2016), Volpe (2015)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The first numerical simulations, based on the stationary bulb model, studied in great detail the mechanisms under which neutrinos first synchronised, then underwent bipolar oscillations and finally spectral splits19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' In the literature, authors commonly refer to fla- vor modes due to νν interactions as collective neutrino oscillations20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 19 These are often called slow modes, as we shall discuss (section II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 20 Note however, that the effects of νν interactions are not neces- Aug 22, 2017 Aug 26, 2017 Aug 28, 20177 Moreover, Malkus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2012) showed that in black hole accretion disks the interplay between νν and mat- ter interactions produced a new mechanism later called neutrino matter resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' This was studied in the context of merging compact objects (black hole-neutron star, neutron star-neutron star) (Frensel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Malkus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2014, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Vaananen and McLaughlin, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Vlasenko and McLaughlin, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2016) and of core-collapse supernovae, with non-standard in- teractions (Stapleford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Earlier than the bulb model, Sawyer (2005) showed that the neutrino-neutrino interaction could trigger sig- nificant flavor conversion on very short scales (see also Sawyer (2009)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' It was only ten years after (Sawyer, 2016), when he considered different neutrinospheres for νe and ¯νe and found flavor modes with nm scale, that his findings triggered excitement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Indeed, for long, theo- rists had searched for mechanisms that could take place behind the shock wave and impact the explosion dynam- ics of core-collapse supernovae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' These modes were called fast, in contrast with the ones found in the bulb model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Subsequent studies showed that a sufficient but not necessary condition for the occurrence of fast modes is that the neutrinospheres of νe and ¯νe cross each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The conditions to have fast modes and their impact is ac- tively investigated (see for example (Abbar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2019, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Abbar and Volpe, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Chakraborty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2016b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Dasgupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' George et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Glas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Just et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Nagakura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Wu and Tamborra, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2017) and the review by Tam- borra and Shalgar (2021)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' While most of the developments focussed on the novel flavor mechanisms and their impact, numerous studies concentrated on the neutrino evolution equations them- selves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Indeed the majority of the literature employs mean-field equations, derived by several authors (Bal- antekin and Pehlivan, 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Friedland and Lunardini, 2003b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Samuel, 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Sawyer, 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Serreau and Volpe, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Sirera and Perez, 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Volpe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Ya- mada, 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' However, in very dense stellar regions, or in the early universe, where collisions matter, neutrino quantum kinetic equations are necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Such equa- tions were obtained using different approaches (Blaschke and Cirigliano, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Froustey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' McKellar and Thomson, 1994;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rudzsky, 1990;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Sigl and Raffelt, 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Vlasenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2014a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' In principle, the theoretical framework should consistently evolve from the collision- dominated to the mean-field regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The full solution of neutrino quantum kinetic equa- tions is achievable, if the medium is homogeneous and isotropic as in the early universe (Bennett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Froustey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Unfortunately, in dense stellar environments this becomes a formidable numerical task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Not only the number of degrees of freedom is 7, but also, sarily collective, neither oscillatory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' when νν interactions are sizable, we face a non-linear many-body problem!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' There are currently serious efforts to explore the role of collisions and their interplay with flavor mechanisms (see for example (Capozzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Hansen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Richers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Xiong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2022)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' One should keep in mind that, even in models with reduced degrees of freedom and approximations, the de- scription of neutrino propagation requires the solution of a large number of stiff and coupled non-linear equa- tions (in presence of νν interactions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' In order to avoid it, Banerjee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2011) proposed to linearize the equations of motion and replace the solution of the full non-linear problem by eigenvalue equations, as first pointed out by Sawyer (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Vaananen and Volpe (2013) provided an alternative derivation inspired by the Random-Phase- Approximation used, for example, in the study of collec- tive vibrations in atomic nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Izaguirre et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2017) cast the linearized equations in a dispersion-relation ap- proach, commonly used in the study of fast modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Moreover, new numerical methods based on deep learning techniques (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (Armstrong, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Arm- strong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rrapaj et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2021)) have been re- cently employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' They go beyond theoretical approaches using forward integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' But are mean-field equations enough when neutrinos start free-streaming?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' This aspect is actively debated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Balantekin and Pehlivan (2007) discussed corrections to the mean-field approximation using the coherent-state path-integral approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Volpe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2013) pointed out that the most general mean-field equations include both pairing correlations, analogous to those of Bardeen- Cooper-Schrieffer (Bardeen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 1957), and wrong- helicity contributions due to the absolute neutrino mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Vlasenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2014a) derived neutrino quantum kinetic equations including wrong-helicity contributions, termed spin coherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Serreau and Volpe (2014) referred to them as helicity coherence when they derived the most general mean-field equations for anisotropic and inhomo- geneous media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' In addition, Cherry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2012) showed that a small fraction of backward neutrinos (neutrino halo) can influ- ence the neutrino flavor content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' This result questioned the validity of descriptions of neutrino flavor evolution as an initial value problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Halo effects were further studied in O-Ne-Mg (Cherry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2013) and iron core-collapse supernovae (see for example (Sarikas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2012b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Pehlivan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2011) showed that the use of algebraic methods and of the Bethe ansatz opens the way to exact solutions of the quantum many-body problem of neutrino propagation in dense media (without the matter term and collisions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Further investigated, this has uncovered the importance of many-body correlations (Birol et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Pehlivan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2014) and brought exciting con- nections to quantum information theory (Cervia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lacroix et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Patwardhan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rog- gero, 2021a,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Roggero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2022) and quantum devices 8 (Amitrano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Hall et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Other intriguing connections have been established be- tween a weakly interacting neutrino gas in a dense envi- ronment and other many-body systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Pehlivan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2011) showed that the neutrino Hamiltonian can be re- lated to the (reduced) Bardeen-Cooper-Schrieffer Hamil- tonian of Cooper pairs in superconductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Volpe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2013) applied the Born-Bogoliubov-Green-Kirkwood- Yvon (BBGKY) hierarchy to an interacting neutrino gas in a medium and established a formal connection with atomic nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Also, a futher connection exist be- tween flavor evolution of an interacting neutrino gas with the transition from laminar to turbulent regimes in non- linear fluids, as pointed out by Mirizzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Besides these aspects flavor evolution in dense objects is also interesting because it is tightly connected to neu- trino properties and non-standard physics or particles, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' sterile neutrinos (Fetter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' McLaughlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Tamborra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Xiong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2019), non-standard interactions (Blennow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Chatelain and Volpe, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Das et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Esteban-Pretel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2007a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Stapleford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Wolfenstein, 1978), the neutrino mass ordering (Barger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Dighe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Engel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Gava et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Serpico et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2012) or CP violation (Akhme- dov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Balantekin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Gava and Volpe, 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Pehlivan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Popov and Studenikin, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' There are numerous reviews on core-collapse super- nova neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' They focus on oscillations in media (Kuo and Pantaleone, 1989), on the diffuse supernova ν back- ground (Ando and Sato, 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Beacom, 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lunardini, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Mathews et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2020), on pre-supernova neutri- nos (Kato et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2020), on νν interactions (Duan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Tamborra and Shalgar, 2021), on νν interactions and turbulence (Duan and Kneller, 2009), on supernova ν detection (Scholberg, 2012), on observations (Horiuchi and Kneller, 2018), on production, oscillations and detec- tion (Mirizzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2016) and on the neutrino evolution equations (Volpe, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The goal of this review is to highlight the richness and complexity of neutrino flavor evolution in dense media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The review encompasses in particular two aspects of this fascinating problem, namely flavor mechanisms and the theoretical frameworks, and discusses aspects of super- nova neutrino observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Since fifteen years this is a developing field, where new approaches, exciting connec- tions and interesting ideas keep being proposed which makes the writing of this review challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The structure of the manuscript is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Section II is focussed on flavor mechanisms in media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Section III focusses on the theoretical description of neutrino prop- agation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Section IV is on past and future observations of supernova neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Section V presents conclusions and perspectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' NEUTRINO FLAVOR MECHANISMS IN ENVIRONMENTS Neutrino flavor mechanisms are quantum mechanical phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Suggested by Pontecorvo (1957), vacuum oscillations are analogous to Rabi oscillations in atomic physics, or K0 − ¯K0 oscillations in meson systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Oscillations in vacuum arise because the flavor (or in- teraction) and mass (or propagation) bases do not coin- cide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' This produces an interference phenomenon among the mass eigenstates when neutrinos propagate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The two bases are related by the PMNS matrix21 U, that is |να⟩ = U ∗ αk |νk⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (1) with k = 1, 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', N and α = e, µ, τ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', N the mass and flavor indices22 respectively, N being an arbitrary number of neutrino families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The matrix is unitary (U = U †).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' For antineutrinos the same relation holds, with Uαk instead of U ∗ αk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The massive neutrino states are eigenstates of the free Hamiltonian H0 = diag(Ek), with eigenenergies Ek = � ⃗p2 k + m2 k , (2) momentum ⃗pk and mass mk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' In the flavor basis, the free Hamiltonian reads Hf vac = UHvacU †.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (3) Let us recall the neutrino equations of motion mostly used to determine how flavor changes in media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Based on such equations several mechanisms emerge, that I shall highlight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Mean-field equations Neutrino flavor states evolve according to the Schr¨odinger-like equation23 i d dt|να(t)⟩ = H |να(t)⟩ , (4) with the initial condition |να(0)⟩ = |να⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' When neutrinos traverse dense matter, the neutrino Hamiltonian receives different contributions, namely H = Hf vac + Hmat + Hνν + HNSI , (5) where Hmat comes from neutrino interactions with mat- ter and Hνν from νν interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The last term HNSI is 21 From now on I use ℏ = c = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 22 A sum on repeated indices is subtended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 23 Unless differently stated, I shall consider neutrinos as plane waves and neglect space-time curvature due to the presence of strong gravitational fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 9 present, if non-standard interactions exist between neu- trinos and neutrinos, or neutrinos and matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' We now have a closer look at the different contributions to the neutrino Hamiltonian Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='(5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' When neutrinos tra- verse a dense astrophysical medium, they interact with the background electrons, protons, neutrons and neutri- nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The mean-field is the simplest widely used approxi- mation to implement such interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' A number of authors have derived mean-field evolu- tion equations including neutrino interactions with mat- ter and neutrinos (Balantekin and Pehlivan, 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Fried- land and Lunardini, 2003b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Samuel, 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Sawyer, 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Serreau and Volpe, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Sigl and Raffelt, 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Sirera and Perez, 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Vlasenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2014a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Volpe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Yamada, 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' In words, the mean-field approximation consists in adding the amplitudes associated with neu- trino scattering on a background particle, weighted by the quantum expectation value of the particle number operator over the background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Integrating such quantity, over the degrees of freedom of the background particle, generates a potential that acts on the neutrino propagat- ing through the medium (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Let us first consider neutrino-matter interactions whose contribution can be derived from the charged- and neutral-current interactions terms of the GWS model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Associated with charged-current ν-e scattering is the well known mean-field Hamiltonian Hmat = √ 2GF(ne − n¯e) , (6) GF being the Fermi coupling constant and ne (n¯e) the electron (positron) number density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Neutral-current in- teractions of νe, νµ and ντ on neutrons, which give equal mean-field contributions, do not influence oscillations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' whereas charged-current interactions on electrons and protons cancel each other, in a neutral medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Equation (6) holds for a homogenous, isotropic and unpolarized medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' This is a good approximation for example for the Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' If the assumptions are relaxed, more mean-field terms appear due for example to polar- isation, as discussed by e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (Nunokawa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 1997b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' In dense media, like supernovae or compact objects rem- nants, interesting features arise due to anisotropy, as we shall see.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' For 2ν flavors, the evolution equations (4) with the vacuum and matter terms read i d dt � |να⟩ |νβ⟩ � = H � |να⟩ |νβ⟩ � , (7) where H = Hc + � - ∆m2 4E cos2 2θ + √ 2GFne ∆m2 4E sin2 2θ ∆m2 4E sin2 2θ ∆m2 4E cos2 2θ � , (8) after subtracting a term common to all flavors Hc = � E + (m2 1 + m2 2)/4E � 1 , (9) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 3 Mean-field approximation: tadpole diagrams for neutrino-electron (upper) and neutrino-neutrino scattering (lower diagrams) that contribute to the neutrino evolution equations in media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (Figure adapted from Volpe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2013)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' proportional to the identity matrix, with the neutrino energy E = |p|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The quantity ω = ∆m2/(2E) is the vacuum oscillation frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Let us remind that, to investigate flavor evolution, one often evolves neutrino amplitudes, effective spins or den- sity matrices, instead of neutrino states (see for example (Giunti and Kim, 2007)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' It was Harris and Stodolsky (1982) who first discussed the density matrix (and po- larization vector) approach in relation to neutrinos, to describe the coherence properties of a two state system undergoing random fluctuations in a medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' For 3ν flavors the neutrino density matrix reads ϱ = � � � ⟨a† e,iae,i⟩ ⟨a† µ,jae,i⟩ ⟨a† τ,kae,i⟩ ⟨a† e,iaµ,j⟩ ⟨a† µ,jaµ,j⟩ ⟨a† µ,jaτ,k⟩ ⟨a† e,iaτ,k⟩ ⟨a† τ,kaµ,j⟩ ⟨a† τ,kaτ,k⟩ � � � , (10) where the quantum expectation values24 is over the back- ground that neutrinos are traversing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Here the indices i, j, k indicate the quantum numbers (usually momen- tum and helicity), which characterize neutrino states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' A similar expression holds for antineutrinos, but with ¯ϱij = ⟨b† ibj⟩25 instead of ϱij = ⟨a† jai⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The diagonal en- tries of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (10) are the quantum expectation values of the occupation number operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Instead of evolving neutrino states (4), one can solve the Liouville Von Neumann equation for the neutrinos or 24 The operators a† and a are the particle creation and an- nihilation operators that satisfy the equal-time anticommuta- tion rules {a(p, h), a†(p′, h′)} = (2π)3δ3(p − p′)2Epδh,h′ and {a(p, h), a(p′, h′)} = {a†(p, h), a†(p′, h′)} = 0 (h, h′ are helici- ties).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Similar rules hold for the antiparticle creation and annihi- lation operators, b† and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 25 Note that with this convention ϱ and for ¯ϱ have the same equa- tions formally;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' inversely to ¯ϱij = ⟨b† jbi⟩ which introduces com- plex conjugates of ¯ϱ in the equations of motion (see for example (Sigl and Raffelt, 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Volpe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2013)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (p) V(R) (p) (d)_a e-(p) v(R) (k)vB(p) (Vg(k) Ve(p) Vα(p) Ve(p) Va(ki) Vg(k)10 the antineutrino density matrix26, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' idϱ dt = [H, ϱ] id ˙¯ϱ dt = [ ¯H, ¯ϱ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (11) Dense media have the peculiarity that neutral-current νν interactions are sizable and contribute in the mean- field approximation through the Hamiltonian Hνν = √ 2GF � p (1 − ˆp · ˆp′) [ϱ(p) − ¯ϱ(p)] , (12) where � p = � dp/(2π)3 ˆp = p/|p|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (13) The term ˆp · ˆp′, coming from the V -A structure of the weak interactions, contributes in anisotropic media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' More generally, the mean-field equations of motion that describe neutrino (and similarly antineutrinos) propagation in dense environments read i(∂t + v · ∇x + F · ∇p)ϱ = [H, ϱ] (14) with v = p/E the neutrino velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The second term is an advective term which contributes in presence of spatial inhomogeneities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The third term depends on a possible external force F, such as the gravitational one, that acting on the neutrinos could change its momentum or energy (because of trajectory bending for example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Since the Liouville operator depends on time, coordinate and momentum, the problem one faces in determining neutrino flavor is 7-dimensional, and therefore extremely challenging numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The solution of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (11) with the mean-field terms Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (6) and (12), the vacuum term Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (3) reveals flavor mechanism that mostly arise from the interplay between the different contributions, as we now describe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The MSW effect The MSW effect is a reference phenomenon for flavor evolution studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Several of the uncovered mechanisms are either MSW-like or multiple MSW phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' To clarify this link, we remind some basics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The MSW effect arises when a resonance condition is satisfied, the resonance width is large and evolution through it is adiabatic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' It is equivalent to the two-level problem in quantum mechanics (Cohen-Tannoudji et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Let us introduce the matter basis which, by definition, instantaneously diagonalizes the neutrino Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 26 Note that in the equation of motion for antineutrinos the vacuum contribution to the Hamiltonian has a minus sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Here we consider only the vacuum and the matter con- tributions27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The flavor basis is related to the matter basis through the relation |να⟩ = ˜U ∗ αk |˜νk⟩ , (15) with k = 1, 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' In the unitary matrix ˜U, effective mixing parameters in matter replace the vacuum ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' From Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (4) and (15), one gets the following equation of motion for the matter basis i d dt|˜ν(t)⟩ = ˜H |˜ν(t)⟩ = � K + i ˜U † d ˜U dt � |˜ν(t)⟩ , (16) where K = diag(˜k1, ˜k2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' ˜kN) depends on the matter eigenvalues ˜ki (i = 1, N) and the matter Hamiltonian ˜H now includes the derivatives of the effective mixing parameters in matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' These depend on the specific en- vironment neutrinos are traversing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Let us consider the explicit expressions for 2ν flavors for which equation (15) reads � |νe⟩ |νx⟩ � = � 1 0 0 eı ˜β �� cos ˜θ sin ˜θ − sin ˜θ cos ˜θ � � |˜ν1⟩ |˜ν2⟩ � , (17) with ˜θ and ˜β the effective mixing angle and phase re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Neglecting the phase, the evolution equation of the matter basis (16) reads i d dt � |˜ν1⟩ |˜ν2⟩ � = � ˜k1 i ˙˜θ i ˙˜θ ˜k2 � � |˜ν1⟩ |˜ν2⟩ � , (18) where ˜k1, ˜k2 are given by ˜k2 − ˜k1 = � (∆m2 cos 2θ − A)2 + ∆m2 sin2 2θ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (19) In matter neutrinos acquire an effective mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Figure 4 shows how ˜k1 and ˜k2 evolve as a function of the electron number density in an environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The effective mixing angle diagonalizing the 2 × 2 ma- trix given by equation (8) satisfies sin2 2˜θ = ∆m2 sin2 2θ (∆m2 cos2 2θ − A)2 + ∆m2 sin2 2θ , (20) with A = 2EHmat Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' One can see that, when the following equality holds √ 2GFne = ∆m2 2E cos 2θ , (21) the matter mixing angle Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (20) is maximal, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' ˜θ = π/4, and the distance between the matter eigenvalues minimal (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 27 More generally, a ”matter” basis can be introduced whatever terms are included in the Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 11 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 4 The Mikheev-Smirnov-Wolfenstein effect: the figure shows the matter eigenvalues as a function of matter number density (solid lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The dashed lines are the matter eigen- values in absence of mixings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' At the MSW resonance location, the matter mixing angle is maximal and the eigenvalues ”re- pel” each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The MSW effect is a two-level problem in quantum mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (Figure from (Akhmedov, 1999)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Relation (21) corresponds to the difference of the diag- onal elements in the Hamiltonian Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (8) being equal to zero (or to a minimal distance of the matter eigenvalues in the matter basis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' It is the MSW resonance condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Its fulfillment gives the sign of the mass-squared differ- ences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' In fact, if θ < π/4 (first octant), cos 2θ > 0 and the condition holds for ∆m2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' If θ > π/4 (second octant), cos 2θ < 0 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (21) requires ∆m2 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' When the resonance condition is satisfied and its width ΓMSW = ∆m2 sin2 2θ , (22) large, the fate of neutrinos depends on the adiabaticity of the neutrino evolution through the resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' This can be quantified by the adiabaticity parameter γ−1 = | ˙˜θ| ˜k2 − ˜k1 = ∆m2 2E sin2 2θ cos 2θ ne |dne/dr| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (23) Quantitatively, if γ−1 ≫ 1, evolution is fully non- adiabatic, the matter eigenstates strongly mix at the res- onance and νe exits as ν1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' On the contrary, if γ−1 ≪ 1, the mixing of matter eigenstates at the resonance is sup- pressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Then, neutrinos evolve independently, just ac- quiring a phase, from the inner regions to the surface of the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Hence, if νe ≈ ˜νB at π/2 (high density region) and evolves through the MSW resonance adiabatically, it exits the star as a ν2 and is mostly detected as νx on Earth (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Since we now know that the MSW effect suppresses 8B solar neutrinos to 1/3 the value predicted by the Standard Solar Model, this gives us the sign of the corresponding mass-squared difference: ∆m2 12 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The concept of adiabaticity is general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The adiabatic- ity parameter, defined as the ratio of the modulus of the off-diagonal term of the neutrino Hamiltonian over the difference of the diagonal ones, can be generalized in pres- ence, for example, of νν interactions as done in (Galais FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 5 Spin formalism: picture of the neutrino evolution in flavor space for 2ν flavors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' When electron neutrinos are pro- duced in the Sun, the vector µ precess around the matter vector n and, if evolution is adiabatic, it follows n until it be- comes the vacuum vector n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Note that the third component of the neutrino spin vector corresponds either to νe (upward) or to νµ (downward).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (Figure from Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (1988).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=') et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' In this case it also involves the deriva- tives of the mixing phase that arise because the neutrino Hamiltonian Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (12) is complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' If evolution through the resonance(s) is adiabatic, neu- trinos adjust to a smooth density profile during propa- gation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' This contrasts with what happens when steep variations of the density profiles are present, as e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' in presence of shock waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' With the spin formalism, we look at these phenomena with different eyes, since we follow neutrinos through the evolution of effective spin28, subject to effective magnetic fields (Cohen-Tannoudji et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' In this context, vacuum oscillation is a precession of neutrino spins Pp in flavor space, around the vacuum (effective) magnetic field Bvac tilted by 2θ (Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 1988, 1987) (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' From the evolution of the Pz component, one recovers the vacuum oscillation formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' As for the MSW effect, it takes place in matter when Pp goes through the x-y plane, since the MSW reso- nance condition corresponds to Pz = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Adiabatic evolu- tion occurs when the precession frequency of Pp around B = Bvac+Bmat is fast compared to the rate with which B changes, so that spins follow the magnetic field dur- ing propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' On the contrary, if evolution is non- adiabatic, Pp ”lags behind”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The description in terms of neutrino isospins has been largely exploited to study neutrino flavor evolution in dense environments and in particular when νν interac- tions are sizable (see for example the review by Duan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2010)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 28 Also called neutrino ”isospins” or ”polarization” vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Ve Ve (α) (b)12 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 6 MSW effect in core-collapse supernovae: effective neu- trino masses as a function of the electron number density in absence (dashed) or in presence (solid lines) of neutrino mixings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The diagram corresponds to normal mass ordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (Figure adapted from (Dighe and Smirnov, 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=') C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The MSW effect in dense media Neutrinos face more than one resonance if the density of the astrophysical or cosmological environment is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Dighe and Smirnov (2000) pointed out that there are three such resonances in supernovae: the high (H-), the low (L-) and the Vµτ (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' As the MSW resonance condition shows, the resonance location depends on the neutrino energy and mixing pa- rameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (21) one finds that the H- and L- resonances, associated with (θ31, ∆m2 31) and (θ21, ∆m2 21) respectively, take place at ρres ≃ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='4 × 106 g cm3 � ∆m2 1 eV2 ��15 MeV E ��0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='5 Ye � cos 2θ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (24) Therefore for a 10 MeV neutrino ρres = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='7 × 103 g/cm3 for the H–resonance and ρres = 10 g/cm3 for the L- resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Moreover radiative corrections that differ- entiate νµ and ντ (Botella et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 1987), introduce the potential Vµτ = 10−5Ve which becomes relevant if the medium is very dense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The Vµτ resonance is associated with the atmospheric mixing parameters (θ32, ∆m2 32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' A characteristic feature of flavor phenomena is that they induce spectral modifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Let us consider the neutrino spectra at the neutrinosphere, φ0 ¯νe and φ0 ¯νx, and assume ν evolution through H- and L-resonances only in terms of probabilities29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' For example, if ¯νe, produced in 29 This simple approach assumes that the evolution at each res- onance is factorizable and neglects the role of phases from the neutrino amplitudes and of the neutrino mixing matrix U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' the inner stellar regions, traverse the three resonances, their spectra become φ¯νe = ¯pφ0 ¯νe + (1 − ¯p)φ0 ¯νx , (25) where ¯p is the spectral swapping probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' In partic- ular, ¯p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='68 and ¯p = 0 for normal and inverted mass ordering respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' As supernova simulations show, the neutrino spectra at the neutrinospheres, φ0 ¯νe and φ0 ¯νx, are well described by pinched Fermi-Dirac distributions (Dighe and Smirnov, 2000) or by power laws (Keil et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Because of their microscopic interactions, the neutrino average energies (approximately) satisfy the hierarchy ⟨Eνe⟩ < ⟨E¯νe⟩ < ⟨Eνx⟩, with typical energies Eνe ∈ [8, 14] MeV, E¯νe ∈ [14, 18] MeV and Eνx ∈ [16, 20] MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' In fact, νx undergo neutral current interactions and decouple from deeper hotter regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Unlike νx, νe and ¯νe interact via charged- and neutral-current interactions and decouple from colder outer shells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (25) one sees that, due to the MSW effect, ¯νe acquire hotter spectra in case of inverted mass or- dering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Similarly for νe, if the mass ordering is nor- mal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' These (and other) spectral modifications will im- pact charged-current events associated with inverse-β decay or neutrino-nucleus interactions, in a scintillator, Cherenkov, lead, or liquid argon detector, if a new su- pernova event occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' On the contrary, neutral-current events are ”flavor blind” and therefore are not sensitive to spectral swapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' It is important to keep in mind that if, under some con- ditions, the supernova fluxes of the different neutrino fla- vors become (practically) degenerate, then spectral dis- tortions due to flavor mechanisms, according e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (25), impact neither directly nor indirectly observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' For this reason, the possibility of flavor equilibration is often discussed in the literature and theorists have been actively looking for this simplifying possibility (particu- larly when νν interactions are sizable).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' A few more remarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' First, in core-collapse super- novae, for typical matter profiles (in absence of shock waves), the evolution through the L-resonance is adia- batic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Second, since φ0 ¯ντ = φ0 ¯νµ at three level, the Vµτ res- onance, which mixes νµ and ντ, does not produce spectral modifications and therefore observables effects30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Third, the only unknown parameter that impacts the standard MSW effect is the neutrino mass ordering, since the sign of ∆m2 23 ≈ ∆m2 13 is not determined yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Therefore, the detection of the neutrino signal from a future galactic su- pernova could inform us as on this key property31, as we 30 At least at the current status of our knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Note that recent calculations including muons in supernova simulations yield νµ and ντ fluxes with small differences (Bollig et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 31 Similarly, numerous studies investigated ways to identify θ13 with a supernova neutrino signal, until it was measured by the Daya- Bay (An et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2012), the RENO (Ahn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2012) and the Double Chooz (Abe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2012) experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=" H Ve H Vt' 3m Vt' L V1m e e13 FIG." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 7 Shock waves: matter density profiles, at different post- bounce times, as a function of distance in an exploding core- collapse supernova.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Front and reverse shocks are visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The yellow and blue bands correspond to the densities and neu- trino energies that fulfill the high (H-) and low (L-) MSW resonance conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (Figure from (Tomas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=') shall see.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Let us now discuss multiple MSW resonances and MSW-like mechanisms that arise in dense and sometimes explosive environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Shock wave effects Shock-related structures in supernova neutrino obser- vations could inform us on shock reheating and propa- gation, a unique observation of the explosion mechanism on its becoming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The availability of large scale observa- tories and a close supernova would offer the possibility to observe such structures and other deviations from the expected exponential cooling of the newly formed neu- tron star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Even if there are variations among models, some features appear as sufficiently generic to deserve investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' In an exploding supernova, shock waves constitute a major perturbation of the electron fraction, defined as Ye = (ne− − ne+)/(nn + np) (the ni with i = n, p the neutron and the proton number densities), and of the pre-supernova matter density profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The shock wave reaches the H-resonance region in about 2 s after core- bounce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Tomas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2004) showed the presence of both a front and a reverse shock, due to the earlier slower ejecta meeting a hot supersonically expanding neutrino- driven wind (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Schirato and Fuller (2002) first pointed out that shock waves could ”shut off” flavor evolution when passing through an MSW resonance region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Because of the steep- ness of the density profile, νµ,τ ⇋ νe would be suppressed due to non-adiabatic evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Hence the νe, ¯νe fluxes would become colder producing dips in the supernova neutrino time signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The passage of shock waves in MSW resonance regions engenders two effects: it makes the resonance temporar- ily non-adiabatic and induces multiple MSW-resonances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The evolution through the different resonance locations can be treated as incoherent or as coherent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' In the for- mer, the MSW resonances are independent, in the latter coherent evolution produces interference effects among the matter eigenstates called phase effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Since the MSW resonance condition is necessary for shock wave effects, they occur either in the νe signal (for normal), or in the ¯νe signal (for inverted mass ordering).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The change in the adiabaticity at the MSW resonance locations impacts the evolution in the H-resonance re- gion32 and modifies the neutrino average energies creat- ing dips or bumps in the supernova time signal and the corresponding rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' These features were investigated in a series of works (Fogli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2003, 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Kneller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lunardini and Smirnov, 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Takahashi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2003), see the review by Duan and Kneller (2009)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Fogli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2003) pointed out that multiple resonances could produce phase effects that would average out for large values of θ13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Dasgupta and Dighe (2007) investi- gated them in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Phase effects require semi-adiabatic and coherent evolution at the resonances33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' They are dif- ficult to be seen because even when the coherence con- dition is met, the associated oscillations are smeared by the energy resolution of detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Let us consider the presence of a dip in a supernova density profile as an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' A neutrino of energy E encounters two resonances at locations x1 and x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' If |νh⟩ and |νl⟩ are the heavier and lighter matter eigenstates respectively, at x < x1, one has |νh⟩(x ≪ x1) ≈ |νe⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' While evolution before the resonance is adiabatic, the resonance mixes the matter eigenstates just before the crossing x < x1− yielding new matter eigenstates � |νh(x1+)⟩ |νl(x1+)⟩ � = � cos χ1 sin χ1eiφ − sin χ1e−iφ cos χ1 � � |νh(x1−)⟩ |νl(x1−)⟩ � , (26) where Pi = sin2 χ1 is the hopping probability for an iso- lated resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The matter eigenstates acquire a rela- tive phase up to the second resonance at x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' After the latter the νe survival probability is (far from x2) Pνe→νe = cos2(χ1−χ2)−sin 2χ1 sin 2χ2 sin2 � � x2 x1 ∆ ˜m2 4E dx � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (27) The last term, due to the interference between the mat- ter eigenstates, oscillates with the neutrino energy and 32 Note that the shock wave also influences the neutrino evolution through the L-resonance region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' However its impact (at low en- ergies and at late times) is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 33 In a wave-packet description in flat spacetime, decoherence arises at distances larger than the coherence length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' For a typical wave- packet width at production, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' σ ≈ 10−11-10−12 cm−1 and E ∈ [5, 80] MeV (average energy between two matter eigenstates) one gets Lcoh ≈ 104 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 21012 L EL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='1 sec @1010 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='0 sec Q 109 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='0 sec 108E 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='0 sec 107 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='7 sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 106 105 104 103 PH 102E 10 PL 1 10 LLuL Ll 107 108 109 10 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='10 R (cm)14 with the resonance locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' It produces fast oscilla- tions (phase effects) as a function of energy or, for a given energy, as a function of time because the shock wave propagation slightly shifts such locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' In ab- sence of coherence, the interference term averages out and the two resonances at x1 and at x2 are independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Two studies implemented shock wave effects and νν in- teractions (in the bulb model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Using a consistent treat- ment that retains phase information, Gava et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2009) showed that, depending on the neutrino energy, dips or bumps are present in the positron time signal of scintil- lators or Cherenkov detectors (inverted mass ordering).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Similar features are present in the electron time signal for example of argon-based detectors such as DUNE for nor- mal mass ordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' In contrast, the detailed analysis of the time signal for the lead detector HALO-2 performed by Ekinci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2021) showed changes at the level of a few percent for the one-neutron and two-neutron emis- sion rates in neutrino-lead interactions, so too small to be seen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Besides shock waves, turbulence can play a significant role in supernova explosions (see for example Radice et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2018a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The influence of turbulence on the neutrino flavor content has features in common with shock wave effects, as we shall now discuss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Turbulence effects Noisy media, originating e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' from helioseismic g- modes or temperature fluctuations, influence neutrino flavor evolution, as pointed out in relation with the so- lar neutrino problem (see for example Balantekin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (1996);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Nunokawa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (1996);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' and Sawyer (1990)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' In particular, Loreti and Balantekin (1994) showed that randomly fluctuating matter density and magnetic fields tend to depolarize neutrinos, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' the survival probability averages to one-half.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Neutrino propagation in stochastic media was also discussed in Burgess and Michaud (1997) and Torrente-Lujan (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Interestingly, solar neutrino and KamLAND data con- strain matter density fluctuations in our Sun at a few per- cent level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' This result holds for delta correlated (white) noise, and correlation lengths of 10 km (see Balantekin and Yuksel (2003b) and Guzzo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2003)) to 100 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Hence, one can extract the solar oscillation parameters independently from fluctuations (Burgess et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Simulations of exploding core-collapse supernovae show that non-radial turbulent flows associated with convection and SASI have explosion supportive effects (Couch and Ott, 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Janka, 2012, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Mezzacappa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Radice et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2018a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Hydrodynamic in- stabilities generate large scale anisotropies between the proto-neutron star and the supernova envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' There- fore, supernova neutrinos reaching the Earth ”see” stochastic matter density profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Noisy media might influence the supernova neu- trino flavor content significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' First investigations evolved fluctuations-averaged density matrices, or prob- abilities34, with delta-correlated fluctuations and static (Loreti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 1995) or dynamic density profiles with front and reverse shocks (Fogli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Friedland and Gruzinov (2006) argued for Kolmogorov correlated fluc- tuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Kneller and Volpe (2010) evolved neutrino amplitudes and built a statistical ensemble of instantiations for the neutrino survival probabilities using one-dimensional simulations and Kolmogorov fluctuations added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Retain- ing the phase information, the approach revealed the presence of multiple MSW resonances from turbulence and a transition, when increasing the fluctuations ampli- tude, from phase effects due to shock waves to a fluctua- tions dominated regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lund and Kneller (2013) inves- tigated the interplay between neutrino-neutrino interac- tions, shock waves and turbulence using one-dimensional dynamical simulations for three progenitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' These stud- ies (Fogli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Friedland and Gruzinov, 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Kneller and Volpe, 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Loreti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 1995) showed that large amplitude fluctuations resulted into depolarization of the neutrino probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Borriello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2014) came to different conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The authors performed the first investigation exploiting fluctuations from high resolution two-dimensional super- nova simulations down to scales smaller than typical mat- ter oscillation lengths35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' These fluctuations followed bro- ken power laws (with exponents 5/3 and 3)36 in agree- ment with two-dimensional Kolgomorov-Kraichnan the- ory of turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Their analysis showed small damping of the neutrino probabilities due to matter fluctuations and absence of strong or full depolarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Clearly further work is needed to determine the impact of turbulence on flavor evolution and to assess if matter fluctuations introduce a loss of memory effects, or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' MSW-like mechanisms The MSW effect arises from the cancellation of the vac- uum and the matter contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' New resonance con- ditions emerge from the interplay of the different terms of the neutrino Hamiltonian Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (5) describing neutrino propagation in a dense medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Thus, various types of MSW-like phenomena have been uncovered, in particu- lar the matter-neutrino resonance, helicity coherence and the I-resonance that I shall now discuss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 34 This gives a generalized Parke’s formula with a damping factor (Burgess and Michaud, 1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 35 Note that small scale fluctuations (less than 10 km) have smaller scales than what can be resolved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 36 Three-dimensional simulations should bring turbulence spectra with a Kolmogorov exponent of 5/3 at all scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 15 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 8 Matter-neutrino resonances: potentials of the neu- trino Hamiltonian, as a function of distance, in an accretion disk model of a black hole-neutron star merger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The νν inter- action (Vν) and the matter (Ve) terms cross at the locations of the symmetric and standard MNRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The nutation region is due to a cancellation between the νν interactions and the vacuum (∆32) terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The MSW region is also shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (Figure adapted from (Malkus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=') 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Matter-neutrino resonance Accretion disks around compact objects – binary neu- tron star merger remnants or black holes37 – produce large amounts of neutrinos with luminosities and aver- age energies similar to those of core-collapse supernovae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' An important difference is that, in these environments, matter is neutron rich which produces an excess of the ¯νe flux over the νe one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Computationally, even the sim- plest models require spherical symmetry breaking which is numerically more involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' It is to be noted that, in the context of core-collapse supernovae, spherical sym- metry was assumed in numerous studies which yielded interesting results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' In a collapsar type disk, Malkus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2012) found a novel conversion mechanism called the matter-neutrino resonance (MNR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The MNR arises in regions above the disk when the νν and ν-matter interactions cancel each other Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (6) and (12) (Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 8 and 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Indeed, the excess of the ¯νe flux over the νe one gives a different sign to the two contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Moreover, because of the geometry of the disks and the ¯νe decoupling deeper than νe, the Hνν sign can flip (at some point).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' If the flip in sign is not present the phenomenon is called standard MNR (Malkus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2014);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' whereas if it is present, the process is called the symmetric MNR (Malkus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2012, 2016) (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Adiabatic evolution through the MNRs produces efficient conversion of νe into νµ, ντ in the former;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' or of νe and ¯νe in the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' This can influence the electron fraction Ye and favor disk wind nucleosynthesis of r-process elements (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 37 From collapsing stars or from black hole-neutron star binaries FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 9 Matter-neutrino resonances: the stars indicate the lo- cations where the resonance condition is fulfilled along differ- ent trajectories above a binary neutron star merger remnant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (Figure from (Frensel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=') Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2016) and Frensel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2017) showed that patterns of flavor evolution depend on the neutrino path38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Both studies were based on astrophysical inputs from the detailed two-dimensional simulations of a binary neutron star merger remnant by Perego et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Frensel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2017) also found that the neutrino capture rates on neutrons and protons for different initial condi- tions and azimuthal angles showed variations by tens of percent due to flavor mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' In these studies the flavor history of one neutrino is taken as representative of all trajectories39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' A consistent treatment of the neutrino-neutrino interaction term of the Hamiltonian Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (12) should implement the neutrino evolution along different paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Vlasenko and McLaugh- lin (2018) showed that the MNRs take place even in a more consistent treatment, leading to significant neutrino conversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' What is the underlying mechanism of the matter- neutrino resonances?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2016), with a schematic model, and Chatelain and Volpe (2017), with detailed BNS simulations, showed the MNRs are multiple MSW resonances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The crossing of the potentials shows the loca- tion where the MNR starts (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Furthermore if one looks at the full evolution of the Hamiltonian, the mat- ter and the νν interaction terms cancel for tens of km, conconmitantly with the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Indeed, using a per- turbative argument, Chatelain and Volpe (2017) showed that the νν interactions adjust to the matter term over long distances: the MNR condition is fulfilled multiple times due to the non-linearity of the equations and non- linear feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 38 First studies fixed the azimuthal angle θ to 45◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 39 This is equivalent to the treatment, in the supernova context, of νν interactions in the single-angle approximation (bulb model, see section II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 10-16 symmetrie MNR region MNR 10-18 region I (erg) Teglon nutation Potential 10-20 MSW 10*22 regicn 10-24 105 10° 10° 108 10° 1010 Position (cm)300 1014 250 1013 1012 200 1011 109 100 108 107 50 106 105 150 100 50 0 50 100 150 reyl [km]16 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 10 Nucleosynthetic abundances: the crosses show the scaled solar abundances in comparison with predictions in a black hole-accretion disk scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The lines correspond to predictions in absence of neutrino oscillations (red), with os- cillations (blue) or without the νν interaction (green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Visible are the second and the third r-process peak as well as the rare elements plateau in-between.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The neutrino mass ordering is normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (Figure from (Malkus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=') 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Spin or helicity coherence The derivation of extended mean-field equations be- yond the ones usually employed in flavor studies Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (11) uncovered new terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Thanks to these, new resonances become possible that can influence the neutrino content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Volpe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2013) derived mean-field equations in- cluding pairing correlators and wrong-helicity contribu- tions, due to the neutrino mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Afterwards Vlasenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2014a) obtained quantum kinetic equations for Majorana neutrinos, using the Closed-Time-Path formal- ism, and pointed to wrong-helicity terms ∼ m/E naming them spin coherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Serreau and Volpe (2014) presented the most general mean-field equations and called such contributions helicity coherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Present in anisotropic media, they couple neutrinos with antineutrinos but are suppressed, as expected, by the ratio m/E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' In a toy 2ν model Vlasenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2014b) first stud- ied if helicity coherence modifies flavor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The authors found that it could trigger significant ν-¯ν transformation through non-linear feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Motivated by such findings, Chatelain and Volpe (2017) investigated these terms in binary neutron star mergers with inputs from detailed simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' In contrast with the previous findings, their results showed that, while the resonance condition for helicity coherence (similar to the MNR one) was fulfilled, adia- batic evolution was absent for the ensemble of trajectories considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Indeed, the authors were able to show that non-linear feedback could not induce multiple matching of the resonance conditions40, contrarily to the MNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 40 Unless peculiar matter density profiles are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Note that The work of Tian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2017) on the role of helicity coherence in core-collapse supernovae reached the same conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' I-resonance Non-standard interactions are present in theories be- yond the Standard Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Limits on non-standard neutrino-neutrino interactions are rather loose (Bilenky and Santamaria, 1999);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' whereas oscillations and scat- tering experiments provide tight constraints on non- standard neutrino-matter interactions (NSI) (see for ex- ample the reviews by Biggio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2009);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Davidson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2003);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Farzan and Tortola (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' and Ohlsson (2013)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' After decades of attempts, Akimov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2017) (CO- HERENT Collaboration) measured coherent neutrino- nucleus scattering, giving, among others, new constraints (Coloma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Giunti, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' NSI are often evoked in the interpretation of neu- trino oscillation experiments, as possible explanations of anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' If NSI exist, mixing angles and mass- squared differences inferred by experiments are modified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' In dense astrophysical environments, NSI were studied by several authors (Chatelain and Volpe, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Esteban- Pretel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2007b, 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Fogli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Stapleford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2016): they significantly impact flavor evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Moreover the role of non-standard neutrino-neutrino in- teractions in supernovae was also studied by Blennow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' In the context of primordial nucleosynthe- sis, NSI give a subleading contribution to the effective number of degrees of freedom (Mangano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Esteban-Pretel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2007b, 2010) explored the com- bined effect of νν interactions and ν-matter NSI in core- collapse supernovae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' For NSI couplings |ϵ| ≥ 10−2, the I-resonance41, an MSW-like phenomenon, emerges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' It takes place when the standard and non-standard matter terms cancel each other, for ν and ¯ν simultaneously, and independently from the neutrino energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The I-resonance triggers efficient conversions of νe → νµ,τ and ¯νe → ¯νµ,τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Stapleford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2016) performed an extensive investi- gation of NSI effects as a function of their couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The authors found that, even for NSI strengths well below bounds, NSIs produce (symmetric and standard) MNRs in core-collapse supernovae and impact νν interactions effects (in the bulb model) and the MSW-H resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The first investigation of NSI effects in BNS was per- formed by Chatelain and Volpe (2018) (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' They showed that neutrino-neutrino interactions play a role on the I-resonance, contrarily to previous findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' In- deed, when the νν interactions matter, the I-resonance becomes a synchronized MSW effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The investigation the argument holds for supernovae as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 41 I stands for ”internal”, since the phenomenon occurs close to the neutrinosphere, in the most deleptonized inner layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 3 4 Y(A) 6 8 100 150 200 A17 400 200 0 200 400 x (km) 0 100 200 300 400 500 600 z (km) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='40 Ye FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 11 I-resonances: locations where the I-resonance condi- tion is fulfilled above a binary neutron star merger remnant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Values of the electron fraction Ye, before flavor evolution, are coded with colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (Figure from (Chatelain and Volpe, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=') of a large ensemble of trajectories, above a BNS remnant, uncovered that, as in core-collapse supernovae, even very small values of NSI parameters produce intricate patterns of mechanisms, including MNRs, I- and synchronized I- resonances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Neutrino-neutrino interactions Dense environments have sizable νν standard neutral- current interactions because neutrinos are emitted in large amounts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' In a seminal work, Pantaleone (1992) pointed out that the νν interactions introduce off- diagonal potentials42 and make ν evolution a non-linear many-body problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' In the last fifteen years theorists have tirelessly worked to understand the neutrino-neutrino refraction effects, the novel flavor mechanisms, how they arise and their impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' They have established connections with other many-body systems and figured out new approaches to deal with such interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Several reviews dress a de- tailed picture of these developments (Duan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Duan and Kneller, 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Horiuchi and Kneller, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Mi- rizzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Tamborra and Shalgar, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Volpe, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Here I shall highlight aspects emerged from the efforts to solve this complex problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Flavor mechanisms due to νν interactions are cur- rently classified as slow or as fast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Slow modes occur at typical distances of O(102-103) km from the neutri- nosphere, whereas fast modes have scales of O(1) m or much less and frequencies as large as µ ∼ √ 2GFnν, nν being the neutrino number density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Their rate can ex- ceed the vacuum oscillation frequency by large factors, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' µ/ω = 105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 42 The νν Hamiltonian has off-diagonal complex contributions, be- cause of its dependence on the neutrino and antineutrino density matrices Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (12) (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 12 Neutrino-neutrino interactions in the bulb model: ge- ometry of the neutrino emission from a newly formed proto- neutron star (of radius R) in a core-collapse supernova ex- plosion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The bulb model assumes both spherical symmetry and azimuthal symmetry along the radial direction r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Each neutrino state is characterized by the the momentum p and the angle θR, the angle θ at the intersection point between a neutrino emitted at θR (red line) and a radially propagating neutrino (dashed line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The blue lines show the maximal angle that contributes to the νν interaction Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (Figure adapted from (Duan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2010)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Slow modes Already twenty years ago Samuel (1993) showed that νν interactions stimulated new flavor effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Studies in the cosmological context (see for example (Kostelecky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Kostelecky and Samuel, 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Pastor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2002)) uncovered a stunning mechanism where neutrino spins ”sticked together”, precessing collectively around an effective magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Synchronized oscillations were included for example in investigations of cosmologi- cal neutrino-antineutrino asymmetries (Abazajian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Dolgov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2002), also with CP violation (Gava and Volpe, 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Duan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2006b) uncovered collective flavor modes43 in supernovae, due to νν forward scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Using the bulb model (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 12) Duan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2006a) per- formed ”single-angle” and demanding ”multi-angle” sim- ulations44 which showed large-scale modes and spectral splits45 (Duan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2006a)46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Moreover, Balantekin and 43 The modes were named slow after the identification of fast modes (see Section G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 44 Computationally, one can treat Hνν Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (12) in two ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' In the simplified ”single-angle” approximation, the flavor history of a neutrino, at a given angle with respect to the radial direction, is representative of all angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (Authors took π/4 and often 0◦, which strictly speaking corresponds to ”non-interacting” neutri- nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' In contrast, multi-angle simulations include the full angu- lar dependence of the νν potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Concerning binary compact objects, in which calculations are particularly challenging, it is also common to treat Hνν as in ”single-angle” approximation (see for example (Frensel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Malkus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2014, 2012, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2016)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 45 Splits are sharp boundary features at the edges of spectral swap intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 46 Note that in these early works, the neutrino-neutrino and matter number densities were such that self-induced flavor conversion was intertwined with matter effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' PNS R18 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 13 Bulb model: example of the neutrino spectral swap- ping in supernovae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Due to νν interactions, the initial (quasi- thermal) νe and νµ spectra undergo significant modification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' As the figure shows, there is a critical energy (Ecrit = 8 MeV) for which, if E < Ecrit the νe flux is unchanged, whereas for E > Ecrit the νx and the νe fluxes interchange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (Figure adapted from (Duan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2010)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Yuksel (2005) found that the neutrino-neutrino refraction impacted the equilibrium electron-fraction Ye, important for r-process nucleosynthesis in neutrino-driven winds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' For several years the bulb model has received peculiar attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Simulations showed puzzling flavor behaviors which triggered intense theoretical work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Named collec- tive neutrino oscillations, these phenomena occur for siz- able νν interactions and for non-zero mixings (even for extremely small values of θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Three regimes emerge when neutrinos travel from the neutrinospheres, where the neutrino number densities are large, to regions where matter dominates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' They are called the synchronization, the bipolar instability (Duan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2006a, 2007a, 2006b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Hannestad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2006) and the spectral splits (Dasgupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Duan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2006a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Fogli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Galais et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Raffelt and Smirnov, 2007) (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The spin formalism in flavor space gives an image of these three phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' First, as in the early universe at the epoch of primordial nucleosynthesis, neutrino spins syn- chronize in a stable collective mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Second, neutrino spins experience an instability where νe¯νe pairs convert into νx¯νx ones, due to lepton-number conservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' They perform precession and nutation around B and behave like a pendulum (Duan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2007a, 2006b, 2007b), or a gyroscopic pendulum (Hannestad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Third, they undergo either full or no conversion, depending on the neutrino energy, generating spectral swapping and splits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' A pictorial image of these modes for 3ν flavors was given in the e3-e8 triangle diagram by Dasgupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2009) who employed Bloch vectors and the SU(3) alge- bra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Other approaches to these phenomena brought further insight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Using the matter basis, Galais et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2012) found that with the νν refraction the adiabaticity parameters depend on the matter angle and phase derivative and bipolar oscillations start when the latter diverges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' More- over, Galais and Volpe (2011) showed that spectral splits arise from a magnetic resonance phenomenon: the swap- ping emerges because the spins satisfy (or do not satisfy) a magnetic resonance condition, depending on the neu- trino energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Pehlivan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2011) used a different angle of attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' With an algebraic many-body approach and the Bethe ansatz, they demonstrated that the splits emerged in the transition from a quasi-particle to a particle description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' In a subsequent study Pehlivan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2017) established that the emergence of the splits, from the regions where neutrinos strongly interact to those where they weakly interact, is similar to the behavior of Cooper pairs in the BEC-BCS crossover in experiments with ultra cold atomic gases (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' While first studies revealed instabilities only in in- verted mass ordering, Dasgupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2009) showed that more plausible ratios of the νe, ¯νe, νx fluxes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' different from one) produced single and multiple spec- tral splits in both hierarchies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Moreover, Esteban-Pretel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2008) argued that large matter densities intro- duced decoherence in multi-angle calculations because of the angular factor (1 − ˆp · ˆp′) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='(12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Following these findings, Chakraborty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2011b,c) showed, us- ing one-dimensional supernova simulations, that matter suppressed collective effects when the matter exceeded the νν number density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' This finding should be confirmed by multi-angle calculations using multi-dimensional su- pernova simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The studies assumed stationarity and homogeneity of the medium where neutrinos propagate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Moreover, they used the mean-field approximation and neglected col- lisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Thanks to these approximations, the full 7- dimensional problem of neutrino flavor evolution reduces to a more tractable one, typically in two- or three- dimensions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (E, r) or (E, r, θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' But even schematic models with a reduced number of degrees of freedom are often quite challenging to solve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' A step towards higher spatial dimensionality was pro- vided by the so-called neutrino line model with two spa- tial dimensions, either with only two neutrino beams from each initial condition (Duan and Shalgar, 2015), or with multi-angles at each point source (Abbar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2015), showing inhomogeneous modes located at larger neutrino densities than homogenous ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Abbar and Duan (2015) and Dasgupta and Mirizzi (2015) identi- fied temporal instabilities arising in non-stationary mod- els since time can cancel a (albeit) constant matter term, producing instabilities (quite) deep in the supernova.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Moreover, Cherry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2012, 2013) uncovered that a neutrino halo – a small amount of back-scattered neutri- nos due to collisions – could completely reshape the fla- vor patterns produced by forward scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' This find- ing cast doubts on the treatment of νν interactions as an initial value problem and showed limitations of the mean- a Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' initial , initial Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' final final 20 40 60 E (MeV)19 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 14 Neutrino-neutrino interactions in a supernova: cor- respondence between neutrinos propagating from dense to di- lute regions and the BEC to BCS limits in ultra cold atomic gases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (Figure from (Pehlivan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2017)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' field approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The halo effect was further studied by Sarikas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2012b) and Cherry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Furthermore, Raffelt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2013) found, in a linearized study of the bulb model, azymuthal-angle-instabilities, showing that solutions do not necessarily inherit the sym- metries of the initial or boundary conditions (unless en- forced).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' This fact was dubbed spontaneous symmetry breaking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Several works (Chakraborty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Mi- rizzi, 2013) confirmed symmetry breaking solutions with linearized analysis in supernovae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Flavor evolution can even reveal chaotic behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' This was mentioned by Raffelt and de Sousa Seixas (2013) in a stationary model with two opposite neutrino momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' And, in fact, Hansen and Hannestad (2014) clearly identified in the same model the exponential di- vergence of Liapounov exponents of infinitely close initial trajectories of the neutrino spin vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' So, after the discovery of novel phenomena in the bulb model, models increased in complexity and included e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' non-stationarity, inhomogeneities, unconstrained sym- metries and more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' These in-depth investigations kept uncovering new features and the richness of flavor evo- lution in dense media, due to the νν refraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' While theorists thought they were shaping a solid understand- ing, fast modes arrived ”on the stage”, triggering another ”runaway” of studies .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Fast modes In a 3ν two-beam model Sawyer (2005, 2009) found that νν interactions could ”speed-up” flavor transfor- mation and produce counterintuitive modifications on a short time scale of t = (2 √ 2GFnν)−1 of order O(1) m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Much later Sawyer (2016) considered non-trivial angu- lar distributions at the neutrinospheres, with ¯νe emitted deeper than νe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' He identified modes with nm scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Sawyer’s findings triggered excitement again: fast modes took place close to the neutrinosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' They could influence the supernova dynamics and nucleosyn- thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Theorists had found the short scale modes they were looking for, finally!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Contrarily to slow modes, fast modes have the pecu- liarity that they are not triggered neither by the mix- ings Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (3) nor by the matter contribution Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='(6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Since only the neutrino emission matters, Izaguirre et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2017) introduced the angle distribution of the electron lepton number (ELN) which for 2ν flavors reads Gv = √ 2GF � ∞ 0 dE E2 2π2 [φνe(E, v) − φ¯νe(E, v)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (28) As indicated by the results of Sawyer (2016) and pointed out by Dasgupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2017) and Izaguirre et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2017), fast modes take place when the angular distribu- tions of νe and ¯νe cross each other along a given direction (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Gv changes sign): this is an ELN crossing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Its pres- ence is a necessary but not a sufficient condition for fast modes to occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' In the last years many studies of fast modes have been realized based on the linearized approach (see for exam- ple (Abbar and Duan, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Abbar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2019, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Capozzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Chakraborty and Chakraborty, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Chakraborty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2016a,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Delfan Azari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' George et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Padilla-Gay et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Shal- gar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Tamborra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2014a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Wu and Tam- borra, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Xiong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2020) and Tamborra and Shal- gar (2021) for a review).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' It was already advocated by Sawyer (2005) that fast modes could bring flavor equilibration of the different neutrino species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The ansatz has been admitted in the literature for some time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' In fact it has the advantage that it simplifies the problem since the neutrino spectra emerge identical from a supernova.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' However, in a two- beam model, Abbar and Volpe (2019) evolved for the first time fast modes in the full non-linear regime, showing they do not necessarily lead to flavor equilibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Fast modes behave differently in three-flavors with re- spect to two-flavors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' For the former the concept of ELN crossing needs to be generalized to µLN and τLN cross- ings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Chakraborty and Chakraborty (2020) first inves- tigated such effects with the dispersion relation treating both time and space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The authors pointed out the im- portance of three-flavor effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The further analysis of Capozzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2020), that went up to the non-linear regime, showed instabilities over tens of ns scale which were either absent (in 2ν flavors), or got damped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Clearly these findings emphasize the need for three-flavor analy- sis of fast modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Since the scale for fast modes is so short, there can be regions requiring the treatment of flavor mechanisms and collisions when the medium is very dense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Indeed the fast rate exceeds the collision rate even within a su- pernova core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Using a one-dimensional model, with two momentum modes, Capozzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2019) analyzed the interplay between collisions and fast modes and showed that collisions can trigger the conditions for fast conver- sions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Significant efforts are made to study flavor con- version modes in presence of collisions (see for example BCS BEC Weakly interacting regime Strongly interacting regime V Proto-neutron star V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='20 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 15 Fast modes in multidimensional supernova simula- tions: Mollweide projection of the ¯νe-over-νe ratio at a dis- tance r = 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='6 km in a snapshot at post-bounce time of 200 ms of a 3D supernova model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The white crosses indicate the location where fast modes occur, based on a linearized anal- ysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (Figure from (Abbar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2020)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Hansen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Richers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' and Xiong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2022))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' It is necessary to keep in mind, though, that a consistent treatment in multidimensional simula- tions is numerically very challenging and therefore far ahead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Fast modes are present in schematic models, but do they occur in detailed supernova simulations?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The first investigation by Tamborra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2017), based on one- dimensional simulations, concluded for the absence of fast modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Searches in multidimensional simulations re- vealed the presence of fast modes, in contrast with this early finding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Abbar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2020) identified fast-growing modes in two- and three-dimensional simulations when α = nνe/n¯νe is of the order of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' A linear stability analysis confirmed the presence of fast modes in correspondence with the angular crossings (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 15), even deep in the supernova core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Their influence on the neutrino spec- tra was found to be small, since the neutrino spectra are already very similar at the location of the crossings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Glas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2020) and Delfan Azari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2020) also found ELN crossings nearby the neutrinosphere and con- firmed the presence of fast modes in detailed supernova simulations (with full Boltzmann transport) in three- and two-dimensions respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' It is to be noted that multi- dimensional supernova simulations do not provide full in- formation of the neutrino angular distributions as a func- tion of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Therefore methods have been developed, as in Dasgupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2018), to identify fast modes using the moments of the angular distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Beyond the studies in the supernova context, Wu and Tamborra (2017) performed the first analysis of fast modes in accretion disks resulting from binary compact objects mergers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' They found the conditions for fast modes to be generically met because of the excess of ¯νes over νes and of the geometry of such environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' As for nucleosynthesis, Xiong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2020) studied the influence of fast oscillations in neutrino-driven winds in a low and a high mass core-collapse supernova.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' They showed that partial (or total) flavor equilibration creates more proton-rich conditions (Ye > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='5) enhancing the νp process and mass ejection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2017) considered the impact of fast modes on the r-process in a neutrino driven wind nearby a black hole remnant from compact binary mergers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Under the approximate assumption of flavor equilibration, fast modes produced an increase of lanthanides (more generally nuclei with A > 130) up to a factor of 103, due to the decrease of Ye, showing a potentially high impact on kilonova light-curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' On the contrary, in an analysis for an hyper-massive BNS merger remnant, George et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2020) finds lan- thanides to be little affected by fast pairwise conversion (under the same equilibration hypothesis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The subject of fast modes and their impact undergoes a fast development where interesting aspects keep being uncovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Clearly, these short scale modes will keep attracting attention in the coming years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' FLAVOR EVOLUTION: THEORETICAL FRAMEWORKS Neutrinos propagating in a dense environment consti- tute a unique, weakly interacting, many-body system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Its description benefits of a lucky situation in some respects, since one does not have to deal with phenomenological interactions as for atomic nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' It is a specific case also, because theoretical approaches developed for many-body systems need to be extended to particles with mixings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The literature is rich with theoretical approaches for this system (see the review by Volpe (2015)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' These range from the mean-field and the extended mean-field equa- tions, the linearized equations and a dispersion relation approach to the neutrino quantum kinetic equations, as I shall now describe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The mean-field approximation The mean-field constitutes the simplest approximation to describe neutrino propagation in astrophysical envi- ronments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Neutrino mean-field equations were derived by several authors (Balantekin and Pehlivan, 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Fried- land and Lunardini, 2003b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Samuel, 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Sawyer, 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Serreau and Volpe, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Sirera and Perez, 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Volpe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Yamada, 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' It is common to determine neutrino propagation using the formalisms of Green’s functions, of density matri- ces, neutrino (iso)spins, or of neutrino amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The density matrix formalism is widely used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The ν and ¯ν one-body densities are defined as ϱ1,ij(p, h, p′, h′) = ⟨a† j(p′, h′)ai(p, h)⟩ , ¯ϱ1,ij(p, h, p′, h′) = ⟨b† i(p, h)bj(p′, h′)⟩ , (29) where the quantum expectation value are over the astro- physical or cosmological background;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' i, j ∈ [1, N] with N the number of neutrino families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The diagonal ele- ments of the one-body density matrix correspond to the expectation value of the number operator and are the 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='6 km 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='821 only contributions for particles without mixings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The off-diagonal elements (i ̸= j) implement coherence due to the mixings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Sometimes it is said that neutrino evo- lution requires a ”matrix of densities” quoting Sigl and Raffelt (1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' A simple way to derive mean-field evolution equations is through the Ehrenfest theorem i ˙ϱ1,ij = ⟨[a† jai, H]⟩ , (30) where H is the neutrino Hamiltonian Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The mean-field approximation consists in neglecting the correlated part of the two-body density47 ϱ12 = ϱ1ϱ2 − c12 , (31) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' setting c12 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Since only the uncorrelated part is retained, the two particles propagate independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Thus the full many-body system evolves as made up of independent particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Another way to define the mean-field approximation is saying that one replaces two-body interaction terms by one-body ones as I1I2 → I1⟨I2⟩ + ⟨I1⟩I2 + ⟨I1⟩⟨I2⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (32) If one takes the charged- or the neutral-current interac- tion terms of the GWS model,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' the corresponding mean- field Hamiltonian has the general bilinear form (Serreau and Volpe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 2014) HMF(t) = � dx ¯ψi(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' x)Γij(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' x)ψj(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (33) where Γij(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' x) is the interaction kernel that depends on the specific interaction terms that one considers,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' ψi de- notes the neutrino field in the mass basis (for the ith mass eigenstate)48 ψj(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' x) = � h � p [uj(p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' h)aj(p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' h)e−ip·x+vj(p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' h)b† je+ip·x] (34) with p · x = pµxµ and uj(p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' h) and vj(p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' h) the four- components complex spinors,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' solution of the Dirac equa- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' In the mean-field approximation, the interaction kernel is quadratic (and not quartic) in the creation and annihilation operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' A theoretical framework to treat the evolution of many-body systems is given by the Born and Green (1946), Bogoliubov (1946), Kirkwood (1935), Yvon (1935) (BBGKY) hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The hierarchy, introduced 47 Note that the (reduced) density matrices are also referred to as one-particle, two-particle, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' , instead of one-body, two-body and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 48 Here we write down expressions considering neutrino Dirac fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The generalization to Majorana fields is straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' for a non-relativistic many-body system, replaces the ex- act evolution of the quantum many-body system by a hierarchy of integro-differential equations for n-body den- sity matrices49 ϱ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='n = ⟨a† n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' a† 1a1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' an⟩ (35) that can be truncated at different levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The mean-field approximation corresponds to truncating the hierarchy at the lowest level, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' assuming c12 = 0 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' While BBGKY was originally for a non-relativistic many-body system, Calzetta and Hu (1988) generalized it for relativistic many-body systems which involves, in particular, an infinite hierarchy of equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The mean-field Hamiltonian: a derivation To determine the contribution from a mean-field Hamiltonian, one has to add the scattering amplitudes for the corresponding scattering process as Vkr(ρ) = � s,p v(kp,rs)ρsp , (36) and sum (or integrate) over the background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The quan- tity ρsp is ρsp = ⟨a† pas⟩ , (37) where k, p, r, s are, each, a set of single-particle in- dices like (⃗p, h), characterizing the single-particle neu- trino states of a Fock space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Let us derive the mean-field term of the matter Hamil- tonian, associated with a tadpole diagram (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 3) as an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The charged-current interaction term associ- ated with νe-e scatting reads HCC =GF √ 2[ ¯ψνe(t, x)γµ(1 − γ5)ψe(t, x)] × [ ¯ψe(t, x)γµ(1 − γ5)ψνe(t, x)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (38) The first step in determining Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (36) is to evaluate the matrix elements v(kp,rs) ≡ ⟨k, p|HCC|r, s⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (39) More explicitly one needs to calculate v(kp,rs) =GF √ 2⟨νe, e| � dx [ ¯ψe(t, x)γµ(1 − γ5)ψe(t, x)] × [ ¯ψνe(t, x)γµ(1 − γ5)ψνe(t, x)]|νe, e⟩ , (40) where the Fierz transformation has been applied to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 49 Note that Wang and Cassing (1985) reformulated the hierarchy as a set of equations for n-body correlation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 22 By introducing the Fourier expansions of the electron and the neutrino quantum fields Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (34), one can simplify the matrix element using the general relation (from the equal-time anti-commutation rules) ⟨e(1)|a†(2)a(3)|e(4)⟩ =(2π)3δ3(p1 − p2)2Ep1δh1,h2 (2π)3δ3(p3 − p4)2Ep3δh3,h4 , (41) where here the labels (1,2,3,4) stand for a set of single particle quantum numbers (p, h) for the two incoming and the two outgoing particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Using this relation one gets v(kp,rs) =GF √ 2 � dx [¯uνe(k′)γµ(1 − γ5)uνe(k)] × [¯ue(p′)γµ(1 − γ5)ue(p)]ei(p+k−p′−k′)·x , (42) where the first two factors in the integral depend on spinorial products, whereas the last one ensures momen- tum conservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The second step to determine the mean-field term is to evaluate V (ρ) =GF √ 2 � he,h′e � pe,p′e � dx [¯uνe(k′)γµ(1 − γ5)uνe(k)] × [¯ue(p′)γµ(1 − γ5)ue(p)]ei(p+k−p′−k′)·x × ρ(pe,he,p′e,h′e) , (43) and perform the integration over the degrees of freedom of the electron background (at finite temperature T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' One very often makes the following ansatz ρ(pe,he,p′e,h′e) = ⟨ψ(T)|a† e(p′ e, h′ e)ae(pe, he)|ψ(T)⟩ = (2π)3δ3(pe − p′ e)δhe,h′e2Epρp , (44) that is one assumes that the background particles are uncorrelated (independent) and that the medium is ho- mogeneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' This hypothesis corresponds to considering forward-scattering only, where the electrons (and there- fore neutrino) momenta are unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' By plugging Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (44) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (43), the spinorial prod- ucts in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (43) can be evaluated thus giving 8pµ(kµ − msµ) = 8EpEk(1 − ˆp · ˆk)(1 − hν) , (45) where we have introduced the 4-vector sµ = hν �|k| m , Ek m|k| � , (46) and already imposed momentum conservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Therefore one gets V (ρe) =GF √ 2 (2π)3δ3(k − k′)8Ek(1 − hν) � p (1 − ˆp · ˆk)ρp , =2a √ 2GF � dp (2π)3 (1 − ˆp · ˆk)ρp , (47) with50 a = (2π)3δ3(k − k′)Ek(1 − hν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' If the medium is isotropic the angular dependence in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (47) averages out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Since the total number of electrons in the medium is given by Ne ≡ � he � p ⟨a† e(p, he)ae(p, he)⟩ = 2V � dp (2π)3 ρp , (48) equation (47) becomes V (ρe) = √ 2GFne (49) where ne = N/V is the electron number density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' This is the well known mean-field Hamiltonian responsible for the MSW effect in matter Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Following the procedure just outlined, one can derive any mean-field contribution to the neutrino Hamiltonian, such as the those coming from νν interactions, or from NSI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Beyond the usual mean-field The mean-field equations Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (11) with vacuum mix- ings, the standard charged- and neutral-current ν-matter and νν interactions have been widely used in studies of flavor evolution in dense astrophysical environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' A few works explored extensions of such equations to estab- lish the robustness of the mean-field approximation and the possible necessity to go beyond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' It was pointed out by several authors that the Hamil- tonian with νν interactions is analogous to an interacting system of spins that have a spin-exchange interaction and feel an external magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Balantekin and Pehlivan (2007) provided a derivation of the mean field equations using the algebraic formulation of the neutrino Hamilto- nian51 H = � ω ωB0 · Jω + µ � p,q (1 − cos θpq′)Jp · Jq , (50) where the last terms depends on the generators of SU(2) algebra(s)52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The authors used a coherent-state path- integral approach and showed that the mean-field equa- tions correspond to the saddle point approximation of the path-integral for the full many-body system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Moreover, they pointed out contributions beyond the mean-field as corrections to the saddle-point solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 50 Note that the factor a goes away when calculating the neutrino mean-field evolution equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 51 Here µ = √ 2GF/V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 52 They depend on the operators J+(p) = a† x(p)ae(p), J−(p) = a† e(p)ax(p), J0(p) = 1 2 � a† x(p)ax(p) − a† e(p)ae(p) � , which sat- isfy the commutation relations [J0(p), J±(p)] = ±δ3(p − q)J±(p) and [J+(p), J−(q)] = 2δ3(p − p)J0(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 23 Volpe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2013) used the BBGKY hierarchy to de- rive mean-field equations53for the ν and ¯ν one-body den- sity matrices Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (29) in a dense astrophysical environ- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Moreover, thanks to the hierarchy, the authors pointed out that the neutrino evolution equations had further terms at the mean-field level, namely two-point correlators from wrong-helicity contributions due to neu- trino masses and from pairing (or abnormal) densities54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' For Dirac neutrinos the latter read κij(t, q, h, q′, h′) = ⟨bj(t, q, h′)ai(t, q, h)⟩ , (51) and their hermitian conjugates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Volpe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2013) showed that in presence of pairing correlators one can cast the extended mean-field evolution equations, simi- larly to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (11) as i ˙R = [H, R] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (52) The quantities H and R are the generalized Hamilto- nian and density that includes both the ν and ¯ν density matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' By introducing a Bogoliubov transformation, such a system of ν and ¯ν with pairing correlators can be described in terms of independent quasi-particles (Vaana- nen and Volpe, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Furthermore, in their derivation of neutrino quantum kinetic equations, Vlasenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2014a) pointed out contributions from the correlators, named spin coherence ζij(t, q) = ⟨a† j(t, q, +)ai(t, q, −)⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (53) They are due to the neutrino mass and suppressed by the factor m/E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Serreau and Volpe (2014) derived the most general mean-field equations for inhomogeneous and anisotropic media considering Dirac as well as Majorana neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Such equations include contributions either from pair- ing or from wrong-helicity correlators – helicity coher- ence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Using the approach of Serreau and Volpe (2014), Kartavtsev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2015) also included contributions from neutrino electromagnetic properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Spin or helicity coherence requires anisotropy of the medium to be non-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The corresponding generalized Hamiltonian can again be cast in the form Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (52) but this time it has both flavor and helicity structure (Serreau and Volpe, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Vlasenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2014a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Helicity coher- ence couples ν with ¯ν, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' active and sterile neutrinos if ν are Dirac particles, or neutrinos and antineutrinos if ν are Majorana particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 53 In the context of atomic nuclei, the neutrino mean-field equa- tions correspond to the so-called Time Dependent Hartree-Fock approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' If the initial state for the many-body system is a Slater determinant, it remains a Slater determinant at all times (Ring and Schuck, 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 54 Sigl and Raffelt (1993) mentioned such correlations but dis- carded them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Note that neutrino-antineutrino correlations were included in the neutrino evolution equations in the context of baryogenesis via leptogenesis by Fidler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The impact on flavor evolution of the supplementary terms from the correlators (51) and (53) was investigated as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Kartavtsev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2015) pointed out that the pairing correlators do not influence flavor because the large kinetic contributions cannot be removed55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' This fact makes the influence of the off-diagonal contributions from pairing correlators tiny.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Concerning helicity coherence (see section F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='II), simu- lations in 3ν flavors with detailed astrophysical inputs from binary neutron star merger remnants (Chatelain and Volpe, 2017) or supernovae (Tian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2017) showed that non-linear feedback does not operate in detailed set- tings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' As a consequence, helicity coherence does not seem to influence the neutrino flavor in media as for now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Linearization Linearization is a widespread approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' It is used in many domains of physics, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' in nuclear physics, con- densed matter or in hydrodynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The linearization procedure transforms the solution of the equations of motion into eigenvalue equations, making the numerical problem more tractable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The first application of linearization to the neutrino mean-field equations in a supernova was done by Sawyer (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Afterwards, Banerjee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2011) derived a lin- earized version of the equations of motion in the bulb model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Since then, the procedure has been widely em- ployed in the study of both slow and fast modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Vaana- nen and Volpe (2013) provided an alternative derivation of the linearized equations, by generalizing the random- phase-approximation (RPA) commonly used in the study of atomic nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Subsequently Izaguirre et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2017) re- cast the linearized equations in a dispersion relation ap- proach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The linearised equations Let us have a closer look at the linearized version of the equations of motion for supernova neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Here we follow Banerjee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2011) and consider the bulb model, which includes νν neutral-current interactions (section II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' For 2ν flavors, the neutrino flux matrices can be rewritten as ρℓ,r = φνe,ℓ,r + φνx,ℓ,r 2 + gℓ � sℓ,r Sℓ,r S∗ ℓ,r - sℓ,r � , (54) where ℓ = (ω, u), Sℓ,r is an Hermitian matrix and gℓ,r = (φνe,ℓ,r − φνx,ℓ,r) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (55) 55 Usually, the diagonal contributions, proportional to the identity matrix, are substracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' They do not impact neutrino flavor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 24 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 16 Linearisation: example of the solution of the eigen- value equations for neutrino Fermi-Dirac distributions at the neutrinosphere (upper figure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The positive frequencies corre- spond to neutrinos, whereas the negative ones to antineutri- nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The quantity κ1, as a function of the νν interaction cou- pling constant µ, is the imaginary part of one of the two un- stable solutions (lower figure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (Figure adapted from Banerjee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=') The elements of Sℓ satisfy the normalization condition s2 ℓ,r + |Sℓ,r|2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The quantities (1 + s)/2 are the sur- vival probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The indices (ω, u, r) are the vacuum oscillation frequency ω = ∆m2/2E, u = sin2 θR with u ∈ [0, 1] characterizes the neutrino emission at the neu- trinosphere (at r = R), r is the distance defining the νν intersection point along the symmetry direction (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' In the linearization procedure one considers the ini- tial state as ”quasi-static” and performs small variations around it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' In our case this amounts to considering the initial states at the neutrinosphere Sℓ,R = diag(1, −1) and ρℓ,R = diag(φ0 νe, φ0 νx) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (54) and consider a small amplitude approximation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' sℓ,r ≃ 1 |Sℓ,r| ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (56) Under this hypothesis, the neutrino mean-field equations Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (11) with the contributions from vacuum Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (3), matter Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (6), and νν interaction Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (12) become i∂rSℓ,r = [ω + u(λ + ϵ)] Sℓ,r − µr � 1 0 du′ � +∞ −∞ dω′(u + u′)gℓ′Sℓ′,r , (57) when r ≫ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (57) the third term on the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' is the total lepton number ϵ = � du dω gℓ,r (normalized to φ0 ¯νe);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' whereas the second and the last terms on the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' are the matter and the νν terms, with the following coupling constants λ = √ 2GF(ne − n¯e) µr = √ 2GFφ0 ¯νeR2 8πr4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (58) One seeks for solutions of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (57) of the type Sℓ,r = Qℓ,re−iΩr , (59) which leads to the eigenvalue equations [ω + (λ + ϵ)u − Ω]Qℓ,r = −µr � 1 0 du′ � +∞ −∞ dω′(u + u′)gℓ′Qℓ′,r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (60) If the eigenvalue Ω ∈ ℜ, the initial condition is stable and the system performs small oscillations around it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' If Ω ∈ C, one faces an instability in flavor space56: the system deviates exponentially from the initial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' This is often called a runaway solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Figure 16 gives an example of the application of linearized equations in the supernova context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Therefore, a complex eigenvalue indicates the start of flavor modification when neutrinos depart from the neu- trinosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' However, it is worth emphasizing that lin- earization does not provide any information on the full non-linear regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Indeed the linearized equations are inherently based on the small amplitude approximation Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='(56).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Only the full numerical solution of the equa- tions of motion tells us how significant flavor conversion is, at large scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' In the study of atomic nuclei or metallic clusters, lin- earized equations are obtained with RPA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' With this ap- proach one determines small variations of the matter den- sity around the initial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' If the eigenvalues are real that indicates that the initial state is a true ground state;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' if they are complex, then the initial state is not a ground state of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The latter situation is in fact what one looks for in the neutrino case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Also, in RPA, one can face ”spurious” solutions that are numerical artifacts (see for example (Ring and Schuck, 2004)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' These were also found in the neutrino context e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' when multi-angle calculations of the νν interaction do not include a suffi- ciently large number of angle bins (Sarikas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2012a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Here is how Vaananen and Volpe (2013) generalized RPA to neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' At initial time, the density matrices ϱ and ¯ϱ correspond to a stationary state [h0, ϱ0] = 0 [¯h0, ¯ϱ0] = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (61) Since RPA is a small amplitude approximation, one per- forms small variations δϱ(t) of the density57 around ϱ0: δϱ = ρ0 + δϱ(t) = ϱ0 + ϱ′e−iΩt + ϱ′†eiΩ∗t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (62) 56 Note that the linearized equations admit the pair (Ω, Ω∗) as solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 57 Similarly for δ¯ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 3 F 2 [ 1 1[ 2E 3 E 2 1 0 1 2 Frequencyo0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='10 V800 K1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='06 00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='00 0 50 100 150 200 μ25 where ϱ′ here stands for the off-diagonal terms of the density matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The mean-field Hamiltonian around this solution changes accordingly h(ϱ) = h0 + δh δϱ ��� ϱ0δϱ + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' ¯h(¯ϱ) = ¯h0 + δ¯h δ¯ϱ ��� ¯ϱ0δ¯ϱ + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (63) By implementing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (62) and retaining terms at lowest order, one obtains linearized equations of motion58 which can be cast in the following matrix form59 � A B ¯B ¯A � � ϱ′ ¯ϱ′ � = Ω � ϱ′ ¯ϱ′ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (64) The condition for their applicability, that the system is initially in a ”quasi-static” state Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (29), is satisfied in the matter basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Therefore, the linearized equations of motion Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (64) are applicable at any time of the ν evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' A dispersion-relation approach A third formulation of the linearized equations was suggested by Izaguirre et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2017), for fast modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The authors cast them in a dispersion-relation approach where the neutrino modes are neutrino flavor (iso)spin waves, described by a four vector c and a ”polarization” vector, in matter and neutrino backgrounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Instead of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (60) one seeks for plane waves (in an homogeneous and stationary background) Sv(t, r) = Qv(Ω, K)e−i(Ωt−K·r) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (65) After linearizing the mean-field equations (11), one gets the following equation vµkµQv = − � dv 4π vµv′ µGvQv′ (66) where vµ = (1, v), and (Ω, K) is replaced60 by (ω, k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' By considering Qv = aµkµ/vµkµ , (67) 58 The detailed derivation and the explicit expressions for the A, B, ¯ A, ¯B matrices can be found in Vaananen and Volpe (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 59 In the application of the RPA approach to atomic nuclei, the ini- tial state is the nucleus ground state, while the variations around it determine the excited states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The quantities ρ′ and ¯ρ′ are called the forward and backward amplitudes and correspond to particle-hole and hole-particle excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The RPA and its nu- merous variants (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' QRPA, CRPA, SRPA) are used to study the excited states of atomic nuclei, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' the Giant Resonances, or to calculate the transition matrix elements of single β, 2β(2ν) and 2β(0ν) decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 60 Going to a comoving frame, to get rid of the background contri- bution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 17 Dispersion-relation approach: results for a two-beam model with angular modes G1 and G2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The red region cor- responds to complex values of ω or k for real k or ω with k = (0, 0, kz) for which fast modes are grow either in time (temporal) or in space (spatial instability).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (Figure adapted from Izaguirre et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=') one can recast Equation (66) as a dispersion relation ω = ω(k) as Πµνaν = 0 (68) seeking for non-trivial solutions, such that det[Πµν] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The ”polarisation tensor” in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (68) reads Πµν = ηµν + � dv 4π Gv vµvν ω − v · k , (69) with ηµν = diag(+, −, −, −).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Equation (68) is a quartic equation in ω whose roots give four possible dispersion- relation branches: a) (ω, k) ∈ ℜ for stable solutions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' b) ω or k ∈ C for an unstable solution that grows in time, or in space (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 17);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' c) ω, k ∈ C, for a mode growing in space and time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' A classification of instabilities was suggested by Capozzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2017) following the one for plasmas by Sturrock (1958) and Briggs (1964).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' According to this classification, the linear instabilities can be of four cate- gories: the completely stable or stable with damping cor- respond to spatially stable modes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' the absolute and con- vective grow either spatially or temporally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Note that most of flavor evolution studies (for slow or fast modes) were performed by evolving either space or time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The work was further deepened by Yi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2019) who pointed out the complex dispersion relation branches to be bounded by its critical points and their useful- ness in identifying fast modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Among the applications is the study by Martin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2020) in a dynamic one- dimensional model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The authors showed that fast modes evolve coherently in the non-linear regime in space and in time, when the corresponding ELN crossings undergo absolute or convective instabilities in the linear regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 4F cos(8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=') = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='9 Gi = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='5 cos(62) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='3 G2 = +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='5 2 Complex w for real kz E Complexk, for real w Frequency 0 4 2 0 2 4 Wave number k26 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Towards neutrino quantum kinetic equations The study of the interplay between collisions and fla- vor modes is numerically challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Thus its investi- gation is still at its premises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' First it requires a con- sistent theoretical framework where one goes from the dense collision-dominated regime to the dilute regions in which mean-field equations are sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Second, the di- mensionality remains high in astrophysical environments;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' whereas in the early universe isotropy and homogeneity reduces the dimensionality of the problem making it nu- merically tractable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Numerous authors derived neutrino quantum kinetic equations (QKEs) for the early universe (Blaschke and Cirigliano, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Froustey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' McKellar and Thomson, 1994;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rudzsky, 1990;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Sigl and Raffelt, 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Stodolsky, 1987;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Vlasenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2014a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Such QKEs are being used for the study of neutrino flavor evolution in dense astrophysical environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The general form of QKEs, including collision terms (proportional to G2 F) and mean-field contributions (linear in GF) read i(∂t + v · ∇)ϱp(t) = [h, ϱ] + C[ϱ, ¯ϱ] , (70) and similarly for ¯ϱ, with C the collision term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' While the mean-field Hamiltonian introduces coherence, the colli- sion term is responsible for production and absorption of neutrinos and of kinematical decoherence among neutri- nos with different momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rudzsky (1990) gave an early derivation of the Boltz- mann equation for relativistic distribution functions with mixings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Sigl and Raffelt (1993) derived quantum kinetic equations for a matrix of densities for the early universe, implementing antineutrinos for the first time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' They in- cluded (anti)neutrino scattering on neutrons, protons, electrons, positrons, ν and pair annihilation in a per- turbative approach using the assumption of molecular chaos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' In this approximation, the collision timescale is short enough that correlations do not develop between collisions: the incoming and outgoing particles in the col- lision integrals are free single-particle states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Vlasenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2014a) gave an alternative derivation of the flavored quantum Boltzmann equations for Majo- rana neutrinos using the Closed Time Path (or ”in-in”) formalism and the 2 Particle-Irreducible (2PI) effective action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Blaschke and Cirigliano (2016) extended their re- sults and obtained the full collision term for neutrinos for anisotropic media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Extending the work by Volpe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2013) for supernova neutrinos, Froustey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2020) rederived the neutrino quantum kinetic equations with the BBGKY hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' A separation of scales?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' In dense matter, an important length-scale is the neu- trino mean-free path, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' λ = (σρ)−1, with ρ the matter number density and σ the interaction cross section of a neutrino with a particle of the medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Close to matter saturation density, at ρ = 3 × 1014 g/cm3, and for a typ- ical cross section σ = 6 × 10−41 cm2, a 10 MeV neutrino has a mean free path of about a meter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' At densities of ρ = 1010 g/cm3, the idealized location where neutrinos start free streaming, λ is of tens of kilometers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The flavor length-scale is another important quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' For a long time, the MSW resonance(s) have provided the only flavor length-scale in flavor studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' As previ- ously discussed, considering for example typical super- nova matter profiles, the H-resonance is approximately located at ρ = 103 g/cm3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' whereas the L-resonance is at about ρ = 1 g/cm3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' If the MSW effect was the only flavor phenomenon, the mean-free path and the flavor length-scales would be well separated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' For many years, this argument has supported the use of two distinct theoretical treatments, one for the dense region, where the particles of the medium acts as random scatterer, and one for the dilute region, where neutrinos free stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' This translates, formally, in the use of rel- ativistic Boltzmann transport equations (Bruenn, 1985;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Cardall et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lindquist, 1966) for the former, and of mean-field equations for the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' It is common practice numerically to simplify the seven-dimensional transport neglecting, in particular, the mixing and the mean-field terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' On the other hand, flavor studies usually separate the two regions of ap- plicability treating the neutrinosphere as an idealized, sharp, surface;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' although the neutrino decoupling region61 is build up by collisions, and is energy and flavor depen- dent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Since the work of Duan, Fuller, Carlson and Qian (Duan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2006a), fifteen years of investigations have introduced new scales in the problem showing that the separation of the collision and of the flavor scales does not necessarily hold in presence of neutrino-neutrino in- teractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' This is particularly true in presence of fast modes, that take place very close to the neutrinosphere and have wavelengths shorter than the collision one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' In the supernova context, studies have started of the inclusion of collisions as well as the contributions from the mixings and the mean-field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' In particular, the influence of collisions on fast modes is being investigated (Capozzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Hansen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Martin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Shal- gar and Tamborra, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Capozzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2019) high- lighted collisions can trigger fast modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Martin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2021) studied, in the linear and non-linear regimes, a homogeneous gas model with mono-energetic neutrino undergoing direction changing ν-nucleon elastic scatter- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' They showed that collisions can suppress fast modes and the ELN distributions tend asymptotically to be- come isotropic62 as a function of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Hansen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2022) found that collisions can enhance or suppress fast 61 This is conventionally defined as the region where the opacity is 2/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 62 For an isotropic ELN, there are no fast modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 27 flavor conversions depending on the hypothesis that neu- trino emission is close to isotropic, or forward peaked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (Richers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2019) studied the influence of collisions on slow modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' It is interesting to note that steps towards a consis- tent solution of the full QKEs were also done in the early universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Indeed, for the off-diagonal contributions of the collision term, a damping approximations was extensively used by authors (see for example (Dolgov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Gava and Volpe, 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Mangano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Froustey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2020) and Bennett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2021) recently performed the first calculations with the full collision term, includ- ing the mixings and mean-field terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' With radiative corrections to the plasma equation of state, this has given a very precise value of the effective number of degrees of freedom Neff = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='0440 at the epoch of primordial nucle- osynthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Moreover Cirigliano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2017) investigated the role of anisotropies in a dense neutrino gas with two spatial dimensions, in presence of νν interactions and collisions, and showed instabilities are not necessarily suppressed by kinematical decoherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Hansen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2021) showed that ν-¯ν asymmetry can significantly grow due to the non-linearity of the evolution and influence Neff in pres- ence of small anisotropies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Much more work is needed to achieve definite under- standing of the impact of collisions on flavor evolution in dense astrophysical environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' A unified and consis- tent solution of the full neutrino QKEs clearly represents a longterm goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Neutrinos in presence of strong gravitational fields The theoretical description of neutrino propagation discussed so far are in flat spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' In core-collapse supernovae, accretion disks around black holes or com- pact binary mergers, there is a compact central object producing a strong gravitational field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Gravity modifies neutrino propagation and impacts flavor evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' To investigate its role, the neutrino equations of motion need to be extended to curved spacetime, as done by Cardall and Fuller (1997);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Chatelain and Volpe (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Deaton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Dvornikov (2013);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Dvornikov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2005a);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' and Piriz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' So far, the influence of gravity has received limited at- tention, although the first works exploring its role date back to the eighties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' It was Stodolsky (Stodolsky, 1979) who first considered the problem of finding the quan- tum mechanical phase acquired by a particle propagating along a classical trajectory in presence of gravitational fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' In order to discuss and compare matter and light interferometry, he introduced the quantum mechanical phase along a path, from the spacetime point A to the spacetime point B, Φ = � B A mds, (71) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 18 Decoherence in curved spacetime: drawing of neu- trino wave-packet propagation from a production point P to a ”detection” point D in presence of strong gravitational fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The widths correspond to the trajectories distributions due to the finite wave-packet width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Each wave-packet associ- ated with a mass eigenstate follows a trajectory close to null geodesics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (Figure from Chatelain and Volpe (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=') where m is the particle mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The quantity ds is the infinitesimal line element along the particle worldline ds2 = gµνdxµdxν, (72) with gµν the metric tensor and xµ a coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The covariant phase Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (71) can be rewritten as Φ = � B A pµdxµ, (73) pµ = mgµν dxν ds being the particle canonical momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Afterwards, the impact of gravitational fields on the vacuum oscillation phase was investigated by many au- thors63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' In particular Ahluwalia and Burgard (1996, 1998);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' and Bhattacharya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (1999), Cardall and Fuller (1997);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Chatelain and Volpe (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Fornengo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (1997);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' and Godunov and Pastukhov (2011) considered the case of a Schwatzschild metric of a static and spher- ically symmetric gravitational field, and Lambiase et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2005);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Mosquera Cuesta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Visinelli (2015);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' and Wudka (2001) focussed on the Kerr-Newman met- ric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Dvornikov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2005b) pointed out a new mecha- nism called ”spin light” that neutrinos emit in presence of gravitational fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' In core-collapse supernovae the role of trajectory bend- ing and energy redshift was studied by few authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' For example, Fuller and Qian (1996) argued that the differ- ence in the gravitational redshift between ¯νe and νe can increase the electron fraction and impact r-process nucle- osysnthesis above the nascent proto-neutron star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Yang and Kneller (2017) found that trajectory bending nearby a very compact source in a supernovae produces a neu- trino ”halo” similar to the one identified by Cherry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 63 It is to be noted that the discussions, in some of the early works, about the possibility to separate the two contributions to the oscillation phase, from the mixings and the gravitational field, are of academic interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 28 (2012) due to νν interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Caballero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2012) pro- vided the full nucleosynthetic outcomes of r-process ele- ments in black hole accretion disks models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Their results clearly show the importance of the inclusion of trajectory bending and neutrino energy redshift when determining element abundances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Decoherence is also an important aspect of neutrino flavor evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Indeed, in a wave-packet descrip- tion of neutrino propagation the wave-packets associ- ated with neutrino mass eigenstates can decohere sup- pressing flavor oscillations (see for example (Giunti and Kim, 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' In vacuum one quantifies decoherence by wave-packet separation through the ”coherence length”, Lcoh ≃ (4 √ 2E2/|∆m2|)σx for Gaussian wave-packets, with σx the intrinsic wave-packet dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Note that the Daya-Bay experiment investigated the effects of a wave-packet description of vacuum oscillations and set the first limit on its width, finding it not significant (An et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' For supernova neutrinos, since neutrinos travel over large distances, decoherence effects by wave-packet sep- aration can be sizable as discussed e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' in Kersten and Smirnov (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Akhmedov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2017) investi- gated such effects in the density matrix formalism and showed that, in vacuum, they induce a damping of the off-diagonal terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Moreover they studied decoherence effects in presence of dense matter and neutrino back- grounds for the cases of adiabatic and non-adiabatic evo- lution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Extending the formalism to curved-spacetime, Chate- lain and Volpe (2017) investigated the impact of wave- packet decoherence in a Schwarzschild metric (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' They pointed out that, in curved space-time, instead of the coherence length a coherence proper time τcoh quan- tifies decoherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' This is defined as the time at which the difference between the proper times at a ”detection” point D satisfies τ = σt � B(rD)64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Neglecting matter and νν interactions, decoherence was found to produce modifications of the proper time by several tens of a per- cent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Connections: from atomic nuclei to quantum devices Several authors have unravelled exciting connections between a weakly interacting neutrino gas and other many body systems (Mirizzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Pehlivan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Vaananen and Volpe, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Volpe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2013), or investigated the role of many-body correlations and of entanglement (Amitrano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Bell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2003, 2002a,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Birol et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Cervia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Fried- land and Lunardini, 2003a,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lacroix et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Mar- tin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2023,?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Patwardhan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Pehli- 64 Here B(r) = 1 − rs/r and rs = 2M are the Schwarzschild radius and M the mass of the central object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 19 Probability that each neutrino is found in the 11 state, [Pν]11 when µ ≤ ω0 for a number of neutrinos N = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Shown are the initial configuration (a νe at ωi with i = 1, 8 and νx at ωi with i = 8, 16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The many-body results (purple) are compared with the mean-field ones (green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The entan- glement entropy, that encodes information on the deviation from the mean-field limit, peaks at the spectral split frequen- cies ω/ω0 = 2 and 7 (ω0 is the vacuum oscillation frequency).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (Figure adapted from (Patwardhan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2021)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' van et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Roggero, 2021a,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Roggero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Besides being interesting from the theoretical point of view, these studies brought novel ways to approach the problem of neutrino propagation in dense environments and opened new numerical treatments, in particular us- ing quantum devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Pehlivan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2011) showed that the neutrino Hamil- tonian in dense environments Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (50) (without matter and for constant µ) is formally equivalent to the reduced Bardeen-Cooper-Schieffer (BCS) Hamiltonian in the the- ory of superconductivity (Bardeen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 1957) HBCS = � k ϵktz k + GT +T − , (74) with the quasi-spin operators t+ k = c† k,↓c† k,↑ t− k = ck,↑ck,↓ tz k = (c† k,↓ck,↓ −c† k,↑ck,↑ −1) , (75) which describe Cooper pairs of of valence electrons in a lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' This result highlighted that the neutrino Hamiltonian is exactly solvable since, as pointed out by Richardson (1966), the BCS Hamiltonian has analytical solutions thanks to the algebraic Bethe ansatz method;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' whereas Gaudin (1976) showed the exact solvability of the model because of the number of quantum invariants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The work of Pehlivan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2011) was further elaborated in Pehlivan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2014) in presence of neutrino magnetic moments coupling to magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' These developments have brought the first compar- isons of mean-field and exact results (for a small num- ber of particles), showing in some cases significant differ- ences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Cervia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2019) and Patwardhan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2021) have employed concepts from quantum information the- ory, in particular entanglement entropy, to quantify the Menn-field Inny-body log(2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='2 7 8 10 12 14 16 m/m29 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 20 Analogy between non-linear fluids and supernova neutrinos with νν interactions: as the translation symme- try is broken in a two-dimensional model, the streamlines of the νe flux along the vertical direction become irregular, showing large variations and converging towards preferred di- rections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' This behavior is analogous to the transition from laminar to turbulent regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (Figure adapted from (Mirizzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2015)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' entanglement between neutrino states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The entanglement entropy of a neutrino with frequency ω with the rest of the neutrinos is defined as S(ω) = −Tr[ϱ(red) ω log(ϱ(red) ω )] = − � s=± λs,ωlog(λs,ω) (76) with the reduced density matrix obtained by tracing over all other neutrinos ϱ(red) ω = Trω′̸=ωϱ (77) and the eigenvalues given by65 λ±,ω = 1 2(1 ± |Pω|) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (78) with P the polarization vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' If the neutrino mode is maximally entangled with its environment, then |Pω| = 0, and the entanglement entropy S(ω) = log(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' In the mean-field approximation, the many-body wave function is the factorized product of single-particle wave functions, giving |Pω| = 1 and S(ω) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Therefore the entan- glement entropy provides information on the deviations from the mean-field limit due to many-body effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Figure 19 shows the entanglement entropy for a sys- tem of the order of 10 particles, the results showing it is the highest for the neutrinos whose energies are the closest to the spectral split (see section II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Roggero (2021a,b) and Martin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2022) and Roggero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2022) performed further investigations of the entangle- ment entropy for neutrino systems, following the real- time dynamics of systems of larger size, up to 102 and 103 particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 65 We remind that ϱ(red) ω = 1 2 (1 + σ · Pω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Furthermore, as previously mentioned, Volpe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2013) established a formal connection between neutri- nos propagating in dense media and atomic nuclei, or metallic clusters, through the BBGKY hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' More- over it is to be noted that the pairing correlators Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (51) are formally analogous to pairing correlations in the BCS theory for superconductivity in condensed matter or pairing in nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' An analogy with fluids was pointed out by Mirizzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2015) in a two-dimensional model of supernova neutri- nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The authors discussed that the instability produced by νν interactions, breaking the spatial symmetry, has a nice analogy with nonlinear fluid instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' In par- ticular, the transition from a coherent to an incoher- ent regime in flavor behaves like a streaming flow that changes from laminar to turbulent regime (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' New developments also concern numerical methods that are at variance with forward integration techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' An example are the recent efforts to employ an inference procedure, as in the statistical data assimilation explored by Armstrong (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Armstrong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' and Rra- paj et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2021), that looks for the optimization of a cost function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The method does not require knowledge of the initial conditions, but rather constraints at some locations of the coordinate axis (not necessarily at the bounds) that parametrize the model equations of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Finally, recent studies are opening the exciting pos- sibility to investigate correlations and entanglement of strongly correlated neutrinos on quantum computers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Hall et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2021) studied the evolution, the entanglement entropy (of a pair) and concurrence of a 4 particle neu- trino system using, for the first time, a quantum device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Amitrano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2023) considered trapped-ion qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Having highlighted aspects of our understanding of neutrino flavor evolution at dense, we now turn to ob- servations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' PAST AND FUTURE OBSERVATIONS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' SN1987A On the 23rd of February 1987 Sk-69◦202 exploded in the Large Magellanic Cloud, a satellite galaxy of the Milky Way, producing SN1987A (Arnett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 1989;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Bethe, 1990;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Raffelt, 1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' It was the first naked-eye since Kepler’s supernova (Ia) in 1604.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The supernova was at 50 ± 5 kpc from the Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Measurements based on the expanding photosphere method agreed within 10 % (Schmidt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' SN1987A was unique in many respects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Thirty years after, there are finally indications for a compact object, likely a neutron star, at its location (Alp et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Cigan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Page et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The earlier SN1054, seen by Chinese astronomers, left a pulsar in the Crab nebula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' SN1987A progenitor, the first known, was a blue supergiant, whereas supernovae progenitors were thought to be red supergiants (the blue 12 10 10 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='8 [2元/k] x30 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 21 SN1987A events at Kamiokande, IMB and Baksan: energies correspond to secondary positrons produced in in- verse β-decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Events have been shifted at same t=0 (clock relative offsets are unknown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (Figure from Fiorillo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=') problem) (Arnett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 1989;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Bethe, 1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The inner ring, large mixing and asymmetrical ejecta of SN1987A (Arnett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 1989;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Janka, 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Podsiadlowski, 1992) indicated strong asphericity in the explosion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' SN1987A, the closest supernova in the last hundreds of years, was observed in all wavelengths from gamma rays to radio, and for the first time, neutrinos from the collapse of the stellar core were detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Suzuki (2008), at its 20th anniversary, gave a lively description of this pioneering observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The water C¸erenkov detector KII (Hirata et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 1987) observed a neutrino burst66 of 11 events of energy 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='5 to 36 MeV in 13 s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' while IMB (Bionta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 1987) measured 8 events in 6 s with 20 MeV to 40 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' BST (Alekseev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 1988) detected a burst of 5 events in 9 s and, about 5 h before, the Mont Blanc Liquid Scintillator Detector (LSD) (Aglietta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 1987) recorded 5 events during 7 s with energy ≥ 7 MeV (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Since no signal has been found correspondingly in the other detectors, the LSD events remain controversial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 66 The probability that the observed burst is a random fluctuation over a constant background is of about 6 × 10−7 (Raffelt, 1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Baade (1934) first suggested that in supernovae the tremendous energy comes from the gravitational collapse of the inner core into a neutron star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Hoyle and Fowler (1960) proposed stars could dye due to thermonuclear runaway (SN Ia) of degenerate material or implosion of the stellar core (SN II and Ib/c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Colgate and Johnson (1960) pointed out that the collapse could be followed by core bounce and shock formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The shock would ex- pel most of the star mass by propagating into the man- tle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' To support the prompt-shock model, Colgate and White (1966) hypothesized that most of the gravitational binding energy of the imploding core, Eb ∼ GM 2 NS/RNS, namely [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='5, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='5]×10−53 erg, would be emitted with neu- trinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' A few percent of this energy deposited by neutri- nos back into matter energy, could drive the supernova explosion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Bethe and Wilson (1985) shaped it into the delayed neutrino-heating mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Is the delayed neutrino-heating mechanism that drives the explosion of most (type II and Ib/c) supernovae?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Early one-dimensional67 calculations faced shock stag- nation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' SN1987A observations of asymmetries and the presence of strong hydrodynamic mixing processes dur- ing the explosion gave momentum to the development of multi-dimensional simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Nowadays, although some explosions are overemphasized by the imposed sym- metry, two-dimensional simulations of different progeni- tors successfully expel the supernova mantle, aided by re- alistic neutrino transport, convection, turbulence and hy- drodynamic instabilities in particular the SASI (Blondin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2003) and LESA (Tamborra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2014a)(see the reviews (Burrows, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Foglizzo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Janka, 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Mezzacappa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Radice et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2018a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Taki- waki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2021)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Weaker than two-dimensional, three- dimensional calculations show indications for successful explosions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The answer to this longstanding unresolved question seem to lie in a foreseeable future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The observation of the 24 events from SN1987A is im- portant for fundamental physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The neutrino signal gave the start of the explosion, the energy released in the gravitational collapse, the temperature at the neu- trinosphere, information on the explosion mechanism on the one hand and limits on neutrino properties and non- standard physics on the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Assuming energy equipartition among the neutrino species, analysis show the total energy associated with SN987A events to be 5 × 10−52 erg at best-fit value (Vissani, 2015) and (Loredo and Lamb, 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Pagliaroli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2009b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Sato and Suzuki, 1987), confirming Col- gate and White’s hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The spectra agree reason- ably well with thermal expectations with ¯νe tempera- tures T = 4 MeV (Arnett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 1989;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Loredo and Lamb, 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Pagliaroli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2009b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Schramm and Truran, 1990;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 67 A comparison of one-dimensional simulations show an encourag- ing agreement among groups, in spite of the differences in the numerical methods and approximations (O’Connor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 50 Kamiokande 10 n 50 40 IMB ++ 50 18 Baksan 20 0 2 4 6 8 10 Time after first event []31 Vissani, 2015), the neutrino time signal confirms the ex- pected supernova pulse duration and neutron star cool- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Therefore, SN1987A events confirmed the global picture68 of the neutrino emission during a gravitational core-collapse supernova explosion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Moreover, the Bayesian analysis (Loredo and Lamb, 2002) and the one of Pagliaroli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2009b) of the time signal supported the delayed-shock over the favored prompt-shock model by unambiguously showing the pres- ence of an accretion phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' From nuclear matter equa- tions of state, the formation of a neutron star (instead of a black hole) was favored, with mass (Sato and Suzuki, 1987) and radius (Loredo and Lamb, 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Pagliaroli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2009b) compatible with expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The radius of the neutrino emitting surface was found to be 18 km (Raffelt, 1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Analyses of the SN1987A events have produced a wealth of information on non-standard neutrino prop- erties, interactions or particles (Raffelt, 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Tanabashi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' For example, neutrinos flew through space for 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='6 ×105 years, yielding a bound on the neutrino life- time of τνe/m > 5×105 s/eV (rest frame) (Bethe, 1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The non-observation of a γ-ray signal over background, in correspondence with the neutrino time signal, gave stringent bounds on the neutrino lifetime from radiative decays (Mohapatra and Pal, 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Payez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Raffelt, 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Tanabashi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Optical brighten- ing followed neutrino emission by a few hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Neutrino propagation through space, at nearly the same speed as photons, gave a tight constraint on the neutrino speed cν, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' |(c − cν)/c| < 2 × 10−9 (Longo, 1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Moreover, the absence of a dispersion of the neutrino pulse gave upper limits on the neutrino charge and on the νe mass (about 20 eV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Energy-loss arguments on the shortening of ν time signal, associated with the neutron star cooling, gave limits on axions (Payez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2015), on right-handed neutrinos or currents, and on the neutrino magnetic mo- ment (Barbieri and Mohapatra, 1988).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Concerning flavor modification, Jegerlehner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (1996) studied the sensi- tivity of SN1987A events to the MSW effect and found that the ¯νe getting hotter they would be marginally com- patible with observations69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Using modern supernova fluxes, likelihood analysis by Vissani (2015) show that the MSW impact on the two dozen events is small and comparable to variations due to other parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Clearly, the historical SN1987A is representative of how much, patience for such rare events, can be rewarded in knowledge and progress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 68 Note that the KII and IMB events showed a forward-peaked angular distribution, instead of isotropic, which is likely to be due to a statistical fluctuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 69 Indeed the ⟨E¯νe⟩ (and ⟨E¯νµ,τ ⟩ with MSW) were marginally com- patible with SN1987A events because of incomplete microphysics in the supernova simulations of that epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' From the next supernova Neutrinos emitted in the first instants of the core- collapse will be detected several hours before optical emission and guide optical instruments (Abe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' If the supernova is nearby, pre-supernova neutrinos from thermal (Odrzywolek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2004) and weak processes (Patton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2017) in the late stages of the stellar evo- lution could be observed preceding core-collapse and give advanced warning (Yoshida et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2016) as well as in- formation on the supernova progenitor (see the review by Kato et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2020)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' SK could detect about 200 pre- supernova neutrinos 12 hours before collapse of a 15-25 M star at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='2 kpc (such as Betelgeuse);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' whereas SK+Gd and KamLAND could reach 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='6 kpc (Abe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2016) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='69 kpc (Asakura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2016) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' More- over the measurement of the full neutrino lightcurve, up to 100 s, if the supernova is close enough would yield interesting information on the late cooling phases of the proto-neutron star formed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The observation of the next supernova will benefit of the SNEWS network (Al Kharusi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' From the complementarity of the technologies available, we shall measure the time and energy of neutrino flavors through inverse β-decay, neutrino-nucleus scattering, neutral cur- rent scattering on electrons as well as on protons (Beacom et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' For inverse β-decay the cross sections are precisely known (Ricciardi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Strumia and Vissani, 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' On the contrary, the cross sections associated with charged-current ν-nucleus interactions are still affected by theoretical uncertainties, with the exception of heavy water whose cross sections are known with a few percent precision (Balantekin and Yuksel, 2003a) and 12C (Hayes and Towner, 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Volpe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Volpe (2004, 2007) suggested to use a novel technique, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' the low energy beta-beam, to perform measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' These will finally be performed at the Spallation Neutron Source (Avi- gnone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Barbeau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Besides pro- viding a better knowledge of the spin and spin-isospin weak nuclear response to neutrinos, in the energy range of interest for the detection of supernova neutrinos, such measurements could shed further light on the issue of the quenching of the axial-vector coupling constant, also for forbidden states, as pointed out by Volpe (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The observation of the supernova time signal will be rich of fundamental lessons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The νe flux from the 50 ms neutronization-burst represent less than 1% of the to- tal neutrino luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The accretion phase that lasts about 500 ms and the neutron star cooling about 10 sec- onds take away most of the gravitational binding energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' If close enough, the neutrino time signal from a future su- pernova will confirm the delayed-neutrino heating mecha- nism which is the current paradigm for most of supernova explosions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' If the supernova is close enough, the precise measure- ment of the time signal will be crucial to definitely assess 32 the explosion mechanism through the identification of os- cillations with high frequencies, correlated with SASI, whose measurement requires a very precise time resolu- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' In this respect, modifications from flavor evolution should not swamp the signature (see for example (M¨uller and Janka, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Tamborra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2014b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Walk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2020)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The measurement of early (< 20 ms) stages of neu- trino emission would give information on the bounce time (Halzen and Raffelt, 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Note that this is key to esti- mate the burst time of the gravitational waves (Pagliaroli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2009a) which are mainly produced by the oscilla- tions of the newly formed proto-neutron star (Abdika- malov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The concomitant multimessenger event of neutrino and gravitational waves from a core- collapse supernova was discussed also for example by Halim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' From the point of view of flavor evolution, the neutronization-burst represents a unique phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Only the MSW effect appears to influence the neutrino spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Neither fast nor slow modes are at work, as we under- stand them now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The former requires crossings in the neutrinos and antineutrinos angular distributions, the latter ν¯ν pairs (in the bulb model70).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Moreover there are neither effects from shock waves, since shock waves reach the MSW region after 1-2 s, nor from turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Therefore, the neutronization-burst appears to be a good laboratory to explore non-standard properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' These include for example non-standard νν interactions (Das et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2017) or neutrino non-radiative decay (Ando, 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' de Gouvˆea et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Since flavor mechanisms produce neutrino spectral modifications (see Section II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='B), an important question to ask is with which precision we will be able to recon- struct the supernova neutrino fluxes when the next su- pernova blows off (see for example (Gallo Rosso, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lujan-Peschard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Vaananen and Volpe, 2011)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The answer obviously depends on which observatories will be operating at that time and on the supernova distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' It is interesting to note that a precise determination and reconstruction of the supernova neutrino spectra might not be trivial in likelihood analysis where the en- semble of the parameters are left free to vary, even in the simplest case with the MSW effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Indeed Minakata et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2008) pointed out the presence of parameter de- generacies that can in principle be broken by combining detection channels (Gallo Rosso et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' However, while most of the neutrino parameters appear to be pre- cisely measurable (for a supernova at 10 kpc) identifying the neutrino pinching for some of the flavors might be more tricky (Gallo Rosso et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 70 In more complex model as well, the νν interaction did not appear to influence this early phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' As for unknown neutrino properties, the neutrino sig- nal from the next supernova could be a good laboratory to determine the neutrino mass ordering for which there are currently hints with a low statistical significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The passage of the shock wave can be pictured as it goes through the MSW region (see section II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Shock waves effects can be important and produce distortions of the positron (or electron) time signals, depending on the neu- trino energy and mass ordering (see for example (Fogli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2003, 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Kneller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lunardini and Smirnov, 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Takahashi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2003) and the reviews by Duan and Kneller (2009) and Horiuchi and Kneller (2018)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Also, the rise time of the neutronization-burst can be used to determine the neutrino mass ordering in a detector like IceCube (Serpico et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Although these signatures are interesting, it is likely that Earth- based experiments like JUNO, DUNE or Hyper-K will measure the neutrino mass ordering before the next su- pernova blows off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' In particular the latter should measure it at about 3 σ after 6 (An et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2016) or 10 years (Abe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2011) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' As for CP violation in the lepton sector, hints for sin δ < 0 (90 % CL) indicates that the CP violating phase should be discovered soon from the DUNE and Hyper-K experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The effects of the Dirac CP vio- lating phase was studied in the context of core-collapse supernovae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Akhmedov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2002) concluded that there should be no impact of the Dirac phase on the νe fluxes in a supernova, even if the νµ and ντ fluxes are unequal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' In contrast with such findings Balantekin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2008) demonstrated that the Dirac phase can impact the elec- tron neutrino fluxes if the muon and tau neutrino fluxes differ, because of e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' radiative corrections or of non- standard interactions, such as flavor-changing neutral currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The result relies on a factorization condition of the neutrino Hamiltonian, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' H(δ) = S†H(δ = 0)S with S†(δ) = diag(1, 1, eiδ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' These findings were generalized in presence of νν inter- actions by Gava and Volpe (2008), beyond the mean-field to the full many-body problem (Pehlivan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2014) and on the neutrino degeneracy parameter in the early uni- verse (Gava and Volpe, 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Numerical calculations showed the impact of the phase to be small (Balantekin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' However, the combined effect of the Ma- jorana CP violating phase(s) and the neutrino magnetic moment could trigger sizable effects, opening the possi- bility for new resonances, as pointed out by (Popov and Studenikin, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Obviously, even when the mass ordering and CP vi- olation will be precisely measured, supernova neutrinos will remain interesting probes for non-standard physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Indeed there are numerous flavor mechanisms related to other key unknown neutrino properties71 that have been 71 These are not the main focus of this review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 33 extensively discussed in the literature, such as sterile neu- trinos, non-standard interactions or the neutrino mag- netic moment (see for example Giunti and Studenikin (2015);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Nunokawa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (1997a);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Pehlivan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2014);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' and Sasaki and Takiwaki (2021)) which can give sizable modification of the neutrino spectra in presence of strong magnetic fields, as in core-collapse supernovae or nearby compact objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The discovery of the diffuse supernova neutrino background Complementary to the observations from one super- nova is the DSNB, made of neutrinos emitted by past core-collapse supernovae, and which is nearly isotropic and constant in time (see the reviews by Ando and Sato (2004);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Beacom (2010);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lunardini (2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' and Mathews et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2020)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The DSNB depends on cosmological, as- trophysical and particle physics aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The DSNB flux, including a progenitor dependence, reads φνα(Eν) = c � � dM dz ���dtc dz ��� RSN(z, M) φνα(E′ ν, M) , (79) where z ∈ [0, zmax] is the cosmological redshift, c is the speed of light, E′ ν is the neutrino energy at the star location at redshift z, related to the energy Eν on Earth through E′ ν = Eν(1 + z), φνα(E′ ν, M) is the time- integrated neutrino flux (fluence) for a progenitor of mass M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (79) usually z ∈ [0, 5] and M ∈ [8, 125] M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' However only the lowest redshifts, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' z ∈ [0, 2] give the most important contribution to the DSNB flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' More- over considering 100 M⊙ instead of 125 M⊙ does not in- troduce any significant difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The progenitor mass dependence of the DSNB flux was pointed out by Lunardini and Tamborra (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' It is to be noted that the most general expression for the DSNB flux should also have an explicit dependence on the galaxy metallicity as considered for example by Nakazato et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The first factor in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (79) is the cosmological time that depends on the cosmological model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Usually the ΛCDM model is assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The expansion history of the universe is then ��� dz dtc ��� = H0(1 + z) � ΩΛ + (1 + z)3Ωm , (80) with Ωm and ΩΛ the matter and the dark energy cosmic energy densities, H0 = 70 km s−1 Mpc−1 is the Hub- ble constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' DSNB predictions show that he DSNB is not sensitive to variations compatible with the Hubble tension72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Barranco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2018) investigated the influ- ence of cosmological models other than the ΛCDM on the DSNB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 72 There is currently a tension between the Hubble constant value The second important input in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (79) is the evolving core-collapse supernova rate73 RSN(z, M) that is related to the star-formation rate history ˙ρ∗(z) according to RSN(z, M) = ˙ρ∗(z) φ(M)dM � 125 M⊙ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='5 M⊙ φ(M)MdM , (81) where φ(M) is the initial mass function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' With his seminal work Salpeter (1955) introduced the power law initial mass function φ(M) ∼ Mχ , (82) for M ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='5, 1] M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The quantity φ(M)d(M) gives the number of stars in the mass interval [M, M + dM].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Since, the Salpeter IMF has been employed, χ being determined with an uncertainty of about 10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Baldry and Glaze- brook (2003) introduced a modified broken power law for the IMF with χ = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='5 at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='1 M⊙ ≤ M ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='5 M⊙ and χ = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='12 for M > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='5 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Note that such a mod- ified IMF gives a similar result for RSN(z, M) (Horiuchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The universality of the IMF at high masses can be questioned, with respect to the local environment and the cosmic time, as discussed for example by Ziegler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The cosmic star-formation history can be deduced from observations (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (Hopkins and Beacom, 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Reddy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rujopakarn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2010)) and is de- scribed by a piecewise continuous form of a broken power law (Yuksel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2008) (see also (Madau and Dickinson, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Singh and Rentala, 2021)) ˙ρ∗(z) = ˙ρ0 � (1 + z)αη + �1 + z B �βη + �1 + z C �γη�−1/η , (83) with α = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='4, β = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='3, γ = −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='5 the logarithmic slopes at low, intermediate and high redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The quantity η = −10 is the smoothing function and the constants defining the redshift breaks are B = 5000, C = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Currently, the local core-collapse supernova rate is known with the following precision RSN(0) = � 125 M⊙ 8 M⊙ RSN(0, M)dM = (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='25 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='5) × 10−4yr−1Mpc−3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (84) It constitutes one of the largest uncertainties in the DSNB predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Indeed there is a disagreement by a factor 2 at 0 ≤ z ≤ 1 between the core-collapse super- nova rate deduced from the star-formation rate history extracted with the ”distance ladder method”, H0 = 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='03 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='42 km s−1Mpc−1, and the one obtained from the Cosmo- logical Microwave Background (CMB), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' H0 = 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='5 km s−1Mpc−1 (Di Valentino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 73 Number per unit time per unit comoving volume 34 and the one from direct core-collapse supernova obser- vations (Horiuchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2011)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' This is known as the ”supernova rate problem”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Several parametrization of the cosmic star-formation rate history are available in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The one given by equation (84) from Horiuchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2009) and Yuksel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2008) includes GRB data z > 4 and is commonly employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The parametrization in (Fogli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2004) is outdated, whereas the one in (Priya and Lunardini, 2017) present kinks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (Mathews et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2014) suggested an alternative parametrization by including only the subset of the star-formation rate data corrected for extinction by dust74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The third and last important factor is the neutrino flux from one single-supernova with progenitor mass M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lu- nardini (Lunardini, 2009) pointed out the relic supernova background can receive a significant contribution from failed supernovae (directly collapsing into a black hole).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Indeed, due to the compression of baryonic matter during black hole formation, the supernova generates large neu- trino fluxes with higher average energies and larger differ- ences among flavors than optical supernovae, depending on the (soft or stiff) equation of state, as pointed out by Sumiyoshi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Note that Schilbach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2019) investigated the DSNB only coming form black hole accretion disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Although the fraction of supernovae that turn into a black hole is sub-leading, this contribution influence the tail of the DSNB spectrum and contributes substantially to the DSNB rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' If one includes the neutrino spectra from core-collapse supernovae that leave a neutron star or a black hole, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (79) becomes φνα(Eν) =c � dz (1 + z) ���dtc dz ��� × �� Ω dM RSN(z, M) φNS να (E′ ν, M) + � Σ dM RSN(z, M) φBH να (E′ ν, M) � , where Ω and Σ correspond to the range of masses for which the collapse gives a NS or a BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Thus the BH fraction is given by fBH = � Σ dMφ(M) � 125M⊙ 8M⊙ dMφ(M) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (85) The fraction of failed supernovae is currently debated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' It constitutes another important factor of uncertainty in the DSNB predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' It has been argued by Hori- uchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2014, 2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' O’Connor and Ott (2011);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' and 74 Moreover the authors argued that the ”supernova rate problem” could be solved by the inclusion of contributions from binaries, from failed supernovae and from electron-capture ONeMg super- novae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Ugliano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2016) that the star compactness could be a good indicator of the fraction of supernovae leaving black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' This is in contrast with Ertl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2016) who suggested as indicators two parameters, M4 and µ4, giving the enclosed mass and its derivative (s = 4, dimen- sionless entropy per nucleon), to better predict successful explosions in the neutrino driven wind mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Predictions of the DSNB flux and rates have different levels of sophistication either with respect to astrophys- ical inputs, neutrino flavor mechanism, neutrino proper- ties and new physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Concerning the astrophysical de- pendence, Ivanez-Ballesteros and Volpe (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Moeller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' and Tabrizi and Horiuchi (2021) included a progenitor mass dependence and the fraction of failed supernovae based on one-dimensional supernova simula- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Note that according to the detailed simulations by Kresse et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2021) the BH fraction75 ranges from 17 % to 41 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Horiuchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2021, 2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Kresse et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' and Moeller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2018) performed extensive su- pernova simulations to include a detailed progenitor mass dependence, and also the contribution from binary sys- tems (which is very uncertain).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' It is to be noted that Fukugita and Kawasaki (2003);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lunardini (2006);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Vissani and Pagliaroli (2011);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' and Yuksel and Beacom (2007) furnished DSNB predictions based on SN1987A observa- tions for the relic neutrino spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Apart from the progenitor dependence, predictions on the DSNB neutrino spectra and rates are influenced by flavor conversion mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The established MSW ef- fect is routinely implemented in predictions (Ando and Sato, 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Chakraborty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2011a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' De Gouvea et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Ekanger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Galais et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Horiuchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Ivanez-Ballesteros and Volpe, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Kresse et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Moeller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Priya and Lunardini, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Tabrizi and Horiuchi, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' So far only a few stud- ies implemented flavor effects beyond the MSW mecha- nism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Galais et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2010) investigated shock waves and νν interactions effects in the bulb model and found vari- ations up to 10%-20% due to the shock waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Nakazato (2013) found that the DSNB rates also depend on the shock wave revival time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Interestingly, the DSNB is also sensitive to non- standard physics (Farzan and Palomares-Ruiz, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Goldberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' de Gouvˆea et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Reno et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' In particular the DSNB is sensitive to non-radiative two-body decay in the window τ/m ∈ [109, 1011) s/eV (see Ando (2003);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' De Gouvea et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Fogli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2004);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Ivanez-Ballesteros and Volpe (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' and Tabrizi and Horiuchi (2021)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' This window is unique compared to terrestrial experiments, astrophys- ical sources like the Sun or a supernova, and cosmological probes such as BBN or the CMB (for the latter see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', 2022)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 75 Priya and Lunardini (2017) also considered a more conservative value of fBH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 35 Clearly, if the current hint is borne out, the DSNB will become a unique laboratory for astrophysics, particle physics and for the search of new physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' CONCLUSIONS AND PERSPECTIVES In our journey from the beginnings of neutrino physics and astronomy we went through some of the discover- ies that paved our knowledge of these elusive particles and opened new horizons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' After the breakthroughs of the evidence for neutrino oscillations and of the solution of the solar neutrino problem, experiments and theory put milestones in our understanding of neutrino masses and mixings, of neutrinos from stellar and cosmological environments and set important limits on new physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Still, neutrino physics and astrophysics remain nowadays a very active domain of research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Among the most challenging unsolved issues is the evo- lution and flavor modification of neutrinos from dense compact objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' What makes this problem so intrigu- ing and challenging is that, besides shock waves and turbulence inherent to exploding environments, nearby compact objects one has sizable neutrino-neutrino inter- actions that render neutrino flavor evolution a complex non-linear many-body problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Efforts to solve it are motivated not only by theoretical interest but also, obvi- ously, by observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' So far, investigations of shock wave effects have mostly used parametric matter density profiles of one- dimensional supernova simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Dips, or bumps, are characteristic features of the neutrino time signals due to the shock wave passage in MSW regions and, in partic- ular, in the H-resonance one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The identification of such structures offers a mean to identify the neutrino mass or- dering;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' normal if the shock wave passage is ”seen” in the νe time signals, inverted, if ”seen” in the ¯νe one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Since experiments like JUNO, DUNE or Hyper-K are likely to unambiguously measure the neutrino mass ordering be- fore next supernova, the imprint of the shock waves in the time signals will give a picture of the explosion dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' It is useful to keep in mind that multi-dimensional supernova simulations present strong anisotropies which can produce large angular variations of the front and the reverse shocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Moreover, down-flows colliding with hot matter that expands due to convection can induce multi- ple shocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' As a consequence, the exact structures might be direction dependent and possibly evolve chaotically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Therefore, further investigations are necessary to assess if the generic features of the shock wave passage, iden- tified in one-dimensional studies, remain, when one im- plements information from multi-dimensional supernova simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Turbulence also contributes to the not-yet understood core-collapse supernova explosion mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' It is an- other important aspect that impacts flavor evolution since it introduces matter density fluctuations which might produce neutrino depolarization, as early pointed out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Their characteristics – amplitude, scale and power spectrum – should be extracted from multi-dimensional supernova simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Since this is a difficult numerical task, most of the available studies have used parametric matter profiles where fluctuations are superimposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' So far, only one investigation has exploited information from a two-dimensional supernova simulation, finding weak in- dications that depolarization takes place, in contrast with all previous findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Clearly, new studies are called for, with inputs from two- and three-dimensional simulations, to establish if neutrino probabilities do have a loss of memory effect, or not, due to turbulence, in an explod- ing supernova.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Interestingly, studies have uncovered that, besides the established MSW effect present in compact objects, mul- tiple MSW resonances are produced by shock waves or turbulence in supernovae, or more generally because of non-linear feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' In particular the MSW-like phenomena that were pointed out include the matter- neutrino resonances, the resonance due to helicity coher- ence, or the I- and synchronized I-resonances triggered by non-standard neutrino-matter interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Another feature that impacts the neutrino flavor in dense media are neutrino-neutrino interactions which were first studied in the nineties in the context of the early universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Their investigation in core-collapse su- pernovae and compact binary mergers has triggered an intense theoretical activity in the last fifteen years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' In fact, novel unexpected flavor phenomena, located much deeper than the MSW region, have attracted enthusiasm because of the potential impact on the supernova explo- sion mechanism and on nucleosynthesis, besides the one on future observations of supernova neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Neutrino evolution in presence of neutrino-neutrino in- teractions is still an unsolved problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' From the studies performed so far we have learnt that, relaxing an approx- imation, or going beyond approaches, unforeseen aspects emerge that can overturn how we represent the picture of neutrino flavor evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The first, widely investigated, bulb model revealed collective slow modes that are trig- gered by mixings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' With frequencies √µω at typical dis- tances of O(102-103) km from the neutrinosphere, such modes occur in regions where they cannot induce extra heating to help explosions, whereas they can influence the r-process, as shown by numerous studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Moreover the interplay of νν interactions with other contributions, such as the standard and non-standard neutrino-matter interactions, opens the way to new MSW-like phenomena (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' the matter-neutrino and the I-resonances).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' With time we have learnt that the in- clusion of new degrees of freedom, as in non-stationary models, or in models with two-dimensional spatial de- grees of freedom like the line model, opens up new re- gions for flavor instabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' There are also situations where small initial perturbations, that do not pertain the same symmetries as the initial neutrino emission, give 36 solutions that spontaneously break symmetries (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' the azymuthal one).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' And in some cases, even chaotic flavor evolution can emerge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' If, instead of only forward-scattering neutrinos, one includes a small amount of back-scattered neutrinos or a better description of the ν angular emission, then the whole picture can be overturned as it came unexpectedly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The first option cast doubts on the treatment of neutrino evolution as an initial value problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' For the second, crossings of the νe and ¯νe angular distributions turned out to trigger short scale flavor modes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' O(1) m or much less, very close to the neutrinosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' These fast modes are currently very actively investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' It is now established that they occur in two- and three- dimensional supernova simulations, nearby the neutri- nosphere and even inside the proton-neutron star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' If the neutrino spectra are similar, at the fast mode location, as it appears, their influence on the spectra is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' So, far, only a couple of studies have evolved fast modes to the full non-linear regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' There are indications that fast modes can influence the r-process in binary neutron star mergers and the νp-process in core-collapse super- novae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Also three-flavor effects were shown to be impor- tant to determine when flavor evolution is modified on large scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The study of fast modes, the conditions for their occurrence and impact is at present a fast develop- ing field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Their understanding will require more work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' With a few exceptions, all the findings concerning fla- vor evolution in dense environments available in the lit- erature use the mean-field approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Linearized mean-field equations and a dispersion relation approach for fast modes are commonly used to study when neutrino flavor modification is triggered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' This has the advantage of solving an eigenvalue equation close to the initial quasi- static condition but looses the long-term evolution of the full non-linear problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' A special effort was devoted to check the validity of the mean-field equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' This lead to new evolution equa- tions, to the re-derivation of quantum kinetic equations and to the first attempts to solve kinetic equations with the inclusion of mixings in schematic models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Neutrinos in the early universe, where neutrino kinetic equations are needed, represent a different case in many respects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The homogeneity and isotropy of the medium made possible the first consistent calculations of neutrino evolution with the full collision term, the mixings and mean-field term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' In the supernova context, extended mean-field evo- lution equations were derived using in particular the coherent-state path integral, the closed time path inte- gral and the BBGKY hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Such equations included, in particular, contributions by supplementary two-point correlators, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' helicity coherence and pairing correla- tors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' For the former calculations based on detailed sim- ulations of binary compact mergers and core-collapse su- pernovae showed that they do not trigger significant fla- vor evolution due to non-linear feedback, as perturbative arguments also show.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' For the latter, no flavor modifica- tion appears because the kinetic terms dominate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The impact of collisions on flavor evolution is currently an open problem which is numerically very challenging because of its high dimensionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' For a long time, the argument of the separation of scales between flavor mech- anisms (the MSW effect) and the collision-dominated re- gion justified the use of mean-field equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' With the advent of νν interaction studies for dense astrophysical environments, the identification of slow and then fast modes has deeply changed our vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The interplay between the collisions and fast modes is receiving particular attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Studies of models with lower dimensionality and approximate treatment of colli- sions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' direction changing and neutrino-nucleon only) uncovered the possibility that collisions can trigger fast modes, or suppress them, or enhance them, depending for example on the angular distribution of the neutrino emis- sion at the neutrinosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' While the models studied so far necessarily have many approximations and limita- tions, there are clear hints that we need to go towards fur- ther complexity since even if the collision rate is smaller than flavor scale, collisions are likely to be important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' And, in fact, it goes without saying that even the cross- ings between the neutrino and antineutrino angular dis- tributions, associated with the occurrence of fast modes, should emerge from collisions in a fully consistent treat- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' All these developments are based on theoretical ap- proaches in flat spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' However, strong gravitational fields are present nearby compact objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Their impact on flavor evolution is still in an exploratory phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' An extension of the equations of motion in curved spacetime has been discussed by a few authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Several studies investigated the impact on the vacuum oscillation phase for different metrics and recently on the decoherence by wave packet separation, in a wave- packet treatment of neutrino evolution in curved space- time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' A ”halo” effect was found in an exploding super- nova, whereas it was clearly shown that the inclusion of gravity effects (trajectory bending, energy redshift) influ- ence r-process nucleosynthesis in accretion disks around black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Clearly, gravitational effects on neutrino propagation and flavor evolution should deserve more at- tention in the coming years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Intriguing connections between a system of weakly in- teracting neutrinos and other domains have been uncov- ered, often opening new unforeseen possibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' In par- ticular, an algebraic formulation and the Bethe ansatz showed the νν many-body Hamiltonian to be solvable (under some conditions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' This and further works have yielded the first comparison between mean-field and many-body results highlighting the role of many-body correlations in particular through the entanglement en- tropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Moreover first calculations based on an inference procedure and on quantum devices are appearing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The latter open exciting new possibilities that are and will 37 certainly attract a lot of interest in the coming years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' After the 24 ¯νe events of SN1987A, we are eagerly awaiting for the next supernova to precisely measure the neutrino light-curves hopefully this time, if the supernova is close and we are patient enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' It goes without say- ing that this observation is crucial both for astrophysics and for particle physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' We will learn a lot on the super- nova explosion, possibly having definite evidence for the explosion mechanism and the favored neutrino heating mechanism, the onset time of the explosion, important for gravitational wave detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' We will get a picture of the shock wave passage in the MSW region from the time and energy signals, and eventually signatures of the SASI instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' With the advent of SNEWS 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='0 we will be able lo locate the exploding star through its neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Besides, the upcoming discovery of the diffuse super- nova neutrino background will be crucial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' It will be the second time ever we observe neutrinos from core-collapse supernovae, with a unique sensitivity to the evolving core-collapse supernova rate, the fraction of failed super- novae and binaries, flavor mechanisms and non-standard neutrino properties such as neutrino decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' The diffuse supernova neutrino background promise to remain for many years an incredible laboratory for astrophysics and particle physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' In this journey, I highlighted aspects of our current un- derstanding of flavor evolution in dense media, setting it in the context of the historical developments in neutrino physics, of what we now know and of what we would like to discover in the coming years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' With a bit of an historical perspective also on the theoretical progress in this field, I have discussed numerous aspects that appear now as clear (although with its limitations and approx- imations) and the numerous theoretical challenges still ahead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Serious progress has been done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' But one lesson we have learnt, that new possibilities can always come up and completely change the way we look at this complex prob- lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Clearly, the novel developments recently emerged and future ones might give, once more, a completely new insight on this fascinating subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' ACKNOWLEDGMENTS Along the years I had interesting discussions with nu- merous researchers in the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' In particular, I wish to thank Baha Balantekin, Carlo Giunti, Thomas Janka, Gail McLaughlin, George Fuller, Manfred Lindner, Kate Scholberg, Francesco Vissani, and also Evgeny Akhme- dov, Alessandro Mirizzi, Georg Raffelt, Alexey Smirnov, Rebecca Surman, Irene Tamborra as well as many others with whom I discussed at workshops and conferences, or I exchanged correspondence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Moreover I would like to thank very warmly to two of my former PhD students, Am´elie Chatelain and J´erˆome Gava, for the enthusiasm we shared working together on this intriguing domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' REFERENCES Aartsen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (IceCube) (2014), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 113, 101101, arXiv:1405.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='5303 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Aartsen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (IceCube) (2018), Science 361 (6398), 147, arXiv:1807.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='08794 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Abazajian, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Beacom, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Bell (2002), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 66, 013008, arXiv:astro-ph/0203442.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Abbar, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2021), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 103 (4), 045014, arXiv:2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='13655 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Abbar, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Duan (2015), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' B 751, 43, arXiv:1509.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='01538 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Abbar, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Duan (2018), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 98 (4), 043014, arXiv:1712.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='07013 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Abbar, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Duan, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Shalgar (2015), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 92 (6), 065019, arXiv:1507.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='08992 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Abbar, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Duan, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Sumiyoshi, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Takiwaki, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Volpe (2019), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 100 (4), 043004, arXiv:1812.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='06883 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Abbar, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Duan, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Sumiyoshi, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Takiwaki, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Volpe (2020), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 101 (4), 043016, arXiv:1911.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='01983 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Abbar, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Volpe (2019), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' B 790, 545, arXiv:1811.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='04215 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Abbott, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (LIGO Scientific, Virgo) (2016), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 116 (6), 061102, arXiv:1602.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='03837 [gr-qc].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Abbott, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2017a), Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 848 (2), L12, arXiv:1710.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='05833 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Abbott, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (LIGO Scientific, Virgo) (2017b), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 119 (16), 161101, arXiv:1710.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='05832 [gr-qc].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Abdikamalov, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Pagliaroli, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Radice (2020), arXiv:2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='04356 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='SR].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Abe, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2011), arXiv:1109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='3262 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Abe, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (Super-Kamiokande) (2016), Astropart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 81, 39, arXiv:1601.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='04778 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Abe, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (Hyper-Kamiokande) (2018), arXiv:1805.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='04163 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='ins-det].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Abe, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (Super-Kamiokande) (2021), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 104 (12), 122002, arXiv:2109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='11174 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Abe, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (Double Chooz) (2012), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 108, 131801, arXiv:1112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='6353 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Abi, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (DUNE) (2021), Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' C 81 (5), 423, arXiv:2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='06647 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Aglietta, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (1987), Europhys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 3, 1315.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Agostini, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Benato, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Detwiler, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Men´endez, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Vissani (2022), arXiv:2202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='01787 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Agostini, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (Borexino) (2021), Astropart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 125, 102509, arXiv:1909.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='02422 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Aguilar-Arevalo, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (MiniBooNE) (2018), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 121 (22), 221801, arXiv:1805.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='12028 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Aharmim, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (SNO) (2006), Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 653, 1545, arXiv:hep-ex/0607010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Ahluwalia, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Burgard (1996), Gen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Grav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 28, 1161, arXiv:gr-qc/9603008 [gr-qc].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Ahluwalia, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Burgard (1998), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D57, 4724, arXiv:gr-qc/9803013 [gr-qc].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Ahmad, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (SNO) (2001a), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 87, 071301, arXiv:nucl-ex/0106015 [nucl-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Ahmad, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (SNO) (2001b), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 87, 071301, arXiv:nucl-ex/0106015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Ahmad, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (SNO) (2002), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 89, 011301, arXiv:nucl-ex/0204008 [nucl-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 38 Ahn, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (RENO) (2012), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 108, 191802, arXiv:1204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='0626 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Aker, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (KATRIN) (2022), Nature Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 18 (2), 160, arXiv:2105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='08533 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Akhmedov, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Kopp, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lindner (2017), JCAP 09, 017, arXiv:1702.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='08338 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Akhmedov, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (1999), in ICTP Summer School in Particle Physics, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 103–164, arXiv:hep-ph/0001264.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Akhmedov, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lunardini, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Smirnov (2002), Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' B 643, 339, arXiv:hep-ph/0204091.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Akimov, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (COHERENT) (2017), Science 357 (6356), 1123, arXiv:1708.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='01294 [nucl-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Al Kharusi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (SNEWS) (2021), New J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 23 (3), 031201, arXiv:2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='00035 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Alekseev, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Alekseeva, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Krivosheina, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Volchenko (1988), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' B205, 209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Alp, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2018), Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 864 (2), 174, arXiv:1805.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='04526 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Altarelli, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Feruglio (2010), Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 82, 2701, arXiv:1002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='0211 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Amitrano, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Roggero, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Luchi, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Turro, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Vespucci, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Pederiva (2023), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 107 (2), 023007, arXiv:2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='03189 [quant-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' An, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (JUNO) (2016), J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' G 43 (3), 030401, arXiv:1507.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='05613 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='ins-det].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' An, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (Daya Bay) (2012), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 108, 171803, arXiv:1203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='1669 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' An, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (Daya Bay) (2017), Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' C 77 (9), 606, arXiv:1608.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='01661 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Ando, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2003), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' B 570, 11, arXiv:hep- ph/0307169.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Ando, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2004), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 70, 033004, arXiv:hep- ph/0405200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Ando, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Beacom, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Yuksel (2005), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 95, 171101, arXiv:astro-ph/0503321.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Ando, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Sato (2004), New J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 6, 170, arXiv:astro-ph/0410061.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Arg¨uelles, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Esteban, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Hostert, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Kelly, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Kopp, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Machado, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Martinez-Soler, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Perez-Gonzalez (2022), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 128 (24), 241802, arXiv:2111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='10359 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Armstrong, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2022), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 105 (8), 083012, arXiv:2111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='07412 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='comp-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Armstrong, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Patwardhan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Ahmetaj, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Sanchez, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Miskiewicz, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Ibrahim, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Singh (2022), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 105 (10), 103003, arXiv:2201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='08505 [astro- ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Armstrong, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Patwardhan, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rrapaj, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Ardizi, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Fuller (2020), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 102 (4), 043013, arXiv:2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='07725 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Arnett, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Bahcall, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Kirshner, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Woosley (1989), Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 27, 629.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Asakura, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (KamLAND) (2016), Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 818 (1), 91, arXiv:1506.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='01175 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Avignone, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Gudkov, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Chatterjee, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Efremenko, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Strayer (2001), Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' B Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Suppl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 91, 113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Baade, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (1934), On super-novae, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Bahcall, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (1964), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 12, 300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Bahcall, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Bahcall, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Shaviv (1968), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 20, 1209, [,45(1968)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Balantekin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Fetter, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Loreti (1996), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 54, 3941, arXiv:astro-ph/9604061.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Balantekin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Pehlivan (2007), J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' G 34, 47, arXiv:astro-ph/0607527.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Balantekin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Yuksel (2005), New J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 7, 51, arXiv:astro-ph/0411159.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Balantekin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Gava, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Volpe (2008), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' B 662, 396, arXiv:0710.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='3112 [astro-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Balantekin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Yuksel (2003a), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' C 68, 055801, arXiv:hep-ph/0307227.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Balantekin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Yuksel (2003b), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 68, 013006, arXiv:hep-ph/0303169.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Baldry, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Glazebrook (2003), Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 593, 258, arXiv:astro-ph/0304423.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Banerjee, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Dighe, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Raffelt (2011), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 84, 053013, arXiv:1107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='2308 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Barbeau, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Efremenko, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Scholberg (2021), arXiv:2111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='07033 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Barbieri, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Mohapatra (1988), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 61, 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Bardeen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Cooper, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Schrieffer (1957), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 108, 1175.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Barger, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Huber, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Marfatia (2005), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' B 617, 167, arXiv:hep-ph/0501184.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Barinov, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2022), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 128 (23), 232501, arXiv:2109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='11482 [nucl-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Barranco, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Bernal, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Delepine (2018), J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' G 45 (5), 055201, arXiv:1706.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='03834 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='CO].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Baxter, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (SNEWS) (2022), Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 925 (2), 107, arXiv:2109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='08188 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='IM].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Beacom, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2010), Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 60, 439, arXiv:1004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='3311 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Beacom, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Farr, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Vogel (2002), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 66, 033001, arXiv:hep-ph/0205220.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Beacom, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Vagins (2004), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 93, 171101, arXiv:hep-ph/0309300 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Beacom, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Vogel (1998), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 58, 053010, arXiv:hep-ph/9802424.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Bell, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rawlinson, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Sawyer (2003), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' B 573, 86, arXiv:hep-ph/0304082.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Bell, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Sawyer, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Volkas (2002a), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' A 65, 052105, arXiv:quant-ph/0106082.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Bell, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Sawyer, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Volkas, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Wong (2002b), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' A 65, 042328, arXiv:quant- ph/0109014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Bennett, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Buldgen, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' De Salas, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Drewes, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Gari- azzo, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Pastor, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Wong (2021), JCAP 04, 073, arXiv:2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='02726 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Bethe, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (1986), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 56, 1305.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Bethe, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (1939), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 55, 434.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Bethe, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (1990), Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 62, 801.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Bethe, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Wilson (1985), Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 295, 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Bhattacharya, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Habib, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Mottola (1999), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 59, 067301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Biggio, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Blennow, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Fernandez-Martinez (2009), JHEP 08, 090, arXiv:0907.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='0097 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Bilenky, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Santamaria (1999), in Neutrino Mix- ing: Meeting in Honor of Samoil Bilenky’s 70th Birthday, arXiv:hep-ph/9908272.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Bionta, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (1987), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 58, 1494.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Birol, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Pehlivan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Balantekin, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Kajino (2018), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 98 (8), 083002, arXiv:1805.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='11767 [astro- ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Blaschke, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Cirigliano (2016), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 94 (3), 033009, arXiv:1605.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='09383 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Blennow, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Mirizzi, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Serpico (2008), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 78, 113004, arXiv:0810.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='2297 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Blondin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Mezzacappa, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' DeMarino (2003), Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 584, 971, arXiv:astro-ph/0210634.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 39 Bogoliubov, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (1946), Journal of Physics USSR 10, 1175.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Bollig, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Janka, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lohs, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Martinez-Pinedo, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Horowitz, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Melson (2017), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 119 (24), 242702, arXiv:1706.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='04630 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Born, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Green (1946), Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' A188 10, 1175.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Borriello, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Chakraborty, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Janka, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lisi, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Mi- rizzi (2014), JCAP 11, 030, arXiv:1310.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='7488 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='SR].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Botella, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lim, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Marciano (1987), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 35, 896.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Bouchez, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Cribier, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rich, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Spiro, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Vignaud, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Hampel (1986), Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' C 32, 499.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Briggs, (1964), (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='T Press, Cambridge).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Bruenn, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (1985), Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Suppl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 58, 771.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Burbidge, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Burbidge, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Fowler, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Hoyle (1957), Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 29, 547.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Burgess, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Dzhalilov, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Maltoni, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rashba, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Semikoz, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Tortola, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Valle (2004), JCAP 01, 007, arXiv:hep-ph/0310366.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Burgess, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Michaud (1997), Annals Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 256, 1, arXiv:hep-ph/9606295.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Burrows, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2013), Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 85, 245, arXiv:1210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='4921 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='SR].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Burrows, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Radice, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Vartanyan, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Nagakura, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Skinner, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Dolence (2020), Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 491 (2), 2715, arXiv:1909.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='04152 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Caballero, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' McLaughlin, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Surman (2012), Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 745, 170, arXiv:1105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='6371 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Calzetta, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Hu (1988), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 37, 2878.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Cameron, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (1958), Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 8, 299.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Capozzi, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Chakraborty, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Chakraborty, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Sen (2020), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 125, 251801, arXiv:2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='14204 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Capozzi, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Dasgupta, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lisi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Marrone, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Mi- rizzi (2017), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 96 (4), 043016, arXiv:1706.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='03360 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Capozzi, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Dasgupta, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Mirizzi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Sen, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Sigl (2019), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 122 (9), 091101, arXiv:1808.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='06618 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Capozzi, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Di Valentino, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lisi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Marrone, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Mel- chiorri, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Palazzo (2021), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 104 (8), 083031, arXiv:2107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='00532 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Cardall, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Endeve, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Mezzacappa (2013), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 88, 023011, arXiv:1305.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='0037 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Cardall, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Fuller (1997), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D55, 7960, arXiv:hep-ph/9610494 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Cervia, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Patwardhan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Balantekin, d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Copper- smith, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Johnson (2019), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 100 (8), 083001, arXiv:1908.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='03511 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Chakraborty, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Chakraborty (2020), JCAP 01, 005, arXiv:1909.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='10420 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Chakraborty, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Choubey, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Kar (2011a), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' B 702, 209, arXiv:1006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='3756 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Chakraborty, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Fischer, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Mirizzi, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Saviano, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Tomas (2011b), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 84, 025002, arXiv:1105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='1130 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Chakraborty, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Fischer, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Mirizzi, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Saviano, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Tomas (2011c), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 107, 151101, arXiv:1104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='4031 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Chakraborty, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Hansen, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Izaguirre, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Raffelt (2016a), Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' B 908, 366, arXiv:1602.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='02766 [hep- ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Chakraborty, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Hansen, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Izaguirre, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Raffelt (2016b), JCAP 03, 042, arXiv:1602.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='00698 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Chakraborty, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Mirizzi, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Saviano, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Seixas (2014), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 89 (9), 093001, arXiv:1402.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='1767 [hep- ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Chatelain, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Volpe (2017), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 95 (4), 043005, arXiv:1611.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='01862 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Chatelain, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Volpe (2018), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 97 (2), 023014, arXiv:1710.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='11518 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Chatelain, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Volpe (2020), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' B801, 135150, arXiv:1906.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='12152 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Chen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Oldengott, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Pierobon, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Wong (2022), Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' C 82 (7), 640, arXiv:2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='09075 [hep- ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Cherry, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Carlson, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Friedland, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Fuller, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Vlasenko (2012), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 108, 261104, arXiv:1203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='1607 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Cherry, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Carlson, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Friedland, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Fuller, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Vlasenko (2013), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 87, 085037, arXiv:1302.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='1159 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Cherry, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Fuller, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Horiuchi, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Kotake, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Taki- waki, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Fischer (2020), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 102 (2), 023022, arXiv:1912.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='11489 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Choubey, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Harries, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Ross (2007), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 76, 073013, arXiv:hep-ph/0703092.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Christenson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Cronin, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Fitch, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Turlay (1964), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 13, 138.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Cigan, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2019), Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 886, 51, arXiv:1910.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='02960 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Cirigliano, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Paris, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Shalgar (2017), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' B 774, 258, arXiv:1706.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='07052 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Cohen-Tannoudji, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Diu, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lal¨oe (1998), Hermann, Editeurs des sciences et des arts 1-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Colgate, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Johnson (1960), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 5, 235.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Colgate, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' White (1966), Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 143, 626.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Coloma, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Esteban, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Gonzalez-Garcia, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Mal- toni (2020), JHEP 02, 023, [Addendum: JHEP 12, 071 (2020)], arXiv:1911.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='09109 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Couch, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Ott (2015), Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 799 (1), 5, arXiv:1408.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='1399 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Cowan, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Reines, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Harrison, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Kruse, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' McGuire (1956), Science 124, 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Cowan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Sneden, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lawler, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Aprahamian, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Wiescher, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Langanke, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Mart´ınez-Pinedo, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='- K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Thielemann (2021), Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 93 (1), 15002, arXiv:1901.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='01410 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Cusinato, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Guercilena, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Perego, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lo- goteta, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Radice, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Bernuzzi, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Ansoldi (2021), 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='1140/epja/s10050-022-00743-5, arXiv:2111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='13005 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Cˆot´e, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2019), Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 875 (2), 106, arXiv:1809.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='03525 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Das, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Dighe, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Sen (2017), JCAP 05, 051, arXiv:1705.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='00468 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Dasgupta, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Dighe (2007), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 75, 093002, arXiv:hep-ph/0510219.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Dasgupta, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Dighe, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Raffelt, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Smirnov (2009), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 103, 051105, arXiv:0904.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='3542 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Dasgupta, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Mirizzi (2015), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 92 (12), 125030, arXiv:1509.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='03171 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Dasgupta, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Mirizzi, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Sen (2017), JCAP 02, 019, arXiv:1609.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='00528 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Dasgupta, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Mirizzi, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Sen (2018), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 98 (10), 103001, arXiv:1807.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='03322 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Davidson, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Pena-Garay, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rius, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Santamaria 40 (2003), JHEP 03, 011, arXiv:hep-ph/0302093.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Davis, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', Raymond, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Harmer, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Hoffman (1968), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 20, 1205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Davis, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (1964), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 12, 303.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' De Gouvea, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Martinez-Soler, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Perez-Gonzalez, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Sen (2020), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 102, 123012, arXiv:2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='13748 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Deaton, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' O’Connor, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Zhu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Bohn, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Jesse, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Foucart, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Duez, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' McLaughlin (2018), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 98 (10), 103014, arXiv:1806.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='10255 [astro- ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Delfan Azari, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Yamada, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Morinaga, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Iwakami, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Okawa, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Nagakura, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Sumiyoshi (2019), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 99 (10), 103011, arXiv:1902.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='07467 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Delfan Azari, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Yamada, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Morinaga, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Nagakura, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Furusawa, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Harada, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Okawa, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Iwakami, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Sumiyoshi (2020), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 101 (2), 023018, arXiv:1910.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='06176 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Di Valentino, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Mena, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Pan, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Visinelli, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Yang, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Melchiorri, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Mota, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Riess, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Silk (2021), Class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Quant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Grav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 38 (15), 153001, arXiv:2103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='01183 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='CO].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Dighe, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Keil, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Raffelt (2003), JCAP 06, 005, arXiv:hep-ph/0303210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Dighe, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Smirnov (2000), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 62, 033007, arXiv:hep-ph/9907423.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Dolgov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Hansen, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Pastor, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Petcov, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Raffelt, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Semikoz (2002), Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' B 632, 363, arXiv:hep-ph/0201287.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Duan, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Fuller, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Carlson, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='-Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Qian (2006a), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 74, 105014, arXiv:astro-ph/0606616.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Duan, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Fuller, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Carlson, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='-Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Qian (2007a), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 75, 125005, arXiv:astro-ph/0703776.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Duan, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Fuller, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='-Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Qian (2006b), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 74, 123004, arXiv:astro-ph/0511275.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Duan, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Fuller, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='-Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Qian (2007b), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 76, 085013, arXiv:0706.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='4293 [astro-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Duan, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Fuller, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='-Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Qian (2010), Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 60, 569, arXiv:1001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='2799 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Duan, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Kneller (2009), J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' G 36, 113201, arXiv:0904.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='0974 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Duan, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Shalgar (2015), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' B 747, 139, arXiv:1412.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='7097 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Dvornikov, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2013), JCAP 06, 015, arXiv:1306.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='2659 [hep- ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Dvornikov, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Grigoriev, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Studenikin (2005a), Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 14, 309, arXiv:hep-ph/0406114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Dvornikov, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Grigoriev, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Studenikin (2005b), Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 14, 309, arXiv:hep-ph/0406114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Eguchi, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (KamLAND) (2003), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 90, 021802, arXiv:hep-ex/0212021 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Ekanger, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Horiuchi, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Kotake, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Sumiyoshi (2022), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 106 (4), 043026, arXiv:2206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='05299 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Ekinci, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Pehlivan, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Patwardhan (2021), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 103 (4), 043016, arXiv:2101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='01797 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Engel, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' McLaughlin, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Volpe (2003), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 67, 013005, arXiv:hep-ph/0209267.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Ertl, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Janka, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Woosley, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Sukhbold, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Ugliano (2016), Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 818 (2), 124, arXiv:1503.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='07522 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='SR].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' ESA/Hubble,, and NASA (2011), https://esahubble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='org/images/potw1142a/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Esteban-Pretel, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Mirizzi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Pastor, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Tomas, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Raffelt, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Serpico, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Sigl (2008), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 78, 085012, arXiv:0807.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='0659 [astro-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Esteban-Pretel, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Tomas, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Valle (2007a), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 76, 053001, arXiv:0704.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='0032 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Esteban-Pretel, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Tomas, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Valle (2007b), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 76, 053001, arXiv:0704.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='0032 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Esteban-Pretel, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Tomas, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Valle (2010), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 81, 063003, arXiv:0909.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='2196 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Farzan, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Palomares-Ruiz (2014), JCAP 06, 014, arXiv:1401.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='7019 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Farzan, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Tortola (2018), Front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' in Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 6, 10, arXiv:1710.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='09360 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Fetter, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' McLaughlin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Balantekin, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Fuller (2003), Astropart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 18, 433, arXiv:hep-ph/0205029.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Fidler, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Herranen, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Kainulainen, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rahkila (2012), JHEP 02, 065, arXiv:1108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='2309 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Fiorillo, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Raffelt, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Vitagliano (2022), arXiv:2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='11773 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Fogli, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lisi, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Montanino, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Mirizzi (2003), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 68, 033005, arXiv:hep-ph/0304056.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Fogli, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lisi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Marrone, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Mirizzi (2007), JCAP 12, 010, arXiv:0707.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='1998 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Fogli, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lisi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Mirizzi, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Montanino (2002), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 66, 013009, arXiv:hep-ph/0202269.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Fogli, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lisi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Mirizzi, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Montanino (2004), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 70, 013001, arXiv:hep-ph/0401227.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Fogli, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lisi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Mirizzi, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Montanino (2005), JCAP 04, 002, arXiv:hep-ph/0412046.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Fogli, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lisi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Mirizzi, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Montanino (2006), JCAP 06, 012, arXiv:hep-ph/0603033.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Foglizzo, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2015), Publ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Austral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 32, e009, arXiv:1501.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='01334 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Fornengo, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Giunti, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Kim, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Song (1997), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D56, 1895, arXiv:hep-ph/9611231 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Frensel, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Wu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Volpe, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Perego (2017), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 95 (2), 023011, arXiv:1607.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='05938 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Friedland, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Gruzinov (2006), arXiv:astro- ph/0607244.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Friedland, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lunardini (2003a), JHEP 10, 043, arXiv:hep-ph/0307140.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Friedland, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lunardini (2003b), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 68, 013007, arXiv:hep-ph/0304055.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Frohlich, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Martinez-Pinedo, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Liebendorfer, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Thielemann, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Bravo, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Hix, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Langanke, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Zinner (2006), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 96, 142502, arXiv:astro- ph/0511376.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Froustey, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Pitrou, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Volpe (2020), JCAP 12, 015, arXiv:2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='01074 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Fukuda, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (Super-Kamiokande) (1998), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 81, 1562, arXiv:hep-ex/9807003 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Fukugita, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Kawasaki (2003), Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' As- tron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 340, L7, arXiv:astro-ph/0204376.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Fuller, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='-Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Qian (1996), Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' A 606, 167, arXiv:astro-ph/9611126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Galais, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Kneller, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Volpe (2012), J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' G 39, 035201, arXiv:1102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='1471 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='SR].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Galais, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Kneller, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Volpe, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Gava (2010), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 81, 053002, arXiv:0906.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='5294 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Galais, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Volpe (2011), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 84, 085005, arXiv:1103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='5302 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='SR].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Gallo Rosso, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2021), JCAP 06, 046, arXiv:2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='12579 [hep- ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Gallo Rosso, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Vissani, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Volpe (2017), JCAP 11, 036, arXiv:1708.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='00760 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Gallo Rosso, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Vissani, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Volpe (2018), JCAP 04, 040, arXiv:1712.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='05584 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 41 Gando, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (KamLAND) (2012), Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 745, 193, arXiv:1105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='3516 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Gaudin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (1976), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Gava, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Kneller, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Volpe, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' McLaughlin (2009), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 103, 071101, arXiv:0902.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='0317 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Gava, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Volpe (2008), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 78, 083007, arXiv:0807.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='3418 [astro-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Gava, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Volpe (2010), Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' B 837, 50, [Er- ratum: Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='B 957, 115035 (2020)], arXiv:1002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='0981 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' George, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Wu, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Tamborra, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Ardevol-Pulpillo, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Janka (2020), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 102 (10), 103015, arXiv:2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='04046 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Giunti, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2020), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 101 (3), 035039, arXiv:1909.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='00466 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Giunti, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Kim (2007), Fundamentals of Neutrino Physics and Astrophysics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Giunti, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Kouzakov, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Li, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lokhov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Studenikin, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Zhou (2016), Annalen Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 528, 198, arXiv:1506.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='05387 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Giunti, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Laveder (2011), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' C 83, 065504, arXiv:1006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='3244 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Giunti, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Li, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Ternes, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Xin (2022), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' B 829, 137054, arXiv:2110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='06820 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Giunti, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Studenikin (2015), Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 87, 531, arXiv:1403.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='6344 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Glas, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Janka, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Capozzi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Sen, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Dasgupta, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Mirizzi, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Sigl (2020), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 101 (6), 063001, arXiv:1912.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='00274 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Glashow, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (1961), Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 22, 579.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Godunov, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Pastukhov (2011), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 74, 302, arXiv:0906.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='5556 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Goldberg, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Perez, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Sarcevic (2006), JHEP 11, 023, arXiv:hep-ph/0505221.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Goldhaber, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Grodzins, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Sunyar (1958), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 109, 1015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Goriely, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Bauswein, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Just, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Pllumbi, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Janka (2015), Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 452 (4), 3894, arXiv:1504.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='04377 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='SR].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' de Gouvˆea, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Martinez-Soler, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Perez-Gonzalez, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Sen (2022), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 106 (10), 103026, arXiv:2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='01102 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' de Gouvˆea, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Martinez-Soler, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Sen (2020), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 101 (4), 043013, arXiv:1910.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='01127 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Guzzo, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' de Holanda, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Reggiani (2003), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' B 569, 45, arXiv:hep-ph/0303203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Halim, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Casentini, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Drago, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Fafone, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Scholberg, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Vigorito, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Pagliaroli (2021), JCAP 11, 021, arXiv:2107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='02050 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Hall, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Roggero, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Baroni, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Carlson (2021), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 104 (6), 063009, arXiv:2102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='12556 [quant-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Halzen, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Raffelt (2009), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 80, 087301, arXiv:0908.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='2317 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Hannestad, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Raffelt, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Sigl, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Wong (2006), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 74, 105010, [Erratum: Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='D 76, 029901 (2007)], arXiv:astro-ph/0608695.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Hansen, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Hannestad (2014), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 90 (2), 025009, arXiv:1404.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='3833 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Hansen, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Shalgar, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Tamborra (2021), JCAP 07, 017, arXiv:2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='03948 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='CO].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Hansen, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Shalgar, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Tamborra (2022), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 105 (12), 123003, arXiv:2204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='11873 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Harris, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Stodolsky (1982), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' B 116, 464.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Haxton, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (1986), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 57, 1271.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Haxton, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Hamish Robertson, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Serenelli (2013), Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 51, 21, arXiv:1208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='5723 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='SR].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Hayes, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Towner (2000), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' C 61, 044603, arXiv:nucl-th/9907049.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Hirata, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (Kamiokande-II) (1987), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 58, 1490.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Hopkins, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Beacom (2006), Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 651, 142, arXiv:astro-ph/0601463.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Horiuchi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Beacom, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Dwek (2009), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 79, 083013, arXiv:0812.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='3157 [astro-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Horiuchi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Beacom, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Kochanek, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Prieto, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Stanek, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Thompson (2011), Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 738, 154, arXiv:1102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='1977 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='CO].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Horiuchi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Kinugawa, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Takiwaki, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Takahashi, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Kotake (2021), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 103 (4), 043003, arXiv:2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='08524 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Horiuchi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Kneller (2018), J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' G 45 (4), 043002, arXiv:1709.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='01515 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Horiuchi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Nakamura, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Takiwaki, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Kotake, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Tanaka (2014), Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 445, L99, arXiv:1409.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='0006 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Horiuchi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Sumiyoshi, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Nakamura, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Fischer, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Summa, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Takiwaki, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Janka, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Ko- take (2018), Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 475 (1), 1363, arXiv:1709.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='06567 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Hoyle, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Fowler (1960), Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 132, 565, [Erratum: Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 134, 1028 (1961)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Ivanez-Ballesteros, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Volpe (2022), arXiv:2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='12465 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Izaguirre, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Raffelt, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Tamborra (2017), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 118 (2), 021101, arXiv:1610.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='01612 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Janka, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2012), Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 62, 407, arXiv:1206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='2503 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='SR].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Janka, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2017), 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='1007/978-3-319-21846-5 109, arXiv:1702.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='08825 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Jegerlehner, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Neubig, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Raffelt (1996), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 54, 1194, arXiv:astro-ph/9601111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Just, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Abbar, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Wu, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Tamborra, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Janka, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Capozzi (2022), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 105 (8), 083024, arXiv:2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='16559 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Kajino, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Aoki, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Balantekin, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Diehl, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Famiano, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Mathews (2019), Prog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 107, 109, arXiv:1906.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='05002 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Kartavtsev, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Raffelt, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Vogel (2015), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 91 (12), 125020, arXiv:1504.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='03230 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Kato, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Ishidoshiro, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Yoshida (2020), Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 70, 121, arXiv:2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='02519 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Keehn, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lunardini (2012), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 85, 043011, arXiv:1012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='1274 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='CO].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Keil, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Raffelt, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Janka (2003), Astro- phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 590, 971, arXiv:astro-ph/0208035.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Kersten, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Smirnov (2016), Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' C 76 (6), 339, arXiv:1512.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='09068 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Kim, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Kim, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Sze (1988), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 37, 1072.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Kim, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Sze, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Nussinov (1987), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 35, 4014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' King, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2015), J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' G 42, 123001, arXiv:1510.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='02091 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Kirkwood, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (1935), J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 3 300, 1175.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Kneller, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' McLaughlin, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Brockman (2008), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 77, 045023, arXiv:0705.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='3835 [astro-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Kneller, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Volpe (2010), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 82, 123004, arXiv:1006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='0913 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 42 Kostelecky, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Pantaleone, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Samuel (1993), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' B 315, 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Kostelecky, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Samuel (1995), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 52, 621, arXiv:hep-ph/9506262.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Kotake, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Sato, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Takahashi (2006), Rept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Prog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 69, 971, arXiv:astro-ph/0509456.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Kresse, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Ertl, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Janka (2021), Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 909 (2), 169, arXiv:2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='04728 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Kulkarni, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2005), arXiv:astro-ph/0510256 [astro-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Kuo, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='-K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Pantaleone (1989), Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 61, 937.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lacroix, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Balantekin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Cervia, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Patward- han, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Siwach (2022), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 106 (12), 123006, arXiv:2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='09384 [nucl-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lambiase, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Papini, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Punzi, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Scarpetta (2005), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 71, 073011, arXiv:gr-qc/0503027.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Langanke, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Martinez-Pinedo, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Sieverding (2019), 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='22661/AAPPSBL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='41, arXiv:1901.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='03741 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lattimer, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Schramm (1974), Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 192, L145.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lattimer, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Schramm (1976), Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 210, 549.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lee, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='-N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Yang (1956), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 104, 254.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lindquist, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (1966), Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' of Physics 37, 487.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Longo, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (1987), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D36, 3276.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Loredo, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lamb (2002), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D65, 063002, arXiv:astro-ph/0107260 [astro-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Loreti, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Balantekin (1994), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 50, 4762, arXiv:nucl-th/9406003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Loreti, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Qian, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Fuller, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Balantekin (1995), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 52, 6664, arXiv:astro-ph/9508106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Li, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Zhou (2016), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 94 (2), 023006, arXiv:1605.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='07803 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lujan-Peschard, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Pagliaroli, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Vissani (2014), JCAP 07, 051, arXiv:1402.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='6953 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='SR].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lunardini, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2006), Astropart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 26, 190, arXiv:astro- ph/0509233.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lunardini, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2009), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 102, 231101, arXiv:0901.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='0568 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='SR].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lunardini, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2016), Astropart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 79, 49, arXiv:1007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='3252 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='CO].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lunardini, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Peres (2008), JCAP 08, 033, arXiv:0805.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='4225 [astro-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lunardini, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Smirnov (2003), JCAP 06, 009, arXiv:hep-ph/0302033.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lunardini, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Tamborra (2012), JCAP 07, 012, arXiv:1205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='6292 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='SR].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lund, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Kneller (2013), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 88 (2), 023008, arXiv:1304.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='6372 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Madau, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Dickinson (2014), Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' As- trophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 52, 415, arXiv:1403.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='0007 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='CO].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Majorana, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (1937), Nuovo Cim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 14, 171.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Malek, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (Super-Kamiokande) (2003), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 90, 061101, arXiv:hep-ex/0209028 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Malkus, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Friedland, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' McLaughlin (2014), arXiv:1403.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='5797 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Malkus, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Kneller, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' McLaughlin, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Surman (2012), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 86, 085015, arXiv:1207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='6648 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Malkus, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' McLaughlin, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Surman (2016), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 93 (4), 045021, arXiv:1507.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='00946 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Mangano, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Miele, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Pastor, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Pinto, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Pisanti, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Serpico (2005), Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' B 729, 221, arXiv:hep- ph/0506164.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Mangano, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Miele, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Pastor, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Pinto, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Pisanti, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Serpico (2006), Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' B 756, 100, arXiv:hep- ph/0607267.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Martin, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Perego, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Arcones, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='-K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Thielemann, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Ko- robkin, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rosswog (2015), Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 813 (1), 2, arXiv:1506.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='05048 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='SR].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Martin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Carlson, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Cirigliano, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Duan (2021), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 103, 063001, arXiv:2101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='01278 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Martin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Roggero, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Duan, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Carlson (2023), arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='07049 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Martin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Roggero, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Duan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Carlson, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Cirigliano (2022), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 105 (8), 083020, arXiv:2112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='12686 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Martin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Yi, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Duan (2020), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' B 800, 135088, arXiv:1909.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='05225 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Mathews, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Boccioli, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Hidaka, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Kajino (2020), Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' A 35 (25), 2030011, arXiv:1907.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='10088 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Mathews, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Hidaka, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Kajino, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Suzuki (2014), Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 790, 115, arXiv:1405.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='0458 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='CO].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' McKellar, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Thomson (1994), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 49, 2710.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' McLaughlin, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Fetter, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Balantekin, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Fuller (1999), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' C 59, 2873, arXiv:astro-ph/9902106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Mention, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Fechner, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lasserre, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Mueller, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lhuil- lier, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Cribier, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Letourneau (2011), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D83, 073006, arXiv:1101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='2755 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Metzger, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Martinez-Pinedo, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Darbha, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Quataert, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Arcones, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Kasen, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Thomas, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Nugent, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Panov, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Zinner (2010), Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 406, 2650, arXiv:1001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='5029 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Mezzacappa, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Endeve, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Bronson Messer, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Bruenn (2020), Liv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 6, 4, arXiv:2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='09013 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Mezzacappa, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2015), arXiv:1501.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='01688 [astro- ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='SR].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Mikheev, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Smirnov (1986), Nuovo Cim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' C9, 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Minakata, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Nunokawa, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Tomas, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Valle (2008), JCAP 12, 006, arXiv:0802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='1489 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Mirizzi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2013), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 88 (7), 073004, arXiv:1308.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='1402 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Mirizzi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Mangano, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Saviano (2015), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 92 (2), 021702, arXiv:1503.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='03485 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Mirizzi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Tamborra, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Janka, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Saviano, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Schol- berg, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Bollig, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Hudepohl, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Chakraborty (2016), Riv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Nuovo Cim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 39 (1-2), 1, arXiv:1508.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='00785 [astro- ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Moeller, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Suliga, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Tamborra, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Denton (2018), JCAP 05, 066, arXiv:1804.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='03157 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Mohapatra, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Pal (1998), World Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Notes Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 60, 1, [World Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Notes Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='72,1(2004)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Mosquera Cuesta, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lambiase, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Pereira (2017), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 95 (2), 025011, arXiv:1701.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='00431 [gr- qc].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' M¨uller, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Janka (2014), Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 788, 82, arXiv:1402.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='3415 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='SR].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Nagakura, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Johns, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Burrows, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Fuller (2021), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 104 (8), 083025, arXiv:2108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='07281 [astro- ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Nakazato, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2013), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 88 (8), 083012, arXiv:1306.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='4526 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Nakazato, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Mochida, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Niino, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Suzuki (2015), Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 804 (1), 75, arXiv:1503.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='01236 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Nedora, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Bernuzzi, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Radice, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Daszuta, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' En- 43 drizzi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Perego, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Prakash, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Safarzadeh, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Schi- anchi, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Logoteta (2021), Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 906 (2), 98, arXiv:2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='04333 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Nunokawa, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Qian, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Fuller (1997a), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 55, 3265, arXiv:astro-ph/9610209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Nunokawa, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rossi, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Semikoz, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Valle (1996), Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' B 472, 495, arXiv:hep-ph/9602307.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Nunokawa, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Semikoz, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Smirnov, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Valle (1997b), Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' B 501, 17, arXiv:hep- ph/9701420.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' O’Connor, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Ott (2011), Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 730, 70, arXiv:1010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='5550 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' O’Connor, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2018), J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' G 45 (10), 104001, arXiv:1806.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='04175 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Odrzywolek, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Misiaszek, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Kutschera (2004), Astropart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 21, 303, arXiv:astro-ph/0311012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Ohlsson, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2013), Rept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Prog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 76, 044201, arXiv:1209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='2710 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Padilla-Gay, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Shalgar, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Tamborra (2021), JCAP 01, 017, arXiv:2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='01843 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Page, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Beznogov, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Garibay, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lattimer, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Prakash, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Janka (2020), Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 898 (2), 125, arXiv:2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='06078 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Pagliaroli, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Vissani, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Coccia, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Fulgione (2009a), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 103, 031102, arXiv:0903.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='1191 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Pagliaroli, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Vissani, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Costantini, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Ianni (2009b), Astropart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 31, 163, arXiv:0810.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='0466 [astro- ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Pantaleone, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (1992), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' B 287, 128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Parke, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (1986), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 57, 1275.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Pastor, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Raffelt, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Semikoz (2002), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 65, 053011, arXiv:hep-ph/0109035.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Patton, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lunardini, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Farmer, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Timmes (2017), Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 851 (1), 6, arXiv:1709.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='01877 [astro- ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Patwardhan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Cervia, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Balantekin (2021), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 104 (12), 123035, arXiv:2109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='08995 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Pauli, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (1930), Wissenschaftlicher Briefwechsel mit Bohr, Einstein, Heisenberg,, Letter 259.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Payez, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Evoli, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Fischer, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Giannotti, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Mirizzi, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Ringwald (2015), JCAP 02, 006, arXiv:1410.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='3747 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Pehlivan, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Balantekin, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Kajino (2014), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 90 (6), 065011, arXiv:1406.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='5489 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Pehlivan, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Balantekin, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Kajino, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Yoshida (2011), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 84, 065008, arXiv:1105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='1182 [astro- ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='CO].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Pehlivan, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Suba¸sı, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Ghazanfari, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Birol, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Y¨uksel (2017), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 95 (6), 063022, arXiv:1603.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='06360 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Perego, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rosswog, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Cabez´on, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Korobkin, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' K¨appeli, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Arcones, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Liebend¨orfer (2014), Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 443 (4), 3134, arXiv:1405.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='6730 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Piriz, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Roy, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Wudka (1996), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 54, 1587, arXiv:hep-ph/9604403.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Podsiadlowski, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (1992), Astronomical Society of the Pacific 104, 717.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Pontecorvo, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (1957), Sov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' JETP 6, 429, [Zh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Eksp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Teor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Fiz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='33,549(1957)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Pontecorvo, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (1958), Sov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' JETP 7, 172, [Zh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Eksp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Teor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Fiz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='34,247(1957)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Popov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Studenikin (2021), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 103 (11), 115027, arXiv:2102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='07991 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Priya, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lunardini (2017), JCAP 11, 031, arXiv:1705.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='02122 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Qian, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='-Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2014), J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' G 41, 044002, arXiv:1310.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='4462 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='SR].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Radice, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Abdikamalov, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Ott, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' M¨osta, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Couch, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Roberts (2018a), J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' G 45 (5), 053003, arXiv:1710.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='01282 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Radice, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Perego, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Hotokezaka, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Fromm, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Bernuzzi, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Roberts (2018b), Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 869 (2), 130, arXiv:1809.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='11161 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Raffelt, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Sarikas, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' de Sousa Seixas (2013), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 111 (9), 091101, [Erratum: Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 113, 239903 (2014)], arXiv:1305.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='7140 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Raffelt, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' de Sousa Seixas (2013), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 88, 045031, arXiv:1307.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='7625 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Raffelt, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (1996), Stars as laboratories for fundamental physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Raffelt, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Smirnov (2007), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 76, 081301, [Erratum: Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='D 77, 029903 (2008)], arXiv:0705.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='1830 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Reddy, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Steidel, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Pettini, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Adelberger, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Shapley, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Erb, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Dickinson (2008), As- trophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Suppl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 175, 48, arXiv:0706.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='4091 [astro-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Reno, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Jeong, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Palomares-Ruiz, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Sarcevic (2021), PoS ICHEP2020, 603, arXiv:2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='05380 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Ricciardi, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Vignaroli, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Vissani (2022), JHEP 08, 212, arXiv:2206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='05567 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Richardson, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (1966), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 141, 949.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Richers, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' McLaughlin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Kneller, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Vlasenko (2019), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 99 (12), 123014, arXiv:1903.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='00022 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Ring, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Schuck (2004), The nuclear many body prob- lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Roggero, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2021a), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 104 (12), 123023, arXiv:2103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='11497 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Roggero, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2021b), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 104 (10), 103016, arXiv:2102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='10188 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Roggero, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rrapaj, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Xiong (2022), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 106 (4), 043022, arXiv:2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='02783 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rozwadowska, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Vissani, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Cappellaro (2021), New Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 83, 101498, arXiv:2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='03438 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rrapaj, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Patwardhan, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Armstrong, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Fuller (2021), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 103 (4), 043006, arXiv:2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='00695 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rudzsky, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (1990), Astrophysics and Space Science 165, 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rujopakarn, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2010), Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 718, 1171, arXiv:1006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='4359 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='CO].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Salam, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (1957), Nuovo Cim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 5, 299.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Salpeter, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (1955), Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 121, 161.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Samuel, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (1993), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 48, 1462.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Sarikas, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' de Sousa Seixas, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Raffelt (2012a), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 86, 125020, arXiv:1210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='4557 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Sarikas, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Tamborra, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Raffelt, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Hudepohl, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Janka (2012b), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 85, 113007, arXiv:1204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='0971 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Sasaki, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Takiwaki (2021), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 104 (2), 023018, arXiv:2106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='02181 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Sato, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Suzuki (1987), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' B196, 267.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Sawyer, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (1990), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 42, 3908.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Sawyer, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2005), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 72, 045003, arXiv:hep- ph/0503013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Sawyer, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2009), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 79, 105003, arXiv:0803.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='4319 [astro-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 44 Sawyer, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2016), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 116 (8), 081101, arXiv:1509.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='03323 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Schilbach, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Caballero, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' McLaughlin (2019), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 100 (4), 043008, arXiv:1808.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='03627 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Schirato, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Fuller (2002), arXiv:astro- ph/0205390.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Schmidt, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Kirshnerm, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Eastman (1992), Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 395, 366, arXiv:astro-ph/9204004 [astro-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Scholberg, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2000), AIP Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 523 (1), 355, arXiv:astro-ph/9911359.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Scholberg, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2012), Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 62, 81, arXiv:1205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='6003 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='IM].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Schramm, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Truran (1990), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 189, 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Serpico, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Chakraborty, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Fischer, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Hudepohl, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='- T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Janka, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Mirizzi (2012), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 85, 085031, arXiv:1111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='4483 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='SR].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Serreau, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Volpe (2014), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 90 (12), 125040, arXiv:1409.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='3591 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Shalgar, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Padilla-Gay, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Tamborra (2020), JCAP 06, 048, arXiv:1911.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='09110 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Shalgar, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Tamborra (2021), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 103, 063002, arXiv:2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='00004 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Sigl, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Raffelt (1993), Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' B 406, 423.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Singh, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rentala (2021), JCAP 08, 019, arXiv:2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='02951 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Sirera, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Perez (1999), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 59, 125011, arXiv:hep-ph/9810347.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Stapleford, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' V¨a¨an¨anen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Kneller, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' McLaughlin, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Shapiro (2016), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 94 (9), 093007, arXiv:1605.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='04903 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Stodolsky, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (1979), Gen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Grav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 11, 391.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Stodolsky, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (1987), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 36, 2273.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Strumia, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Vissani (2003), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' B 564, 42, arXiv:astro-ph/0302055.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Sturrock, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (1958), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 112, 1488.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Suliga, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Beacom, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Tamborra (2022), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 105 (4), 043008, arXiv:2112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='09168 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Sumiyoshi, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Yamada, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Suzuki (2007), Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 667, 382, arXiv:0706.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='3762 [astro-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Surman, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' McLaughlin, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Hix (2006), As- trophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 643, 1057, arXiv:astro-ph/0509365.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Surman, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' McLaughlin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Ruffert, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Janka, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Hix (2008), Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 679, L117, arXiv:0803.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='1785 [astro-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Suzuki, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2008), Topics in astroparticle and underground physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Proceedings, 10th International Conference, TAUP 2007, Sendai, Japan, September 11-15, 2007, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 120, 072001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Tabrizi, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Horiuchi (2021), JCAP 05, 011, arXiv:2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='10933 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Takahashi, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Sato, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Dalhed, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Wilson (2003), Astropart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 20, 189, arXiv:astro-ph/0212195.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Takiwaki, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Kotake, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Foglizzo (2021), Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 508 (1), 966, arXiv:2107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='02933 [astro- ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Tamborra, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Hanke, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Janka, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' M¨uller, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Raf- felt, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Marek (2014a), Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 792 (2), 96, arXiv:1402.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='5418 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='SR].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Tamborra, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Huedepohl, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Raffelt, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Janka (2017), Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 839, 132, arXiv:1702.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='00060 [astro- ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Tamborra, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Raffelt, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Hanke, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Janka, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Mueller (2014b), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 90 (4), 045032, arXiv:1406.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='0006 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='SR].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Tamborra, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Raffelt, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Hudepohl, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Janka (2012), JCAP 01, 013, arXiv:1110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='2104 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='SR].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Tamborra, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Shalgar (2021), Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 71, 165, arXiv:2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='01948 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Tanabashi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (Particle Data Group) (2018), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 98 (3), 030001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Tanaka, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2018), Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 852 (2), 109, arXiv:1708.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='09101 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Telescope, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2017), https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='nasa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='gov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Tian, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Patwardhan, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Fuller (2017), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 95 (6), 063004, arXiv:1610.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='08586 [astro- ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Tomas, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Kachelriess, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Raffelt, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Dighe, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Janka, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Scheck (2004), JCAP 09, 015, arXiv:astro- ph/0407132.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Torrente-Lujan, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (1999), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 59, 073001, arXiv:hep-ph/9807361.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Turck-Chieze, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lopes (1993), Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 408, 347.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Ugliano, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Janka, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Marek, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Arcones (2016), Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 821, 69, arXiv:1602.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='06327 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Vaananen, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' McLaughlin (2016), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 93 (10), 105044, arXiv:1510.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='00751 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Vaananen, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Volpe (2011), JCAP 10, 019, arXiv:1105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='6225 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='SR].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Vaananen, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Volpe (2013), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 88, 065003, arXiv:1306.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='6372 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Vigorito, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Bruno, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Fulgione, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Molinario (LVD) (2021), PoS ICRC2021, 1111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Visinelli, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2015), Gen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Grav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 47 (5), 62, arXiv:1410.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='1523 [gr-qc].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Vissani, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2015), J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' G42, 013001, arXiv:1409.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='4710 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Vissani, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Fuller, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Cirigliano (2014a), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 89 (10), 105004, arXiv:1309.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='2628 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Vlasenko, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Fuller, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Cirigliano (2014b), arXiv:1406.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='6724 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Vlasenko, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' McLaughlin (2018), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 97 (8), 083011, arXiv:1801.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='07813 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Volpe, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2004), J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' G 30, L1, arXiv:hep-ph/0303222.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Volpe, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2005), J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' G 31, 903, arXiv:hep-ph/0501233.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Volpe, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2007), J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' G34, R1, arXiv:hep-ph/0605033 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Volpe, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2015), Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' E 24 (09), 1541009, arXiv:1506.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='06222 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='SR].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Volpe, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Auerbach, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Colo, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Suzuki, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Van Giai (2000), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' C 62, 015501, arXiv:nucl-th/0001050.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Volpe, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Vaananen, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Espinoza (2013), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 87 (11), 113010, arXiv:1302.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='2374 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Walk, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Tamborra, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Janka, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Summa, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Kresse (2020), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 101 (12), 123013, arXiv:1910.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='12971 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Wang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Cassing (1985), Annals Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 159, 328.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Weinberg, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (1967), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 19, 1264.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Wolfenstein, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (1978), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D17, 2369, [,294(1977)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Wolfenstein, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (1979), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 20, 2634.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Wright, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Kneller, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Ohlmann, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Roepke, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Scholberg, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Seitenzahl (2017), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 95 (4), 043006, arXiv:1609.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='07403 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Wu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Ambler, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Hayward, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Hoppes, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Hudson (1957), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 105, 1413.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Wu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Duan, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='-Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Qian (2016), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' B 45 752, 89, arXiv:1509.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='08975 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Wu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Fischer, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Huther, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Mart´ınez-Pinedo, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='-Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Qian (2014), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 89 (6), 061303, arXiv:1305.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='2382 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Wu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Tamborra (2017), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 95 (10), 103007, arXiv:1701.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='06580 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Wu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Tamborra, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Just, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Janka (2017), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 96 (12), 123015, arXiv:1711.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='00477 [astro- ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Wudka, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2001), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 64, 065009, arXiv:gr- qc/0010077.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Xiong, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Sieverding, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Sen, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='-Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Qian (2020), As- trophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 900 (2), 144, arXiv:2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='11414 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Xiong, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Wu, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Mart´ınez-Pinedo, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Fis- cher, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' George, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lin, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Johns (2022), arXiv:2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='08254 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Xiong, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Wu, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='-Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Qian (2019), 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='3847/1538- 4357/ab2870, arXiv:1904.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='09371 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Yamada, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (2000), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 62, 093026, arXiv:astro- ph/0002502.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Yang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Kneller (2017), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 96 (2), 023009, arXiv:1705.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='09723 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Yi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Ma, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Martin, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Duan (2019), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 99 (6), 063005, arXiv:1901.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='01546 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Yoshida, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Takahashi, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Umeda, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Ishidoshiro (2016), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 93 (12), 123012, arXiv:1606.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='04915 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Yuksel, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Beacom (2007), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 76, 083007, arXiv:astro-ph/0702613.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Yuksel, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Kistler, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Beacom, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Hopkins (2008), Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 683, L5, arXiv:0804.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='4008 [astro- ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Yvon, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (1935), Act.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' et Ind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Zhang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (Super-Kamiokande) (2015), Astropart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' 60, 41, arXiv:1311.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='3738 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Zhu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Perego, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' McLaughlin (2016), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D 94 (10), 105006, arXiv:1607.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='04671 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Ziegler, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Edwards, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Suliga, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Tam- borra, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Horiuchi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Ando, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Freese (2022), arXiv:2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='07845 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content='GA].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' Zyla, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} +page_content=' (Particle Data Group) (2020), PTEP 2020 (8), 083C01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFKT4oBgHgl3EQfcC5D/content/2301.11814v1.pdf'} diff --git a/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf b/HdE2T4oBgHgl3EQf_Alo/content/2301.04244v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..3bc5f5165aed415105a686ff0be716b8d228ea86 --- /dev/null +++ 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a/ItE0T4oBgHgl3EQfiAFl/content/tmp_files/2301.02439v1.pdf.txt b/ItE0T4oBgHgl3EQfiAFl/content/tmp_files/2301.02439v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..dd18bde1404a6cbba6474d2a5a33d9de5e6b1577 --- /dev/null +++ b/ItE0T4oBgHgl3EQfiAFl/content/tmp_files/2301.02439v1.pdf.txt @@ -0,0 +1,5963 @@ +arXiv:2301.02439v1 [math.AG] 6 Jan 2023 +Motivic Higman’s Conjecture +Jesse Vogel +Mathematical Institute, Leiden University, j.t.vogel@math.leidenuniv.nl +Abstract +The G-representation variety RG(Σg) parametrizes the representations of the fundamental +groups of surfaces π1(Σg) into an algebraic group G. Taking G to be the groups of n × n upper +triangular or unipotent matrices, we compare two methods for computing algebraic invariants of +RG(ΣG). Using the geometric method initiated by Gonz´alez-Prieto, Logares and Mu˜noz, based +on a Topological Quantum Field Theory (TQFT), we compute the virtual classes of RG(Σg) +in the Grothendieck ring of varieties for n = 1, . . . , 5, extending the results of [7]. Introducing +the notion of algebraic representatives we are able to efficiently compute the TQFT. Using the +arithmetic method initiated by Hausel and Rodriguez-Villegas, we compute the E-polynomials +of RG(Σg) for n = 1, . . . , 10. For both methods, we describe how the computations can be +performed algorithmically. +Furthermore, we discuss the relation between the representation +varieties of the group of unipotent matrices and Higman’s conjecture. The computations of this +paper can be seen as positive evidence towards a generalized motivic version of the conjecture. +1 +Introduction +Let X be a closed connected manifold, and G an algebraic group over a field k. The G-representation +variety of X is the set of group homomorphisms +RG(X) = Hom(π1(X), G). +As π1(X) is finitely generated, RG(X) can be seen as a closed subvariety of Gn for some n ≥ 0. For +X = Σg a closed surface of genus g, the G-representation variety takes the explicit form +RG(Σg) = +� +(A1, B1, . . . , Ag, Bg) ∈ G2g | +g +� +i=1 +[Ai, Bi] = 1 +� +. +(1.1) +Closely related is the G-character variety of Σg, given by the GIT quotient χG(Σg) = RG(Σg) � G, +where G acts on RG(Σg) by conjugation. The G-character variety, also known as the Betti moduli +space, plays a central role in non-abelian Hodge theory [22, 23]. In particular, for C a complex smooth +projective curve, the Betti moduli space is isomorphic to the moduli space of G-flat connections on +C by the Riemann–Hilbert correspondence, and for G = GLn it is diffeomorphic to the moduli space +of polystable G-Higgs bundles of rank n and degree 0 by the non-abelian Hodge correspondence. +1 + +Invariants. Various algebraic invariants of the G-representation variety and G-character variety +have been studied in the literature, for instance [6, 9, 12] amongst many, and in the recently resolved +P = W conjecture [10, 15]. One of these invariants is the E-polynomial, also known as the Serre +polynomial, which for a complex variety X is given by +e(X) = +� +k,p,q +(−1)k hk;p,q +c +(X) upvq ∈ Z[u, v], +where hk;p,q +c +(X) are mixed Hodge numbers of the compactly supported cohomology of X. +The +E-polynomial can be shown [9, 12] to satisfy +e(X) = e(Z) + e(X \ Z) +and +e(X ×C Y ) = e(X) e(Y ), +for complex varieties X and Y , where Z ⊂ X is a closed subvariety with open complement X \ Z. +The Grothendieck ring of varieties K(VarC), which will be defined in Section 2, is the universal ring +for invariants with these properties, so in particular there is a morphism +e : K(VarC) → Z[u, v] +(1.2) +which sends the class [X] of a complex variety X to its E-polynomial e(X). In this sense, the class +of a variety X in K(VarC), also known as the virtual class of X, is a more refined invariant than +the E-polynomial. +Another such invariant is the point count over a finite field, that is, there is a morphism +# : K(VarFq) → Z +which sends the class [X] of a variety X over Fq to |X(Fq)|. A remarkable theorem by Katz [9, +Theorem 6.1.2] states that if X is a complex variety with a spreading-out ˜X over a finitely generated +Z-algebra R ⊂ C, such that |( ˜X ×R Fq)(Fq)| is a polynomial in q for all ring morphisms R → Fq, +then the E-polynomial of X is precisely this polynomial in q = uv. For example, for the affine line +X = A1 +C, we have |A1 +Fq(Fq)| = q and e(X) = uv, and for this reason we usually write q = uv and +q = [A1 +k], depending on the context. +Previous work. Over the last years, various methods have been developed and used to compute +algebraic invariants of the G-representation varieties of Σg. In [9], Hausel and Rodriguez–Villegas +initiated the arithmetic method in order to compute the E-polynomials of the GLn-representation +varieties and twisted GLn-character varieties. This method makes use of Katz’ theorem to reduce +the problem to counting the points of RG(Σg) over finite fields Fq. Frobenius’ formula, as proven +in [9, Equation 2.3.8], gives an expression for the point count of the representation variety for finite +groups G, +|RG(Σg)| = |G| +� +χ∈ � +G +� |G| +χ(1) +�2g−2 +reducing the problem to the study of the irreducible (complex) characters χ ∈ �G of G over finite fields +Fq. In [16], this method was used to compute the E-polynomials of twisted SL2-character varieties, +and in [2] it was used to compute the E-polynomials of the GL3- and SL3-character varieties, and +those of the GL2- and SL2-character varieties of non-orientable surfaces. +2 + +Logares, Mart´ınez, Mu˜noz and Newstead used geometric arguments in order to compute E-polynomials +of the SL2-character varieties [12, 14] and PGL2-character varieties [13]. This approach was refor- +mulated and generalized by Gonz´alez-Prieto, Logares and Mu˜noz [6], who constructed a Topological +Quantum Field Theory (TQFT) that computes the virtual class of RG(Σg) in K(Vark). Using this +TQFT, they computed the virtual class of the AGL1-representation variety, and in [4] it was used +to compute the virtual class of the SL2-representation variety and the twisted SL2-character variety. +Concretely, this method defines a lax monoidal functor Z : Bord2 → K(Vark)-Mod, from the cate- +gory of (pointed) 2-bordisms to the category of K(Vark)-modules, such that Z(Σg)(1) = [RG(Σg)]. +Using functoriality, it suffices to compute +Z( +) : K(Var/G) → K(Vark), +Z( +) : K(Var/G) → K(Var/G), +and +Z( +) : K(Vark) → K(Var/G) +to obtain Z(Σg) by composition. +Outline. The goal of this paper is to compute the E-polynomials and virtual classes of the G- +representation varieties of Σg for G equal to the group Tn of upper triangular n × n matrices or the +group Un of unipotent n × n matrices, making use of both the TQFT method and the arithmetic +method. +Therefore, this work is an extension of [7], in which the virtual class of RTn(Σg) was +computed for G = Tn and 1 ≤ n ≤ 4 using the TQFT method. In particular, in Section 3 we will +compute the virtual classes of RTn(Σg) and RUn(Σg) for 1 ≤ n ≤ 5. +In this paper, we aim for an organized and concise framework. +Rather than working with the +explicit description of the TQFT as done in [4, 5, 6], we will be working with the essential maps as in +Definition 3.1, avoiding the need for a ‘reduction’ or ‘simplification’ of the TQFT, as appears in [4, 7]. +Introducing the notion of algebraic representatives in Subsection 2.3, we can take maximal advantage +of the lack of monodromy in the conjugacy classes of Tn and Un in order to simplify the computations. +Moreover, the computations are organized in such a way that most of the computations for Tn can +be reused for Un. +Furthermore, in Section 4, we will apply the arithmetic method to compute E-polynomials of +RTn(Σg) and RUn(Σg) for 1 ≤ n ≤ 10. In particular, we compute the representation theory of +Tn and Un over finite fields Fq, encoded in the representation zeta function. These zeta functions +can be computed using a recursive algorithm, Algorithm 4.14. +Results. In Theorem 3.12, the virtual class of the T5-representation variety RT5(Σg) is given, and in +Theorem 3.14, the virtual classes of the Un-representation varieties RUn(Σg) are given for 1 ≤ n ≤ 5. +In Theorem 4.17, the representation zeta functions of Un are given for 1 ≤ n ≤ 10, and in Theorem +4.18 those of Tn for 1 ≤ n ≤ 10. These directly determine the E-polynomials of RUn(Σg) and +RTn(ΣG), and for 1 ≤ n ≤ 5 these E-polynomials can be seen to agree with the virtual classes +through the map (1.2). +The computation of the representation zeta functions of Tn and Un can be seen as a generalization +of [24] and [19], where the number of conjugacy classes of Un(Fq), denoted k(Un(Fq)), was computed +as a polynomial in q for 1 ≤ n ≤ 13 and 1 ≤ n ≤ 16, respectively. Namely, since the number of +3 + +irreducible representations of a finite group equals its number of conjugacy classes, evaluating the +representation zeta function ζG(s) in s = 0 yields k(G). This relates the zeta functions ζUn(Fq)(s) to +a conjecture by Higman first posed in [11]. +Conjecture 1.1 (Higman). For every n ≥ 1, the number of conjugacy classes of Un(Fq) is a +polynomial in q. +As mentioned, this conjecture has been verified for 1 ≤ n ≤ 16. However, as shown in [8], there +exist pattern subgroups of Un whose number of conjugacy classes over Fq is not polynomial in q. +In [19], a concrete such pattern subgroup P ⊂ U13 is given, and moreover it is shown that k(U59) +can be expressed as a Z[q]-linear combination of k(P) and k(P ′) for other pattern subgroups P ′. +This suggests the conjecture might fail for n = 59, unless the non-polynomial terms in the linear +combination cancel. +Looking at the results of Theorem 4.17 and 4.18, we pose a generalized version of Higman’s conjec- +ture. +Conjecture 1.2 (Generalized Higman). For any n ≥ 1, the representation zeta function ζUn(Fq)(s) +is a polynomial in q and q−s, and ζTn(Fq)(s) is a polynomial in q, q−s and (q − 1)−s. +Similarly, looking at the virtual classes of RTn(Σg) and RUn(Σg), we pose a motivic version of the +conjecture. Note that Conjecture 1.3 does not imply Conjecture 1.2, since a variety X could have +virtual class in Z[q] without having polynomial count, e.g. X = {(x, y) | x2 + y2 = 1}. +Conjecture 1.3 (Motivic Higman). For any g ≥ 0 and n ≥ 1, the virtual classes of the Tn- and +Un-representation variety of Σg are polynomial in q = [A1 +C]. +2 +Grothendieck ring of varieties +Let k be a field. By a variety over k we understand a reduced separated scheme of finite type over +a field k, not necessarily irreducible. All varieties are assumed to be over k. +Definition 2.1. Let S be a variety. The Grothendieck ring of varieties over S, denoted K(Var/S), +is the quotient of the free abelian group on isomorphism classes of varieties over S, by relations of +the form +[X] = [Z] + [U], +for all closed subvarieties Z ⊂ X with open complement U = X \ Z. Multiplication on K(Var/S) +is defined by the fiber product over S, +[X] · [Y ] = [X ×S Y ], +and this is easily seen to give a ring structure on K(Var/S). In particular, the unit element is the +class 1S = [S] of S over itself, and the zero element is the class 0S = [∅]. The class [X] of a variety +X in K(Var/S) is also called its virtual class. +4 + +In order to distinguish between classes over different bases, we write [X]S for the class of X in +K(Var/S), and when the base is S = Spec k, we simply write [X]. Furthermore, we denote by +q = [A1 +k] ∈ K(Vark) the class of the affine line. +For any variety S, the ring K(Var/S) naturally has the structure of a K(Vark)-module, where scalar +multiplication is given by +[T ] · [X]S = [T × X]S +for all varieties T and varieties X over S, and extended linearly. +For any morphism f : T → S of varieties, there are induced maps +f! : K(Var/T ) → K(Var/S), +[X]T �→ [X]S, +f ∗ : K(Var/S) → K(Var/T ), +[X]S �→ [X ×S T ], +which are seen to be K(Vark)-module morphisms. Moreover, the map f ∗ is a ring morphism. +For any variety S, let c : S → Spec k denote the final morphism. In particular, for any variety X +over S, we have +c![X]S = [X]. +2.1 +Stratifications +Definition 2.2. Let X be a variety. A stratification of X is a collection of locally closed subvarieties +{Xi}i∈I such that � +i∈I Xi = X and Xi ∩ Xj = ∅ for i ̸= j. +Lemma 2.3. Let X be a variety over S, with stratification {Xi}i∈I. Then only finitely many of the +Xi are non-empty, and � +i∈I[Xi]S = [X]S. +Proof. Proof by induction on the dimension of X. If dim X = 0, then X is a finite set of points, +and the result is clear. Now assume that dim X > 0 and that the result holds for all varieties of +dimension less than dim X. +First consider the case where X is irreducible. Some U = Xi contains the generic point of X, and +is therefore open. The complement Z = X \ U is of smaller dimension than X and is stratified by +the other Xi. Since [X]S = [Z]S + [U]S, the result follows from the induction hypothesis. +Now consider the case where X is reducible. Take an irreducible component and remove the inter- +sections with the other irreducible components, which gives an irreducible open subset U ⊂ X. The +complement Z = X \ U is a closed subvariety with fewer irreducible components than X. Since +{Z ∩ Xi}i∈I is a stratification of Z and {U ∩ Xi}i∈I a stratification of U, apply induction on the +number of irreducible components of X to find that only finitely many are non-empty, and we have +[U]S = +� +i∈I +[U ∩ Xi]S +and +[Z]S = +� +i∈I +[Z ∩ Xi]S. +Since [Xi]S = [U ∩ Xi]S + [Z ∩ Xi]S for all i, it follows that [X]S = [U]S + [Z]S = � +i∈I[Xi]S. +5 + +A stratification {Si}i∈I of S induces a decomposition of the Grothendieck ring K(Var/S). This is +a decomposition as K(Vark)-algebras. +Proposition 2.4. Let S be a variety with stratification {Si}i∈I. Then +K(Var/S) ∼= +� +i∈I +K(Var/Si) +as K(Vark)-algebras. +Proof. Writing fi : Si → S for the inclusions, the isomorphism is explicitly given by +X �→ (f ∗ +i X)i∈I +with inverse +� +i∈I +(fi)∗Xi ←� (Xi)i∈I. +Note that the sum is well-defined since by Lemma 2.3 only finitely many Si are non-empty. These +maps are easily seen to be inverse to each other, and morphisms of K(Vark)-algebras. +Notation 2.5. For any X ∈ K(Var/S), we will write X|Si ∈ K(Var/Si) for the components of the +image of X under this isomorphism. +2.2 +Special algebraic groups +Special algebraic groups were first introduced by Serre [20]. +Definition 2.6. An algebraic group G over k is special if any ´etale G-torsor is locally trivial in the +Zariski topology. Equivalently, G is special if for all k-schemes X, the natural map H1 +Zar(X, G) → +H1 +´et(X, G) is an isomorphism. +Lemma 2.7. Let G be a special algebraic group. Then for every G-torsor of varieties X → S, we +have [X]S = [G] · 1S in K(Var/S). +Proof. Since G is special, the G-torsor X → S is Zariski-locally trivial, so there exists a stratification +{Si}i∈I of S such that X×SSi ∼= G×Si. Now, [X]S = � +i∈I[G×Si]S = [G]·� +i∈I[Si]S = [G]·1S. +Proposition 2.8. Let 1 → N → G → H → 1 be an exact sequence of algebraic groups. If N and +H are special, then so is G. +Proof. Since G being special is equivalent to H1 +´et(X, G) = H1 +Zar(X, G) for all k-schemes X, the result +follows from long exact sequences in cohomology, and the five lemma. +Alternatively, any G-torsor X → S can be written as the composition of the N-torsor X → X/N and +the H-torsor X/N → X/G ∼= S. As H is special, there exist opens Si ⊂ S such that (X/N) ×S Si ∼= +H × Si. Pulling back the N-torsor X ×S Si → H × Si along Si +(1,id) +−−−→ H × Si gives an N-torsor +Yi → Si, which is also Zariski-locally trivial as N is special. Hence, there exist opens Sij ⊂ Si such +that Yi ×Si Sij ∼= N × Sij. There is now a natural morphism G × Sij → X ×S Sij of G-torsors over +Sij, which must be an isomorphism. Therefore, X → S is Zariski-locally trivial. +6 + +Example 2.9. +■ By Hilbert’s Theorem 90, the general linear groups GLn are special [18, Propo- +sition III.4.9, Lemma III.4.10]. +■ From the exact sequence 0 → SLn → GLn +det +−−→ Gm → 1 follows by Proposition 2.8 that SLn +is also special. +■ The group Z/2Z is not special. For example, for char(k) ̸= 2, the Z/2Z-torsor A1 +k \ {0} → +A1 +k \ {0} given by x �→ x2 is not locally trivial in the Zariski topology. +Lemma 2.10. Let k be an algebraically closed field. Any unipotent group U ⊂ Un over k is special. +Proof. By [17, Theorem 17.24], U can be realized as an extension of copies of Ga. +From [18, +Proposition III.3.7] follows that Ga is special, so the result follows from Proposition 2.8. +Lemma 2.11 (Motivic Orbit-Stabilizer Theorem). Let G be an algebraic group acting on a variety +X over a field k. If ξ ∈ X is a point such that Stab(ξ) is special, then +[G] = [Stab(ξ)] · [Orbit(ξ)]. +Proof. The map G → Orbit(ξ) given by g �→ gξg−1 is a Stab(ξ)-torsor, hence Zariski-locally trivial, +so the result follows from Lemma 2.7. +Example 2.12. The above lemma does not hold without the condition that Stab(ξ) is special. +Namely, for char(k) ̸= 2, consider G = Gm acting on X = A1 +k via t · x = t2x. Then, Orbit(1) = +A1 +k \ {0} and Stab(1) = Z/2Z, but [G] ̸= 2(q − 1). +2.3 +Algebraic representatives +Definition 2.13. Let G be an algebraic group over k, and let X be a variety with a transitive +G-action. A point ξ ∈ X is an algebraic representative for X if, Zariski-locally on X, there exists a +morphism of varieties γ : X → G such that x = γ(x) · ξ for all x ∈ X. Equivalently, ξ ∈ X is an +algebraic representative if the Stab(ξ)-torsor +Pξ = {(x, g) ∈ X × G | x = g · ξ} +πX +−−→ X +is Zariski-locally trivial. +Remark 2.14. If there exists an algebraic representative ξ ∈ X, then all ξ′ ∈ X are algebraic +representatives, with γ′ equal to γ post-composed with some translation on G. +Example 2.15. Algebraic representatives need not always exist. Suppose char(k) ̸= 2 and consider +G = Gm acting on X = A1 +k \ {0} by t · x = t2x. Then X does not have an algebraic representative. +Namely, without loss of generality we may take ξ = 1, but then the Z/2Z-torsor +Pξ = {(x, t) ∈ X × G | x = t2} +πX +−−→ X +is non-trivial. +7 + +Example 2.16. Suppose char(k) ̸= 2 and consider the element A = +�−1 0 +0 +1 +� +in PGL2, whose +stabilizer +��x 0 +0 1 +�� +⊔ +��0 x +1 0 +�� +∼= Gm ⋊ Z/2Z is not special. +Namely, the conjugacy class of A +consists of all elements with trace 0, and since [PGL2] = q3 − q is not divisible by [{A ∈ PGL2 | +tr(A) = 0}] = q2, the stabilizer of A is not special by Lemma 2.11. +Definition 2.17. Let G be an algebraic group over k, and let X be a variety with a G-action. +A family of algebraic representatives for X is a morphism of varieties ρ : X → T with a section +ξ : T → X such that, Zariski-locally on X, there exists a morphism of varieties γ : X → G such that +x = γ(x) · ξ(ρ(x)) for all x ∈ X. +Example 2.18. Consider G = T2 acting on X = +��a b +0 1 +� +| a ̸= 0, 1 +� +by conjugation. Then X +has a family of representatives given by T = +��a 0 +0 1 +� +| a ̸= 0, 1 +� +, with ξ and ρ the inclusion and +projection, and +γ +� +a +b +0 +1 +� += +� +1 +b +1−a +0 +1 +� +. +Proposition 2.19. Let k be an algebraically closed field. Consider G = Tn for some n ≥ 1, acting +by conjugation on a unipotent conjugacy class U ⊂ Tn. Then U has an algebraic representative. +Proof. Pick any point ξ ∈ U. We need show that the Stab(ξ)-torsor +P = {(u, g) ∈ U × G : u = gξg−1} +πU +−−→ U +is locally trivial in the Zariski topology. In particular, it suffices to show that Stab(ξ) is a special +group. Since Tn is triangularizable, the subgroup Stab(ξ) is so as well, hence we can write +1 → U → Stab(ξ) → D → 1, +where U is the maximal normal unipotent subgroup, which is special by Lemma 2.10. Furthermore, +there is a natural splitting D → Stab(ξ), and any d ∈ D must satisfy dii = djj whenever ξij ̸= 0. +Hence, D is a product of copies of Gm, which is also special. Now the result follows from Proposition +2.8. +The following lemma shows why it is useful to have algebraic representatives. +Proposition 2.20. Let S be a variety with a G-action and a family of algebraic representatives +ξ : T → S and ρ : S → T . Then for any morphism f : Y → S and G-equivariant morphism +g : X → S, we have +[X ×S Y ]S = [(X ×S T ) ×T Y ]S ∈ K(Var/S), +where (X ×S T ) ×T Y is seen as a variety over S via the composition f ◦ πY . +Proof. Zariski-locally on X, there is a commutative diagram +X ×S Y +(X ×S T ) ×T Y +S +ϕ +ψ +f◦πY +8 + +where ϕ(x, y) = (γ(f(y))·x, ρ(f(y)), y) and ψ(x, t, y) = (γ(f(y))·x, y). Note that ϕ and ψ are easily +seen to be well-defined over S and inverse to each other. +In the case of algebraic representatives, i.e. when T is a point, we obtain the following corollaries. +Corollary 2.21. Let S be a variety with a G-action and an algebraic representative ξ ∈ S. Then +for any morphism f : Y → S and G-equivariant morphism g : X → S, we have +[X ×S Y ]S = [X|ξ] · [Y ]S ∈ K(Var/S). +Corollary 2.22. Let S be a variety with G-action and algebraic representative ξ ∈ S. Then for any +G-equivariant map f : X → S, we have +[X]S = [X] +[S] · 1S ∈ K(Var/S). +Proof. Apply the previous corollary with f = idS to find that +[X]S = [X|ξ] · 1S. +Looking at this equality in K(Vark), we see that [X|ξ] = [X]/[S]. +2.4 +Algorithmic computations in K(Vark) +Let A = {a1, . . . , an} be a finite set, and let F and G be finite subsets of k[A]. Then we write +X(A, F, G) +for the reduced locally closed variety of An +k given by f = 0 for all f ∈ F and g ̸= 0 for all g ∈ G. +In this subsection we will describe strategies for computing the virtual class of varieties of this form +in K(Vark). These strategies are combined into a recursive algorithm, Algorithm 2.24, which will +be used in Section 3 to compute the virtual class of the representation variety RG(Σg). We remark +already that the algorithm will not be a general recipe for computing virtual classes, since it is +allowed to fail. In fact, whenever the algorithm does not fail, it will return the virtual class of +X(A, F, G) as a polynomial in q = [A1 +k], and it is clear that not all virtual classes are of this form. +However, it turns out that the algorithm is sufficiently general for the purposes of this paper. +The same algorithm was used in [7, Appendix A] to compute the virtual class of RTn(Σg) for +1 ≤ n ≤ 4. An implementation of this algorithm can be found at [25]. +Notation 2.23. For a ∈ A, f ∈ k[A] and u ∈ k[A \ {a}], we write eva(f, u) for the evaluation +of f in a = u. When u, v ∈ k[A \ {a}], we denote by eva(f, u/v) the evaluation of f in a = u/v +multiplied by vdega(f), so that eva(f, u/v) ∈ k[A \ {a}]. For subsets F ⊂ k[A], we write eva(F, −) = +{eva(f, −) : f ∈ F}. +Algorithm 2.24. Input: Finite sets A, F and G as above. +Output: The virtual class of X(A, F, G) as a polynomial in q = [A1 +k]. +9 + +1. If F contains a non-zero constant or if 0 ∈ G, then X = ∅, so return [X] = 0. +2. If F = G = ∅ or A = ∅, then X = A#A +k +, so return [X] = q#A. +3. If all f ∈ F and g ∈ G are independent of some a ∈ A, then X ∼= A1 +k × X′ with X′ = +X(A \ {a}, F, G), so return [X] = q[X′]. +4. If f = un (with n > 1) for some f ∈ F and u ∈ k[A], then we can replace f with u, not +changing X. That is, X = X(A, (F \ {f}) ∪ {u}, G). Similarly, if g = un (with n > 1) for +some g ∈ G and u ∈ k[A], then X = X(S, F, (G \ {g}) ∪ {u}). +5. If some f ∈ F is univariate in a ∈ A, and factors as f = (a − α1) · · · (a − αm) for some αi ∈ k, +then return [X] = �m +i=1[Xi] with +Xi = X(A \ {a}, eva(F \ {f}, αi), eva(G, αi)). +6. Suppose f = uv for some f ∈ F and non-constant u, v ∈ k[A]. Then X can be stratified by +X ∩ {u = 0} and X ∩ {u ̸= 0, v = 0}, so return [X] = [X1] + [X2] with +X1 = X(A, (F \ {f}) ∪ {u}, G), +X2 = X(A, (F \ {f}) ∪ {v}, G ∪ {u}). +7. Suppose f = au + v for some f ∈ F, a ∈ A and u, v ∈ k[A] with u and v independent of a and +u non-zero. Then X can be stratified by X ∩ {u = 0, v = 0} and X ∩ {u ̸= 0, a = −v/u}, so +return [X] = [X1] + [X2] with +X1 = X(A, (F \ {f}) ∪ {u, v}, G), +X2 = X(A, eva(F \ {f}, −v/u), eva(G, −v/u) ∪ {u}). +8. If char(k) ̸= 2, suppose f = a2u+av+w for some f ∈ F, a ∈ S and u, v, w ∈ k[A] with u, v and +w independent of a, and u non-zero. Moreover, suppose that the discriminant D = v2 − 4uw +is a square, that is, D = d2 for some d ∈ k[A]. Then return [X] = [X1] + [X2] + [X3] + [X4], +with +X1 = X(A, (F \ {f}) ∪ {u, av + w}, G), +X2 = X(A, eva(F \ {f}, −v/2u) ∪ {D}, eva(G, −v/2u) ∪ {u}), +X3 = X(A, eva(F \ {f}, (−v − d)/2u), eva(G, (−v − d)/2u) ∪ {u, D}), +X4 = X(A, eva(F \ {f}, (−v + d)/2u), eva(G, (−v + d)/2u) ∪ {u, D}). +9. If G ̸= ∅, pick any g ∈ G, and return [X] = [X1] − [X2] with +X1 = X(A, F, G \ {g}), +X2 = X(A, F ∪ {g}, G). +10. Fail. +10 + +3 +TQFT method +The idea of the TQFT method is to define a Topological Quantum Field Theory (TQFT), that is, +a lax monoidal functor Z : Bordn → K(Vark)-Mod from the category of (pointed) n-bordisms to +the category of modules over K(Vark), such that the invariant Z(M)(1) ∈ K(Vark) associated to a +closed manifold X is precisely the class [RG(X)]. Then, functoriality allows for the closed manifold +X to be seen as a composition of simpler bordisms, so that computing the image under Z of the +simpler bordisms gives the invariant of X. +For the purpose of this paper, it suffices to take n = 2 and define maps Z( +), Z( +) and Z( +) +that mimic the TQFT, rather than giving the full description of the TQFT. Although these maps do +not define an actual TQFT, they compute the same invariant (Theorem 3.2) and have the advantage +of being easier to define and to deal with. For an elaborate discussion on the TQFT method, we +refer to [4, 12]. +Definition 3.1. Let G be an algebraic group over k. Define the following K(Vark)-module mor- +phisms, +Z( +) : K(Vark) → K(Var/G), +1 +�→ + + +{1} +G + + , +Z( +) : K(Var/G) → K(Vark), + + +X +G +f + + �→ +� +f −1(1) +� +, +Z( +) : K(Var/G) → K(Var/G), + + +X +G +f + + �→ + + +X × G2 +(x, A, B) +G +f(x)[A, B] + + . +Theorem 3.2. For any algebraic group G and g ≥ 0, we have +[RG(Σg)] = Z( +) ◦ Z( +)g ◦ Z( +)(1). +Proof. This follows directly from the explicit form of the G-representation variety of Σg (1.1). +This theorem shows that, in order to compute [RG(Σg)], we must understand the map Z( +). In +the following subsections, we will take G = Tn with 1 ≤ n ≤ 5 and describe how to compute the +matrix associated to Z( +) when restricted to the K(Vark)-submodule of K(Var/G) generated +by the unipotent conjugacy classes. To be precise, we will take G equal to +˜Tn = + + + + + + + + + + + + + + + + + + + + + + + + + + +a1,1 +a1,2 +· · · +a1,n−1 +a1,n +0 +a2,2 +· · · +a2,n−1 +a2,n +... +... +... +... +... +0 +0 +· · · +an−1,n−1 +an−1,n +0 +0 +· · · +0 +1 + + + + + + + + + +| ai,i ̸= 0 + + + + + + + + + + + + + + + + + +, +11 + +and using the isomorphism Tn ∼= Gm × ˜Tn, it follows that +[RTn(Σg)] = [RGm(Σg)][R˜Tn(Σg)] = (q − 1)2g[R˜Tn(Σg)]. +(3.1) +The reason to focus on ˜Tn rather than Tn is to simplify the equations a little bit by reducing the +number of variables. +In Subsection 3.1 we will give an example with G = ˜T2 in order to demonstrate the method. For +higher n, the process will be analogous. In particular, we will work with a K(Vark)-submodule +of K(Var/G) generated by the unipotent conjugacy classes of G. Then the matrix representing +Z( +) is computed with respect to these generators. +To accomplish this, we need to determine the unipotent conjugacy classes of G, equations describing +them, and an algebraic representatives for each one. This data can be computed automatically, and +is done in Subsection 3.2. +In Subsection 3.4 we will compute the matrix representing Z( +). +The final results will be +presented in Subsection 3.5. +3.1 +Example G = ˜T2 +In order to demonstrate the method, we will compute the TQFT, that is, the maps in Definition +3.1, for the group G = ˜T2 by hand. This group is also known as the group of affine transformations +of the line +AGL1 = +�� +a +b +0 +1 +� +| a ̸= 0 +� +. +In particular, the corresponding representation variety RAGL1(Σg) parametrizes flat rank one affine +bundles over Σg. +As the computations below will show, the map Z( +) restricts to an endomorphism of the +K(Vark)-submodule of K(Var/G) generated by the classes of +I = {1} → G +and +J = +�� +1 +x +0 +1 +� +| x ̸= 0 +� +→ G. +Therefore, we will represent the maps as matrices with respect to these generators. In particular, it +is clear that +Z( +) = +� +1 +0 +� +and +Z( +) = +� +1 +0 +� +. +Proposition 3.3. The map Z( +) is, with respect to the generators I and J , represented by the +matrix +Z( +) = q2 +� +q − 1 +(q − 2)(q − 1) +q − 2 +q2 − 3q + 3 +� +. +Proof. First, let us compute Z( +)(I)|I = +�� +(A, B) ∈ ˜T2 +2 | [A, B] = 1 +�� +. Writing A = +� +a +b +0 +1 +� +12 + +and B = +� +x +y +0 +1 +� +, the equation [A, B] = 1 translates to ay − bx + b − y = 0. We distinguish the +following cases: +■ If a = 1, then b(x − 1) = 0, so either b = 0 or x = 1. Hence, the virtual class of this stratum is +q +(y) +((q − 1) +(b=0) ++ +q +(x=1) +− +1 +(b=0 +x=1) +) = 2q(q − 1). +■ If a ̸= 1, then y = b(x − 1)/(a − 1), yielding a virtual class of q(q − 2)(q − 1). +Together, Z( +)(I)|I = q2(q − 1)I. Next, note that +c!Z( +)(I)|I + [J ]c!Z( +)(I)|J = [˜T2]2 = q2(q − 1)2, +from which follows that Z( +)(I)|J = q2(q − 2)J , using Corollary 2.22 and the fact that J has +an algebraic representative. +The next coefficient can be computed from the previous one as +Z( +)(J )|I = ([J ]c!Z( +)(I)|J )I = q2(q − 2)(q − 1)I. +Finally, for the last coefficient, note that +c!Z( +)(J )|I + [J ]c!Z( +)(J )|J = [J ][˜T2]2 = q2(q − 1)3, +from which follows that Z( +)(J )|J = q2(q2 − 3q + 3)J . +In order to apply Theorem 3.2, it is convenient to diagonalize the map Z( +) as PDP −1 with +P = +� +1 − q +1 +1 +1 +� +and +D = +� +q2 +0 +0 +q2(q − 1)2 +� +. +Now from Theorem 3.2, and straightforward matrix multiplication, it follows that +[R˜T2(Σg)] = q2g−1(q − 1)2g + q2g−1(q − 1). +3.2 +Describing conjugacy classes +The aim of this subsection is to describe the conjugacy classes of ˜Tn, with a focus on the unipotent +conjugacy classes. In particular, we will find algebraic representatives for the unipotent conjugacy +classes, and families of algebraic representatives for the non-unipotent conjugacy classes. Further- +more, we will find equations describing the unipotent conjugacy classes, and finally, compute the +virtual class of the unipotent conjugacy classes as well as that of the stabilizers of their representa- +tives. All of this data is needed in the later subsections in order to compute the matrix of Z( +) +with respect to the generators given by the unipotent classes. +Representatives. In order to find the number of unipotent conjugacy classes of ˜Tn, and represen- +tatives for each one, one can use Belitskii’s algorithm as described in [3], which works for 1 ≤ n ≤ 5. +13 + +Note that the representatives will be automatically algebraic by Proposition 2.19. Denote the re- +sulting unipotent conjugacy classes by U1, . . . , UM and the respective algebraic representatives by +ξ1, . . . , ξM. The number M of unipotent conjugacy classes, depending on n, is given by the following +table. +n +1 +2 +3 +4 +5 +M +1 +2 +5 +16 +61 +Alternatively, one can use the qualitative result of Belitskii’s algorithm that for 1 ≤ n ≤ 5 every +unipotent matrix can be conjugated in ˜Tn to a matrix containing only 0’s and 1’s. This gives a +finite number of matrices (2n(n−1)/2) which can easily be partitioned based on whether or not they +are conjugate. Now take one representative out of each conjugacy class. +Next, we describe the non-unipotent conjugacy classes of ˜Tn in terms of families depending on +their diagonal. Define a diagonal pattern to be a partition of the set {1, 2, . . ., n}. Then, for any +matrix A ∈ ˜Tn, the diagonal pattern δA of A is the partition such that i and j are equivalent if +Aii = Ajj. Note that two matrices A and B in ˜Tn are conjugate only if their diagonals coincide, +but not necessarily if. Now, we look at the following families of conjugacy classes: +Cδ,i = {A ∈ ˜Tn | δA = δ and A ∼ diag(A) + ξi − 1}, +for any diagonal pattern δ and 1 ≤ i ≤ M, where diag(A) denotes the diagonal part of A. Any such +Cδ,i has a family of representatives +ξδ,i : Cδ,i = {A ∈ ˜Tn | δA = δ and A = diag(A) + ξi − 1} → Cδ,i +given by inclusion. Of course, for some diagonal matrices D it may happen that D + ξi − 1 and +D + ξj − 1 are conjugate while i ̸= j, but such duplicates are easily filtered out. After this filtering, +we obtain families of conjugacy classes C1, . . . , CN, where the number N is given, depending on n, +by the following table. +n +1 +2 +3 +4 +5 +N +2 +3 +12 +61 +372 +Note that we can choose our indices in such a way that the ξi coincide with the unipotent represen- +tatives for 1 ≤ i ≤ M. +Equations. +Next, we want to find equations describing the unipotent conjugacy classes Ui for +1 ≤ i ≤ M. +For simplicity we will compute equations for the closure Ui rather than Ui. +In +Subsection 3.3 it will be shown that this is sufficient. The closure Ui is the closure of the image of +the map +fi : G → G, +g �→ gξig−1. +Since G is affine, fi can equivalently be described by a morphism of rings +f # +i +: OG(G) → OG(G). +In particular, the closure Ui corresponds to the ideal Ii ⊂ OG(G) which is the kernel of f # +i . Gen- +erators for such these ideals can be computed using Gr¨obner basis [1], and this gives us the desired +equations. In particular, we use [1, Theorem 2.4.2] in order to compute the kernel of f # +i . +14 + +Orbits and stabilizers. For any A ∈ ˜Tn, in order to compute the virtual class of the orbit of A, +we can use Lemma 2.11. Indeed, as we have seen in Proposition 2.19, the stabilizer Stab(A) of any +A ∈ ˜Tn is special. In order to compute the virtual class of the stabilizer of A, we use Algorithm +2.24, since we have an explicit description +Stab(A) = {B ∈ ˜Tn | AB − BA = 0}. +Regarding the families of conjugacy classes Ci, we will make use of the following lemma. +Lemma 3.4. For every 1 ≤ i ≤ N, there is an isomorphism +Ci ∼= Ci × Orbit(A) +for every A ∈ Ci. +Proof. Note that the fibers of Ci → Ci are given by Orbit(A) ∼= G/Stab(A), so it suffices to show +that Stab(A) is constant for all A ∈ Ci. As there are only a finite number of cases to consider, +this can easily be verified explicitly. Alternatively, note that B ∈ Stab(A) if and only if for all +1 ≤ i ≤ j ≤ n, +Bij(Aii − Ajj) + +j +� +k=i+1 +BkjAik − +j−1 +� +k=i +BikAkj = 0. +(∗) +We claim that Bij = 0 for all i ≤ j such that Aii ̸= Ajj. The result follows from this claim, as +Aij ∈ {0, 1} for i ̸= j, so the solutions to (∗) will be independent of A. We proof the claim by +induction on j − i, the case j − i = 0 being trivial. For the general case, take i ≤ j such that +Aii ̸= Ajj. if there exists a k ∈ {i + 1, . . . , j} such that Aki ̸= 0, then Akk = Aii ̸= Ajj, so Bkj = 0. +Similarly, if there exists a k ∈ {i, . . . , j − 1} such that Akj ̸= 0, then Akk = Ajj ̸= Aii so Bki = 0. +Therefore, (∗) reduces to Bij = 0. +3.3 +Transition matrix +Let X be a variety with stratification Xi, and let Y be a variety over X. The goal of this subsection +is to show that, in order to compute the classes [Y ∩ Xi]X, it is sufficient to compute the classes +[Y ∩ Xi]X instead, making use of an inclusion–exclusion principle. +Example 3.5. Suppose X is stratified by X0 ⊂ X closed and its open complement X1 = X \ X0. +If we want to compute [Y ∩ X0] and [Y ∩ X1], then computing the latter would likely result in the +computation [Y ∩ X] − [Y ∩ X0], which shows that the result of the computation of [Y ∩ X0] can be +reused. Therefore, instead of computing [Y ∩X0] and [Y ∩X1], one can compute [Y ∩X0] = [Y ∩X0] +and [Y ∩ X1] = [Y ∩ X] = [Y ], from which formally follows that [Y ∩ X1] = [Y ∩ X1] − [Y ∩ X0]. +Lemma 3.6. Let X be a topological space with stratification {Xi}i∈I. Then Xi = Xj if and only if +i = j. +Proof. For each i ∈ I, write Xi = Zi ∩ Ui for some closed Zi ⊂ X and open Ui ⊂ X. Without loss +of generality, we may assume Zi = Xi. Now, if Xi = Xj for some i, j ∈ I, then both Xi and Xj are +15 + +open and dense in Xi = Xj, so they must intersect. But this contradicts the assumption that Xi +and Xj are disjoint, since they are part of the stratification. +Definition 3.7. Let X be a variety over S with a stratification {Xi}i∈I. Put a partial ordering +on I where i ≤ j if and only if Xi ⊂ Xj. Reflexivity and transitivity are clear, and anti-symmetry +follows from the above lemma. Hence, the equalities +[Xi]S = [Xi]S − +� +j j} of n × n unipotent +matrices over a finite field Fq. Define the subgroup N ⊂ Un(Fq) as the kernel of +Un(Fq) → Un−1(Fq), +A �→ (Aij)n−1 +i,j=1, +so that the quotient Un(Fq)/N is isomorphic to Un−1(Fq). Now we have a split exact sequence +0 +N +Un(Fq) +Un−1(Fq) +0, +which yields a semidirect decomposition Un(Fq) = N ⋊Un−1(Fq), where N is abelian. Moreover, for +any unipotent subgroup U ⊂ Un(Fq), the above exact sequence can be intersected with U to obtain +0 +U ∩ N +U +U ∩ Un−1(Fq) +0, +yielding a semidirect decomposition U = (U ∩ N) ⋊ (U ∩ Un−1(Fq)). +Identifying N ∼= Gn−1 +a +(Fq), the irreducible characters χα ∈ X = Hom(N, C∗) of N are given by +tuples α = (α1, . . . , αn−1) ∈ Gn−1 +a +(Fq), via +χα(x) = ζ⟨α,x⟩ +p +for all x ∈ N, +23 + +where ⟨α, x⟩ = � +i,j αijxij ∈ Fp, with αij and xij are the coefficients of αi and xi, viewing Gn−1 +a +(Fq) +as vector space over Fp. Since Un−1(Fq) acts on N ∼= Gn−1 +a +(Fq) by left multiplication, it acts on +X ∼= Gn−1 +a +(Fq) by right multiplication. +From now on, we will omit the field Fq from the group, simply writing G instead of G(Fq). +Example 4.10. Consider U3 ∼= G2 +a ⋊ U2, for which X = Hom(G2 +a, C∗) ∼= G2 +a, and H acts on +� +α +β +� +∈ X by right-multiplication, that is, +� +α +β +� � +1 +a +0 +1 +� += +� +α +β + aα +� +. +Hence, the orbits in X under H are given by +�� +α +β +� +: β ∈ Fq +� +for all α ̸= 0 and +�� +0 +β +�� +for +all β ∈ Fq. We choose the following representatives: +■ χα = +� +α +0 +� +, which yields Hα = {1}, so we get a contribution to the zeta function equal to +(q − 1) · ζ{1}(s) · [H : Hα]−s = (q − 1)q−s. +■ χβ = +� +0 +β +� +, which yields Hβ = U2, so we get a contribution to the zeta function equal to +q · ζU2(s) · [H : Hβ]−s = q2. +Adding up the contributions, it follows that ζU3(s) = q2 + (q − 1)q−s. +Example 4.11. Consider U4 ∼= G3 +a ⋊ U3, for which X = Hom(G3 +a, C∗) ∼= G3 +a, and H acts on +� +α +β +γ +� +∈ X by right-multiplication, that is, +� +α +β +γ +� + + + +1 +a +b +0 +1 +c +0 +0 +1 + + + = +� +α +β + aα +γ + bα + cβ +� +. +Hence, the orbits in X under H are given by +�� +α +β +γ +� +: β, γ ∈ Fq +� +for all α ̸= 0, +�� +0 +β +γ +� +: γ ∈ Fq +� +for all β ̸= 0, and +�� +0 +0 +γ +�� +for all γ ∈ Fq. We choose the following representatives: +■ χα = +� +α +0 +0 +� +yields Hα ∼= Ga, contributing (q − 1) · ζGa(s) · [H : Hα]−s = q1−2s(q − 1). +■ χβ = +� +0 +β +0 +� +yields Hβ ∼= G2 +a, contributing (q − 1) · ζG2a(s) · [H : Hβ]−s = q2−s(q − 1). +■ χγ = +� +0 +0 +γ +� +yields Hγ = U3, contributing q · ζU3 · [H : Hγ]−s = q3 + (q − 1)q1−s. +In total, ζU4(s) = q3 + q1−s(q − 1)(q + 1) + q1−2s(q − 1). +The construction as described above works more generally for a connected algebraic subgroup G ⊂ +Tn. Namely, let G′ be the image of the map G → ˜Tn given by A �→ A/Ann. Then either G ∼= G′ +or G ∼= Gm × G′, because the only connected subgroups of Gm are {1} and Gm itself. +Since +24 + +ζGm(s) = q − 1 is known, we may assume G ⊂ ˜Tn. Now let H and N be the image and kernel, +respectively, of the map +G → Tn−1, +A �→ (Aij)n−1 +i,j=1, +so that G = N ⋊ H with N abelian and H ⊂ Tn−1, and we can repeat the above arguments. +Example 4.12. Consider G = T2 ∼= GmטT2 with ˜T2 ∼= Ga⋊Gm, for which X = Hom(Ga, C∗) ∼= Ga +and H acts on α ∈ X by multiplication. Hence, the orbits in X under H are given by {0} and +{α : α ̸= 0}. We choose the following representatives: +■ χ0 = 0 yields H0 = Gm, contributing ζGm(s) = q − 1, +■ χ1 = 1 yields H1 = {1}, contributing (q − 1)−s. +In total, ζT2(s) = ζGm(s) ζ˜T2(s) = (q − 1)((q − 1) + (q − 1)−s) = (q − 1)2 + (q − 1)1−s. +From these examples we see that the representation zeta functions can be computed in a recursive +manner using Proposition 4.8. +We will treat one more example, in order to illustrate that the +stabilizers Hi need not be constant along families of representatives. This suggests that one needs +to consider parametrized families of algebraic groups H. +Example 4.13. Consider G = G3 +a ⋊ H with H = +�� 1 0 a +0 1 b +0 0 1 +�� +acting naturally on G3 +a. Then H acts +on X ∼= G3 +a by +� +α +β +γ +� + + + +1 +0 +a +0 +1 +b +0 +0 +1 + + + = +� +α +β +γ + aα + bβ +� +. +Hence, the orbits in X under H are given by +�� +0 +0 +γ +�� +for all γ ∈ Fq and +�� +α +β +γ +� +: γ ∈ Fq +� +for all α ̸= 0 or β ̸= 0. We choose the following representatives: +■ χγ = +� +0 +0 +γ +� +yields Hγ = H, contributing q · ζH(s) = q3. +■ χα,β = +� +α +β +0 +� +yields Hα,β = +� + +1 0 +xβ +0 1 −xα +0 0 +1 + + : x ∈ Fq +� +∼= Ga, contributing +(q2 − 1) · ζGa(s) · [H : Hα,β]−s = q1−s(q − 1)(q + 1). +In total, we obtain ζG(s) = q3 + q1−s(q − 1)(q + 1). +4.4 +Algorithmically computing ζG(s) +In this subsection, we will describe an algorithm to compute ζG(s) for connected algebraic groups +G ⊂ Tn, in the style of examples 4.10, 4.11 and 4.13 An implementation of this algorithm can be +found at [25], together with the code for computing ζUn(s) and ζTn(s) for 1 ≤ n ≤ 10. The resulting +zeta functions are given in Theorem 4.18 and Theorem 4.17. +Before discussing the algorithm, let us give some remarks. +25 + +The algorithm is divided into two parts. The main part, Algorithm 4.14, finds a semidirect decom- +position G ∼= N ⋊ H and applies Corollary 4.9 in order to compute ζG(s). Finding representatives +for the orbits X/H is a more intricate step, and is described as a separate part, in Algorithm 4.15. +As highlighted in Example 4.13, it is possible for the stabilizers Hi to vary along the families of +representatives χi. Therefore, in order for the algorithm to work recursively, we allow the input of +the algorithm to be a family of algebraic groups G ⊂ Tn parametrized by a variety Y over Fq. We +then understand the representation zeta function of G to be +ζG(s) = +� +y∈Y (Fq) +ζGy(s). +As we want the computations to hold over general a ground field Fq, we will practically work over +Z. Then |Y (Fq)| can be computed as a polynomial in q whenever [Y ] ∈ K(Var/Z) can be computed +as a polynomial in q = [A1 +Z] using Algorithm 2.24. +If any decision in the algorithm depends on whether some function f on Y is zero, we distinguish +cases and continue working over the closed subvariety Y ′ = Y ∩ {f = 0} and its open complement +Y ′′ = Y ∩ {f ̸= 0}. +Algorithm 4.14. Input: A family of connected algebraic groups G ⊂ Tn, parametrized by a +variety Y . +Output: The representation zeta function ζG(s) as a polynomial in q, q−s and (q − 1)−s. +1. If n = 0, then G is trivial, so that ζG(s) = |Y (Fq)|. Hence, we can assume n ≥ 1. +2. Since G is connected, the image of the map G → Gn +m given by A �→ (Aii)i is isomorphic to +Gd +m for some 0 ≤ d ≤ n. If d = n, then there is an isomorphism G ∼= Gm × G′ with G′ ⊂ ˜Tn +given by A �→ (Ann, A/Ann), so that ζG(s) = (q − 1)ζG′(s). If d < n, then G ∼= G′ ⊂ ˜Tn via +the map A �→ A/Ann. Either way, we can assume G ⊂ ˜Tn. +3. Write G = N ⋊ H as discussed in Subsection 4.3. The group H can be obtained as the group +of minors H = +� +(Aij)n−1 +i,j=1 : A ∈ G +� +, and N can be obtained as the closed subgroup of G given +by Aij = 0 for 1 ≤ i, j ≤ n − 1 and Aii = 1 for 1 ≤ i ≤ n − 1. +4. Identify N ∼= Gr +a for some 0 ≤ r ≤ n − 1, and consider induced action of H on the characters +Hom(N, C∗) ∼= Gr +a. +5. Use Algorithm 4.15 to find families of representatives χi, parametrized by varieties Zi, for the +orbits X/H, together with their stabilizer Hi and index [H : Hi]. Note that the stabilizers Hi +are generally parametrized by X × Zi. +6. Repeat the algorithm to compute ζHi(s) for all i, from which ζG(s) can be computed using +Corollary 4.9. +Algorithm 4.15. Input: Let H ⊂ Tn be an algebraic group, parametrized by a variety X, acting +linearly on a subvariety Z ⊂ Gr +a of the form Z = �r +i=1 Zi with Zi ∈ {{0}, {1}, Ga}. Write z1, . . . , zr +for the coordinates on Z. The variety X is assumed to be a variety over Z. +26 + +Output: Families of representatives χi together with stabilizers Hi, and the index [H : Hi] as a +polynomial in q. +1. Repeat steps 2 and 3 until every zi is invariant under H. When this is the case, return a single +family of representatives given by χ = (z1, . . . , zr), parametrized over Z, with stabilizer H and +index [H : H] = 1. If both step 2 and 3 do not apply, return failure. +2. If zi +H�→ azi for some coordinate a of H, then a must be a diagonal entry of H. Distinguish +between the following cases: +(a) If zi = 0, then continue with the action of H restricted to Z′ = Z ∩ {zi = 0}. +(b) If zi ̸= 0, then choose representatives with zi = 1 using an appropriate choice of a. +Continue with the action of H′ = H∩{a = 1} restricted to Z′ = Z∩{zi = 1}, remembering +the index [H : H′] = q − 1. +3. Write zi +H�→ � +j ajfj, where aj are coordinates of H and fj are functions on X which are not +identically zero. If some fℓ is invariant under the action of H, then distinguish between the +following cases: +(a) If fℓ = 0, then continue with the restriction to the closed subvariety X′ = X ∩ {fℓ = 0}. +(b) If fℓ ̸= 0, then choose representatives with zi = 0 using an appropriate choice of aℓ. +Continue with the action of H′ = H∩ +� +aℓ = −f −1 +ℓ +� +j̸=ℓ ajfj +� +restricted to Z′ = Z∩{zi = +0}, remembering the index [H : H′] = q. +Remark 4.16. Note that some steps in this algorithm might fail. In fact, if this algorithm were +to never fail, then this would show the representation zeta function ζG(s) is always a polynomial in +q, q−s and (q − 1)−s. Then, evaluating at s = 0 would imply that the number of conjugacy classes +of G is a polynomial in q. In particular, this would imply Higman’s Conjecture 1.1. For us, the +algorithm does not fail when applied to G = Un or G = Tn for 1 ≤ n ≤ 10. +4.5 +Results +The representation zeta functions of Un and Tn, as computed using Algorithm 4.14, are given in +Theorem 4.17 and Theorem 4.18 below. +One can evaluate these zeta functions at s = 0 in order to obtain the number of conjugacy classes +of the groups over finite fields Fq. The resulting polynomials in q can be seen to agree with [19, +Appendix A], where t = q − 1. In this sense, these zeta functions can be viewed as a generalization +of the polynomials k(Un(Fq)) as in [19]. +Furthermore, the E-polynomials of RG(Σg) can be obtained through Theorem 4.1 and Corollary +4.7. Indeed, one can verify that for 1 ≤ n ≤ 5 these E-polynomials agree with the virtual classes as +given by Theorem 3.14 and Theorem 3.12, via the map (1.2). +Theorem 4.17. The representation zeta functions ζUn(s) for 1 ≤ n ≤ 10 are given by +ζU1(s) = 1 +27 + +ζU2(s) = q +ζU3(s) = q−s (q − 1) + q2 +ζU4(s) = q1−s (q − 1) (q + 1) + q1−2s (q − 1) + q3 +ζU5(s) = q1−2s (q − 1) (q + 1) (2q − 1) + q2−s (q − 1) (2q + 1) + q1−3s (q − 1) (2q − 1) ++ q−4s (q − 1)2 + q4 +ζU6(s) = q2−2s (q − 1) (q + 2) �q2 + q − 1� + q2−3s (q − 1) (q + 1) (4q − 3) ++ q−4s (q − 1) �2q2 − 1� �q2 + q − 1� + q3−s (q − 1) (3q + 1) + q1−5s (q − 1)2 (2q + 1) ++ q1−6s (q − 1)2 + q5 +ζU7(s) = q3−2s (q − 1) (q + 1) �2q2 + 3q − 3� ++ q1−4s (q − 1) (2q − 1) �q4 + 5q3 − 3q − 1� + q4−s (q − 1) (4q + 1) ++ q2−3s (q − 1) �3q4 + 6q3 − 2q2 − 5q + 1� ++ q1−5s (q − 1) �q5 + 7q4 − 2q3 − 9q2 + 3q + 1� + q1−6s (q − 1)2 �4q3 + 7q2 − 3q − 1� ++ q1−8s (q − 1)2 (3q − 2) + q−7s (q − 1)2 � +5q3 − 3q + 1 +� ++ q−9s (q − 1)3 + q6 +ζU8(s) = q4−2s (q − 1) (3q + 2) �q2 + 2q − 2� + q5−s (q − 1) (5q + 1) ++ q3−3s (q − 1) +� +q5 + 5q4 + 10q3 − 7q2 − 8q + 3 +� ++ q3−6s (q − 1) �q5 + 7q4 + 16q3 − 24q2 − 14q + 15� ++ q2−4s (q − 1) +� +12q5 + 9q4 − 16q3 − 9q2 + 6q + 1 +� ++ q1−5s (q − 1) �2q7 + 8q6 + 13q5 − 23q4 − 9q3 + 12q2 − 1� ++ q1−7s (q − 1)2 � +6q5 + 18q4 + 4q3 − 19q2 + q + 3 +� ++ q1−8s (q − 1)2 �q5 + 13q4 + 8q3 − 14q2 − 4q + 3� + q1−11s (q − 1)3 (3q + 1) ++ q−9s (q − 1)2 � +4q5 + 10q4 − 7q3 − 8q2 + 3q + 1 +� ++ q−10s (q − 1)2 � +5q4 + q3 − 6q2 + 1 +� ++ q1−12s (q − 1)3 + q7 +ζU9(s) = q5−2s (q − 1) (2q + 1) +� +2q2 + 5q − 5 +� ++ q6−s (q − 1) (6q + 1) ++ q4−3s (q − 1) �2q5 + 9q4 + 14q3 − 15q2 − 11q + 6� ++ q4−4s (q − 1) +� +4q5 + 19q4 + 11q3 − 34q2 − 10q + 14 +� ++ q2−5s (q − 1) �q8 + 5q7 + 29q6 + q5 − 53q4 − 2q3 + 27q2 − 3q − 2� ++ q2−6s (q − 1) +� +10q7 + 33q6 − 9q5 − 68q4 + 10q3 + 38q2 − 11q − 1 +� ++ q1−7s (q − 1) �2q9 + 8q8 + 27q7 + 2q6 − 87q5 + 20q4 + 46q3 − 15q2 − 3q + 1� ++ q1−8s (q − 1)2 � +9q7 + 33q6 + 40q5 − 45q4 − 40q3 + 21q2 + 5q − 1 +� ++ q1−9s (q − 1)2 �2q7 + 30q6 + 42q5 − 44q4 − 48q3 + 25q2 + 7q − 1� ++ q1−11s (q − 1)2 � +4q6 + 25q5 + 5q4 − 48q3 + 7q2 + 9q + 1 +� ++ q1−12s (q − 1)2 �10q5 + 18q4 − 32q3 − 10q2 + 18q − 3� + q1−15s (q − 1)3 (4q − 3) ++ q−10s (q − 1)2 � +2q8 + 13q7 + 38q6 − 24q5 − 49q4 + 20q3 + 11q2 − 3q − 1 +� ++ q−13s (q − 1)3 �12q4 + 10q3 − 13q2 + q + 1� + q−14s (q − 1)3 �9q3 − 2q2 − 5q + 2� ++ q−16s (q − 1)4 + q8 +ζU10(s) = q6−2s (q − 1) (5q + 2) �q2 + 3q − 3� + q7−s (q − 1) (7q + 1) ++ q5−3s (q − 1) +� +3q5 + 15q4 + 19q3 − 28q2 − 13q + 10 +� ++ q4−4s (q − 1) �q7 + 7q6 + 32q5 + 12q4 − 65q3 − 6q2 + 27q − 3� ++ q3−5s (q − 1) +� +2q8 + 21q7 + 42q6 − 16q5 − 103q4 + 24q3 + 50q2 − 13q − 3 +� ++ q2−6s (q − 1) �6q9 + 27q8 + 64q7 − 73q6 − 118q5 + 64q4 + 70q3 − 39q2 + q + 1� ++ q2−7s (q − 1) +� +2q10 + 5q9 + 39q8 + 74q7 − 130q6 − 133q5 + 128q4 + 74q3 − 60q2 + 2q + 1 +� ++ q2−8s (q − 1) �q10 + 12q9 + 39q8 + 67q7 − 137q6 − 172q5 + 200q4 + 63q3 − 80q2 + 2q + 6� +28 + ++ q2−9s (q − 1)2 �10q8 + 65q7 + 117q6 − 36q5 − 221q4 + 18q3 + 98q2 − 11q − 6� ++ q1−11s (q − 1)2 � +6q9 + 31q8 + 109q7 + 8q6 − 240q5 − 10q4 + 135q3 − 17q2 − 8q − 1 +� ++ q1−12s (q − 1)2 �2q9 + 22q8 + 77q7 + 46q6 − 217q5 − 48q4 + 156q3 − 12q2 − 20q + 1� ++ q1−13s (q − 1)2 � +10q8 + 50q7 + 60q6 − 138q5 − 110q4 + 146q3 + 8q2 − 25q + 2 +� ++ q1−15s (q − 1)3 �4q6 + 42q5 + 46q4 − 51q3 − 44q2 + 23q + 5� + q1−19s (q − 1)4 (4q + 1) ++ q−10s (q − 1)2 �2q11 + 8q10 + 50q9 + 112q8 − 29q7 − 227q6 + 17q5 + 123q4 − 24q3 − 12q2 + q + 1� ++ q−14s (q − 1)2 �2q9 + 24q8 + 53q7 − 52q6 − 127q5 + 84q4 + 49q3 − 32q2 − 3q + 3� ++ q−16s (q − 1)3 �10q6 + 37q5 − 9q4 − 42q3 + 6q2 + 10q − 1� ++ q−17s (q − 1)3 �12q5 + 14q4 − 21q3 − 8q2 + 6q + 1� ++ q−18s (q − 1)3 �9q4 − q3 − 9q2 + q + 1� + q1−20s (q − 1)4 + q9. +Theorem 4.18. The representation zeta functions ζTn(s) for 1 ≤ n ≤ 10 are given by +ζT1(s) = q − 1 +ζT2(s) = (q − 1)1−s + (q − 1)2 +ζT3(s) = q−s (q − 1)2−s + 2 (q − 1)2−s + (q − 1)1−2s + (q − 1)3 +ζT4(s) = 3q−s (q − 1)2−2s + 2q−s (q − 1)3−s + q−2s (q − 1)3−s + q−2s (q − 1)2−2s + q−s (q − 1)1−3s ++ 3 (q − 1)3−s + 3 (q − 1)2−2s + (q − 1)1−3s + (q − 1)4 +ζT5(s) = 8q−s (q − 1)3−2s + 7q−2s (q − 1)3−2s + 7q−2s (q − 1)2−3s + 7q−s (q − 1)2−3s ++ 3q−3s (q − 1)3−2s + 3q−s (q − 1)4−s + 2q−2s (q − 1)4−s + 2q−2s (q − 1)1−4s ++ 2q−3s (q − 1)2−3s + 2q−s (q − 1)1−4s + q−3s (q − 1)4−s + q−4s (q − 1)3−2s + 6 (q − 1)3−2s ++ 4 (q − 1)4−s + 4 (q − 1)2−3s + (q − 1)1−4s + (q − 1)5 +ζT6(s) = q−2s (q − 1)1−5s (q + 7) + 29q−2s (q − 1)3−3s + 24q−3s (q − 1)3−3s + 23q−2s (q − 1)2−4s ++ 21q−s (q − 1)3−3s + 17q−3s (q − 1)2−4s + 16q−2s (q − 1)4−2s + 15q−4s (q − 1)3−3s ++ 15q−s (q − 1)4−2s + 13q−3s (q − 1)4−2s + 13q−s (q − 1)2−4s + 10q−4s (q − 1)2−4s ++ 7q−4s (q − 1)4−2s + 5q−5s (q − 1)3−3s + 4q−3s (q − 1)1−5s + 4q−s (q − 1)5−s ++ 3q−2s (q − 1)5−s + 3q−5s (q − 1)4−2s + 3q−s (q − 1)1−5s + 2q−3s (q − 1)5−s ++ 2q−4s (q − 1)1−5s + 2q−5s (q − 1)2−4s + q−4s (q − 1)5−s + q−6s (q − 1)4−2s ++ q−6s (q − 1)3−3s + 10 (q − 1)4−2s + 10 (q − 1)3−3s + 5 (q − 1)5−s + 5 (q − 1)2−4s + (q − 1)1−5s ++ (q − 1)6 +ζT7(s) = 2q−4s (q − 1)1−6s (q + 8) + q−2s (q − 1)2−5s (2q + 53) + q−2s (q − 1)1−6s (2q + 13) ++ q−3s (q − 1)2−5s (2q + 71) + q−3s (q − 1)1−6s (3q + 19) + q−4s (q − 1)2−5s (3q + 67) ++ q−5s (q − 1)1−6s (q + 12) + 107q−3s (q − 1)3−4s + 104q−4s (q − 1)3−4s + 87q−2s (q − 1)3−4s ++ 79q−3s (q − 1)4−3s + 73q−4s (q − 1)4−3s + 73q−5s (q − 1)3−4s + 71q−2s (q − 1)4−3s ++ 49q−5s (q − 1)4−3s + 48q−5s (q − 1)2−5s + 46q−s (q − 1)4−3s + 44q−s (q − 1)3−4s ++ 42q−6s (q − 1)3−4s + 30q−6s (q − 1)4−3s + 28q−2s (q − 1)5−2s + 27q−3s (q − 1)5−2s ++ 24q−s (q − 1)5−2s + 23q−6s (q − 1)2−5s + 22q−4s (q − 1)5−2s + 21q−s (q − 1)2−5s ++ 15q−7s (q − 1)3−4s + 13q−5s (q − 1)5−2s + 12q−7s (q − 1)4−3s + 7q−6s (q − 1)5−2s ++ 5q−7s (q − 1)2−5s + 5q−s (q − 1)6−s + 4q−2s (q − 1)6−s + 4q−6s (q − 1)1−6s ++ 4q−8s (q − 1)4−3s + 4q−s (q − 1)1−6s + 3q−3s (q − 1)6−s + 3q−7s (q − 1)5−2s ++ 3q−8s (q − 1)3−4s + 2q−4s (q − 1)6−s + q−5s (q − 1)6−s + q−8s (q − 1)5−2s + q−9s (q − 1)4−3s ++ 20 (q − 1)4−3s + 15 (q − 1)5−2s + 15 (q − 1)3−4s + 6 (q − 1)6−s + 6 (q − 1)2−5s + (q − 1)1−6s ++ (q − 1)7 +ζT8(s) = 14q−3s (q − 1)2−6s (q + 14) + 6q−7s (q − 1)1−7s (q + 7) + 4q−3s (q − 1)3−5s (q + 87) +29 + ++ 2q−2s (q − 1)2−6s (3q + 50) + 2q−7s (q − 1)2−6s (3q + 89) + q−2s (q − 1)3−5s (3q + 208) ++ q−2s (q − 1)1−7s (3q + 20) + q−3s (q − 1)1−7s � +q2 + 11q + 47 +� ++ q−4s (q − 1)3−5s (9q + 457) + q−4s (q − 1)2−6s (20q + 261) ++ q−4s (q − 1)1−7s (12q + 61) + q−5s (q − 1)3−5s (6q + 485) ++ q−5s (q − 1)2−6s (24q + 305) + q−5s (q − 1)1−7s �2q2 + 20q + 79� ++ q−6s (q − 1)3−5s (6q + 415) + q−6s (q − 1)2−6s (19q + 250) ++ q−6s (q − 1)1−7s �q2 + 13q + 60� + q−8s (q − 1)2−6s (3q + 85) + q−8s (q − 1)1−7s (q + 15) ++ 410q−4s (q − 1)4−4s + 398q−5s (q − 1)4−4s + 340q−6s (q − 1)4−4s + 332q−3s (q − 1)4−4s ++ 297q−7s (q − 1)3−5s + 238q−7s (q − 1)4−4s + 229q−2s (q − 1)4−4s + 192q−4s (q − 1)5−3s ++ 174q−3s (q − 1)5−3s + 171q−5s (q − 1)5−3s + 168q−8s (q − 1)3−5s + 147q−8s (q − 1)4−4s ++ 139q−2s (q − 1)5−3s + 136q−6s (q − 1)5−3s + 110q−s (q − 1)4−4s + 90q−7s (q − 1)5−3s ++ 85q−s (q − 1)5−3s + 80q−s (q − 1)3−5s + 73q−9s (q − 1)3−5s + 71q−9s (q − 1)4−4s ++ 56q−8s (q − 1)5−3s + 45q−3s (q − 1)6−2s + 43q−2s (q − 1)6−2s + 42q−4s (q − 1)6−2s ++ 35q−s (q − 1)6−2s + 34q−5s (q − 1)6−2s + 31q−s (q − 1)2−6s + 30q−9s (q − 1)2−6s ++ 27q−10s (q − 1)4−4s + 26q−9s (q − 1)5−3s + 22q−6s (q − 1)6−2s + 21q−10s (q − 1)3−5s ++ 13q−7s (q − 1)6−2s + 11q−10s (q − 1)5−3s + 7q−8s (q − 1)6−2s + 7q−11s (q − 1)4−4s ++ 6q−s (q − 1)7−s + 5q−2s (q − 1)7−s + 5q−10s (q − 1)2−6s + 5q−s (q − 1)1−7s ++ 4q−3s (q − 1)7−s + 4q−9s (q − 1)1−7s + 4q−11s (q − 1)5−3s + 3q−4s (q − 1)7−s ++ 3q−9s (q − 1)6−2s + 3q−11s (q − 1)3−5s + 2q−5s (q − 1)7−s + q−6s (q − 1)7−s ++ q−10s (q − 1)6−2s + q−12s (q − 1)5−3s + q−12s (q − 1)4−4s + 35 (q − 1)5−3s + 35 (q − 1)4−4s ++ 21 (q − 1)6−2s + 21 (q − 1)3−5s + 7 (q − 1)7−s + 7 (q − 1)2−6s + (q − 1)1−7s + (q − 1)8 +ζT9(s) = 10q−8s (q − 1)4−5s (q + 186) + 10q−8s (q − 1)3−6s (7q + 204) ++ 6q−3s (q − 1)4−5s (q + 182) + 6q−11s (q − 1)3−6s (q + 81) + 4q−2s (q − 1)4−5s (q + 149) ++ 4q−2s (q − 1)1−8s (q + 7) + 4q−6s (q − 1)3−6s (27q + 598) ++ 4q−9s (q − 1)3−6s (11q + 377) + 3q−7s (q − 1)4−5s (4q + 739) ++ 2q−3s (q − 1)2−7s �q2 + 23q + 212� + 2q−4s (q − 1)3−6s (31q + 754) ++ 2q−7s (q − 1)2−7s � +4q2 + 103q + 730 +� ++ 2q−8s (q − 1)2−7s � +3q2 + 72q + 586 +� ++ 2q−10s (q − 1)3−6s (7q + 491) + 2q−11s (q − 1)1−8s (2q + 19) ++ 2q−12s (q − 1)2−7s (q + 38) + q−2s (q − 1)3−6s (12q + 425) ++ q−2s (q − 1)2−7s (12q + 167) + q−3s (q − 1)3−6s (31q + 912) ++ q−3s (q − 1)1−8s � +2q2 + 21q + 85 +� ++ q−4s (q − 1)4−5s (15q + 1658) ++ q−4s (q − 1)2−7s �2q2 + 89q + 758� + q−4s (q − 1)1−8s �4q2 + 45q + 165� ++ q−5s (q − 1)4−5s (16q + 2111) + q−5s (q − 1)3−6s (92q + 2099) ++ q−5s (q − 1)2−7s �9q2 + 157q + 1138� ++ q−5s (q − 1)1−8s � +q3 + 12q2 + 83q + 262 +� ++ q−6s (q − 1)4−5s (21q + 2302) ++ q−6s (q − 1)2−7s �6q2 + 180q + 1339� + q−6s (q − 1)1−8s �10q2 + 97q + 316� ++ q−7s (q − 1)3−6s (101q + 2451) ++ q−7s (q − 1)1−8s �2q3 + 22q2 + 131q + 369� + q−8s (q − 1)1−8s �9q2 + 81q + 277� ++ q−9s (q − 1)2−7s � +3q2 + 80q + 823 +� ++ q−9s (q − 1)1−8s � +2q2 + 39q + 181 +� ++ q−10s (q − 1)2−7s (39q + 536) + q−10s (q − 1)1−8s �2q2 + 25q + 119� ++ q−11s (q − 1)2−7s (11q + 222) + 1395q−9s (q − 1)4−5s + 1254q−6s (q − 1)5−4s ++ 1224q−5s (q − 1)5−4s + 1123q−7s (q − 1)5−4s + 1061q−4s (q − 1)5−4s +30 + ++ 923q−8s (q − 1)5−4s + 911q−10s (q − 1)4−5s + 776q−3s (q − 1)5−4s + 670q−9s (q − 1)5−4s ++ 505q−11s (q − 1)4−5s + 494q−2s (q − 1)5−4s + 436q−10s (q − 1)5−4s + 394q−5s (q − 1)6−3s ++ 381q−4s (q − 1)6−3s + 369q−6s (q − 1)6−3s + 319q−3s (q − 1)6−3s + 299q−7s (q − 1)6−3s ++ 251q−11s (q − 1)5−4s + 242q−12s (q − 1)4−5s + 239q−2s (q − 1)6−3s + 230q−8s (q − 1)6−3s ++ 230q−s (q − 1)5−4s + 225q−s (q − 1)4−5s + 208q−12s (q − 1)3−6s + 154q−9s (q − 1)6−3s ++ 141q−s (q − 1)6−3s + 132q−s (q − 1)3−6s + 126q−12s (q − 1)5−4s + 98q−10s (q − 1)6−3s ++ 89q−13s (q − 1)4−5s + 67q−3s (q − 1)7−2s + 67q−4s (q − 1)7−2s + 61q−2s (q − 1)7−2s ++ 61q−5s (q − 1)7−2s + 58q−13s (q − 1)3−6s + 53q−13s (q − 1)5−4s + 51q−11s (q − 1)6−3s ++ 50q−6s (q − 1)7−2s + 48q−s (q − 1)7−2s + 43q−s (q − 1)2−7s + 34q−7s (q − 1)7−2s ++ 25q−12s (q − 1)6−3s + 25q−14s (q − 1)4−5s + 22q−8s (q − 1)7−2s + 18q−14s (q − 1)5−4s ++ 13q−9s (q − 1)7−2s + 12q−13s (q − 1)2−7s + 11q−13s (q − 1)6−3s + 9q−14s (q − 1)3−6s ++ 8q−12s (q − 1)1−8s + 7q−10s (q − 1)7−2s + 7q−s (q − 1)8−s + 6q−2s (q − 1)8−s ++ 6q−s (q − 1)1−8s + 5q−3s (q − 1)8−s + 5q−15s (q − 1)5−4s + 4q−4s (q − 1)8−s ++ 4q−14s (q − 1)6−3s + 4q−15s (q − 1)4−5s + 3q−5s (q − 1)8−s + 3q−11s (q − 1)7−2s ++ 2q−6s (q − 1)8−s + q−7s (q − 1)8−s + q−12s (q − 1)7−2s + q−15s (q − 1)6−3s ++ q−16s (q − 1)5−4s + 70 (q − 1)5−4s + 56 (q − 1)6−3s + 56 (q − 1)4−5s + 28 (q − 1)7−2s ++ 28 (q − 1)3−6s + 8 (q − 1)8−s + 8 (q − 1)2−7s + (q − 1)1−8s + (q − 1)9 +ζT10(s) = q−6s (q − 1)1−9s (3q + 17) �2q2 + 13q + 63� + 12q−13s (q − 1)3−7s (14q + 349) ++ 10q−14s (q − 1)3−7s (7q + 219) + 7q−5s (q − 1)4−6s (30q + 1201) ++ 7q−12s (q − 1)4−6s (12q + 967) + 5q−2s (q − 1)5−5s (q + 282) ++ 5q−10s (q − 1)5−5s (3q + 1366) + 4q−2s (q − 1)4−6s (5q + 333) ++ 4q−15s (q − 1)1−9s (q + 11) + 3q−2s (q − 1)3−7s (10q + 259) ++ 3q−7s (q − 1)4−6s (118q + 4475) + 3q−8s (q − 1)3−7s � +9q2 + 385q + 4636 +� ++ 3q−11s (q − 1)3−7s �4q2 + 205q + 3159� + 2q−3s (q − 1)4−6s (27q + 1496) ++ 2q−4s (q − 1)2−8s � +8q2 + 139q + 889 +� ++ 2q−5s (q − 1)5−5s (13q + 3104) ++ 2q−5s (q − 1)2−8s �q3 + 25q2 + 293q + 1620� + 2q−8s (q − 1)5−5s (19q + 4337) ++ 2q−12s (q − 1)3−7s � +3q2 + 176q + 3381 +� ++ 2q−13s (q − 1)4−6s (13q + 2180) ++ 2q−14s (q − 1)4−6s (5q + 1223) + 2q−15s (q − 1)2−8s (11q + 170) ++ 2q−16s (q − 1)3−7s (2q + 161) + q−2s (q − 1)2−8s (20q + 257) ++ q−2s (q − 1)1−9s (5q + 37) + q−3s (q − 1)5−5s (8q + 2717) ++ q−3s (q − 1)3−7s � +3q2 + 117q + 2039 +� ++ q−3s (q − 1)2−8s � +6q2 + 104q + 791 +� ++ q−3s (q − 1)1−9s �3q2 + 33q + 134� + q−4s (q − 1)5−5s (21q + 4418) ++ q−4s (q − 1)4−6s (122q + 5413) + q−4s (q − 1)3−7s � +4q2 + 273q + 4096 +� ++ q−4s (q − 1)1−9s �q3 + 15q2 + 108q + 342� + q−5s (q − 1)3−7s �19q2 + 545q + 6943� ++ q−5s (q − 1)1−9s � +2q3 + 35q2 + 230q + 659 +� ++ q−6s (q − 1)5−5s (41q + 7704) ++ q−6s (q − 1)4−6s (299q + 11188) + q−6s (q − 1)3−7s �21q2 + 809q + 9883� ++ q−6s (q − 1)2−8s � +2q3 + 84q2 + 953q + 4932 +� ++ q−7s (q − 1)5−5s (36q + 8593) ++ q−7s (q − 1)3−7s �37q2 + 1103q + 12627� ++ q−7s (q − 1)2−8s � +8q3 + 144q2 + 1413q + 6663 +� ++ q−7s (q − 1)1−9s �2q4 + 21q3 + 143q2 + 654q + 1524� + q−8s (q − 1)4−6s (356q + 14191) ++ q−8s (q − 1)2−8s � +6q3 + 135q2 + 1587q + 7639 +� ++ q−8s (q − 1)1−9s �q4 + 20q3 + 165q2 + 790q + 1818� + q−9s (q − 1)5−5s (20q + 8073) +31 + ++ q−9s (q − 1)4−6s (322q + 13751) + q−9s (q − 1)3−7s �30q2 + 1125q + 13721� ++ q−9s (q − 1)2−8s � +4q3 + 125q2 + 1513q + 7536 +� ++ q−9s (q − 1)1−9s �10q3 + 131q2 + 723q + 1771� + q−10s (q − 1)4−6s (227q + 12026) ++ q−10s (q − 1)3−7s � +21q2 + 934q + 12376 +� ++ q−10s (q − 1)2−8s �8q3 + 142q2 + 1407q + 6977� ++ q−10s (q − 1)1−9s �2q4 + 22q3 + 154q2 + 704q + 1669� + q−11s (q − 1)4−6s (136q + 9425) ++ q−11s (q − 1)2−8s �61q2 + 889q + 5143� ++ q−11s (q − 1)1−9s �6q3 + 73q2 + 433q + 1181� + q−12s (q − 1)2−8s �32q2 + 526q + 3607� ++ q−12s (q − 1)1−9s �2q3 + 36q2 + 255q + 805� + q−13s (q − 1)2−8s �10q2 + 249q + 2095� ++ q−13s (q − 1)1−9s �10q2 + 110q + 431� + q−14s (q − 1)2−8s �5q2 + 101q + 983� ++ q−14s (q − 1)1−9s �2q2 + 33q + 171� + q−15s (q − 1)3−7s (20q + 929) ++ q−16s (q − 1)2−8s (2q + 91) + 5331q−11s (q − 1)5−5s + 3802q−12s (q − 1)5−5s ++ 3288q−7s (q − 1)6−4s + 3213q−6s (q − 1)6−4s + 3128q−8s (q − 1)6−4s ++ 2802q−5s (q − 1)6−4s + 2724q−9s (q − 1)6−4s + 2472q−13s (q − 1)5−5s ++ 2228q−4s (q − 1)6−4s + 2216q−10s (q − 1)6−4s + 1657q−11s (q − 1)6−4s ++ 1544q−3s (q − 1)6−4s + 1442q−14s (q − 1)5−5s + 1192q−15s (q − 1)4−6s ++ 1149q−12s (q − 1)6−4s + 937q−2s (q − 1)6−4s + 759q−15s (q − 1)5−5s + 757q−6s (q − 1)7−3s ++ 729q−5s (q − 1)7−3s + 725q−13s (q − 1)6−4s + 700q−7s (q − 1)7−3s + 655q−4s (q − 1)7−3s ++ 607q−8s (q − 1)7−3s + 525q−s (q − 1)5−5s + 524q−3s (q − 1)7−3s + 492q−16s (q − 1)4−6s ++ 480q−9s (q − 1)7−3s + 427q−s (q − 1)6−4s + 426q−14s (q − 1)6−4s + 413q−s (q − 1)4−6s ++ 377q−2s (q − 1)7−3s + 365q−10s (q − 1)7−3s + 346q−16s (q − 1)5−5s ++ 249q−11s (q − 1)7−3s + 225q−15s (q − 1)6−4s + 217q−s (q − 1)7−3s + 203q−s (q − 1)3−7s ++ 163q−12s (q − 1)7−3s + 155q−17s (q − 1)4−6s + 133q−17s (q − 1)5−5s ++ 105q−16s (q − 1)6−4s + 97q−4s (q − 1)8−2s + 94q−5s (q − 1)8−2s + 93q−3s (q − 1)8−2s ++ 92q−13s (q − 1)7−3s + 85q−6s (q − 1)8−2s + 82q−2s (q − 1)8−2s + 74q−17s (q − 1)3−7s ++ 70q−7s (q − 1)8−2s + 63q−s (q − 1)8−2s + 57q−s (q − 1)2−8s + 50q−8s (q − 1)8−2s ++ 50q−14s (q − 1)7−3s + 43q−17s (q − 1)6−4s + 42q−18s (q − 1)5−5s + 35q−18s (q − 1)4−6s ++ 34q−9s (q − 1)8−2s + 25q−15s (q − 1)7−3s + 22q−10s (q − 1)8−2s + 16q−18s (q − 1)6−4s ++ 13q−11s (q − 1)8−2s + 12q−17s (q − 1)2−8s + 11q−16s (q − 1)7−3s + 9q−18s (q − 1)3−7s ++ 9q−19s (q − 1)5−5s + 8q−16s (q − 1)1−9s + 8q−s (q − 1)9−s + 7q−2s (q − 1)9−s ++ 7q−12s (q − 1)8−2s + 7q−s (q − 1)1−9s + 6q−3s (q − 1)9−s + 5q−4s (q − 1)9−s ++ 5q−19s (q − 1)6−4s + 4q−5s (q − 1)9−s + 4q−17s (q − 1)7−3s + 4q−19s (q − 1)4−6s ++ 3q−6s (q − 1)9−s + 3q−13s (q − 1)8−2s + 2q−7s (q − 1)9−s + q−8s (q − 1)9−s ++ q−14s (q − 1)8−2s + q−18s (q − 1)7−3s + q−20s (q − 1)6−4s + q−20s (q − 1)5−5s ++ 126 (q − 1)6−4s + 126 (q − 1)5−5s + 84 (q − 1)7−3s + 84 (q − 1)4−6s + 36 (q − 1)8−2s ++ 36 (q − 1)3−7s + 9 (q − 1)9−s + 9 (q − 1)2−8s + (q − 1)1−9s + (q − 1)10 . +The computation times2 for the representation zeta functions ζUn(s) are given by +n +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +time +0.00s +0.01s +0.08s +0.25s +0.77s +2.41s +8.18s +27.43s +1m44s +6m52s +2As performed on an Intel®Xeon®CPU E5-4640 0 @ 2.40GHz. +32 + +and those for the representation zeta functions ζTn(s) are given by +n +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +time +0.00s +0.03s +0.15s +0.51s +1.81s +6.32s +26.16s +1m53s +6m39s +55m46s +As an example, applying Theorem 4.1 and Corollary 4.7 with G = T6, we obtain +e(RT6(Σg)) = q18g−3 (q − 1)6g+3 + q18g−3 (q − 1)8g+2 + 2q20g−5 (q − 1)4g+4 + 5q20g−5 (q − 1)6g+3 ++ 3q20g−5 (q − 1)8g+2 + 2q22g−7 (q − 1)2g+5 + 10q22g−7 (q − 1)4g+4 + 15q22g−7 (q − 1)6g+3 ++ 7q22g−7 (q − 1)8g+2 + q22g−7 (q − 1)10g+1 + 4q24g−9 (q − 1)2g+5 + 17q24g−9 (q − 1)4g+4 ++ 24q24g−9 (q − 1)6g+3 + 13q24g−9 (q − 1)8g+2 + 2q24g−9 (q − 1)10g+1 ++ q26g−11 (q − 1)2g+5 (q + 7) + 23q26g−11 (q − 1)4g+4 + 29q26g−11 (q − 1)6g+3 ++ 16q26g−11 (q − 1)8g+2 + 3q26g−11 (q − 1)10g+1 + 3q28g−13 (q − 1)2g+5 ++ 13q28g−13 (q − 1)4g+4 + 21q28g−13 (q − 1)6g+3 + 15q28g−13 (q − 1)8g+2 ++ 4q28g−13 (q − 1)10g+1 + q30g−15 (q − 1)12g + q30g−15 (q − 1)2g+5 + 5q30g−15 (q − 1)4g+4 ++ 10q30g−15 (q − 1)6g+3 + 10q30g−15 (q − 1)8g+2 + 5q30g−15 (q − 1)10g+1 . +5 +Conclusion +Comparison of methods. +We have seen two very different methods to compute very similar +invariants of the G-representation variety RG(Σg). +An obvious advantages of the TQFT method is that it produces a more refined invariant, the virtual +class in K(VarC). However, in practice the virtual class and E-polynomial seem to hold the same +information, as the virtual classes in our results are all contained in the subring Z[q] ⊂ K(VarC), +and are therefore completely determined by the corresponding E-polynomials. This appears also to +be the case for G = SL2 [4], so it would be interesting to see if there are linear algebraic groups for +which this is not the case. +A possibly more significant advantage of the TQFT method is that it provides deeper geometric +insight. From the TQFT we can directly understand the fibration +G2g → G, +(A1, B1, . . . , Ag, Bg) �→ +g +� +i=1 +[Ai, Bi], +in K(Var/G), and not only the fiber over the identity, that is, the G-representation variety. This +allows the method to be more flexible. As seen in Subsection 3.6, the TQFT method can be slightly +modified to also study the twisted representation varieties. +In a similar spirit, the TQFT can +naturally be lifted to a G-equivariant setting in order to study the G-character stacks [RG(Σg)/G] +[5]. +On the other hand, the arithmetic method is orders of magnitude faster and the complexity of +the computations remains quite manageable as the rank of the groups Tn and Un increases. Once +33 + +the representation zeta function has been computed, the E-polynomial of RG(Σg) can be obtained +directly by evaluation, whereas for the TQFT method the map Z( +) must be diagonalized, which +is a very expensive operation. +Of course, for general algebraic groups G, the arithmetic method is not as systematic as the TQFT +method, since there is no recipe how to compute the irreducible representations of G(Fq). +Extending the TQFT method. Naturally, one can ask whether the TQFT method can compute +the virtual class of RTn(Σg) for n ≥ 6. Unfortunately, this is not possible without major modifi- +cations to the method for a number of reasons. First of all, we would like to take the conjugacy +classes of G as a generating set of a submodule of K(Var/G), but for n = 6 it is shown [3] that +there are infinitely many unipotent conjugacy classes in G = Tn. For example, an infinite family of +non-conjugate elements in T6 is given by + + + + + + + + +1 1 α 0 0 0 +0 1 0 0 1 0 +0 0 1 1 1 0 +0 0 0 1 0 0 +0 0 0 0 1 1 +0 0 0 0 0 1 + + + + + + + + +for α ∈ k, +and moreover every such element appears as the commutator of elements in T6. Secondly, from +a practical point of view, we do not expect the computations to be feasible, both looking at the +computation times and the prospect of diagonalizing a polynomial-valued matrix of size ≥ 250. +Motivic Higman’s conjecture. As mentioned, it is very remarkable that all virtual classes in +our results are all contained in the subring Z[q] ⊂ K(VarC). Therefore, in order to compute the +virtual classes of RTn(Σg) and RUn(Σg) for n ≥ 6, instead of using the TQFT method, it seems +more natural to try to prove the Motivic Higman’s Conjecture 1.3. Then it suffices to compute the +E-polynomials, which can be done more efficient using the arithmetic method. +34 + +References +[1] +W.W. Adams and P. Loustaunau. An Introduction to Gr¨obner Bases. American Mathematical +Soc., 1994. +[2] +D. Baraglia and P. Hekmati. “Arithmetic of singular character varieties and their E-polynomials”. +Proc. Lond. Math. Soc. (3) 114.2 (2017), pp. 293–332. +[3] +S. Bhunia. “Conjugacy classes of centralizers in the group of upper triangular matrices”. Jour- +nal of Algebra and its Applications 19.1 (2020). +[4] +´A. Gonz´alez-Prieto. “Virtual Classes of Parabolic SL2(C)-Character Varieties”. Adv. Math. +368 (2020), pp. 107–148. +[5] +´A. Gonz´alez-Prieto, M. Hablicsek, and J.T. Vogel. “Virtual Classes of Character Stacks” +(2022). arXiv: 2201.08699 [math.AG]. +[6] +´A. Gonz´alez-Prieto, M. Logares, and V. Mu˜noz. “A Lax Monoidal Topological Quantum Field +Theory For Representation Varieties”. B. Sci. Math. 161 (2020). 102871. +[7] +M. Hablicsek and J.T. Vogel. “Virtual Classes of Representation Varieties of Upper Triangular +Matrices via Topological Quantum Field Theories”. Symmetry, Integrability and Geometry: +Methods and Applications 18.095 (2022). +[8] +Z. Halasi and P.P. P´alfy. “The number of conjugacy classes in pattern groups is not a polyno- +mial function”. Journal of Group Theory 14.6 (2011), pp. 841–854. +[9] +T. Hausel and F. Rodriguez-Villegas. “Mixed Hodge polynomials of character varieties”. In- +vent. Math. 174.3 (2008), pp. 555–624. +[10] +T. Hausel et al. “P = W via H2” (2022). arXiv: 2209.05429 [math.AG]. +[11] +G. Higman. “Enumerating p-groups. I. Inequalities”. Proc. London Math. Soc. (3) 10 (1960), +pp. 24–30. +[12] +M. Logares, V. Mu˜noz, and P.E. Newstead. “Hodge Polynomials of SL2(C)-Character Varieties +for Curves of Small Genus”. Rev. Mat. Complut. 26.2 (2013), pp. 635–703. +[13] +J. Mart´ınez. “E-polynomials of PGL(2, C) character varieties of surface groups” (2017). arXiv: +1705.04649 [math.AG]. +[14] +J. Mart´ınez and V. Mu˜noz. “E-polynomials of the SL(2, C)-character varieties of surface +groups”. Int. Math. Res. Not. IMRN 3 (2016), pp. 926–961. +[15] +D. Maulik and J. Shen. “The P = W conjecture for GLn” (2022). arXiv: 2209.02568 [math.AG]. +[16] +M. Mereb. “On the E-polynomials of a family of character varieties”. PhD thesis. The Uni- +versity of Texas at Austin, 2018. +[17] +J.S. Milne. Algebraic Groups (v2.00). Available at www.jmilne.org/math/. 2015. +[18] +J.S. Milne. Etale Cohomology (PMS-33). Princeton: Princeton University Press, 1980. +[19] +I. Pak and A. Soffer. “On Higman’s k(Un(q)) conjecture” (2015). arXiv: 1507.00411 [math.CO]. +[20] +J.P. Serre. “Espaces fibr´es alg´ebriques”. fr. S´eminaire Claude Chevalley 3 (1958). +35 + +[21] +J.P. Serre. Linear representations of finite groups. Vol. 42. Graduate texts in mathematics. +Springer, 1977. +[22] +C.T. Simpson. “Moduli of representations of the fundamental group of a smooth projective +variety I”. Inst. Hautes ´Etudes Sci. Publ. Math 79 (1994), pp. 47–129. +[23] +C.T. Simpson. “Moduli of representations of the fundamental group of a smooth projective +variety II”. Inst. Hautes ´Etudes Sci. Publ. Math 79 (1994), pp. 5–79. +[24] +A. Vera-L´opez and J.M. Arregi. “Conjugacy classes in unitriangular matrices”. Linear Algebra +Appl. 370 (2003), pp. 85–124. +[25] +J.T. Vogel. Math Code. Available at https://github.com/jessetvogel/math-code. 2022. +36 + diff --git a/ItE0T4oBgHgl3EQfiAFl/content/tmp_files/load_file.txt b/ItE0T4oBgHgl3EQfiAFl/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6291bc1ba07fec6e5b669118f61a89bcbf9b64ef --- /dev/null +++ b/ItE0T4oBgHgl3EQfiAFl/content/tmp_files/load_file.txt @@ -0,0 +1,5153 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf,len=5152 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='02439v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='AG] 6 Jan 2023 Motivic Higman’s Conjecture Jesse Vogel Mathematical Institute, Leiden University, j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='vogel@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='leidenuniv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='nl Abstract The G-representation variety RG(Σg) parametrizes the representations of the fundamental groups of surfaces π1(Σg) into an algebraic group G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Taking G to be the groups of n × n upper triangular or unipotent matrices, we compare two methods for computing algebraic invariants of RG(ΣG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Using the geometric method initiated by Gonz´alez-Prieto, Logares and Mu˜noz, based on a Topological Quantum Field Theory (TQFT), we compute the virtual classes of RG(Σg) in the Grothendieck ring of varieties for n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' , 5, extending the results of [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Introducing the notion of algebraic representatives we are able to efficiently compute the TQFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Using the arithmetic method initiated by Hausel and Rodriguez-Villegas, we compute the E-polynomials of RG(Σg) for n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' , 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' For both methods, we describe how the computations can be performed algorithmically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Furthermore, we discuss the relation between the representation varieties of the group of unipotent matrices and Higman’s conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' The computations of this paper can be seen as positive evidence towards a generalized motivic version of the conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' 1 Introduction Let X be a closed connected manifold, and G an algebraic group over a field k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' The G-representation variety of X is the set of group homomorphisms RG(X) = Hom(π1(X), G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' As π1(X) is finitely generated, RG(X) can be seen as a closed subvariety of Gn for some n ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' For X = Σg a closed surface of genus g, the G-representation variety takes the explicit form RG(Σg) = � (A1, B1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' , Ag, Bg) ∈ G2g | g � i=1 [Ai, Bi] = 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='1) Closely related is the G-character variety of Σg, given by the GIT quotient χG(Σg) = RG(Σg) � G, where G acts on RG(Σg) by conjugation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' The G-character variety, also known as the Betti moduli space, plays a central role in non-abelian Hodge theory [22, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' In particular, for C a complex smooth projective curve, the Betti moduli space is isomorphic to the moduli space of G-flat connections on C by the Riemann–Hilbert correspondence, and for G = GLn it is diffeomorphic to the moduli space of polystable G-Higgs bundles of rank n and degree 0 by the non-abelian Hodge correspondence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' 1 Invariants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Various algebraic invariants of the G-representation variety and G-character variety have been studied in the literature, for instance [6, 9, 12] amongst many, and in the recently resolved P = W conjecture [10, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' One of these invariants is the E-polynomial, also known as the Serre polynomial, which for a complex variety X is given by e(X) = � k,p,q (−1)k hk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='p,q c (X) upvq ∈ Z[u, v], where hk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='p,q c (X) are mixed Hodge numbers of the compactly supported cohomology of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' The E-polynomial can be shown [9, 12] to satisfy e(X) = e(Z) + e(X \\ Z) and e(X ×C Y ) = e(X) e(Y ), for complex varieties X and Y , where Z ⊂ X is a closed subvariety with open complement X \\ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' The Grothendieck ring of varieties K(VarC), which will be defined in Section 2, is the universal ring for invariants with these properties, so in particular there is a morphism e : K(VarC) → Z[u, v] (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='2) which sends the class [X] of a complex variety X to its E-polynomial e(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' In this sense, the class of a variety X in K(VarC), also known as the virtual class of X, is a more refined invariant than the E-polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Another such invariant is the point count over a finite field, that is, there is a morphism # : K(VarFq) → Z which sends the class [X] of a variety X over Fq to |X(Fq)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' A remarkable theorem by Katz [9, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='2] states that if X is a complex variety with a spreading-out ˜X over a finitely generated Z-algebra R ⊂ C, such that |( ˜X ×R Fq)(Fq)| is a polynomial in q for all ring morphisms R → Fq, then the E-polynomial of X is precisely this polynomial in q = uv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' For example, for the affine line X = A1 C, we have |A1 Fq(Fq)| = q and e(X) = uv, and for this reason we usually write q = uv and q = [A1 k], depending on the context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Previous work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Over the last years, various methods have been developed and used to compute algebraic invariants of the G-representation varieties of Σg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' In [9], Hausel and Rodriguez–Villegas initiated the arithmetic method in order to compute the E-polynomials of the GLn-representation varieties and twisted GLn-character varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' This method makes use of Katz’ theorem to reduce the problem to counting the points of RG(Σg) over finite fields Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Frobenius’ formula, as proven in [9, Equation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='8], gives an expression for the point count of the representation variety for finite groups G, |RG(Σg)| = |G| � χ∈ � G � |G| χ(1) �2g−2 reducing the problem to the study of the irreducible (complex) characters χ ∈ �G of G over finite fields Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' In [16], this method was used to compute the E-polynomials of twisted SL2-character varieties, and in [2] it was used to compute the E-polynomials of the GL3- and SL3-character varieties, and those of the GL2- and SL2-character varieties of non-orientable surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' 2 Logares, Mart´ınez, Mu˜noz and Newstead used geometric arguments in order to compute E-polynomials of the SL2-character varieties [12, 14] and PGL2-character varieties [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' This approach was refor- mulated and generalized by Gonz´alez-Prieto, Logares and Mu˜noz [6], who constructed a Topological Quantum Field Theory (TQFT) that computes the virtual class of RG(Σg) in K(Vark).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Using this TQFT, they computed the virtual class of the AGL1-representation variety, and in [4] it was used to compute the virtual class of the SL2-representation variety and the twisted SL2-character variety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Concretely, this method defines a lax monoidal functor Z : Bord2 → K(Vark)-Mod, from the cate- gory of (pointed) 2-bordisms to the category of K(Vark)-modules, such that Z(Σg)(1) = [RG(Σg)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Using functoriality, it suffices to compute Z( ) : K(Var/G) → K(Vark), Z( ) : K(Var/G) → K(Var/G), and Z( ) : K(Vark) → K(Var/G) to obtain Z(Σg) by composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Outline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' The goal of this paper is to compute the E-polynomials and virtual classes of the G- representation varieties of Σg for G equal to the group Tn of upper triangular n × n matrices or the group Un of unipotent n × n matrices, making use of both the TQFT method and the arithmetic method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Therefore, this work is an extension of [7], in which the virtual class of RTn(Σg) was computed for G = Tn and 1 ≤ n ≤ 4 using the TQFT method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' In particular, in Section 3 we will compute the virtual classes of RTn(Σg) and RUn(Σg) for 1 ≤ n ≤ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' In this paper, we aim for an organized and concise framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Rather than working with the explicit description of the TQFT as done in [4, 5, 6], we will be working with the essential maps as in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='1, avoiding the need for a ‘reduction’ or ‘simplification’ of the TQFT, as appears in [4, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Introducing the notion of algebraic representatives in Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='3, we can take maximal advantage of the lack of monodromy in the conjugacy classes of Tn and Un in order to simplify the computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Moreover, the computations are organized in such a way that most of the computations for Tn can be reused for Un.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Furthermore, in Section 4, we will apply the arithmetic method to compute E-polynomials of RTn(Σg) and RUn(Σg) for 1 ≤ n ≤ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' In particular, we compute the representation theory of Tn and Un over finite fields Fq, encoded in the representation zeta function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' These zeta functions can be computed using a recursive algorithm, Algorithm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' In Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='12, the virtual class of the T5-representation variety RT5(Σg) is given, and in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='14, the virtual classes of the Un-representation varieties RUn(Σg) are given for 1 ≤ n ≤ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' In Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='17, the representation zeta functions of Un are given for 1 ≤ n ≤ 10, and in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='18 those of Tn for 1 ≤ n ≤ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' These directly determine the E-polynomials of RUn(Σg) and RTn(ΣG), and for 1 ≤ n ≤ 5 these E-polynomials can be seen to agree with the virtual classes through the map (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' The computation of the representation zeta functions of Tn and Un can be seen as a generalization of [24] and [19], where the number of conjugacy classes of Un(Fq), denoted k(Un(Fq)), was computed as a polynomial in q for 1 ≤ n ≤ 13 and 1 ≤ n ≤ 16, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Namely, since the number of 3 irreducible representations of a finite group equals its number of conjugacy classes, evaluating the representation zeta function ζG(s) in s = 0 yields k(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' This relates the zeta functions ζUn(Fq)(s) to a conjecture by Higman first posed in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='1 (Higman).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' For every n ≥ 1, the number of conjugacy classes of Un(Fq) is a polynomial in q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' As mentioned, this conjecture has been verified for 1 ≤ n ≤ 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' However, as shown in [8], there exist pattern subgroups of Un whose number of conjugacy classes over Fq is not polynomial in q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' In [19], a concrete such pattern subgroup P ⊂ U13 is given, and moreover it is shown that k(U59) can be expressed as a Z[q]-linear combination of k(P) and k(P ′) for other pattern subgroups P ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' This suggests the conjecture might fail for n = 59, unless the non-polynomial terms in the linear combination cancel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Looking at the results of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='17 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='18, we pose a generalized version of Higman’s conjec- ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='2 (Generalized Higman).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' For any n ≥ 1, the representation zeta function ζUn(Fq)(s) is a polynomial in q and q−s, and ζTn(Fq)(s) is a polynomial in q, q−s and (q − 1)−s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Similarly, looking at the virtual classes of RTn(Σg) and RUn(Σg), we pose a motivic version of the conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Note that Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='3 does not imply Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='2, since a variety X could have virtual class in Z[q] without having polynomial count, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' X = {(x, y) | x2 + y2 = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='3 (Motivic Higman).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' For any g ≥ 0 and n ≥ 1, the virtual classes of the Tn- and Un-representation variety of Σg are polynomial in q = [A1 C].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' 2 Grothendieck ring of varieties Let k be a field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' By a variety over k we understand a reduced separated scheme of finite type over a field k, not necessarily irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' All varieties are assumed to be over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Let S be a variety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' The Grothendieck ring of varieties over S, denoted K(Var/S), is the quotient of the free abelian group on isomorphism classes of varieties over S, by relations of the form [X] = [Z] + [U], for all closed subvarieties Z ⊂ X with open complement U = X \\ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Multiplication on K(Var/S) is defined by the fiber product over S, [X] · [Y ] = [X ×S Y ], and this is easily seen to give a ring structure on K(Var/S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' In particular, the unit element is the class 1S = [S] of S over itself, and the zero element is the class 0S = [∅].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' The class [X] of a variety X in K(Var/S) is also called its virtual class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' 4 In order to distinguish between classes over different bases, we write [X]S for the class of X in K(Var/S), and when the base is S = Spec k, we simply write [X].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Furthermore, we denote by q = [A1 k] ∈ K(Vark) the class of the affine line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' For any variety S, the ring K(Var/S) naturally has the structure of a K(Vark)-module, where scalar multiplication is given by [T ] · [X]S = [T × X]S for all varieties T and varieties X over S, and extended linearly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' For any morphism f : T → S of varieties, there are induced maps f!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' : K(Var/T ) → K(Var/S), [X]T �→ [X]S, f ∗ : K(Var/S) → K(Var/T ), [X]S �→ [X ×S T ], which are seen to be K(Vark)-module morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Moreover, the map f ∗ is a ring morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' For any variety S, let c : S → Spec k denote the final morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' In particular, for any variety X over S, we have c!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' [X]S = [X].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='1 Stratifications Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Let X be a variety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' A stratification of X is a collection of locally closed subvarieties {Xi}i∈I such that � i∈I Xi = X and Xi ∩ Xj = ∅ for i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Let X be a variety over S, with stratification {Xi}i∈I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Then only finitely many of the Xi are non-empty, and � i∈I[Xi]S = [X]S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Proof by induction on the dimension of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' If dim X = 0, then X is a finite set of points, and the result is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Now assume that dim X > 0 and that the result holds for all varieties of dimension less than dim X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' First consider the case where X is irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Some U = Xi contains the generic point of X, and is therefore open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' The complement Z = X \\ U is of smaller dimension than X and is stratified by the other Xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Since [X]S = [Z]S + [U]S, the result follows from the induction hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Now consider the case where X is reducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Take an irreducible component and remove the inter- sections with the other irreducible components, which gives an irreducible open subset U ⊂ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' The complement Z = X \\ U is a closed subvariety with fewer irreducible components than X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Since {Z ∩ Xi}i∈I is a stratification of Z and {U ∩ Xi}i∈I a stratification of U, apply induction on the number of irreducible components of X to find that only finitely many are non-empty, and we have [U]S = � i∈I [U ∩ Xi]S and [Z]S = � i∈I [Z ∩ Xi]S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Since [Xi]S = [U ∩ Xi]S + [Z ∩ Xi]S for all i, it follows that [X]S = [U]S + [Z]S = � i∈I[Xi]S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' 5 A stratification {Si}i∈I of S induces a decomposition of the Grothendieck ring K(Var/S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' This is a decomposition as K(Vark)-algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Let S be a variety with stratification {Si}i∈I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Then K(Var/S) ∼= � i∈I K(Var/Si) as K(Vark)-algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Writing fi : Si → S for the inclusions, the isomorphism is explicitly given by X �→ (f ∗ i X)i∈I with inverse � i∈I (fi)∗Xi ←� (Xi)i∈I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Note that the sum is well-defined since by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='3 only finitely many Si are non-empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' These maps are easily seen to be inverse to each other, and morphisms of K(Vark)-algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Notation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' For any X ∈ K(Var/S), we will write X|Si ∈ K(Var/Si) for the components of the image of X under this isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='2 Special algebraic groups Special algebraic groups were first introduced by Serre [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' An algebraic group G over k is special if any ´etale G-torsor is locally trivial in the Zariski topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Equivalently, G is special if for all k-schemes X, the natural map H1 Zar(X, G) → H1 ´et(X, G) is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Let G be a special algebraic group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Then for every G-torsor of varieties X → S, we have [X]S = [G] · 1S in K(Var/S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Since G is special, the G-torsor X → S is Zariski-locally trivial, so there exists a stratification {Si}i∈I of S such that X×SSi ∼= G×Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Now, [X]S = � i∈I[G×Si]S = [G]·� i∈I[Si]S = [G]·1S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Let 1 → N → G → H → 1 be an exact sequence of algebraic groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' If N and H are special, then so is G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Since G being special is equivalent to H1 ´et(X, G) = H1 Zar(X, G) for all k-schemes X, the result follows from long exact sequences in cohomology, and the five lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Alternatively, any G-torsor X → S can be written as the composition of the N-torsor X → X/N and the H-torsor X/N → X/G ∼= S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' As H is special, there exist opens Si ⊂ S such that (X/N) ×S Si ∼= H × Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Pulling back the N-torsor X ×S Si → H × Si along Si (1,id) −−−→ H × Si gives an N-torsor Yi → Si, which is also Zariski-locally trivial as N is special.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Hence, there exist opens Sij ⊂ Si such that Yi ×Si Sij ∼= N × Sij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' There is now a natural morphism G × Sij → X ×S Sij of G-torsors over Sij, which must be an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Therefore, X → S is Zariski-locally trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' 6 Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' ■ By Hilbert’s Theorem 90, the general linear groups GLn are special [18, Propo- sition III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='9, Lemma III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' ■ From the exact sequence 0 → SLn → GLn det −−→ Gm → 1 follows by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='8 that SLn is also special.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' ■ The group Z/2Z is not special.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' For example, for char(k) ̸= 2, the Z/2Z-torsor A1 k \\ {0} → A1 k \\ {0} given by x �→ x2 is not locally trivial in the Zariski topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Let k be an algebraically closed field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Any unipotent group U ⊂ Un over k is special.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' By [17, Theorem 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='24], U can be realized as an extension of copies of Ga.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' From [18, Proposition III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='7] follows that Ga is special, so the result follows from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='11 (Motivic Orbit-Stabilizer Theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Let G be an algebraic group acting on a variety X over a field k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' If ξ ∈ X is a point such that Stab(ξ) is special, then [G] = [Stab(ξ)] · [Orbit(ξ)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' The map G → Orbit(ξ) given by g �→ gξg−1 is a Stab(ξ)-torsor, hence Zariski-locally trivial, so the result follows from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' The above lemma does not hold without the condition that Stab(ξ) is special.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Namely, for char(k) ̸= 2, consider G = Gm acting on X = A1 k via t · x = t2x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Then, Orbit(1) = A1 k \\ {0} and Stab(1) = Z/2Z, but [G] ̸= 2(q − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='3 Algebraic representatives Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Let G be an algebraic group over k, and let X be a variety with a transitive G-action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' A point ξ ∈ X is an algebraic representative for X if, Zariski-locally on X, there exists a morphism of varieties γ : X → G such that x = γ(x) · ξ for all x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Equivalently, ξ ∈ X is an algebraic representative if the Stab(ξ)-torsor Pξ = {(x, g) ∈ X × G | x = g · ξ} πX −−→ X is Zariski-locally trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' If there exists an algebraic representative ξ ∈ X, then all ξ′ ∈ X are algebraic representatives, with γ′ equal to γ post-composed with some translation on G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Algebraic representatives need not always exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Suppose char(k) ̸= 2 and consider G = Gm acting on X = A1 k \\ {0} by t · x = t2x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Then X does not have an algebraic representative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Namely, without loss of generality we may take ξ = 1, but then the Z/2Z-torsor Pξ = {(x, t) ∈ X × G | x = t2} πX −−→ X is non-trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' 7 Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Suppose char(k) ̸= 2 and consider the element A = �−1 0 0 1 � in PGL2, whose stabilizer ��x 0 0 1 �� ⊔ ��0 x 1 0 �� ∼= Gm ⋊ Z/2Z is not special.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Namely, the conjugacy class of A consists of all elements with trace 0, and since [PGL2] = q3 − q is not divisible by [{A ∈ PGL2 | tr(A) = 0}] = q2, the stabilizer of A is not special by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Let G be an algebraic group over k, and let X be a variety with a G-action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' A family of algebraic representatives for X is a morphism of varieties ρ : X → T with a section ξ : T → X such that, Zariski-locally on X, there exists a morphism of varieties γ : X → G such that x = γ(x) · ξ(ρ(x)) for all x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Consider G = T2 acting on X = ��a b 0 1 � | a ̸= 0, 1 � by conjugation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Then X has a family of representatives given by T = ��a 0 0 1 � | a ̸= 0, 1 � , with ξ and ρ the inclusion and projection, and γ � a b 0 1 � = � 1 b 1−a 0 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Let k be an algebraically closed field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Consider G = Tn for some n ≥ 1, acting by conjugation on a unipotent conjugacy class U ⊂ Tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Then U has an algebraic representative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Pick any point ξ ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' We need show that the Stab(ξ)-torsor P = {(u, g) ∈ U × G : u = gξg−1} πU −−→ U is locally trivial in the Zariski topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' In particular, it suffices to show that Stab(ξ) is a special group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Since Tn is triangularizable, the subgroup Stab(ξ) is so as well, hence we can write 1 → U → Stab(ξ) → D → 1, where U is the maximal normal unipotent subgroup, which is special by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Furthermore, there is a natural splitting D → Stab(ξ), and any d ∈ D must satisfy dii = djj whenever ξij ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Hence, D is a product of copies of Gm, which is also special.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Now the result follows from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' The following lemma shows why it is useful to have algebraic representatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Let S be a variety with a G-action and a family of algebraic representatives ξ : T → S and ρ : S → T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Then for any morphism f : Y → S and G-equivariant morphism g : X → S, we have [X ×S Y ]S = [(X ×S T ) ×T Y ]S ∈ K(Var/S), where (X ×S T ) ×T Y is seen as a variety over S via the composition f ◦ πY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Zariski-locally on X, there is a commutative diagram X ×S Y (X ×S T ) ×T Y S ϕ ψ f◦πY 8 where ϕ(x, y) = (γ(f(y))·x, ρ(f(y)), y) and ψ(x, t, y) = (γ(f(y))·x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Note that ϕ and ψ are easily seen to be well-defined over S and inverse to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' In the case of algebraic representatives, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' when T is a point, we obtain the following corollaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Let S be a variety with a G-action and an algebraic representative ξ ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Then for any morphism f : Y → S and G-equivariant morphism g : X → S, we have [X ×S Y ]S = [X|ξ] · [Y ]S ∈ K(Var/S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Let S be a variety with G-action and algebraic representative ξ ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Then for any G-equivariant map f : X → S, we have [X]S = [X] [S] · 1S ∈ K(Var/S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Apply the previous corollary with f = idS to find that [X]S = [X|ξ] · 1S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Looking at this equality in K(Vark), we see that [X|ξ] = [X]/[S].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='4 Algorithmic computations in K(Vark) Let A = {a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' , an} be a finite set, and let F and G be finite subsets of k[A].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Then we write X(A, F, G) for the reduced locally closed variety of An k given by f = 0 for all f ∈ F and g ̸= 0 for all g ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' In this subsection we will describe strategies for computing the virtual class of varieties of this form in K(Vark).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' These strategies are combined into a recursive algorithm, Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='24, which will be used in Section 3 to compute the virtual class of the representation variety RG(Σg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' We remark already that the algorithm will not be a general recipe for computing virtual classes, since it is allowed to fail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' In fact, whenever the algorithm does not fail, it will return the virtual class of X(A, F, G) as a polynomial in q = [A1 k], and it is clear that not all virtual classes are of this form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' However, it turns out that the algorithm is sufficiently general for the purposes of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' The same algorithm was used in [7, Appendix A] to compute the virtual class of RTn(Σg) for 1 ≤ n ≤ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' An implementation of this algorithm can be found at [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Notation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' For a ∈ A, f ∈ k[A] and u ∈ k[A \\ {a}], we write eva(f, u) for the evaluation of f in a = u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' When u, v ∈ k[A \\ {a}], we denote by eva(f, u/v) the evaluation of f in a = u/v multiplied by vdega(f), so that eva(f, u/v) ∈ k[A \\ {a}].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' For subsets F ⊂ k[A], we write eva(F, −) = {eva(f, −) : f ∈ F}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Input: Finite sets A, F and G as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Output: The virtual class of X(A, F, G) as a polynomial in q = [A1 k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' 9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' If F contains a non-zero constant or if 0 ∈ G, then X = ∅, so return [X] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' If F = G = ∅ or A = ∅, then X = A#A k , so return [X] = q#A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' If all f ∈ F and g ∈ G are independent of some a ∈ A, then X ∼= A1 k × X′ with X′ = X(A \\ {a}, F, G), so return [X] = q[X′].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' If f = un (with n > 1) for some f ∈ F and u ∈ k[A], then we can replace f with u, not changing X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' That is, X = X(A, (F \\ {f}) ∪ {u}, G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Similarly, if g = un (with n > 1) for some g ∈ G and u ∈ k[A], then X = X(S, F, (G \\ {g}) ∪ {u}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' If some f ∈ F is univariate in a ∈ A, and factors as f = (a − α1) · · · (a − αm) for some αi ∈ k, then return [X] = �m i=1[Xi] with Xi = X(A \\ {a}, eva(F \\ {f}, αi), eva(G, αi)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Suppose f = uv for some f ∈ F and non-constant u, v ∈ k[A].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Then X can be stratified by X ∩ {u = 0} and X ∩ {u ̸= 0, v = 0}, so return [X] = [X1] + [X2] with X1 = X(A, (F \\ {f}) ∪ {u}, G), X2 = X(A, (F \\ {f}) ∪ {v}, G ∪ {u}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Suppose f = au + v for some f ∈ F, a ∈ A and u, v ∈ k[A] with u and v independent of a and u non-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Then X can be stratified by X ∩ {u = 0, v = 0} and X ∩ {u ̸= 0, a = −v/u}, so return [X] = [X1] + [X2] with X1 = X(A, (F \\ {f}) ∪ {u, v}, G), X2 = X(A, eva(F \\ {f}, −v/u), eva(G, −v/u) ∪ {u}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' If char(k) ̸= 2, suppose f = a2u+av+w for some f ∈ F, a ∈ S and u, v, w ∈ k[A] with u, v and w independent of a, and u non-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Moreover, suppose that the discriminant D = v2 − 4uw is a square, that is, D = d2 for some d ∈ k[A].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Then return [X] = [X1] + [X2] + [X3] + [X4], with X1 = X(A, (F \\ {f}) ∪ {u, av + w}, G), X2 = X(A, eva(F \\ {f}, −v/2u) ∪ {D}, eva(G, −v/2u) ∪ {u}), X3 = X(A, eva(F \\ {f}, (−v − d)/2u), eva(G, (−v − d)/2u) ∪ {u, D}), X4 = X(A, eva(F \\ {f}, (−v + d)/2u), eva(G, (−v + d)/2u) ∪ {u, D}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' If G ̸= ∅, pick any g ∈ G, and return [X] = [X1] − [X2] with X1 = X(A, F, G \\ {g}), X2 = X(A, F ∪ {g}, G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Fail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' 10 3 TQFT method The idea of the TQFT method is to define a Topological Quantum Field Theory (TQFT), that is, a lax monoidal functor Z : Bordn → K(Vark)-Mod from the category of (pointed) n-bordisms to the category of modules over K(Vark), such that the invariant Z(M)(1) ∈ K(Vark) associated to a closed manifold X is precisely the class [RG(X)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Then, functoriality allows for the closed manifold X to be seen as a composition of simpler bordisms, so that computing the image under Z of the simpler bordisms gives the invariant of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' For the purpose of this paper, it suffices to take n = 2 and define maps Z( ), Z( ) and Z( ) that mimic the TQFT, rather than giving the full description of the TQFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Although these maps do not define an actual TQFT, they compute the same invariant (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='2) and have the advantage of being easier to define and to deal with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' For an elaborate discussion on the TQFT method, we refer to [4, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Let G be an algebraic group over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Define the following K(Vark)-module mor- phisms, Z( ) : K(Vark) → K(Var/G), 1 �→ \uf8ee \uf8ef\uf8ef\uf8ef\uf8f0 {1} G \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fb , Z( ) : K(Var/G) → K(Vark), \uf8ee \uf8ef\uf8ef\uf8f0 X G f \uf8f9 \uf8fa\uf8fa\uf8fb �→ � f −1(1) � , Z( ) : K(Var/G) → K(Var/G), \uf8ee \uf8ef\uf8ef\uf8f0 X G f \uf8f9 \uf8fa\uf8fa\uf8fb �→ \uf8ee \uf8ef\uf8ef\uf8ef\uf8f0 X × G2 (x, A, B) G f(x)[A, B] \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' For any algebraic group G and g ≥ 0, we have [RG(Σg)] = Z( ) ◦ Z( )g ◦ Z( )(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' This follows directly from the explicit form of the G-representation variety of Σg (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' This theorem shows that, in order to compute [RG(Σg)], we must understand the map Z( ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' In the following subsections, we will take G = Tn with 1 ≤ n ≤ 5 and describe how to compute the matrix associated to Z( ) when restricted to the K(Vark)-submodule of K(Var/G) generated by the unipotent conjugacy classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' To be precise, we will take G equal to ˜Tn = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed a1,1 a1,2 · · a1,n−1 a1,n 0 a2,2 · · a2,n−1 a2,n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' 0 0 · · an−1,n−1 an−1,n 0 0 · · 0 1 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 | ai,i ̸= 0 \uf8fc \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8fd \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8fe , 11 and using the isomorphism Tn ∼= Gm × ˜Tn, it follows that [RTn(Σg)] = [RGm(Σg)][R˜Tn(Σg)] = (q − 1)2g[R˜Tn(Σg)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='1) The reason to focus on ˜Tn rather than Tn is to simplify the equations a little bit by reducing the number of variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' In Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='1 we will give an example with G = ˜T2 in order to demonstrate the method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' For higher n, the process will be analogous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' In particular, we will work with a K(Vark)-submodule of K(Var/G) generated by the unipotent conjugacy classes of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Then the matrix representing Z( ) is computed with respect to these generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' To accomplish this, we need to determine the unipotent conjugacy classes of G, equations describing them, and an algebraic representatives for each one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' This data can be computed automatically, and is done in Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' In Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='4 we will compute the matrix representing Z( ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' The final results will be presented in Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='1 Example G = ˜T2 In order to demonstrate the method, we will compute the TQFT, that is, the maps in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='1, for the group G = ˜T2 by hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' This group is also known as the group of affine transformations of the line AGL1 = �� a b 0 1 � | a ̸= 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' In particular, the corresponding representation variety RAGL1(Σg) parametrizes flat rank one affine bundles over Σg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' As the computations below will show, the map Z( ) restricts to an endomorphism of the K(Vark)-submodule of K(Var/G) generated by the classes of I = {1} → G and J = �� 1 x 0 1 � | x ̸= 0 � → G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Therefore, we will represent the maps as matrices with respect to these generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' In particular, it is clear that Z( ) = � 1 0 � and Z( ) = � 1 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' The map Z( ) is, with respect to the generators I and J , represented by the matrix Z( ) = q2 � q − 1 (q − 2)(q − 1) q − 2 q2 − 3q + 3 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' First, let us compute Z( )(I)|I = �� (A, B) ∈ ˜T2 2 | [A, B] = 1 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Writing A = � a b 0 1 � 12 and B = � x y 0 1 � , the equation [A, B] = 1 translates to ay − bx + b − y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' We distinguish the following cases: ■ If a = 1, then b(x − 1) = 0, so either b = 0 or x = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Hence, the virtual class of this stratum is q (y) ((q − 1) (b=0) + q (x=1) − 1 (b=0 x=1) ) = 2q(q − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' ■ If a ̸= 1, then y = b(x − 1)/(a − 1), yielding a virtual class of q(q − 2)(q − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Together, Z( )(I)|I = q2(q − 1)I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Next, note that c!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='Z( )(I)|I + [J ]c!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='Z( )(I)|J = [˜T2]2 = q2(q − 1)2, from which follows that Z( )(I)|J = q2(q − 2)J , using Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='22 and the fact that J has an algebraic representative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' The next coefficient can be computed from the previous one as Z( )(J )|I = ([J ]c!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='Z( )(I)|J )I = q2(q − 2)(q − 1)I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Finally, for the last coefficient, note that c!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='Z( )(J )|I + [J ]c!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='Z( )(J )|J = [J ][˜T2]2 = q2(q − 1)3, from which follows that Z( )(J )|J = q2(q2 − 3q + 3)J .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' In order to apply Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='2, it is convenient to diagonalize the map Z( ) as PDP −1 with P = � 1 − q 1 1 1 � and D = � q2 0 0 q2(q − 1)2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Now from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='2, and straightforward matrix multiplication, it follows that [R˜T2(Σg)] = q2g−1(q − 1)2g + q2g−1(q − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='2 Describing conjugacy classes The aim of this subsection is to describe the conjugacy classes of ˜Tn, with a focus on the unipotent conjugacy classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' In particular, we will find algebraic representatives for the unipotent conjugacy classes, and families of algebraic representatives for the non-unipotent conjugacy classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Further- more, we will find equations describing the unipotent conjugacy classes, and finally, compute the virtual class of the unipotent conjugacy classes as well as that of the stabilizers of their representa- tives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' All of this data is needed in the later subsections in order to compute the matrix of Z( ) with respect to the generators given by the unipotent classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Representatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' In order to find the number of unipotent conjugacy classes of ˜Tn, and represen- tatives for each one, one can use Belitskii’s algorithm as described in [3], which works for 1 ≤ n ≤ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' 13 Note that the representatives will be automatically algebraic by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Denote the re- sulting unipotent conjugacy classes by U1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' , UM and the respective algebraic representatives by ξ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' , ξM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' The number M of unipotent conjugacy classes, depending on n, is given by the following table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' n 1 2 3 4 5 M 1 2 5 16 61 Alternatively, one can use the qualitative result of Belitskii’s algorithm that for 1 ≤ n ≤ 5 every unipotent matrix can be conjugated in ˜Tn to a matrix containing only 0’s and 1’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' This gives a finite number of matrices (2n(n−1)/2) which can easily be partitioned based on whether or not they are conjugate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Now take one representative out of each conjugacy class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Next, we describe the non-unipotent conjugacy classes of ˜Tn in terms of families depending on their diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Define a diagonal pattern to be a partition of the set {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=', n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Then, for any matrix A ∈ ˜Tn, the diagonal pattern δA of A is the partition such that i and j are equivalent if Aii = Ajj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Note that two matrices A and B in ˜Tn are conjugate only if their diagonals coincide, but not necessarily if.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Now, we look at the following families of conjugacy classes: Cδ,i = {A ∈ ˜Tn | δA = δ and A ∼ diag(A) + ξi − 1}, for any diagonal pattern δ and 1 ≤ i ≤ M, where diag(A) denotes the diagonal part of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Any such Cδ,i has a family of representatives ξδ,i : Cδ,i = {A ∈ ˜Tn | δA = δ and A = diag(A) + ξi − 1} → Cδ,i given by inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Of course, for some diagonal matrices D it may happen that D + ξi − 1 and D + ξj − 1 are conjugate while i ̸= j, but such duplicates are easily filtered out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' After this filtering, we obtain families of conjugacy classes C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' , CN, where the number N is given, depending on n, by the following table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' n 1 2 3 4 5 N 2 3 12 61 372 Note that we can choose our indices in such a way that the ξi coincide with the unipotent represen- tatives for 1 ≤ i ≤ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Next, we want to find equations describing the unipotent conjugacy classes Ui for 1 ≤ i ≤ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' For simplicity we will compute equations for the closure Ui rather than Ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' In Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='3 it will be shown that this is sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' The closure Ui is the closure of the image of the map fi : G → G, g �→ gξig−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Since G is affine, fi can equivalently be described by a morphism of rings f # i : OG(G) → OG(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' In particular, the closure Ui corresponds to the ideal Ii ⊂ OG(G) which is the kernel of f # i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Gen- erators for such these ideals can be computed using Gr¨obner basis [1], and this gives us the desired equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' In particular, we use [1, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='2] in order to compute the kernel of f # i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' 14 Orbits and stabilizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' For any A ∈ ˜Tn, in order to compute the virtual class of the orbit of A, we can use Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Indeed, as we have seen in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='19, the stabilizer Stab(A) of any A ∈ ˜Tn is special.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' In order to compute the virtual class of the stabilizer of A, we use Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='24, since we have an explicit description Stab(A) = {B ∈ ˜Tn | AB − BA = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Regarding the families of conjugacy classes Ci, we will make use of the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' For every 1 ≤ i ≤ N, there is an isomorphism Ci ∼= Ci × Orbit(A) for every A ∈ Ci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Note that the fibers of Ci → Ci are given by Orbit(A) ∼= G/Stab(A), so it suffices to show that Stab(A) is constant for all A ∈ Ci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' As there are only a finite number of cases to consider, this can easily be verified explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Alternatively, note that B ∈ Stab(A) if and only if for all 1 ≤ i ≤ j ≤ n, Bij(Aii − Ajj) + j � k=i+1 BkjAik − j−1 � k=i BikAkj = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' (∗) We claim that Bij = 0 for all i ≤ j such that Aii ̸= Ajj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' The result follows from this claim, as Aij ∈ {0, 1} for i ̸= j, so the solutions to (∗) will be independent of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' We proof the claim by induction on j − i, the case j − i = 0 being trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' For the general case, take i ≤ j such that Aii ̸= Ajj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' if there exists a k ∈ {i + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' , j} such that Aki ̸= 0, then Akk = Aii ̸= Ajj, so Bkj = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Similarly, if there exists a k ∈ {i, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' , j − 1} such that Akj ̸= 0, then Akk = Ajj ̸= Aii so Bki = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Therefore, (∗) reduces to Bij = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='3 Transition matrix Let X be a variety with stratification Xi, and let Y be a variety over X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' The goal of this subsection is to show that, in order to compute the classes [Y ∩ Xi]X, it is sufficient to compute the classes [Y ∩ Xi]X instead, making use of an inclusion–exclusion principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Suppose X is stratified by X0 ⊂ X closed and its open complement X1 = X \\ X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' If we want to compute [Y ∩ X0] and [Y ∩ X1], then computing the latter would likely result in the computation [Y ∩ X] − [Y ∩ X0], which shows that the result of the computation of [Y ∩ X0] can be reused.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Therefore, instead of computing [Y ∩X0] and [Y ∩X1], one can compute [Y ∩X0] = [Y ∩X0] and [Y ∩ X1] = [Y ∩ X] = [Y ], from which formally follows that [Y ∩ X1] = [Y ∩ X1] − [Y ∩ X0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Let X be a topological space with stratification {Xi}i∈I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Then Xi = Xj if and only if i = j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' For each i ∈ I, write Xi = Zi ∩ Ui for some closed Zi ⊂ X and open Ui ⊂ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Without loss of generality, we may assume Zi = Xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Now, if Xi = Xj for some i, j ∈ I, then both Xi and Xj are 15 open and dense in Xi = Xj, so they must intersect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' But this contradicts the assumption that Xi and Xj are disjoint, since they are part of the stratification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Let X be a variety over S with a stratification {Xi}i∈I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Put a partial ordering on I where i ≤ j if and only if Xi ⊂ Xj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Reflexivity and transitivity are clear, and anti-symmetry follows from the above lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE0T4oBgHgl3EQfiAFl/content/2301.02439v1.pdf'} +page_content=' Hence, the equalities [Xi]S = [Xi]S − � j 3 +2. Then we show that strong enough noise can actually +prevent blow-up with probability 1. Finally, we analyse the effects of weak noise and present conditions +on the initial data that lead to the global existence and the blow-up in finite time of the solutions, and +their associated probabilities are also obtained. +Keywords: Stochastic modified two-component Camassa-Holm system (SMCH2); Pathwise solutions; +Global existence; Blow-up criterion; Blow-up scenatios. +1 +Introduction +Consider the following integrable two-component Camassa-Holm (CH2) shallow water system + + + + + + + +(u − uxx)t + 3uux − 2uxuxx − uuxxx + ρρx = 0, +t > 0, x ∈ R, +ρt + (ρu)x = 0, +t > 0, x ∈ R, +u(0, x) = u0(x), +x ∈ R, +ρ(0, x) = ρ0(x), +x ∈ R. +This system appears initially in [1]. In 2008, it was derived by Constantin and Ivanov in [2], which provided +a demonstration about its derivation in view of the fluid shallow water theory from the hydrodynamic +point of view. +Similar to the Camassa-Holm equation, this system possesses the peakon, multi-kink +solutions and the bi-Hamiltonian structure [3, 4] and is integrable. Well-posedness and wave breaking +mechanism were discussed in [5, 6, 7] and the existence of global solutions was analyzed in [2, 6, 8]. +Obviously, under the constraint of ρ(x, t) = 0, this system reduces to the cerebrated Camassa-Holm +(CH) equation, which was derived physically by Camassa and Holm in [9] (found earlier by Fokas and +Fuchssteiner [10] as a bi-Hamiltonian generalization of the KdV equation) by approximating directly the +Hamiltonian for Euler’s equation in the shallow water region with u(x, t) representing the free surface +above a flat bottom. CH equation is completely integrable [11, 12] and has infinitely many conservation +laws [13]. Local well-posedness for the initial data u0 ∈ Hs with s > 3/2 was proved in [14, 15]. One of +the remarkable features of the CH equation is the presence of breaking waves as well as global solutions +∗Corresponding author +1 + +in time. Wave breaking for a large class of initial data has been established in [14, 15, 16, 17, 18, 19]. +Global solutions were also explored in [14, 16]. The solitary waves of the CH equation are peaked solitons +and are orbitally stable [20]. If ρ(x, t) ̸= 0, this CH2 system is actually an extension of the CH equation. +However, a modified version of the two-component Camassa-Holm (MCH2) system allows a depen- +dence on the average density ρ as well as the pointwise density ρ, and it is written as + + + + + + + +(u − uxx)t + 3uux − 2uxuxx − uuxxx + ρρx = 0, +t > 0, x ∈ R, +ρt + (ρu)x = 0, +t > 0, x ∈ R, +u(0, x) = u0(x), +x ∈ R, +ρ(0, x) = ρ0(x), +x ∈ R, +(1.1) +where u denotes the velocity field, ρ = (1−∂2 +x)(ρ−ρ0) with some constant ρ0. This system was introduced +by Holm et al. in [21], and it does admit peaked solutions in the velocity and average density. Many +authors analytically identified the steepening mechanism that allows the singular solutions to emerge from +smooth spatially confined initial data. They found that wave breaking in the fluid velocity does not imply +singularity in the pointwise density ρ at the point of vertical slope. (1.1) may not be integrable unlike +the CH2 system. The characteristic is that it will amount to strengthening the norm for ρ from L2 to H1 +in the potential energy term. Letting γ = ρ − ρ0, it leads to the conserved quantity +� +R ∥u∥2 +H1 + ∥γ∥2 +H1dx, +which is absent in the CH2 system. This property inspired a series of interesting works for a deep insight +into the MCH2 system in the recent years. The Cauchy problem of (1.1) has been studied in many works +[22, 23, 21, 24]. It has been shown that this system is locally well-posed on the line [22] and on the circle +[24]. Moreover, the authors presented several blow-up results [22, 23, 24, 25]. In addition, basing on a +conserved quantity, the authors established the global existence results for strong solutions to the system +[26]. +Before introducing our model, we recall some theory of infinite dimensional stochastic analysis. Let +S = (Ω, F, P, {Ft}t≥0, W1, W2), +where (Ω, F, P, {Ft}t≥0) is a complete filtration probability space, and W1, W2 are two cylindrical Wiener +process on some separable Hilbert space U and d⟨W1, W2⟩t = κdt, −1 ≤ κ ≤ 1. To be precise, we consider +a separable Hilbert space U as well as a larger Hilbert space U0 such that the canonical embedding U ֒→ U0 +is Hilbert-Schmidt. Therefore we have +Wi = +∞ +� +k=1 +W i +kek ∈ C([0, ∞), U0), +i = 1, 2, +where {W i +k}k≥1 is a sequence of mutually independent one-dimensional Brownian motions and {ek}k∈N +is a complete orthonormal basis of U. +To define the Itˆo stochastic integral +� t +0 +GdWi = +∞ +� +k=1 +� t +0 +GekdW i +k, +i = 1, 2 +on Hs, it is required in [27, 28] for the predictable stochastic process G to take values in the space of +L2(U; Hs), the Hilbert-Schmidt operators from U to Hs. We have +� � t +0 +GdWi, v +� +Hs = +∞ +� +k=1 +� t +0 +(Gek, v)HsdW i +k, +i = 1, 2. +Moreover, the Burkholder-Davis-Gundy inequality +E +� +sup +t∈[0,T ] +���� +� t +0 +GdWi +���� +p +Hs +� +≤ C(p, s)E +� � T +0 +∥G∥2 +L2(U;Hs)ds +� p +2 +, p ≥ 1, i = 1, 2 +2 + +holds for some constant C(p, s) > 0. +In this paper, we are interested in stochastic variants of the MCH2 system to model energy con- +suming/exchanging mechanisms in (1.1) that are driven by external stochastic influences. Adding mul- +tiplicative noise has also been connected to the prevailing hypotheses that the onset of turbulence in +fluid models involves randomness [29, 30, 31]. Precisely, we consider stochastic modified two-component +Camassa-Holm (SMCH2) system + + + + + + + +(u − uxx)t + 3uux − 2uxuxx − uuxxx + ρρx = (1 − ∂2 +x)h1(t, u, ρ) ˙ +W1, +t > 0, x ∈ R, +ρt + (ρu)x = (1 − ∂2 +x)h2(t, u, ρ) ˙ +W2, +t > 0, x ∈ R, +u(0, x) = u0(x), +x ∈ R, +ρ(0, x) = ρ0(x), +x ∈ R, +(1.2) +where h1(t, u, ρ), h2(t, u, ρ) are typically nonlinear functions. +Let γ = ¯ρ − ¯ρ0, then (1 − ∂2 +x)−1ρ = γ. Notice that the deterministic MCH2 type equations with the +weakly dissipative term λ2(1 − ∂2 +x)h1(t, u, ρ), λ2(1 − ∂2 +x)h2(t, u, ρ) have been introduced and studied by +many scholars [32, 33, 34]. In order to model more general random energy exchange, we consider the +possibly nonlinear noise term (1 − ∂2 +x)h1(t, u, ρ) ˙ +W1, (1 − ∂2 +x)h2(t, u, ρ) ˙ +W2 in (1.2), which will be used to +compare with deterministic weakly dissipative MCH2 type equations. +In (1.2), the operator (1 − ∂2 +x)−1 can be expressed by it’s associated Green’s function G(x) = e−|x|/2 +with +[(1 − ∂2 +x)−1f](x) = [G ∗ f](x) = 1 +2 +� +R +e−|x−y|f(y)dy. +So the system (1.2) is equivalent to tha following one + + + + + + + +du + [uux + F1(u, γ)]dt = h1(t, u, γ)dW1, t > 0, x ∈ R, +dγ + [uγx + F2(u, γ)]dt = h2(t, u, γ)dW2, t > 0, x ∈ R, +u(0, x) = u0(x), x ∈ R, +γ(0, x) = γ0(x), x ∈ R, +(1.3) +where F1(u, γ) = ∂x(1 − ∂2 +x)−1(u2 + 1 +2u2 +x + 1 +2γ2 − 1 +2γ2 +x), F2(u, γ) = (1 − ∂2 +x)−1((uxγx)x + uxγ). +The purpose of this paper is as follows: +• The first goal of the present paper is to analyze the existence and uniqueness of pathwise solutions +and to determine possible blow-up criterion for the Cauchy problem (1.3). Under generic assumptions +on h1(t, u, γ), h2(t, u, γ), we will show that (1.3) has a local unique pathwise solution(see Theorem 3.5 +below). +• The second goal of this work is to study the case of strong nonlinear noise and consider its effect. +As we will see in (3.3) below, for the solution to (1.3), its Hs × Hs-norm blows up if and only if its +W 1,∞×W 1,∞-norm blows up. This suggests choosing a noise coefficient involving the W 1,∞×W 1,∞-norm +of (u, γ). Therefore in this work we consider the case that h1(t, u, γ)dW1 = h1(t, u, γ)d(�∞ +i=1 Wi(t)ei) = +a(t)(1+∥u∥W 1,∞ +∥γ∥W 1,∞)θudW1, where {ei}i∈N∗ denote an orthonormal basis, {Wi}i∈N∗ is a family of +independent standard real-valued Wiener processes. Similarly, we consider the case that h2(t, u, γ)dW2 = +a(t)(1 + ∥u∥W 1,∞ + ∥γ∥W 1,∞)θγdW1, where θ > 0, 0 < a∗ ≤ a2(t) ≤ a∗. To simplify the model, we write +W1 as W, and we will try to determine the range of θ and a∗, a∗ such that the solution exists globally in +time. +• The third goal of this paper is to consider weak linear noise effects associated with the phenomenon +of wave breaking. Due to Theorem 3.6 below, we see that if wave breaking occurs, the noise term does not +grow fast. Hence we consider θ = 0 in (1.3), namely a non-autonomous pre-factor depending on time t. +Precisely, we consider the MCH2 equation with linear multiplicative noise. We will study the conditions +that lead to the global existence and the blow-up in finite time of the solution, and then analyze the +associated probabilities. +3 + +2 +Notation and preliminaries +In this section, we begin by introducing some notations and recall some elementary results, for complete- +ness, we list the lemmas and skip their some proofs for conciseness. +Let L2 be the usual space of square-integrable functions on R. For any real number s ∈ R, Ds = +(1 − ∂2 +x)s/2 is defined by � +Dsf(x) = (1 + x2) +s +2 �f(x), where �f is the Fourier transform of f. The Sobolev +space Hs is defined as +Hs ≜ {f ∈ L2(R) : ∥f∥2 +Hs = +� +R +(1 + x2)s| �f(x)|2dx < ∞}, +and the inner product +(f, g)Hs := +� +R +(1 + x2)s �f(x)�g(x)dx = ( � +Dsf, � +Dsg)L2. +In addition, x ≲ y, x, y ∈ R means that there exists C > 0, which may vary from line to line and +depend on various parameters, such that x ≤ Cy. Hereafter, C denotes a positive constant, whose value +may change from one place to another. +Firstly, we summarize some auxiliary results, which will be used to prove our main results. Define the +regularizing operator Tǫ on R as +Tǫf(x) := (1 − ǫ2∆)−1f(x) = +� +R +eiξx ˆf(ξ) +1 + ǫ2|ξ|2 dξ, +ǫ ∈ (0, 1). +(2.1) +Since Tǫ can be characterized by its Fourier multipliers, see [35], it is easy to see that +[Ds, Tǫ] = 0, +(Tǫf, g)L2 = (f, Tǫg)L2, +∥Tǫu∥Hs ≤ ∥u∥Hs. +(2.2) +Where [Ds, Tǫ] = DsTǫ − TǫDs. We therefore have the following lemma. +Lemma 2.1 [35] Let f, g : R → R such that g ∈ W 1,∞ and f ∈ L2. Then for some C > 0, +∥[Tǫ, (g · ∇)]f∥L2 ≤ C∥g∥W 1,∞∥f∥L2. +Furthermore, we also need to recall some useful commutator estimates. +Lemma 2.2 [36] If r > 0, then Hr � L∞ is an algebra. Moreover, ∥uv∥Hr ≲ ∥u∥L∞∥v∥Hr+∥u∥Hr∥v∥L∞. +Lemma 2.3 [36] Let r > 0, if u ∈ Hr � W 1,∞ and v ∈ Hr−1 � L∞, then +∥[Dr, u]v∥L2 ≲ ∥∂xu∥L∞∥Dr−1v∥L2 + ∥Dru∥L2∥v∥L∞, +where [Dr, u]v = Druv − uDrv. +A direct application of Lemma 2.2-2.3 gives the following estimates and we omit the proof here. +Lemma 2.4 For the F1, F2 defined in (1.3) and for any u, γ, u1, u2, γ1, γ2 ∈ Hs with s > 1/2, we have +∥F1(u, γ)∥Hs ≲(∥u∥W 1,∞ + ∥γ∥W 1,∞)(∥u∥Hs + ∥γ∥Hs), +s > 3/2, +∥F2(u, γ)∥Hs ≲(∥u∥W 1,∞ + ∥γ∥W 1,∞)(∥u∥Hs + ∥γ∥Hs), +s > 3/2, +∥F1(u1, γ1) − F1(u2, γ2)∥Hs ≲(∥u1∥Hs + ∥u2∥Hs + ∥γ1∥Hs + ∥γ2∥Hs) +(∥u1 − u2∥Hs + ∥γ1 − γ2∥Hs), +s > 3/2, +∥F1(u1, γ1) − F1(u2, γ2)∥Hs ≲(∥u1∥Hs+1 + ∥u2∥Hs+1 + ∥γ1∥Hs+1 + ∥γ2∥Hs+1) +4 + +(∥u1 − u2∥Hs + ∥γ1 − γ2∥Hs), +1/2 < s < 3/2, +∥F2(u1, γ1) − F2(u2, γ2)∥Hs ≲(∥u1∥Hs + ∥γ2∥Hs)(∥u1 − u2∥Hs + ∥γ1 − γ2∥Hs), +s > 3/2, +∥F2(u1, γ1) − F2(u2, γ2)∥Hs ≲(∥u1∥Hs+1 + ∥γ2∥Hs+1)(∥u1 − u2∥Hs + ∥γ1 − γ2∥Hs), +1/2 < s < 3/2. +In addition, we provide the following algebraic inequality, which will be used in the proof of Theorem 3.6. +Lemma 2.5 Let c, M1, M2 > 0. Assume a, b∗, b∗ > 0, +either η > 1, 0 < +� +b∗ < b(t) < +√ +b∗ and 2b∗ > b∗ +or η = 1, 0 < +� +b∗ < b(t) < +√ +b∗ and 2b∗ > a + b∗. +Then there is a constant C > 0 such that for all 0 ≤ x1 ≤ M1y1 < ∞, 0 ≤ x2 ≤ M2y2 < ∞, +a(x1 + x2)(y2 +1 + y2 +2) + b(t)(1 + x1 + x2)η(y2 +1 + y2 +2) +1 + y2 +1 + y2 +2 +−2b(t)(1 + x1 + x2)η(y2 +1 + y2 +2)2 +(1 + y2 +1 + y2 +2)2 ++ +cb(t)(1 + x1 + x2)η(y2 +1 + y2 +2)2 +(1 + y2 +1 + y2 +2)2(1 + log(1 + y2 +1 + y2 +2)) ≤ C. +Proof: Since 0 ≤ +x1 +M1 ≤ y1 < ∞, 0 ≤ x2 +M2 ≤ y2 < ∞, we obtain +a(x1 + x2)(y2 +1 + y2 +2) + b(t)(1 + x1 + x2)η(y2 +1 + y2 +2) +1 + y2 +1 + y2 +2 +− 2b(t)(1 + x1 + x2)η(y2 +1 + y2 +2)2 +(1 + y2 +1 + y2 +2)2 ++ +cb(t)(1 + x1 + x2)η(y2 +1 + y2 +2)2 +(1 + y2 +1 + y2 +2)2(1 + log(1 + y2 +1 + y2 +2)) +≤a(x1 + x2) + b∗(1 + x1 + x2)η − 2b∗(1 + x1 + x2)η +(y2 +1 + y2 +2)2 +(1 + y2 +1 + y2 +2)2 + +cb∗(1 + x1 + x2)η +1 + log(1 + ( x1 +M1 )2 + ( x2 +M2 )2). +When η > 1 and 2b∗ > b∗ or η = 1 and 2b∗ > a + b∗, we find that the inequality will tend to −∞ for +x1 → ∞ or x2 → ∞ (namely y1 → ∞ or y2 → ∞), which completes the proof. +✷ +Finally, we present the following lemma to establish the Theorem 3.7 on the global existence solutions. +Lemma 2.6 Let α(t) be a deterministic and locally bounded function. Suppose that λ > 0, and x(t) +satisfies +x(t) = e +� t +0 α(t +′ )dWt′ −� t +0 λα2(t +′ )dt +′ +. +For R > 1 define τR = inf{t ≥ 0 : x(t) > R}. Then we have +P{τR = ∞} ≥ 1 − R−2λ. +Proof: Noting that +x(t)2λ = e +� t +0 2λα(t +′ )dWt′ − 1 +2 +� t +0 (2λ)2α2(t +′ )dt +′ +is an exponential martingale, by the martingale stopping theorem, we can derive Ex(t ∧ τR)2λ = 1. This +yields +P(τR = ∞) = lim +n→∞ P(τR > n) = lim +n→∞ P(x(n ∧ τR)2λ < R2λ) ≥ lim +n→∞ +� +1 − Ex(n ∧ τR)2λ +R2λ +� += 1 − R−2λ, +which finishes the proof. +✷ +5 + +Lemma 2.7 Let s > 3/2, F1, F2 and Tǫ be given in (1.3) and (2.1) respectively. Then there is a constant +K = K(s) > 0 such that for all ǫ > 0, +|(Tǫ[uux], Tǫu)Hs| + |(TǫF1(u, γ), Tǫu)Hs| + |(Tǫ[uγx], Tǫγ)Hs| + |(TǫF2(u, γ), Tǫγ)Hs| +≤ +K(∥u∥W 1,∞ + ∥γ∥W 1,∞)(∥u∥2 +Hs + ∥γ∥2 +Hs). +Proof: According to (2.2), we derive +(Tǫ[uux], Tǫu)Hs += +(DsTǫ[uux], DsTǫu)L2 += +([Ds, u]ux, DsT 2 +ǫ u)L2 + ([Tǫ, u]Dsux, DsTǫu)L2 + (uDsTǫux, DsTǫu)L2, +(Tǫ[uγx], Tǫγ)Hs += +(DsTǫ[uγx], DsTǫγ)L2 += +([Ds, u]γx, DsT 2 +ǫ γ)L2 + ([Tǫ, u]Dsγx, DsTǫγ)L2 + (uDsTǫγx, DsTǫγ)L2. +Then by Lemma 2.1, Lemma 2.2, (2.2) and Sobolev embedding Theorem, we have +|(Tǫ[uux], Tǫu)Hs| + |(Tǫ[uγx], Tǫγ)Hs| ≲ (∥u∥W 1,∞ + ∥γ∥W 1,∞)(∥u∥2 +Hs + ∥γ∥2 +Hs). +In addition, using Lemma 2.4 and (2.2), we obtain that +|(TǫF1(u, γ), Tǫu)Hs| + |(TǫF2(u, γ), Tǫγ)Hs| ≲ (∥u∥W 1,∞ + ∥γ∥W 1,∞)(∥u∥2 +Hs + ∥γ∥2 +Hs). +Combining the above two inequalities, we complete the proof. +✷ +3 +Local existence and uniqueness for SMCH2 +In this section, we consider the following stochastic system with multiplicative noise (1.3) in Hs(R) × +Hs(R). +3.1 +Assumptions +For the main results in this paper, we rely on the following different assumptions concerning random per- +turbation term in (1.3). We assume that (h1, h2) : [0, ∞)×(Hs×Hs) ∋ (t, u, γ) → (h1(t, u, γ), h2(t, u, γ)) ∈ +L2(U, Hs × Hs) are continuous in (t, u, γ). Moreover, we assume +Assumption 3.1 (H.1) There exists some non-decreasing function f : [0, ∞) → [0, ∞) with f(0) = 0 +such that for all (u, γ) ∈ Hs × Hs, s > 1/2, +2 +� +i=1 +∥hi(t, u, γ)∥L2(U,Hs) ≤ f(∥u∥W 1,∞ + ∥γ∥W 1,∞)(1 + ∥u∥Hs + ∥γ∥Hs). +(3.1) +(H.2) There exists some non-decreasing function g : [0, ∞) → [0, ∞) such that for all (u1, γ1), (u2, γ2) ∈ +Hs × Hs, s > 1/2, +sup +∥u1∥Hs ,∥γ1∥Hs,∥u2∥Hs ,∥γ2∥Hs≤N +2 +� +i=1 +∥hi(t, u1, γ1) − hi(t, u2, γ2)∥L2(U,Hs) +≤ +g(N) · (∥u1 − u2∥Hs + ∥γ1 − γ2∥Hs), N ≥ 1. +Assumption 3.2 h1(t, u, γ)dW1 = a(t)(1+∥u∥W 1,∞+∥γ∥W 1,∞)θudW, h2(t, u, γ)dW2 = a(t)(1+∥u∥W 1,∞+ +∥γ∥W 1,∞)θγdW for a standard 1-D Brownian motion W and θ > 0, 0 < a∗ ≤ a2(t) ≤ a∗ for all t. +Assumption 3.3 h1(u, γ)dW1 = b(t)udW, h2(u, γ)dW2 = b(t)γdW for a standard 1-D Brownian mo- +tion W, and there are constants b∗, b∗ > 0 such that 0 < b∗ ≤ b2(t) ≤ b∗ for all t. +6 + +3.2 +Definitions of the solutions. +Next, we give the definition of pathwise solution to (1.3). +Definition 3.4 (Pathwise solutions). Let S = (Ω, F, P, {Ft}t≥0, W1, W2) be a fixed stochastic basis. Let +s > 3/2 and z0 = (u0, γ0) be an Hs × Hs-valued F0-measurable random variable. +1.A local pathwise solution to (1.3) is a pair (z, τ), where τ ≥ 0 is a stopping time satisfying P{τ > 0} = 1 +and z = (u, γ) : Ω × [0, τ) → Hs × Hs is an Ft-adapted Hs × Hs-valued process satisfying P − a.s. +z ∈ C([0, τ); Hs × Hs), +(3.2) +and P − a.s., +u(t) − u(0) + +� t +0 +[uux + F1(u, γ)]dt +′ = +� t +0 +h1(t +′, u, γ)dW1, +γ(t) − γ(0) + +� t +0 +[uγx + F2(u, γ)]dt +′ = +� t +0 +h2(t +′, u, γ)dW2, +t ∈ [0, τ). +2. Local pathwise uniqueness: if given any two local pathwise solutions (z1, τ1) and (z2, τ2) with P{z1(0) = +z2(0)} = 1, we have +P{z1(t) = z2(t), +t ∈ [0, τ1 ∧ τ2)} = 1. +3.Additionally, (z, τ ∗) is called a maximal pathwise solution to (1.3) if τ∗ > 0 almost surely and there is +an increasing sequence τn → τ ∗ such that for any n ∈ N, (z, τn) is a pathwise solution to (1.3) and on +the set {τ ∗ < ∞}, +sup +t∈[0,τn] +(∥u∥Hs + ∥γ∥Hs) ≥ n, n ≥ 1. +4. If (z, τ ∗) is a maximal pathwise solution and τ ∗ = ∞ almost surely, then we call that the pathwise +solution exists globally. +3.3 +Main results and remarks. +Now, we summarize our major contributions, such as existence of pathwise solutions, global well-posedness +of (1.3) and the blow-up results, and the concrete proofs will be provided later in the remainder of the +paper. +Theorem 3.5 (Maximal solutions) Let s > 3/2, and h1(t, u, γ), h2(t, u, γ) satisfy Assumption 3.1. For +a given stochastic basis S = (Ω, F, P, {Ft}t≥0, W1, W2), if (u0, γ0) is an Hs × Hs-valued F0-measurable +random variable, then there is a local unique pathwise solution (z, τ) to (1.3) in the sense of Definition +3.4 with +z ∈ C([0, τ); Hs × Hs). +Moreover, (z, τ) can be extended to a unique maximal pathwise solution (z, τ ∗) and the following blow up +scenario satisfies P − a.s. on the set {τ∗ < ∞}, +1{lim sup +t→τ∗(∥u(t)∥Hs+∥γ(t)∥Hs)=∞} = 1{lim sup +t→τ∗(∥u(t)∥W 1,∞+∥γ(t)∥W 1,∞)=∞}. +(3.3) +Remark 3.1 The proof of Theorem 3.5 combines the techniques as used in the papers [35, 37, 38, 39, +40, 41]. By constructing the approximate sequence of the truncation problem of W 1,∞ × W 1,∞. Such a +cut-off means linear growth of u and γ, and guarantees the global existence of an approximate solution. +7 + +Turning to noise-driven regularization effects, the blow-up scenario (3.3) suggests relating the noise +coefficient to the W 1,∞ × W 1,∞ of (u, γ). Therefore we consider scalable noise impact, i.e. we assume +h1(t, u, γ)dW1 = a(t)(1+∥u∥W 1,∞ +∥γ∥W 1,∞)θudW, h2(t, u, γ)dW2 = a(t)(1+∥u∥W 1,∞ +∥γ∥W 1,∞)θγdW +for a standard 1-D Brownian motion W, some θ > 0, 0 < a∗ ≤ a2(t) ≤ a∗ for all t. When a∗, a∗ and θ +satisfy certain stronger conditions, the noise term remove the formation of singularities. +Theorem 3.6 (Global existence for strong nonlinear noise). Let Assumption 3.2 hold and assume that +S = (Ω, F, P, {Ft}t≥0, W) is a fixed stochastic basis. Let s > 5/2, (u0, γ0) ∈ Hs×Hs be an Hs×Hs-valued +F0-measurable random variable. Assume that θ and a∗, a∗(0 < a∗ ≤ a2(t) ≤ a∗) satisfy +either 2a∗ > a∗, θ > 1/2 or 2a∗ > K + a∗, θ = 1/2, +where K = K(s) is the constant introduced in Lemma 2.7. Then the corresponding maximal solution +(z, τ ∗) to (1.3) satisfies +P{τ ∗ = ∞} = 1. +Remark 3.2 Theorem 3.6 means that blow-up of pathwise solutions might only be observed if the noise +is weak. According to Theorem 3.6, we can see that if wave breaking occurs, the noise term will not bring +rapid growth. Therefore, we consider θ = 0 but a non-autonomous pre-factor dependent on time t is +introduced. +To detect such noise, we analyze the simpler form h1(t, u, γ)dW1 = b(t)udW, h2(t, u, γ)dW2 = b(t)γdW, +W is a standard 1-D Brownian motion. Even in this linear noise case the situation is quite interesting +allowing for global existence as well as blow-up of solutions. For global existence, we can identify two +cases. +Using Lemma 2.2-Lemma 2.4 and the integration by parts, we conclude that there is a C = C(s) > 1 +such that +− +� +R +Dsv1Ds(v1v1x)dx − +� +R +Dsv1DsF1(v1, v2)dx − +� +R +Dsv2Ds(v1v2x)dx − +� +R +Dsv2DsF2(v1, v2)dx +≤ 1 +2C(∥v1∥W 1,∞ + ∥v2∥W 1,∞)(∥v1∥2 +Hs + ∥v2∥2 +Hs). +(3.4) +Theorem 3.7 (Global existence for weak noise I). Let s > 3/2, Assumption 3.3 be verified and S = +(Ω, F, P, {Ft}t≥0, W) be a fixed stochastic basis. Assume (u0, γ0) is an Hs×Hs-valued F0-measurable ran- +dom variable. Let Q = Q(s) > 0 be the constant such that the embedding ∥u∥W 1,∞ < Q∥u∥Hs, ∥γ∥W 1,∞ < +Q∥γ∥Hs holds. Let C = C(s) > 1 be in (3.4). If there is a R > 1 and λ1 > 1 satisfying P − a.s. +∥u0∥2 +Hs + ∥γ0∥2 +Hs < +b2 +∗ +4C2Q2λ2 +1R, +then (1.3) has a maximal solution (z, τ ∗) satisfying for any 0 < λ2 < λ1−1 +λ1 +the estimate +P +� +∥u(t)∥2 +Hs + ∥γ(t)∥2 +Hs < +b2 +∗ +C2Q2λ2 +1 +for all t > 0 +� +≥ 1 − R−2λ2. +Remark 3.3 Theorem 3.7 presents a global existence solution with bounded initial data. This result can +not be observed in the deterministic case. +Theorem 3.8 (Global existence for weak noise II) Let s > 5/2, Assumption 3.3 be verified and S = +(Ω, F, P, {Ft}t≥0, W) be a fixed stochastic basis. (u0, γ0) is an Hs × Hs-valued F0-measurable random +variable. If +P{(1 − ∂2 +x)u0(x) > 0, ∀x ∈ R} = p, +P{(1 − ∂2 +x)u0(x) < 0, ∀x ∈ R} = q, +8 + +and there exists some x0 ∈ R such that +P{(1 − ∂2 +x)u0(x) ≤ 0, x ≤ x0 and (1 − ∂2 +x)u0(x) ≥ 0, x ≥ x0} = m, +for some p, q, m ∈ [0, 1], then the corresponding maximal solution (z, τ ∗) to (1.3) satisfies +P{τ ∗ = ∞} ≥ p + q + m. +Remark 3.4 The proof of Theorem 3.8 depends on the analysis of a PDE with random coefficient. When +b(t) = 0 and taking (p, q, m) = (1, 0, 0), (p, q, m) = (0, 1, 0) or (p, q, m) = (0, 0, 1) in Theorem 3.8, we +obtain the global existence for the deterministic MCH2 system. Therefore, in this sense, Theorem 3.8 +covers the deterministic result. +Theorem 3.9 (Wave breaking criterion for weak noise I) Let S = (Ω, F, P, {Ft}t≥0, W) be a fixed +stochastic basis and s > 5/2. Let Assumption 3.3 be verified and (u0, γ0) be an Hs × Hs-valued F0- +measurable random variable. If for some c ∈ (0, 1) and x0 ∈ R, +u0x(x0) < −1 +2 +� +(b∗)2 +c2 ++ 4(∥u0∥2 +H1 + ∥γ0∥2 +H1) − b∗ +2c P − a.s., +(3.5) +then the maximal solution (z, τ ∗) to (1.3) satisfies +P{τ ∗ < ∞} ≥ P +� +e +� t +0 b(t +′ )dWt′ + +� t +0 +b∗−b2(t +′ +) +2 +dt +′ +≥ c for all t +� +> 0. +Remark 3.5 Theorem 3.9 detects the solution singularities in finite time under certain initial data, while +Theorem 3.7 provides a global existence result. We stress that these two results do not contain each other. +In Theorem 3.7, assuming ∥u0∥2 +Hs + ∥γ0∥2 +Hs < +b2 +∗ +2C2Q2λ2 +1R, then z globally exists with probability greater +than 1 − R−2λ2. In Theorem 3.9, (3.5) implies that ∥u0∥2 +Hs > +1 +Q2 ∥u0∥2 +W 1,∞ > (b∗)2 +c2Q2 > +b2 +∗ +2C2Q2λ2 +1R. +Theorem 3.10 (Wave breaking criterion for weak noise II) Let S = (Ω, F, P, {Ft}t≥0, W) be a fixed +stochastic basis and s > 5/2. Let Assumption 3.3 be verified and (u0, γ0) be an Hs × Hs-valued F0- +measurable random variable. If for some c ∈ (0, 1), +� +R +u3 +0x(x)dx < − +� +(b∗)2 +4c2 (∥u0∥2 +H1 + ∥γ0∥2 +H1)2 + 15 +8 (∥u0∥2 +H1 + ∥γ0∥2 +H1)3 +− b∗ +2c(∥u0∥2 +H1 + ∥γ0∥2 +H1) P − a.s., +(3.6) +then the maximal solution (z, τ ∗) to (1.3) satisfies +P{τ ∗ < ∞} ≥ P +� +e +� t +0 b(t +′ )dWt′ +� t +0 +b∗−b2(t′ ) +2 +dt +′ +≥ c for all t +� +> 0. +Remark 3.6 Theorem 3.10 detects the solution singularities in finite time under certain initial data. +(3.6) implies that +� +R u3 +0x(x)dx < − b∗ +c (∥u0∥2 +H1 + ∥γ0∥2 +H1), which combined with minx∈Ru0x(x)(∥u0∥2 +H1 + +∥γ0∥2 +H1) ≤ +� +R u3 +0x(x)dx derives ∥u0∥2 +Hs > +1 +Q2 ∥u0∥2 +W 1,∞ > (b∗)2 +c2Q2 > +b2 +∗ +2C2Q2λ2 +1R. So, the initial value condi- +tions here and those given in Theorem 3.7 do not contain each other. +9 + +4 +Sketch of the Proof of Theorem 3.5 +We consider the initial value problem (1.3). The proof of existence and uniqueness of pathwise solutions +can be carried out by standard procedures used in many works, see [35, 37, 39, 40, 42, 43] for more details. +Therefore we only give a sketch. +1. (Approximation scheme) The first step is to construct a suitable approximation scheme. For any R > 1, +we let χR(x) : [0, ∞) → [0, 1] be a C∞ +0 +function such that χR(x) = 1 for x ∈ [0, R] and χR(x) = 0 for +x > 2R. Then we consider the following cut-off problem on R, + + + + + + + +du + χR(∥u∥W 1,∞ + ∥γ∥W 1,∞)[uux + F1(u, γ)]dt = χR(∥u∥W 1,∞ + ∥γ∥W 1,∞)h1(t, u, γ)dW1, t > 0, +dγ + χR(∥u∥W 1,∞ + ∥γ∥W 1,∞)[uγx + F2(u, γ)]dt = χR(∥u∥W 1,∞ + ∥γ∥W 1,∞)h2(t, u, γ)dW2, t > 0, +u(ω, 0, x) = u0(ω, x), +γ(ω, 0, x) = γ0(ω, x). +(4.1) +From (2.4), we observe that the nonlinear term F1(u, γ), F2(u, γ), preserves the Hs × Hs-regularity of +(u, γ) for any s > 3/2. However, in order to apply the stochastic differential equation (SDE) theory in +Hilbert space to (4.1), we will mollify the transport term uux, uγx since the products uux and uγx lose +one regularity. For this reason, we consider the following approximation scheme: + + + + + + + + + + + +du + G1,ǫ(u, γ)dt = χR(∥u∥W 1,∞ + ∥γ∥W 1,∞)h1(t, u, γ)dW1, t > 0, x ∈ R, +dγ + G2,ǫ(u, γ)dt = χR(∥u∥W 1,∞ + ∥γ∥W 1,∞)h2(t, u, γ)dW2, t > 0, x ∈ R, +G1,ǫ(u, γ) = χR(∥u∥W 1,∞ + ∥γ∥W 1,∞)[Jǫ((Jǫu)(Jǫu)x) + F1(u, γ)], +G2,ǫ(u, γ) = χR(∥u∥W 1,∞ + ∥γ∥W 1,∞)[Jǫ((Jǫu)(Jǫγ)x) + F2(u, γ)], +u(0, x) = u0(x) ∈ Hs, γ(0, x) = γ0(x) ∈ Hs, +(4.2) +where Jǫ is the Friedrichs mollifier. According to the theory of SDE in Hilbert space (see for example +[28, 44]), for a fixed stochastic basis S = (Ω, F, P, {Ft}t≥0, W1, W2) and for (u0, γ0) ∈ Hs × Hs with +s > 5/2, (4.2) admits a unique solution (uǫ, γǫ) ∈ C([0, Tǫ), Hs × Hs). +In addition, the uniform L∞(Ω; W 1,∞ × W 1,∞) condition provided by the cut-off function χR enables +us to split the expectation E(∥uǫ∥2 +Hs∥uǫ∥W 1,∞|F0), E(∥uǫ∥2 +Hs∥γǫ∥W 1,∞|F0) to close a priori L2(Ω, Hs × +Hs, P(·|F0)) estimate for uǫ, γǫ. Then we can go along the lines as we prove Lemma 4.1 to find that for +each fixed ǫ, if Tǫ < ∞, then lim supt→Tǫ(∥uǫ∥W 1,∞ + ∥γǫ∥W 1,∞) = ∞. Due to the cut-off in (4.2) for +P-a.s. ω ∈ Ω, ∥uǫ∥W 1,∞, ∥γǫ∥W 1,∞ are always bounded and hence (uǫ, γǫ) is actually a global in time +solution, that is (uǫ, γǫ) ∈ C([0, ∞), Hs × Hs) P − a.s. +2. (Pathwise solution to the cut-off problem in Hs × Hs with s > 5/2) By applying the stochastic com- +pactness arguments from Prokhorov’s and Skorokhod’s Theorem, we obtain the almost sure convergence +for a new approximation solution (( ˜uǫ, ˜γǫ), +˜ +W1ǫ, +˜ +W2ǫ) defined on a new probability space. By virtue of a +refined martingale representation Theorem [45, Theorem A.1], we may set ǫ → 0 in (( ˜uǫ, ˜γǫ), +˜ +W1ǫ, +˜ +W2ǫ) +to obtain a martingale solution in Hs × Hs with s > 5/2. Here, the Gy¨ongy-Krylov characterization [46] +of the convergence in probability can be used here to prove the convergence of the original approximation +solutions, and one can refer to [35, Theorem 1.7] for more details. +Finally, since F1(u, γ), F2(u, γ) satisfy the estimates as in Lemma 2.4 and h1(t, u, γ), h2(t, u, γ) satisfies +Assumption 3.1, we conclude that G1,ǫ, G2,ǫ, χR(∥u∥W 1,∞ + ∥γ∥W 1,∞)h1, χR(∥u∥W 1,∞ + ∥γ∥W 1,∞)h2 are +Lipschitz continuous. So, one can obtain the pathwise uniqueness easily. Then by the Yamada-Watanabe +principle, we derive the existence and uniqueness of the pathwise solution to (4.2) denoted by zR = +(uR, γR). +3. (Remove the cut-off and extend the range of s to s > 3/2) +Let τR := R ∧ inf{t ≥ 0 : ∥uR(t)∥W 1,∞ + ∥γR(t)∥W 1,∞ > R}. By the pathwise uniqueness to (4.2), we +conclude that zR(t) = z ¯R(t), +t ∈ [0, τ R ∧ τ ¯ +R). In particular, τR is increasing in R. Let τ ∗ = limR→∞ τR +and define +z = +∞ +� +R=1 +1[τR−1,τR)zR. +Then (z, τ ∗) is the unique pathwise solution to (1.3) for s > 5 +2. +10 + +Next, we extend the range of s to s > 3 +2. When z0 ∈ L∞(Ω, Hs × Hs) with s > 3/2, by mollifying +the initial data, we obtain a sequence of regular solutions {zn, ζn}n∈N to (1.3). Motivated by [42], one +can prove that there is some stopping time τ with P(τ > 0) = 1, a subsequence (still denoted by zn) and +some process z such that P − a.s. +lim +n→∞ sup +t∈[0,s] +(∥zn − z∥Hs) = 0 , +s < τ +and +sup +t∈[0,s] +∥z∥Hs×Hs ≤ ∥z0∥Hs×Hs + 2, +s < τ. +(4.3) +Then we can let n → ∞ to prove that (z, τ) is a solution to (1.3). +Besides, a cutting argument as in [37, 39, 42] enables us to remove the L∞(Ω, Hs × Hs) assumption +on (u0, γ0). More precisely, consider the decomposition +Ωm = {m − 1 ≤ ∥u0∥Hs + ∥γ0∥Hs < m}, +m ≥ 1. +We conclude �∞ +m=1 P(Ωm) = 1. Therefore we have P − a.s. +z0(ω, x) = +� +m≥1 +zm +0 (ω, x) := +� +m≥1 +z0(ω, x)1Ωm. +For each initial value zm +0 , we let (zm, ζm) be the pathwise unique solution to (1.3) satisfying (4.3). +Moreover, as Ωm ∩ Ωm′ = ∅, m ̸= m +′, F1(0, 0) = 0, F2(0, 0) = 0 and h1(t, 0, 0) = 0, h2(t, 0, 0) = 0 (see +(3.1)), it follows that +z := +� +m≥1 +zm1Ωm, +ζ = +� +m≥1 +ζm1Ωm +is the unique pathwise solution to (1.3) with corresponding initial condition z0. Since (zm, ζm) satisfies +(4.3), we have P − a.s. +sup +t∈[0,s] +(∥u∥2 +Hs + ∥γ∥2 +Hs) += +∞ +� +m=1 +1Ωm sup +t∈[0,s] +(∥um∥2 +Hs + ∥γm∥2 +Hs) +≤ +C +∞ +� +m=1 +1Ωm(4 + ∥um +0 ∥2 +Hs + ∥γm +0 ∥2 +Hs) += +C(4 + ∥u0∥2 +Hs + ∥γ0∥2 +Hs), +s < ζ. +So, (3.2) holds. Since the passage from (z, ζ) to a unique maximal pathwise solution (z, τ ∗) in the sense +of Definition 3.4 can be carried out as in [37, 42, 47], we omit the details. To finish the proof of Theorem +3.5, we only need to prove the blow-up scenario (3.3). Motivated by [43, 48], we consider the relationship +between the explosion time of ∥z(t)∥Hs×Hs and the explosion time of ∥z(t)∥W 1,∞×W 1,∞ in the next lemma. +Lemma 4.1 (Blow-up scenario 1) Let (z, τ ∗) be the unique maximal solution to (1.3). Then the real- +valued stochastic processes ∥z(t)∥W 1,∞×W 1,∞, ∥z(t)∥Hs×Hs are also Ft-adapted. Besides, for any m, n ∈ +Z+, define +τ1,m = inf{t ≥ 0 : ∥u(t)∥Hs + ∥γ(t)∥Hs ≥ m}, τ2,n = inf{t ≥ 0 : ∥u(t)∥W 1,∞ + ∥γ(t)∥W 1,∞ ≥ n}. +For τ1 := τ ∗ = limm→∞ τ1,m and τ2 = limn→∞ τ2,n, we have +τ1 = τ2 +P − a.s. +11 + +Consequently, 1{limt→τ∗ ∥u(t)∥W 1,∞+∥γ(t)∥W 1,∞=∞} = 1{τ ∗<∞} P − a.s.. +Proof: Since z ∈ C([0, τ ∗); Hs × Hs) almost surely, by the continuous embedding Hs × Hs ֒→ W 1,∞ × +W 1,∞ for s > 3/2, we conclude that ∥z(t)∥W 1,∞×W 1,∞ is Ft-adapted. Moreover, the embedding Hs × +Hs ֒→ W 1,∞ × W 1,∞ for s > 3/2 means that P-a.s. τ1 ≤ τ2 P − a.s. Now we only need to prove P-a.s. +τ2 ≤ τ1. We cannot directly apply the Itˆo formula for ∥u(t)∥2 +Hs + ∥γ(t)∥2 +Hs since we only have u, γ ∈ Hs +and uux, uγx ∈ Hs−1. Therefore, the Itˆo formula in Hilbert space cannot be applied directly, see ([27], +Theorem 4.32) or ([49], Theorem 2.10). Instead, we will use the mollifier operator Tǫ defined in (2.2) +to overcome this difficulty. We apply Tǫ to (1.3), and then use the Itˆo formula for ∥Tǫu∥2 +Hs, ∥Tǫγ∥2 +Hs to +derive +d∥Tǫu(t)∥2 +Hs += +2(Tǫh1(u, γ)dW1, Tǫu)Hs − 2(DsTǫ[uux], DsTǫu)L2dt − 2(DsTǫF1(u, γ), DsTǫu)L2dt ++ +∥Tǫh1(u, γ)∥2 +L2(U,Hs)dt, +d∥Tǫγ(t)∥2 +Hs += +2(Tǫh2(u, γ)dW2, Tǫγ)Hs − 2(DsTǫ[uγx], DsTǫγ)L2dt − 2(DsTǫF2(u, γ), DsTǫγ)L2dt ++ +∥Tǫh2(u, γ)∥2 +L2(U,Hs)dt. +Therefore for any n1, m ≥ 1, r ≥ 0, and t ∈ [0, τ2,n1 ∧ r ∧ τ1,m], +∥Tǫu(t)∥2 +Hs + ∥Tǫγ(t)∥2 +Hs − ∥Tǫu(0)∥2 +Hs − ∥Tǫγ(0)∥2 +Hs +=2 +∞ +� +j=1 +� t +0 +(DsTǫh1(u, γ)ej, DsTǫu)L2dW 1 +j + 2 +∞ +� +i=1 +� t +0 +(DsTǫh2(u, γ)ei, DsTǫγ)L2dW 2 +i +− 2 +� t +0 +(DsTǫ[uux], DsTǫu)L2dt +′ − 2 +� t +0 +(DsTǫF1(u, γ), DsTǫu)L2dt +′ ++ +� t +0 +∥Tǫh1(u, γ)∥2 +L2(U,Hs)dt +′ − 2 +� t +0 +(DsTǫ[uγx], DsTǫγ)L2dt +′ +− 2 +� t +0 +(DsTǫF2(u, γ), DsTǫγ)L2dt +′ + +� t +0 +∥Tǫh2(u, γ)∥2 +L2(U,Hs)dt +′ +=: +� t +0 +∞ +� +j=1 +L1,jdW 1 +j + +� t +0 +∞ +� +i=1 +L2,idW 2 +i + +8 +� +j=3 +� t +0 +Ljdt +′, +where {ek} is the complete orthonormal basis of U. On account of the Burkholder-Davis-Gundy inequality +and (3.1), we obtain that +E +� +sup +t∈[0,τ2,n1∧r∧τ1,m] +������ +� t +0 +∞ +� +j=1 +L1,jdW 1 +j +������ +����F0 +� +≤ CE +�� ∞ +� +j=1 +� τ2,n1∧r∧τ1,m +0 +|L1,j|2dt +� 1 +2 ����F0 +� +≤ 1 +2E +� +sup +t∈[0,τ2,n1∧r∧∧τ1,m] +∥Tǫu∥2 +Hs +����F0 +� ++ Cf 2(n1) +� r +0 +E +� +sup +t′ ∈[0,τ2,n1∧t∧τ1,m] +(1 + ∥u(t +′)∥2 +Hs + ∥γ(t +′)∥2 +Hs) +����F0 +� +dt, +E +� +sup +t∈[0,τ2,n1∧r∧τ1,m] +����� +� t +0 +∞ +� +i=1 +L2,idW 2 +i +����� +����F0 +� +≤ CE +�� ∞ +� +i=1 +� τ2,n1∧r∧τ1,m +0 +|L1,i|2dt +� 1 +2 ����F0 +� +≤ 1 +2E +� +sup +t∈[0,τ2,n1∧r∧τ1,m] +∥Tǫγ∥2 +Hs +����F0 +� ++ Cf 2(n1) +� r +0 +E +� +sup +t′ ∈[0,τ2,n1∧t∧τ1,m] +(1 + ∥u(t +′)∥2 +Hs + ∥γ(t +′)∥2 +Hs) +����F0 +� +dt. +For L3, L6, using integration by part, Sobolev’s inequality and Lemma 2.3, we have +(DsTǫ[uux], DsTǫu)L2 += +([Ds, u]ux, DsT 2 +ǫ u)L2 + ([Tǫ, u]Dsux, DsTǫu)L2 + (uDsTǫux, DsTǫu)L2 +≤ +C∥u∥W 1,∞∥u∥2 +Hs, +(DsTǫ[uγx], DsTǫγ)L2 += +([Ds, u]γx, DsT 2 +ǫ γ)L2 + ([Tǫ, u]Dsγx, DsTǫγ)L2 + (uDsTǫγx, DsTǫγ)L2 +12 + +≤ +C(∥u∥W 1,∞ + ∥γ∥W 1,∞)(∥u∥2 +Hs + ∥γ∥2 +Hs). +For L4, L7, we derive from Lemma 2.4 that +(DsTǫF1(u, γ), DsTǫu)L2 +≤ +C(∥u∥W 1,∞ + ∥γ∥W 1,∞)(∥u∥2 +Hs + ∥γ∥2 +Hs), +(DsTǫF2(u, γ), DsTǫγ)L2 +≤ +C(∥u∥W 1,∞ + ∥γ∥W 1,∞)(∥u∥2 +Hs + ∥γ∥2 +Hs). +For L5, L8, it follows from the Assumption 3.1 that +E +� � τ2,n1∧r∧τ1,m +0 +∥Tǫh1(u, γ)∥2 +L2(U,Hs)dt +′����F0 +� +≤ Cf 2(n1) +� r +0 +E +� +sup +t′∈[0,τ2,n1∧t∧τ1,m] +(1 + ∥u(t +′)∥2 +Hs + ∥γ(t +′)∥2 +Hs) +����F0 +� +dt, +E +� � τ2,n1∧r∧τ1,m +0 +∥Tǫh2(u, γ)∥2 +L2(U,Hs)dt +′����F0 +� +≤ Cf 2(n1) +� r +0 +E +� +sup +t′∈[0,τ2,n1∧t∧τ1,m] +(1 + ∥u(t +′)∥2 +Hs + ∥γ(t +′)∥2 +Hs) +����F0 +� +dt. +Therefore combining the above estimates, we have +E +� +sup +t∈[0,τ2,n1∧r∧τ1,m] +(∥Tǫu(t)∥2 +Hs + ∥Tǫγ(t)∥2 +Hs) +����F0 +� +≤ +C[∥u(0)∥2 +Hs + ∥γ(0)∥2 +Hs] + C +� r +0 +E +� +1 + +sup +t′ ∈[0,τ2,n1∧t∧τ1,m] +(∥u(t +′)∥2 +Hs + ∥γ(t +′)∥2 +Hs) +����F0 +� +dt. +Since the right hand side of the above estimate does not depend on ǫ, and (Tǫu, Tǫγ) tends to (u, γ) in +C([0, τ1,m ∧ r], Hs × Hs) almost surely as ǫ → 0, by Fatou’s Lemma, one can send ǫ → 0 to obtain +E +� +sup +t∈[0,τ2,n1∧r∧τ1,m] +(∥u(t)∥2 +Hs + ∥γ(t)∥2 +Hs) +����F0 +� +≤ C[∥u(0)∥2 +Hs + ∥γ(0)∥2 +Hs] + C +� r +0 +E +� +1 + +sup +t′ ∈[0,τ2,n1∧t∧τ1,m] +(∥u(t +′)∥2 +Hs + ∥γ(t +′)∥2 +Hs)|F0 +� +dt. +Then Gronwall’s inequality shows that for each n1 ∈ Z+, r ∈ R+, there is a constant C = C(n1, r, u0, γ0) > +0 such that +E +� +sup +t∈[0,τ2,n1∧r∧τ1,m] +(∥u(t)∥2 +Hs + ∥γ(t)∥2 +Hs) +����F0 +� +< C(n1, r, u0, γ0). +So, it follows from Chebyshev’s inequality and Fatou’s lemma that +P(τ1 ≤ τ2,n1 ∧ r|F0) ≤ lim +m→∞ P(τ1,m ≤ τ2,n1 ∧ r|F0) +≤ lim +m→∞ P +� +sup +t∈[0,τ2,n1∧r∧τ1,m] +(∥u(t)∥Hs + ∥γ(t)∥Hs) ≥ m +����F0 +� +≤ lim +m→∞ +E[supt∈[0,τ2,n1∧r∧τ1,m](2∥u(t)∥2 +Hs + 2∥γ(t)∥2 +Hs)|F0] +m2 += 0. +Letting r → ∞ first and then n1 → ∞, Fatou’s lemma yields that +P(τ1 ≤ τ2|F0) = 0. +13 + +Therefore, we conclude +P(τ1 ≥ τ2) ≥ 1 − P(τ1 ≤ τ2) = 1 − P[P(τ1 ≤ τ2|F0)] = 1. +We finish the section with the proof of the first blow-up scenario (3.3). +✷ +5 +Proof of Theorem 3.6 +Assume s > 5/2 and let (u0, γ0) be Hs × Hs-valued F0-measurable random variable. Let h1(t, u, γ) = +a(t)(1 + ∥u∥W 1,∞ + ∥γ∥W 1,∞)θu, h2(t, u, γ) = a(t)(1 + ∥u∥W 1,∞ + ∥γ∥W 1,∞)θγ with θ ≥ 1/2 and a(t) ̸= 0. +For s > 3/2, the embedding Hs × Hs ֒→ W 1,∞ × W 1,∞ implies that +sup +∥u1∥Hs,∥γ1∥Hs,∥u2∥Hs,∥γ2∥Hs≤N +2 +� +i=1 +(∥hi(t, u1, γ1) − hi(t, u2, γ2)∥Hs +≤ +g(N)(∥u1 − u2∥Hs + ∥γ1 − γ2∥Hs), +N ≥ 1. +Hence, by Theorem 3.5, we conclude that (1.3) admits a unique pathwise solution z = (u, γ) in Hs × Hs +with s > 5/2 and maximal existence time τ ∗. Define +τm = inf{t ≥ 0 : ∥u∥2 +Hs + ∥γ∥2 +Hs ≥ m}. +Applying the Itˆo formula to ∥Tǫu∥2 +Hs, ∥Tǫγ∥2 +Hs gives +d∥Tǫu∥2 +Hs =2a(t)(1 + ∥u∥W 1,∞ + ∥γ∥W 1,∞)θ(Tǫu, Tǫu)HsdW − 2(Tǫ[uux], Tǫu)Hsdt +− 2(TǫF1(u, γ), Tǫu)Hsdt + a2(t)(1 + ∥u∥W 1,∞ + ∥γ∥W 1,∞)2θ(Tǫu, Tǫu)Hsdt, +d∥Tǫγ∥2 +Hs =2a(t)(1 + ∥u∥W 1,∞ + ∥γ∥W 1,∞)θ(Tǫγ, Tǫγ)HsdW − 2(Tǫ[uγx], Tǫγ)Hsdt +− 2(TǫF2(u, γ), Tǫγ)Hsdt + a2(t)(1 + ∥u∥W 1,∞ + ∥γ∥W 1,∞)2θ(Tǫγ, Tǫγ)Hsdt, +Again, using Itˆo’s formula to log(1 + ∥Tǫu∥2 +Hs + ∥Tǫγ∥2 +Hs) yields +d log(1 + ∥Tǫu∥2 +Hs + ∥Tǫγ∥2 +Hs) = 2a(t)(1 + ∥u∥W 1,∞ + ∥γ∥W 1,∞)θ(Tǫu, Tǫu)Hs +1 + ∥Tǫu∥2 +Hs + ∥Tǫγ∥2 +Hs +dW +− +2(Tǫ[uux], Tǫu)Hs +1 + ∥Tǫu∥2 +Hs + ∥Tǫγ∥2 +Hs dt − +2(TǫF1(u, γ), Tǫu)Hs +1 + ∥Tǫu∥2 +Hs + ∥Tǫγ∥2 +Hs dt ++ a2(t)(1 + ∥u∥W 1,∞ + ∥γ∥W 1,∞)2θ(Tǫu, Tǫu)Hs +1 + ∥Tǫu∥2 +Hs + ∥Tǫγ∥2 +Hs +dt ++ 2a(t)(1 + ∥u∥W 1,∞ + ∥γ∥W 1,∞)θ(Tǫγ, Tǫγ)Hs +1 + ∥Tǫu∥2 +Hs + ∥Tǫγ∥2 +Hs +dW +− +2(Tǫ[uγx], Tǫγ)Hs +1 + ∥Tǫu∥2 +Hs + ∥Tǫγ∥2 +Hs dt − +2(TǫF2(u, γ), Tǫγ)Hs +1 + ∥Tǫu∥2 +Hs + ∥Tǫγ∥2 +Hs dt ++ a2(t)(1 + ∥u∥W 1,∞ + ∥γ∥W 1,∞)2θ(Tǫγ, Tǫγ)Hs +1 + ∥Tǫu∥2 +Hs + ∥Tǫγ∥2 +Hs +dt +− 2a2(t)(1 + ∥u∥W 1,∞ + ∥γ∥W 1,∞)2θ +× [(Tǫu, Tǫu)2 +Hs + (Tǫγ, Tǫγ)2 +Hs + 2(Tǫu, Tǫu)Hs(Tǫγ, Tǫγ)Hs] +(1 + ∥Tǫu∥2 +Hs + ∥Tǫγ∥2 +Hs)2 +dt. +By Lemma 2.1-2.4 and Lemma 2.7, it follows that +E[log(1 + ∥Tǫu(t ∧ τm)∥2 +Hs + ∥Tǫγ(t ∧ τm)∥2 +Hs)|F0] − log(1 + ∥Tǫu0∥2 +Hs + ∥Tǫγ0∥2 +Hs) +14 + += − 2E +� � t∧τm +0 +(Tǫ[uux], Tǫu)Hs +1 + ∥Tǫu∥2 +Hs + ∥Tǫγ∥2 +Hs dt +′����F0 +� +− 2E +� � t∧τm +0 +(TǫF1(u, γ), Tǫu)Hs +1 + ∥Tǫu∥2 +Hs + ∥Tǫγ∥2 +Hs dt +′����F0 +� ++ E +� � t∧τm +0 +a2(t +′)(1 + ∥u∥W 1,∞ + ∥γ∥W 1,∞)2θ(Tǫu, Tǫu)Hs +1 + ∥Tǫu∥2 +Hs + ∥Tǫγ∥2 +Hs +dt +′����F0 +� +− 2E +� � t∧τm +0 +(Tǫ[uγx], Tǫγ)Hs +1 + ∥Tǫu∥2 +Hs + ∥Tǫγ∥2 +Hs dt +′����F0 +� +− 2E +� � t∧τm +0 +(TǫF2(u, γ), Tǫγ)Hs +1 + ∥Tǫu∥2 +Hs + ∥Tǫγ∥2 +Hs dt +′����F0 +� ++ E +� � t∧τm +0 +a2(t +′)(1 + ∥u∥W 1,∞ + ∥γ∥W 1,∞)2θ(Tǫγ, Tǫγ)Hs +1 + ∥Tǫu∥2 +Hs + ∥Tǫγ∥2 +Hs +dt +′����F0 +� +− 2E +� � t∧τm +0 +a2(t +′)(1 + ∥u∥W 1,∞ + ∥γ∥W 1,∞)2θ +× [(Tǫu, Tǫu)2 +Hs + (Tǫγ, Tǫγ)2 +Hs + +2(Tǫu, Tǫu)Hs(Tǫγ, Tǫγ)Hs] +(1 + ∥Tǫu∥2 +Hs + ∥Tǫγ∥2 +Hs)2 +dt +′����F0 +� +≤E +� � t∧τm +0 +K(∥u∥W 1,∞ + ∥γ∥W 1,∞)(∥u∥2 +Hs + ∥γ∥2 +Hs) +1 + ∥Tǫu∥2 +Hs + ∥Tǫγ∥2 +Hs +dt +′����F0 +� ++ E +� � t∧τm +0 +a2(t +′)(1 + ∥u∥W 1,∞ + ∥γ∥W 1,∞)2θ(∥Tǫu∥2 +Hs + ∥Tǫγ∥2 +Hs) +1 + ∥Tǫu∥2 +Hs + ∥Tǫγ∥2 +Hs +dt +′����F0 +� +− 2E +� � t∧τm +0 +a2(t +′)(1 + ∥u∥W 1,∞ + ∥γ∥W 1,∞)2θ +(∥Tǫu∥4 +Hs + ∥Tǫγ∥4 +Hs + 2∥Tǫu∥2 +Hs∥Tǫγ∥2 +Hs) +(1 + ∥Tǫu∥2 +Hs + ∥Tǫγ∥2 +Hs)2 +dt +′����F0 +� +. +Let +Iǫ +1(t′) = K(∥u∥W 1,∞ + ∥γ∥W 1,∞)(∥u∥2 +Hs + ∥γ∥2 +Hs) +1 + ∥Tǫu∥2 +Hs + ∥Tǫγ∥2 +Hs +− K(∥u∥W 1,∞ + ∥γ∥W 1,∞)(∥u∥2 +Hs + ∥γ∥2 +Hs) +1 + ∥u∥2 +Hs + ∥γ∥2 +Hs +, +Iǫ +2(t′) = a2(t +′)(1 + ∥u∥W 1,∞ + ∥γ∥W 1,∞)2θ(∥Tǫu∥2 +Hs + ∥Tǫγ∥2 +Hs) +1 + ∥Tǫu∥2 +Hs + ∥Tǫγ∥2 +Hs +− a2(t +′)(1 + ∥u∥W 1,∞ + ∥γ∥W 1,∞)2θ(∥u∥2 +Hs + ∥γ∥2 +Hs) +1 + ∥u∥2 +Hs + ∥γ∥2 +Hs +, +Iǫ +3(t′) = a2(t +′)(1 + ∥u∥W 1,∞ + ∥γ∥W 1,∞)2θ(∥Tǫu∥4 +Hs + ∥Tǫγ∥4 +Hs + 2∥Tǫu∥2 +Hs∥Tǫγ∥2 +Hs) +(1 + ∥Tǫu∥2 +Hs + ∥Tǫγ∥2 +Hs)2 +− a2(t +′)(1 + ∥u∥W 1,∞ + ∥γ∥W 1,∞)2θ(∥u∥4 +Hs + ∥γ∥4 +Hs + 2∥u∥2 +Hs∥γ∥2 +Hs) +(1 + ∥u∥2 +Hs + ∥γ∥2 +Hs)2 +. +(5.1) +Notice that for any T > 0, (Tǫu, Tǫγ) tends to (u, γ) in C([0, τm ∧ t], Hs × Hs) almost surely as ǫ → 0. It +follows from the dominated convergence theorem that +lim +ǫ→0 E +�� t∧τm +0 +[|Iǫ +1(t′)| + |Iǫ +2(t′)| + |Iǫ +3(t′)|]dt′ +����F0 +� += 0. +Then, by (2.2) and the dominated convergence Theorem, it holds +E[log(1 + ∥u(t ∧ τm)∥2 +Hs + ∥γ(t ∧ τm)∥2 +Hs)|F0] − log(1 + ∥u0∥2 +Hs + ∥γ0∥2 +Hs) +≤E +� � t∧τm +0 +K(∥u∥W 1,∞ + ∥γ∥W 1,∞)(∥u∥2 +Hs + ∥γ∥2 +Hs) +1 + ∥u∥2 +Hs + ∥γ∥2 +Hs +dt +′����F0 +� ++ E +� � t∧τm +0 +a2(t +′)(1 + ∥u∥W 1,∞ + ∥γ∥W 1,∞)2θ(∥u∥2 +Hs + ∥γ∥2 +Hs) +1 + ∥u∥2 +Hs + ∥γ∥2 +Hs +dt +′����F0 +� +15 + +− 2E +� � t∧τm +0 +a2(t +′)(1 + ∥u∥W 1,∞ + ∥γ∥W 1,∞)2θ(∥u∥4 +Hs + ∥γ∥4 +Hs + 2∥u∥2 +Hs∥γ∥2 +Hs) +(1 + ∥u∥2 +Hs + ∥γ∥2 +Hs)2 +dt +′����F0 +� +. +Lemma 2.5 with x1 = ∥u∥W 1,∞, x2 = ∥γ∥W 1,∞, y1 = ∥u∥Hs, y2 = ∥γ∥Hs immediately shows that there +are constants C1, C2 > 0 such that +E[log(1 + ∥u(t ∧ τm)∥2 +Hs + ∥γ(t ∧ τm)∥2 +Hs)|F0] − log(1 + ∥u0∥2 +Hs + ∥γ0∥2 +Hs) +≤ +E +� � t∧τm +0 +C1 − C2 +a2(t +′)(1 + ∥u∥W 1,∞ + ∥γ∥W 1,∞)2θ(∥u∥2 +Hs + ∥γ∥2 +Hs)2 +(1 + ∥u∥2 +Hs + ∥γ∥2 +Hs)2(1 + log(1 + ∥u∥2 +Hs + ∥γ∥2 +Hs))dt +′����F0 +� +, +which means that for some C(u0, γ0, C1, C2, t) > 0, +E +� � t∧τm +0 +a2(t +′)(1 + ∥u∥W 1,∞ + ∥γ∥W 1,∞)2θ(∥u∥2 +Hs + ∥γ∥2 +Hs)2 +(1 + ∥u∥2 +Hs + ∥γ∥2 +Hs)2(1 + log(1 + ∥u∥2 +Hs + ∥γ∥2 +Hs))dt +′����F0 +� +≤ C(u0, γ0, C1, C2, t). +(5.2) +Therefore, for any T > 0, by using Lemma 2.5 and the Burkholder-Davis-Gundy inequality, we find that +E +� +sup +t∈[0,T ∧τm] +log(1 + ∥Tǫu∥2 +Hs + ∥Tǫγ∥2 +Hs) +����F0 +� +− log(1 + ∥Tǫu0∥2 +Hs + ∥Tǫγ0∥2 +Hs) +≤C +� +E +� � T ∧τm +0 +a2(t)(1 + ∥u∥W 1,∞ + ∥γ∥W 1,∞)2θ(∥Tǫu∥2 +Hs + ∥Tǫγ∥2 +Hs)2 +(1 + ∥Tǫu∥2 +Hs + ∥Tǫγ∥2 +Hs)2 +dt +����F0 +�� 1 +2 ++ E +� � T ∧τm +0 +����C1 − C2 +a2(t)(1 + ∥u∥W 1,∞ + ∥γ∥W 1,∞)2θ(∥u∥2 +Hs + ∥γ∥2 +Hs)2 +(1 + ∥u∥2 +Hs + ∥γ∥2 +Hs)2(1 + log(1 + ∥u∥2 +Hs + ∥γ∥2 +Hs)) +���� dt +����F0 +� ++ E +�� T ∧τm +0 +[|Iǫ +1(t)| + |Iǫ +2(t)| + |Iǫ +3(t)|]dt +����F0 +� +≤1 +2E +� +sup +t∈[0,T ∧τm] +(1 + log(1 + ∥u∥2 +Hs + ∥γ∥2 +Hs)) +����F0 +� ++ CE +� � T ∧τm +0 +a2(t)(1 + ∥u∥W 1,∞ + ∥γ∥W 1,∞)2θ(∥Tǫu∥2 +Hs + ∥Tǫγ∥2 +Hs)2 +(1 + ∥Tǫu∥2 +Hs + ∥Tǫγ∥2 +Hs)2(1 + log(1 + ∥u∥2 +Hs + ∥γ∥2 +Hs))dt +����F0 +� ++ C1T + C2E +� � T ∧τm +0 +a2(t)(1 + ∥u∥W 1,∞ + ∥γ∥W 1,∞)2θ(∥u∥2 +Hs + ∥γ∥2 +Hs)2 +(1 + ∥u∥2 +Hs + ∥γ∥2 +Hs)2(1 + log(1 + ∥u∥2 +Hs + ∥γ∥2 +Hs))dt +����F0 +� ++ E +�� T ∧τm +0 +[|Iǫ +1(t)| + |Iǫ +2(t)| + |Iǫ +3(t)|]dt +����F0 +� +≤1 +2E +� +sup +t∈[0,T ∧τm] +(1 + log(1 + ∥u∥2 +Hs + ∥γ∥2 +Hs)) +����F0 +� ++ E +�� T ∧τm +0 +[|Iǫ +1(t)| + |Iǫ +2(t)| + |Iǫ +3(t)|]dt +����F0 +� ++ CE +� � T ∧τm +0 +a2(t)(1 + ∥u∥W 1,∞ + ∥γ∥W 1,∞)2θ(∥Tǫu∥2 +Hs + ∥Tǫγ∥2 +Hs)2 +(1 + ∥Tǫu∥2 +Hs + ∥Tǫγ∥2 +Hs)2(1 + log(1 + ∥u∥2 +Hs + ∥γ∥2 +Hs))dt +����F0 +� ++ C(u0, γ0, C1, C2, T ) + C1T. +Thus, we use the dominated convergence Theorem, (5.2) and (5.1) to obtain +E +� +sup +t∈[0,T ∧τm] +log(1 + ∥u∥2 +Hs + ∥γ∥2 +Hs) +����F0 +� +≤ C(u0, γ0, C1, C2, T ). +Since log(1 + x) is increasing for x > 0, we have that for any m ≥ 1, +P{τm < T |F0} ≤ P +� +sup +t∈[0,T ∧τm] +log(1 + ∥u∥2 +Hs + ∥γ∥2 +Hs) ≥ log(1 + m) +����F0 +� +≤ C(u0, γ0, C1, C2, T ) +log(1 + m) +. +Letting m → ∞ forces P{τ ∗ < T |F0} = 0 for any T > 0, which means P{τ ∗ = ∞} = 1. +16 + +6 +Proof of Theorem 3.7 +In this section, we study (1.3) with linear noise satisfying Assumption 3.3. Depending on the strength +of the noise in (1.3), we provide the global existence of pathwise solutions for the maximal pathwise +solution. Motivated by [35, 37, 47], we introduce +β(ω, t) = e +� t +0 b(t +′ )dWt′ − +� t +0 +b2(t′ ) +2 +dt +′ +. +Proposition 6.1 Let s > 3/2 and h1(t, u, γ) = b(t)u, h2(t, u, γ) = b(t)γ such that b(t) satisfies As- +sumption 3.3. +Let (u0, γ0) be an Hs × Hs-valued F0-measurable random variable and (z, τ ∗) be the +corresponding unique maximal solution to (1.3). Let v1 = β−1u, v2 = β−1γ. Then for t ∈ [0, τ ∗), the +processes v1, v2 solve the following problem + + + +∂tv1 + βv1v1x + β(1 − ∂2 +x)−1∂x(v2 +1 + 1 +2v2 +1x + 1 +2v2 +2 − 1 +2v2 +2x) = 0, +∂tv2 + βv1v2x + β(1 − ∂2 +x)−1((v1xv2x)x + v1xv2) = 0, +v1(ω, 0, x) = u0(ω, x), v2(ω, 0, x) = γ0(ω, x). +(6.1) +Moreover, we have P − a.s. (v1, v2) ∈ C([0, τ ∗); Hs × Hs) ∩ C1([0, τ ∗); Hs−1 × Hs−1). In addition, if +s > 5/2, then it holds +P{∥v1∥H1 + ∥v2∥H1 = ∥u0∥H1 + ∥γ0∥H1 for all t < τ∗} = 1. +(6.2) +Proof: Since b(t) satisfies Assumption 3.3, h1(t, u, γ) = b(t)u, h2(t, u, γ) = b(t)γ satisfy Assumption 3.1, +Theorem 3.5 implies that (1.3) has a unique maximal solution (z, τ ∗). A direct computation with the Itˆo +formula yields +dβ−1 = −b(t)β−1dW + b2(t)β−1dt. +Therefore we have +dv1 += +β−1du + udβ−1 + dβ−1du += +β−1 +� +−uux − (1 − ∂2 +x)−1∂x +� +u2 + 1 +2u2 +x + 1 +2γ2 − 1 +2γ2 +x +�� +dt + b(t)β−1udW ++u[−b(t)β−1dW + b2(t)β−1dt] − b2(t)β−1udt += +−βv1v1x − β(1 − ∂2 +x)−1∂x +� +v2 +1 + 1 +2v2 +1x + 1 +2v2 +2 − 1 +2v2 +2x +� +, +dv2 += +β−1dγ + γdβ−1 + dβ−1dγ += +β−1[−uγx − (1 − ∂2 +x)−1((uxγx)x + uxγ)]dt + b(t)β−1γdW ++γ[−b(t)β−1dW + b2(t)β−1dt] − b2(t)β−1γdt += +−βv1v2x − β(1 − ∂2 +x)−1((v1xv2x)x + v1xv2) +and since v1(ω, 0, x) = u0(ω, x), v2(ω, 0, x) = γ0(ω, x), we see that (v1, v2) satisfies (6.1). +Moreover, +Theorem 3.5 implies (u, γ) ∈ C([0, τ ∗), Hs × Hs) P − a.s., so is (v1, v2). From Lemma 2.4 and (6.1), +we see that for P − a.s., v1t = −βv1v1x − β(1 − ∂2 +x)−1∂x(v2 +1 + 1 +2v2 +1x + 1 +2v2 +2 − 1 +2v2 +2x), v2t = −βv1v2x − +β(1 − ∂2 +x)−1((v1xv2x)x + v1xv2), (v1t, v2t) ∈ C([0, τ ∗), Hs−1 × Hs−1). Hence, it holds P-a.s. (v1, v2) ∈ +C1([0, τ ∗), Hs−1 × Hs−1). +In addition, the first two equations of (6.1) are equivalent to +v1t − v1xxt + 3βv1v1x − 2βv1xv1xx − βv1v1xxx + β(v2 − v2xx)v2x = 0, +(6.3) +v2t − v2xxt + β(v1v2x + v1xv2) − β(v1v2xxx + v1xv2xx) = 0. +(6.4) +17 + +Multiplying both sides of (6.3) by v1 and multiplying both sides of (6.4) by v2, then integrating the +equation on x ∈ R, and finally adding the two derived equations, we arrive at P-a.s. +d +dt +� +R +(v2 +1 + v2 +2 + v2 +1x + v2 +2x)dx = 0, +t < τ ∗, +which implies (6.2). +✷ +Proof of Theorem 3.7. To begin with, we apply the operator Ds to (6.3) and (6.4) , multiply both +sides of the resulting equation by Dsv1, Dsv2 respectively, and then integrate on R to obtain P − a.s. +1 +2 +d +dt(∥v1∥2 +Hs + ∥v2∥2 +Hs) += +−β(ω, t) +� +R +Dsv1Ds(v1v1x)dx − β(ω, t) +� +R +Dsv1DsF1(v1, v2)dx +−β(ω, t) +� +R +Dsv2Ds(v1v2x)dx − β(ω, t) +� +R +Dsv2DsF2(v1, v2)dx. +By (3.4), we conclude that P − a.s. +d +dt(∥v1∥2 +Hs + ∥v2∥2 +Hs) ≤ Cβ(ω, t)(∥v1∥W 1,∞ + ∥v2∥W 1,∞)(∥v1∥2 +Hs + ∥v2∥2 +Hs). +(6.5) +Letting w1 = e− +� t +0 b(t +′ )dWt′ u = e− +� t +0 +b2(t′ ) +2 +dt +′ +v1, w2 = e− +� t +0 b(t +′ )dWt′ γ = e− +� t +0 +b2(t′ ) +2 +dt +′ +v2 and α(ω, t) = +e +� t +0 b(t +′ )dWt′ , we obtain +d +dt(∥w1∥2 +Hs + ∥w2∥2 +Hs) + b2(t)(∥w1∥2 +Hs + ∥w2∥2 +Hs) +≤ Cα(ω, t)(∥w1∥W 1,∞ + ∥w2∥W 1,∞)(∥w1∥2 +Hs + ∥w2∥2 +Hs). +Assume ∥u0∥2 +Hs + ∥γ0∥2 +Hs < +b2 +∗ +2C2Q2λ2 +1R < +b2 +∗ +C2Q2λ2 +1 and define +τ1 = inf +� +t < τ∗ : α(ω, t)(∥w1∥W 1,∞ + ∥w2∥W 1,∞) = (∥u∥W 1,∞ + ∥γ∥W 1,∞) > b(t)2 +Cλ1 +� +. +(6.6) +Then it follows from the embedding (∥u0∥W 1,∞ + ∥γ0∥W 1,∞) ≤ Q(∥u0∥Hs + ∥γ0∥Hs) that P{τ1 > 0} = 1, +and it holds +d +dt(∥w1∥2 +Hs + ∥w2∥2 +Hs) + (λ1 − 1)b2(t) +λ1 +(∥w1∥2 +Hs + ∥w2∥2 +Hs) ≤ 0, +t ∈ [0, τ1). +This implies that for any 0 < λ2 < λ1−1 +λ1 , P − a.s. +∥u(t)∥2 +Hs + ∥γ(t)∥2 +Hs +≤ (∥u0∥2 +Hs + ∥γ0∥2 +Hs)e +� t +0 b(t +′ )dWt′ − +� t +0 +(λ1−1)b2(t +′ +) +λ1 +dt +′ += (∥u0∥2 +Hs + ∥γ0∥2 +Hs)e +� t +0 b(t +′ )dWt′ −λ2 +� t +0 b2(t +′ )dt +′ +e− +� t +0 +(λ1−1)−λ1λ2 +λ1 +b2(t +′ )dt +′ +, +t ∈ [0, τ1). +(6.7) +Define +τ2 = inf{t > 0 : e +� t +0 b(t +′ )dWt′ −λ2 +� t +0 b2(t +′ )dt +′ +> R}. +Notice that P{τ2 > 0} = 1. From (6.7), we have +2∥u(t)∥2 +Hs + 2∥γ(t)∥2 +Hs +≤ +2b2 +∗ +2C2Q2λ2 +1R × R × e− +� t +0 +(λ1−1)−λ1λ2 +λ1 +b2(t +′ )dt +′ +18 + += +b2 +∗ +C2Q2λ2 +1 +e− +� t +0 +(λ1−1)−λ1λ2 +λ1 +b2(t +′)dt +′ +, +t ∈ [0, τ1 ∧ τ2). +(6.8) +By Assumption 3.3, (6.8) and (6.6), we find that on [0, τ1 ∧ τ2), P − a.s. +(∥u∥W 1,∞ + ∥γ∥W 1,∞) ≤ Q(∥u∥Hs + ∥γ∥Hs) ≤ +b∗ +Cλ1 +e− +� t +0 +(λ1−1)−λ1λ2 +2λ1 +b2(t +′)dt +′ +≤ b2(t) +Cλ1 +e− +� t +0 +(λ1−1)−λ1λ2 +2λ1 +b2(t +′)dt +′ +, +which together with λ2 < λ1−1 +λ1 +and b2 > 0 derives +P{τ1 ≥ τ2} = 1. +Therefore it follows from (6.8) and Lemma 2.6 that +P{∥u(t)∥2 +Hs + ∥γ(t)∥2 +Hs ≤ +b2 +∗ +2C2Q2λ2 +1 +for all t > 0} ≥ P{τ2 = ∞} ≥ 1 − R−2λ2, +which completes the proof. +7 +Proof of Theorem 3.8 +7.1 +Proof of Theorem 3.8. +By proposition 6.1, we can proceed to prove Theorem 3.8. +Since Hs ֒→ C2 for s > 5/2, we have +v1, v1x, v2, v2x ∈ C1([0, τ ∗) × R). Then for x ∈ R and P − a.s. ω ∈ Ω, the problem +� +dq(ω,t,x) +dt += β(ω, t)v1(ω, t, q(ω, t, x)), +t ∈ [0, τ ∗), +q(ω, 0, x) = x, +x ∈ R +(7.1) +has a unique solution q(ω, t, x) such that q(ω, t, x) ∈ C1([0, τ ∗) × R) for P − a.s. ω ∈ Ω. Moreover, +differentiating (7.1) with respect to x yields that for P − a.s. ω ∈ Ω, +� +dqx(ω,t,x) +dt += β(ω, t)v1x(ω, t, q(ω, t, x))qx, +t ∈ [0, τ ∗), +qx(ω, 0, x) = 1, +x ∈ R. +For P − a.s. ω ∈ Ω, we solve the above equation to obtain +qx(ω, t, x) = exp +� � t +0 +β(ω, t +′)v1x(ω, t +′, q(ω, t +′, x))dt +′� +, +t ∈ (0, τ ∗). +Thus for P − a.s. ω ∈ Ω, qx(ω, t, x) > 0, (t, x) ∈ [0, τ ∗) × R. +Then the momentum variable V1 = +(1 − ∂2 +x)v1, V2 = (1 − ∂2 +x)v2 satisfy P − a.s. +V1t + βv1V1x + 2βv1xV1 + βv2xV2 = 0, +V2t + +β(v1V2)x = 0. +(7.2) +Applying particle trajectory method (7.1) and the first equation of (7.2), we obtain +d +dt +� +e +� t +0 +β(ω,s)V2v2x(ω,s,x) +V1(ω,s,x) +dsV1(ω, t, q(ω, t, x))q2 +x(ω, t, x) +� +=e +� t +0 +β(ω,s)V2v2x(ω,s,x) +V1(ω,s,x) +dsβ(ω, s)q2 +xV2v2x(ω, s, x) + e +� t +0 +β(ω,s)V2v2x(ω,s,x) +V1(ω,s,x) +dsq2 +x[V1t + βv1V1x + 2βv1xV1] +=e +� t +0 +β(ω,s)V2v2x(ω,s,x) +V1(ω,s,x) +dsq2 +x[V1t + βv1V1x + 2βv1xV1 + βv2xV2] +19 + +=0. +This and qx(ω, 0, x) = 1 imply that +e +� t +0 +β(ω,s)V2v2x(ω,s,x) +V1(ω,s,x) +dsV1(ω, t, q(ω, t, x))q2 +x(ω, t, x) = V1(ω, 0, x). +(7.3) +Consequently, we have sign(V1(ω, t, x))=sign(V1(ω, 0, x)). +The next step, we give the following useful lemma that will be used in the sequel. +Lemma 7.1 (Blow-up scenario 2) Let s > 3/2 and (u0, γ0) be an Hs ×Hs-valued F0-measurable random +variable. Assume that (z, τ ∗) is the corresponding maximal solution. Then z as a W 1,∞ × W 1,∞-valued +process is Ft-adapted for t < τ∗ and P − a.s. on the set {τ ∗ < ∞} +1{lim sup +t→τ∗(∥u(t)∥Hs+∥γ(t)∥Hs)=∞} = 1{lim sup +t→τ∗ ∥u(t)∥W 1,∞=∞}. +(7.4) +Proof: It is clear that {lim sup +t→τ ∗ ∥u(t)∥W 1,∞ = ∞} ⊂ {lim sup +t→τ ∗(∥u(t)∥Hs + ∥γ(t)∥Hs) = ∞}. +It is +sufficient to prove {lim sup +t→τ ∗ ∥u(t)∥W 1,∞ = ∞}C ⊂ {lim sup +t→τ ∗(∥u(t)∥Hs + ∥γ(t)∥Hs) = ∞}C. Notice that +{lim sup +t→τ ∗ ∥u(ω, t)∥W 1,∞ = ∞}C = {∃M(ω) > 0, s.t. ∥u(ω, t)∥W 1,∞ ≤ M(ω), +∀t < τ ∗}. +(7.5) +By the equation (6.4) and using the identity ∂2 +xG ∗ f = ∂2 +x(1 − ∂2 +x)−1f = (1 − ∂2 +x)−1f − f, we have +���� +dv2x(ω, t, q(ω, t, x)) +dt +���� += |v2tx(t, q) + v2xx(t, q)βv1| += | − βv1xv2x − β∂2 +x(1 − ∂2 +x)−1(v1xv2x) − β∂x(1 − ∂2 +x)−1(v1xv2)| += |β(1 − ∂2 +x)−1(v1xv2x) − β∂x(1 − ∂2 +x)−1(v1xv2)| +≤ β∥G∥L∞∥v1xv2x∥L1 + β∥∂xG∥L∞∥v1xv2∥L1 +≤ Cβ(2∥v1x∥L2 + ∥v2x∥L2 + ∥v2∥L2) ≤ Cβ(∥v1(0)∥H1 + ∥v2(0)∥H1). +(7.6) +For m ≥ 1, define +τm = inf{t < τ ∗ : ∥u(t)∥Hs + ∥γ(t)∥Hs ≥ m}. +By (7.6), Sobolev’s embedding and (6.2), we have +∥v2(ω, t, q(ω, t, ·))∥W 1,∞ ≤ C +� t +0 +β(ω, t +′)dt +′(∥u0∥H1 + ∥γ0∥H1) + ∥γ0∥W 1,∞, +t ≤ τm. +(7.7) +In addition, we derive from (6.5) that +d +dt(∥v1∥2 +Hs + ∥v2∥2 +Hs) ≤ C(∥u∥W 1,∞ + β(ω, t)∥v2∥W 1,∞)(∥v1∥2 +Hs + ∥v2∥2 +Hs). +By means of Gronwall’s inequality and (7.5), for any ω ∈ {lim sup +t→τ ∗ ∥u(ω, t)∥W 1,∞ = ∞}C, we obtain +∥v1(T ∧ τm)∥2 +Hs + ∥v2(T ∧ τm)∥2 +Hs +≤ (∥u0∥2 +Hs + ∥γ0∥2 +Hs) exp +�� T ∧τm +0 +C(∥u∥W 1,∞ + β(ω, t)∥v2∥W 1,∞)dt +� +, +≤ (∥u0∥2 +Hs + ∥γ0∥2 +Hs) +× exp +� +C +� +M(T ∧ τm) + +� T ∧τm +0 +β(ω, t) +� � t +0 +β(ω, t +′)dt +′(∥u0∥H1 + ∥γ0∥H1) + ∥γ0∥W 1,∞ +� +dt +�� +. +20 + +This implies on the set {τ ∗ < ∞} ∩ {lim sup +t→τ ∗ ∥u(ω, t)∥W 1,∞ = ∞}C, +∥u(T ∧ τm)∥2 +Hs + ∥γ(T ∧ τm)∥2 +Hs +≤(∥u0∥2 +Hs + ∥γ0∥2 +Hs)β(ω, T ∧ τm) +× exp +� +C +� +Mτm + +� T ∧τm +0 +� +β(ω, t) +� t +0 +β(ω, t +′)dt +′(∥u0∥H1 + ∥γ0∥H1) + ∥γ0∥W 1,∞ +� +dt +�� +<∞, +where we used supt>0 β(ω, t) < ∞ due to supt>0 Eβ(ω, t) = 1 and Doob’s L1-inequality. Hence we can +see that on the set {τ ∗ < ∞}, {lim sup +t→τ ∗ ∥u(t)∥W 1,∞ = ∞}C ⊂ {lim sup +t→τ ∗(∥u(t)∥Hs + ∥γ(t)∥Hs) = ∞}C. +So, we finish the proof. +✷ +Lemma 7.2 (Blow-up scenario 3) Let s > 3/2 and z0 be an Hs × Hs-valued F0-measurable random +variable. Assume that (z, τ ∗) is the corresponding maximal solution. Then z as a W 1,∞ × W 1,∞-valued +process is Ft-adapted for t < τ∗ and P − a.s. on the set {τ ∗ < ∞}, +1{lim sup +t→τ∗(∥u(t)∥Hs +∥γ(t)∥Hs)=∞} = 1{lim inft→τ∗ minx∈R{ux(ω,t,x)}=−∞}. +(7.8) +Proof: It is clear that {lim inft→τ ∗ minx∈R{ux(ω, t, x)} = −∞} ⊂ {lim sup +t→τ ∗(∥u(t)∥Hs +∥γ(t)∥Hs) = ∞}. +The rest of proof is similar to that of Lemma 7.1 by replacing equation (7.5) with +{lim inf +t→τ ∗ min +x∈R{ux(ω, t, x)} = −∞}C = {∃M(ω) > 0, s.t. ux(ω, t, x) > −M(ω), +∀t < τ ∗}. +(7.9) +Without loss of generality, we only need to show that this Lemma holds for s = 2. Multiplying the first +equation in (7.2) by V1 = (1 − ∂2 +x)v1 and integrating by parts, we get +d +dt +� +R +V 2 +1 dx += +−2β(w, t) +� +R +v1V1V1xdx − 4β(w, t) +� +R +V 2 +1 v1xdx − 2β(w, t) +� +R +V1V2v2xdx += +−3β(w, t) +� +R +V 2 +1 v1xdx − 2β(w, t) +� +R +V1V2v2xdx. +(7.10) +Multiplying the second equation in (7.2) by V2 = (1 − ∂2 +x)v2 and integrating by parts, we obtain +d +dt +� +R +V 2 +2 dx += +−β(w, t) +� +R +v1xV 2 +2 dx. +(7.11) +Thus, in view of (7.9), (7.10), (7.11) and (7.7), for any ω ∈ {lim inft→τ ∗ minx∈R{ux(ω, t, x)} = −∞}C, +we obtain +d +dt +� +R +(V 2 +1 + V 2 +2 )dx += −3β(w, t) +� +R +V 2 +1 v1xdx − β(w, t) +� +R +v1xV 2 +2 dx − 2β(w, t) +� +R +V1V2v2xdx +≤ 3M +� +R +(V 2 +1 + V 2 +2 )dx ++ β(ω, t) +� � t +0 +β(ω, t +′)dt +′(∥u0∥H1 + ∥γ0∥H1) + ∥γ0∥W 1,∞ +� � +R +(V 2 +1 + V 2 +2 )dx +By means of Gronwall’s inequality, we arrive at +∥v1(T ∧ τm)∥H2 + ∥v2(T ∧ τm)∥H2 = ∥V1(T ∧ τm)∥L2 + ∥V2(T ∧ τm)∥L2 +21 + +≤ (∥V1(0, ·)∥L2 + ∥V2(0, ·)∥L2) +× exp +� � T ∧τm +0 +� +3M + β(w, t) +� � t +0 +β(ω, t +′)dt +′(∥u0∥H1 + ∥γ0∥H1) + ∥γ0∥W 1,∞ +�� +dt +� +. +Then on the set {τ ∗ < ∞} ∩ {lim inft→τ ∗ minx∈R{ux(ω, t, x)} = −∞}C, +∥u(T ∧ τm)∥H2 + ∥γ(T ∧ τm)∥H2 +≤ β(w, T ∧ τm)(∥V1(0, ·)∥L2 + ∥V2(0, ·)∥L2) +× exp +� � T ∧τm +0 +� +3M + β(w, t) +� � t +0 +β(ω, t +′)dt +′(∥u0∥H1 + ∥γ0∥H1) + ∥γ0∥W 1,∞ +�� +dt +� +< ∞, +where we used supt>0 β(ω, t) < ∞ due to supt>0 Eβ(ω, t) = 1 and Doob’s L1-inequality. +Hence we +can see that {lim inft→τ ∗ minx∈R{ux(ω, t, x)} = −∞}C ⊂ {lim sup +t→τ ∗(∥u(t)∥Hs + ∥γ(t)∥Hs) = ∞}C. This +completes the proof. +✷ +Lemma 7.3 Let V1 = v1 − v1xx and P{V1(ω, 0, x) > 0, ∀x ∈ R} = p, P{V1(ω, 0, x) < 0, ∀x ∈ R} = q, +P +� +V1(ω, 0, x) ≤ 0, x ≤ x0 +and +V1(ω, 0, x) ≥ 0, x ≥ x0 +� += m for some p, q, m ∈ [0, 1]. Then the +maximal solution (z, τ ∗) of (1.3) satisfies +P +� +∥ux(ω, t)∥L∞ ≤ +√ +2 +2 β(ω, t)(∥u0∥H1 + ∥γ0∥H1) +or ux(ω, t) ≥ − +√ +2 +2 β(ω, t, x)(∥u0∥H1 + ∥γ0∥H1), +∀t ∈ [0, τ ∗) +� +≥ p + q + m. +(7.12) +Proof: Denote +Ap = {V1(ω, 0, x) > 0, ∀x ∈ R}, +Aq = {V1(ω, 0, x) < 0, ∀x ∈ R}, +Am = {V1(ω, 0, x) ≤ 0, x ≤ x0 and V1(ω, 0, x) ≥ 0, x ≥ x0}. +Using G(x) = e−|x| +2 +, one can derive that for P-a.s. ω ∈ Ω, and for all (t, x) ∈ [0, τ ∗) × R, +v1(ω, t, x) = 1 +2e−x +� x +−∞ +eξV1(ω, t, ξ)dξ + 1 +2ex +� ∞ +x +e−ξV1(ω, t, ξ)dξ, +v1x(ω, t, x) = −1 +2e−x +� x +−∞ +eξV1(ω, t, ξ)dξ + 1 +2ex +� ∞ +x +e−ξV1(ω, t, ξ)dξ. +Therefore, +v1(ω, t, x) + v1x(ω, t, x) = ex +� ∞ +x +e−ξV1(ω, t, ξ)dξ, +(7.13) +v1(ω, t, x) − v1x(ω, t, x) = e−x +� x +−∞ +eξV1(ω, t, ξ)dξ. +(7.14) +Then one can employ (7.13), (7.14) and sign(V1)=sign(V1(ω, 0, x)), (t, x) ∈ [0, τ∗) × R to obtain that for +all (t, x) ∈ [0, τ ∗) × R, +� −v1(ω, t, x) ≤ v1x(ω, t, x) ≤ v1(ω, t, x), +ω ∈ Ap, +v1(ω, t, x) ≤ v1x(ω, t, x) ≤ −v1(ω, t, x), +ω ∈ Aq. +(7.15) +22 + +In addition, since q(ω, t, ·) is an increasing diffeomorphism of R with qx(ω, t, x) > 0 for all (t, x) ∈ +[0, τ ∗) × R, by (7.3), it follows that for any ω ∈ Am, + + + +V1(ω, t, x) ≤ 0 if x ≤ q(ω, t, x0), +V1(ω, t, x) ≥ 0 if x ≥ q(ω, t, x0), +V1(ω, t, q(ω, t, x0)) = 0. +(7.16) +Therefore, for any ω ∈ Am, when x ≤ q(ω, t, x0), by (7.14) and (7.16), we have v1(ω, t, x) ≤ v1x(ω, t, x); +when x ≥ q(ω, t, x0), by (7.13) and (7.16), we have v1x(ω, t, x) ≥ −v1(ω, t, x), Therefore, it follows from +(6.2) that for any ω ∈ Am, +−v1x(ω, t, x) ≤ |v1(ω, t, x)| ≤ ∥v1(ω, t, x)∥L∞ ≤ +√ +2 +2 (∥u0∥H1 + ∥γ0∥H1), ∀(t, x) ∈ [0, τ ∗) × R +(7.17) +Then for any ω ∈ Am, ux(ω, t) ≥ − +√ +2 +2 β(ω, t, x)(∥u0∥H1 + ∥γ0∥H1). This together with Lemma 7.2 and +supt>0 β(ω, t, x) < ∞ implies that z globally exists. +For any ω ∈ Ap ∪ Aq, it follows from (7.15) that |v1x(ω, t, x)| ≤ |v1(ω, t, x)|, in view of Sobolev +inequality and (6.2), we arrive at +∥v1x(ω, t, x)∥L∞ ≤ ∥v1(ω, t, x)∥L∞ ≤ +√ +2 +2 (∥u0∥H1 + ∥γ0∥H1), ∀(t, x) ∈ [0, τ ∗) × R, ω ∈ Ap ∪ Aq. +(7.18) +Combining Ap ∩ Aq ∩ Am = ∅, (7.17) and (7.18), we derive (7.12). +✷ +Proof of Theorem 3.8. +Note that supt>0 Eβ(ω, t, x) = 1 and Doob’s L1-inequality implies that +supt>0 β(ω, t, x) < ∞. Then we can infer from (7.4), (7.8) and (7.12) that P{τ ∗ = ∞} ≥ p + q + m. This +completes the proof. +7.2 +Proof of Theorem 3.9 +The proof of Theorem 3.9 relies on certain properties of the solution v1, v2 to the equations (6.3) and +(6.4). We first prove the following lemma. +Lemma 7.4 Let s > 5/2 and b(t) satisfy Assumption 3.3. Assume (u0, γ0) is an Hs × Hs-valued F0- +measurable random variable. Let K = +√ +2 +2 (∥u0∥2 +H1 + ∥γ0∥2 +H1) +1 +2 . Then for v1, v2 defined by (6.3), (6.4) +and any x0 ∈ R, +g(ω, t) := v1x(ω, t, q(ω, t, x0)) +satisfies P − a.s. +d +dtg(t) ≤ βK2 − β +2 g2(t), +t < τ ∗. +(7.19) +Moreover, if there exists some x0 ∈ R such that P−a.s. g(0) < − +√ +2K, then P−a.s.. g(t) is non-increasing +on [0, τ ∗) and +g(t) < − +√ +2K, +t ∈ [0, τ ∗). +(7.20) +Proof: For any v1, v2 ∈ H1, by the representation of G ∗ f = (1 − ∂2 +x)−1f, we have +G ∗ +� +v2 +1 + 1 +2v2 +1x +� +(x) = 1 +2 +� x +−∞ +e−x+y +� +v2 +1 + 1 +2v2 +1x +� +(y)dy + 1 +2 +� ∞ +x +ex−y +� +v2 +1 + 1 +2v2 +1x +� +(y)dy. +(7.21) +The following inequality +� x +−∞ +ey +� +v2 +1 + v2 +1x +� +(y)dy ≥ 2 +� x +−∞ +eyv1v1x(y)dy = exv2 +1(x) − +� x +−∞ +eyv2 +1dy +23 + +implies that +1 +2 +� x +−∞ +e−x+y +� +v2 +1 + 1 +2v2 +1x +� +(y)dy ≥ 1 +4v2 +1(x). +(7.22) +Similarly, we get the estimate of the second term in (7.21) as +1 +2 +� ∞ +x +ex−y +� +v2 +1 + 1 +2v2 +1x +� +(y)dy ≥ 1 +4v2 +1(x), +(7.23) +Combining (7.21), (7.22) and (7.23), we deduce G ∗ (v2 +1 + 1 +2v2 +1x)(x) ≥ 1 +2v2 +1(x). In addition, +∥G ∗ v2 +2x∥L∞ ≤ ∥G∥L∞∥v2 +2x∥L1 = 1 +2∥v2 +2x∥L1. +(7.24) +Differentiating the first equation of (6.1) with respect to x, and using (6.2) and (7.24), we have +d +dtv1x(ω, t, q(t, ω, x)) += +v1xt + v1xxβ(ω, t, x)v1(ω, t, q(ω, t, x)) += +−βv2 +1x − β∂2 +x(1 − ∂2 +x)−1 +� +v2 +1 + 1 +2v2 +1x + 1 +2v2 +2 − 1 +2v2 +2x +� +≤ +−1 +2βv2 +1x + 1 +2βv2 +1 + 1 +4βv2 +2 + 3 +4βG ∗ (v2 +2x) +≤ +−1 +2βv2 +1x + β +2 (∥u0∥2 +H1 + ∥γ0∥2 +H1). +In view of the assumptions of Lemma 7.4, we have P − a.s. +d +dtg(t) ≤ −β +2 g2(t) + βK2, +t < τ ∗, +which is (7.19). In order to prove (7.20), define +ζ(w) := inf +� +t ∈ [0, τ ∗) : g(w, t) > − +√ +2K +� +. +If g(0) < − +√ +2K, then P{ζ > 0} = 1. From the definition of ζ(ω), we find that ζ(ω) ≤ τ ∗, for P − a.s. +w ∈ Ω. From (7.19), we have that g(ω, t) is nonincreasing for t ∈ [0, ζ(ω)). Hence by the continuity of +the path of g(ω, t), we obtain that g(ω, t) ≤ g(0) < − +√ +2K, +t ∈ [0, ζ(ω)). In view of the time continuity +of g(ω, t) again, we find that P{ζ = τ ∗} = 1. Hence (7.20) is true. +✷ +Proof of Theorem 3.9. From Lemma 7.4 and (3.5), we rewrite (7.19) as +d +dtg(t) +≤ +−β(t) +2 +� +1 − 2K2 +g2(0) +� +g2(t) − +� g2(t) +g2(0) − 1 +� +β(t)K2 +≤ +−β(t) +2 +� +1 − 2K2 +g2(0) +� +g2(t), +t ∈ [0, τ ∗). +Integrating on both sides leads to P − a.s. +1 +g(t) − +1 +g(0) ≥ +� +1 − 2K2 +g2(0) +� � t +0 +β(t +′) +2 +dt +′, +t ∈ [0, τ ∗). +Assuming Ω +′ = {ω : β(t, ω) ≥ ce− b∗ +2 t for all t}, g(t) ≤ − +√ +2K means that P − a.s. ω ∈ Ω +′ +− 1 +g(0) ≥ +�1 +2 − K2 +g2(0) +� � τ ∗ +0 +β(t +′)dt +′ ≥ +�1 +2 − K2 +g2(0) +��2c +b∗ − 2c +b∗ e− b∗ +2 τ ∗� +. +24 + +If g(0) < − 1 +2 +� +(b∗)2 +c2 ++ 8K2 − b∗ +2c, we obtain on Ω′ +�1 +2 − K2 +g2(0) +�2c +b∗ e− b∗ +2 τ ∗ ≥ 2c +b∗ +�1 +2 − K2 +g2(0) +� ++ +1 +g(0) > 0. +Therefore we have τ ∗ < ∞ P − a.s. on Ω +′, which implies that +P{τ ∗ < ∞} ≥ P{β(t) ≥ ce− b∗ +2 t for all t} = P +� +e +� t +0 b(t +′ )dWt′ + +� t +0 +b∗−b2(t +′ +) +2 +dt +′ +≥ c for all t +� +> 0. +We finish the proof. +7.3 +Proof of Theorem 3.10 +The proof of Theorem 3.10 is similar to that of Theorem 3.9. We first prove the following lemma. +Lemma 7.5 Let s > 5/2 and b(t) satisfy Assumption 3.3. Assume (u0, γ0) is an Hs × Hs-valued F0- +measurable random variable. Let K = +√ +2 +2 (∥u0∥2 +H1 + ∥γ0∥2 +H1) +1 +2 . Then for v1, v2 defined by (6.3), (6.4), +N(ω, t) := +� +R +v3 +1x(ω, t, q(ω, t, x))dx +satisfies P − a.s. +d +dtN(t) ≤ 15β +4 K4 − +β +4K2 N 2(t), +t < τ ∗. +(7.25) +Moreover, if P − a.s. N(0) < − +√ +15K3, then P − a.s.. N(t) is non-increasing on [0, τ ∗) and +N(t) < − +√ +15K3, +t ∈ [0, τ ∗). +(7.26) +Proof: Differentiating the first equation of (6.1) with respect to x, and using the ∂2 +x(1 − ∂2 +x)−1f = +∂2 +xG ∗ f = G ∗ f − f, we have +v1xt + β +2 v2 +1x + βv1v1xx + βG ∗ +� +v2 +1 + 1 +2v2 +1x + 1 +2v2 +2 − 1 +2v2 +2x +� +− β +� +v2 +1 + 1 +2v2 +2 − 1 +2v2 +2x +� += 0. +(7.27) +Let N(t) := +� +R v3 +1x(ω, t, x)dx, t ≥ 0. Multiplying (7.27) with v2 +1x and integrating by parts subsequently, by +G ∗ (v2 +1 + 1 +2v2 +1x)(x) ≥ 1 +2v2 +1(x), we get +1 +3 +dN(t) +dt += − β +6 +� +R +v4 +1xdx − β +� +R +v2 +1xG ∗ (v2 +1 + 1 +2v2 +1x + 1 +2v2 +2 − 1 +2v2 +2x)dx + β +� +R +v2 +1x(v2 +1 + 1 +2v2 +2 − 1 +2v2 +2x)dx +≤ − β +6 +� +R +v4 +1xdx + β +2 +� +R +v2 +1v2 +1xdx + β +2 +� +R +v2 +1xG ∗ v2 +2xdx + β +2 +� +R +v2 +1xv2 +2dx +≤ − β +6 +� +R +v4 +1xdx + β +2 +� +R +v2 +1x(v2 +1 + v2 +2)dx + β +4 ∥v2 +2x∥L1 +� +R +v2 +1xdx. +In view of Sobolev’s embedding and the invariant property of ∥v1(t)∥2 +H1 + ∥v2(t)∥2 +H1 = ∥u0∥2 +H1 + ∥γ0∥2 +H1, +we find that +3 +2 +� +R +v2 +1x(v2 +1 + v2 +2)dx + 3 +4∥v2 +2x∥L1 +� +R +v2 +1xdx ≤ 3 +4(∥u0∥2 +H1 + ∥γ0∥2 +H1)2 + 3 +16(∥u0∥2 +H1 + ∥γ0∥2 +H1)2. +25 + +On the other hand, the Cauchy-Schwarz inequality implies that +���� +� +R +v3 +1xdx +���� ≤ +� � +R +v4 +1xdx +� 1 +2 � � +R +v2 +1xdx +� 1 +2 +, +hence, +� +R +v4 +1xdx ≥ +1 +∥u0∥2 +H1 + ∥γ0∥2 +H1 +� � +R +v3 +1xdx +�2 +. +As defined in Lemma 6.5, K = +√ +2 +2 (∥u0∥2 +H1 + ∥γ0∥2 +H1) +1 +2 , we have the similar Riccati type equation +dN(t) +dt +≤ − β +4K2 N 2(t) + 15β +4 K4, +which is (7.25). In order to prove (7.26), define stopping time +χ(w) := inf +� +t ∈ [0, τ ∗) : N(w, t) > − +√ +15K3 +� +. +If N(0) < − +√ +15K3, then P{χ(ω) > 0} = 1. From the definition of χ(ω), we find that w ∈ Ω, χ(ω) ≤ τ ∗. +From (7.25), we conclude that N(ω, t) is nonincreasing for t ∈ [0, χ(ω)). Hence by the continuity of the +path of N(ω, t), we obtain that N(ω, t) ≤ N(0) < − +√ +15K3. In view of the time continuity of N(ω, t) +again, we find that P{χ = τ ∗} = 1. Therefore, (7.26) is true. +✷ +Proof of Theorem 3.10. From Lemma 7.5 and (3.6), we rewrite (7.25) as +d +dtN(t) +≤ +− β(t) +4K2 +� +1 − 15K6 +N 2(0) +� +N 2(t) − +� N 2(t) +N 2(0) − 1 +� 15β(t) +4 +K4 +≤ +− β(t) +4K2 +� +1 − 15K6 +N 2(0) +� +N 2(t), +t ∈ [0, τ ∗). +Integrating on both sides leads to P − a.s. +1 +N(t) − +1 +N(0) ≥ +� +1 − 15K6 +N 2(0) +� � t +0 +β(t +′) +4K2 dt +′, +t ∈ [0, τ ∗). +Assuming Ω +′ = {ω : β(t, ω) ≥ ce− b∗ +2 t for all t}, also due to N(t) < − +√ +15K3, we get P − a.s. ω ∈ Ω +′ +− +1 +N(0) ≥ +� +1 +4K2 − 15K4 +4N 2(0) +� � τ ∗ +0 +β(t +′)dt +′ ≥ +� +1 +4K2 − 15K4 +4N 2(0) +��2c +b∗ − 2c +b∗ e− b∗ +2 τ ∗� +. +If N(0) < − +� +(b∗)2K4 +c2 ++ 15K6 − b∗K2 +c +, we obtain on Ω′ +� +1 +4K2 − 15K4 +4N 2(0) +�2c +b∗ e− b∗ +2 τ ∗ ≥ 2c +b∗ +� +1 +4K2 − 15K4 +4N 2(0) +� ++ +1 +N(0) > 0. +Therefore we obtain τ∗ < ∞ P − a.s. on Ω +′, which means that +P{τ ∗ < ∞} ≥ P{β(t) ≥ ce− b∗ +2 t for all t} = P +� +e +� t +0 b(t +′ )dWt′ +� t +0 +b∗−b2(t′ ) +2 +dt +′ +≥ c for all t +� +> 0. +So, the proof is finished. +26 + +8 +Acknowledgments +This paper is supported by Fundamental Research Funds for the Central Universities (No. 22D110913). +References +[1] P. Olver, P. Rosenau. Tri-Hamiltonian duality between solitons and solitary-wave solutions having +compact support, Phys. Rev. E 53 (1996) 1900-1906. +[2] A. Constantin, R. Ivanov. On an integrable two-component Camassa-Holm shallow water system, +Phys. Lett. A 372 (2008) 7129-7132. +[3] M. Chen, S. Liu, Y. Zhang. A two-component generalization of the Camassa-Holm equation and its +solutions, Lett. Math. Phys. 75 (2006) 1-15. +[4] G. Falqui. On a Camassa-Holm type equation with two dependent variables, J. Phys. A 39 (2006) +327-342. +[5] J. Escher, O. Lechtenfeld, Z. Yin. Well-posedness and blow-up phenomena for the 2-component +Camassa-Holm equation, Discrete Contin. Dyn. Syst. 19 (2007) 493-513. +[6] G. Gui, Y. Liu. On the global existence and wave-breaking criteria for the two-component Camassa- +Holm system, J. Funct. Anal. 258 (2010) 4251-4278. +[7] Z. Guo, Y. Zhou. On solutions to a two-component generalized Camassa-Holm equation, Stud. Appl. +Math. 124 (2010) 307-322. +[8] Z. Guo. Blow-up and global solutions to a new integrable model with two components, J. Math. +Anal. Appl. 372 (2010) 316-327. +[9] R. Camassa, D. Holm. An integrable shallow water equation with peaked solitons, Phys. Rev. Lett. +71 (1993) 1661-1664. +[10] B. Fuchssteiner, A.S. Fokas. Symplectic structures, their Bcklund transformations and hereditary +symmetries, Phys. D. 4 (1)(1981/1982) 4766. +[11] A. Constantin, H.P. McKean. A shallow water equation on the circle, Comm. Pure Appl. Math. 52 +(1999) 949-982. +[12] A. Constantin, V. Gerdjikov, R. Ivanov. Inverse scattering transform for the Camassa-Holm equation, +Inverse Problems 22(2006) 2197-2207. +[13] J. Lenells. Conservation laws of the Camassa-Holm equation, J. Phys. A 38 (2005) 869-880. +[14] A. Constantin, J. Escher. Well-posedness, global existence and blow-up phenomenon for a periodic +quasi-linear hyperbolic equation, Comm. Pure Appl. Math. 51 (1998) 475-504. +[15] Y. Li, P. Olver. Well-posedness and blow-up solutions for an integrable nonlinear dispersive model +wave equation, J. Differ. Equations. 162 (2000) 27-63. +[16] A. Constantin, J. Escher. Wave breaking for nonlinear nonlocal shallow water equations, Acta Math. +181 (1998) 229-243. +[17] H.P. McKean. Breakdown of a shallow water equation, Asian J. Math. 2 (1998) 767-774. +[18] Y. Zhou. Wave breaking for a shallow water equation, Nonlinear Anal. 57 (2004) 137-152. +27 + +[19] Y. Zhou. Wave breaking for a periodic shallow water equation, J. Math. Anal. Appl. 290 (2004) +591-604. +[20] A. Constantin, W. Strauss. Stability of peakons, Commun. Pure Appl. Math. 53 (2000) 603-610. +[21] D. Holm, L. Naraigh, C. Tronci. Singular solution of a modified two-component Camassa-Holm +equation, Phy. Rev. E. 79 (2009) 1-13. +[22] C. Guan, K.H. Karlsen, Z. Yin. Well-posedness and blow-up phenomena for a modified tow- +component Camassa-Holm equation, Contemp. Math. 526 (2010)199-220. +[23] Z. Guo, M. Zhu. Wave breaking for a modified two-component Camassa-Holm system, J. Differ. +Equations. 252 (2012) 2759-2770. +[24] J. Liu, Z. Yin. On the blow-up phenomena for a modified periodic two-component Camassa-Holm +equation, IMA J. Appl. Math. 77 (2012) 563-577. +[25] Z. Guo, M. Zhu, L. Ni. Blow-up criteria of solutions to a modified two-component Camassa-Holm +system, Nonlinear Anal. Real World Appl. 12 (2011) 3531-3540. +[26] W. Tan, Z. Yin. Global periodic conservative solutions of a periodic modified two-component +Camassa-Holm equation, J Funct Anal. 261 (2011)1204-1226. +[27] D. Prato, G. Zabczyk. Stochastic Equations in Infinite Dimensions, Encyclopedia of Mathematics +and its Applications. 152 (2014). +[28] C. Prevot, M. Rockner. A Concise Course on Stochastic Partial Differential Equations, Lecture Notes +in Mathematics. Berlin (2007). +[29] Z. Brzezniak, M. Capinski, F. Flandoli. Stochastic partial differential equations and turbulence, +Math. Models Methods Appl. Sci. 1 (1991) 41-59. +[30] S. Kuksin, A. Shirikyan. Mathematics of two-dimensional turbulence, Cambridge Tracts in Mathe- +matics, 194 (2012). +[31] E. Weinan. Stochastic hydrodynamics, In Current developments in mathematics, (2000) 109-147. +[32] J. Lenells, M. Wunsch. On the weakly dissipative Camassa-Holm, Degasperis-Procesi, and Novikov +equations, J. Differ. Equations. 255 (2013)441-448. +[33] S. Wu, Z. Yin. Blow-up and decay of the solution of the weakly dissipative Degasperis-Procesi +equation, SIAM J. Math. Anal. 40 (2008) 475-490. +[34] S. Wu, Z. Yin. Global existence and blow-up phenomena for the weakly dissipative Camassa-Holm +equation, J. Differ. Equations. 246 (2009) 4309-4321. +[35] H. Tang. On the pathwise solutions to the Camassa-Holm equation with multiplicative noise, SIAM +J. Math. Anal. 50 (2018) 1322-1366. +[36] T. Kato, G. Ponce. Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure +Appl. Math. 41 (1988) 891-907. +[37] N. Glatt, V. Vicol. Local and global existence of smooth solutions for the stochastic Euler equations +with multiplicative noise, Ann. Probab. 42 (2014) 80-145. +[38] A. Debussche, N. Glatt, R. Temam. Local martingale and pathwise solutions for an abstract fluids +model, Phys. D. 240 (2011) 1123-1144. +28 + +[39] D. Breit, E. Feireisl, M. Hofmanova. Local strong solutions to the stochastic compressible Navier- +Stokes system, Comm. Partial Differential Equations. 43 (2018) 313-345. +[40] D. Breit, E. Feireisl, M. Hofmanova. Stochastically forced compressible fluid flows, De Gruyter Series +in Applied and Numerical Mathematics. 3 (2018). +[41] D. Breit, M. Hofmanova. Stochastic Navier-Stokes equations for compressible fluids, Indiana Univ. +Math. J. 65 (2016) 1183-1250. +[42] N. Glatt, M. Ziane. Strong pathwise solutions of the stochastic Navier-Stokes system, Adv. Differ. +Equ. 14 (2009) 567-600. +[43] C. Rohde, H. Tang. On a stochastic camassa-holm type equation with higher order nonlinearities, +Journal. Dyn Differ. Equ. 4(2020). +[44] G. Kallianpur, J. Xiong. Stochastic differential equations in infinite-dimensional spaces, Expanded +version of the lectures delivered as part of the 1993 Barrett Lectures at the University of Tennessee, +Knoxville, TN, March (1995) 25-27. +[45] M. Hofmanova. Degenerate parabolic stochastic partial differential equations, Stochastic Process. +Appl. 123 (2013) 4294-4336. +[46] I. Gyongy, N. Krylov. Existence of strong solutions for Itˆo stochastic equations via approximations, +Probab. Theory Related Fields. 105 (1996) 143-158. +[47] M. Rockner, R. Zhu, X. Zhu. Local existence and non-explosion of solutions for stochastic fractional +partial differential equations driven by multiplicative noise, Stoch. Proc. Appl. 124 (2014) 1974-2002. +[48] D. Crisan, F. Flandoli, D. Holm. Solution Properties of a 3D Stochastic Euler Fluid Equation. J. +Nonlinear Sci. 29 (2019) 813-870. +[49] L. Gawarecki, V. Mandrekar. Stochastic differential equations in infinite dimensions with applications +to stochastic partial differential equations. Probability and its Applications (NewYork). Springer, +Heidelberg (2011). +29 + diff --git a/LdE2T4oBgHgl3EQfqAhW/content/tmp_files/load_file.txt b/LdE2T4oBgHgl3EQfqAhW/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..523a1edff8b6fabb9c53bb7e01254febbe5fa4d3 --- /dev/null +++ b/LdE2T4oBgHgl3EQfqAhW/content/tmp_files/load_file.txt @@ -0,0 +1,1370 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf,len=1369 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='04034v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='PR] 10 Jan 2023 Existence and Blow-up of solutions for Stochastic Modified Two-component Camassa-Holm System Wujun Lv Department of Statistics, College of Science, Donghua University 201620, Shanghai, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' China lvwujunjaier@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='com Xing Huang∗ Center for Applied Mathematics, Tianjin University 300072, Tianjin, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' China xinghuang@tju.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='cn Abstract: In this paper, we consider the modified two-component Camassa-Holm System with multi- plicative noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' For these SPDEs, we first establish the local existence and pathwise uniqueness of the pathwise solutions in Sobolev spaces Hs×Hs, s > 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Then we show that strong enough noise can actually prevent blow-up with probability 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Finally, we analyse the effects of weak noise and present conditions on the initial data that lead to the global existence and the blow-up in finite time of the solutions, and their associated probabilities are also obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Keywords: Stochastic modified two-component Camassa-Holm system (SMCH2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Pathwise solutions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Global existence;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Blow-up criterion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Blow-up scenatios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' 1 Introduction Consider the following integrable two-component Camassa-Holm (CH2) shallow water system \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 (u − uxx)t + 3uux − 2uxuxx − uuxxx + ρρx = 0, t > 0, x ∈ R, ρt + (ρu)x = 0, t > 0, x ∈ R, u(0, x) = u0(x), x ∈ R, ρ(0, x) = ρ0(x), x ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' This system appears initially in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' In 2008, it was derived by Constantin and Ivanov in [2], which provided a demonstration about its derivation in view of the fluid shallow water theory from the hydrodynamic point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Similar to the Camassa-Holm equation, this system possesses the peakon, multi-kink solutions and the bi-Hamiltonian structure [3, 4] and is integrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Well-posedness and wave breaking mechanism were discussed in [5, 6, 7] and the existence of global solutions was analyzed in [2, 6, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Obviously, under the constraint of ρ(x, t) = 0, this system reduces to the cerebrated Camassa-Holm (CH) equation, which was derived physically by Camassa and Holm in [9] (found earlier by Fokas and Fuchssteiner [10] as a bi-Hamiltonian generalization of the KdV equation) by approximating directly the Hamiltonian for Euler’s equation in the shallow water region with u(x, t) representing the free surface above a flat bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' CH equation is completely integrable [11, 12] and has infinitely many conservation laws [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Local well-posedness for the initial data u0 ∈ Hs with s > 3/2 was proved in [14, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' One of the remarkable features of the CH equation is the presence of breaking waves as well as global solutions ∗Corresponding author 1 in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Wave breaking for a large class of initial data has been established in [14, 15, 16, 17, 18, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Global solutions were also explored in [14, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' The solitary waves of the CH equation are peaked solitons and are orbitally stable [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' If ρ(x, t) ̸= 0, this CH2 system is actually an extension of the CH equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' However, a modified version of the two-component Camassa-Holm (MCH2) system allows a depen- dence on the average density ρ as well as the pointwise density ρ, and it is written as \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 (u − uxx)t + 3uux − 2uxuxx − uuxxx + ρρx = 0, t > 0, x ∈ R, ρt + (ρu)x = 0, t > 0, x ∈ R, u(0, x) = u0(x), x ∈ R, ρ(0, x) = ρ0(x), x ∈ R, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='1) where u denotes the velocity field, ρ = (1−∂2 x)(ρ−ρ0) with some constant ρ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' This system was introduced by Holm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' in [21], and it does admit peaked solutions in the velocity and average density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Many authors analytically identified the steepening mechanism that allows the singular solutions to emerge from smooth spatially confined initial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' They found that wave breaking in the fluid velocity does not imply singularity in the pointwise density ρ at the point of vertical slope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='1) may not be integrable unlike the CH2 system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' The characteristic is that it will amount to strengthening the norm for ρ from L2 to H1 in the potential energy term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Letting γ = ρ − ρ0, it leads to the conserved quantity � R ∥u∥2 H1 + ∥γ∥2 H1dx, which is absent in the CH2 system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' This property inspired a series of interesting works for a deep insight into the MCH2 system in the recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' The Cauchy problem of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='1) has been studied in many works [22, 23, 21, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' It has been shown that this system is locally well-posed on the line [22] and on the circle [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Moreover, the authors presented several blow-up results [22, 23, 24, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' In addition, basing on a conserved quantity, the authors established the global existence results for strong solutions to the system [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Before introducing our model, we recall some theory of infinite dimensional stochastic analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Let S = (Ω, F, P, {Ft}t≥0, W1, W2), where (Ω, F, P, {Ft}t≥0) is a complete filtration probability space, and W1, W2 are two cylindrical Wiener process on some separable Hilbert space U and d⟨W1, W2⟩t = κdt, −1 ≤ κ ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' To be precise, we consider a separable Hilbert space U as well as a larger Hilbert space U0 such that the canonical embedding U ֒→ U0 is Hilbert-Schmidt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Therefore we have Wi = ∞ � k=1 W i kek ∈ C([0, ∞), U0), i = 1, 2, where {W i k}k≥1 is a sequence of mutually independent one-dimensional Brownian motions and {ek}k∈N is a complete orthonormal basis of U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' To define the Itˆo stochastic integral � t 0 GdWi = ∞ � k=1 � t 0 GekdW i k, i = 1, 2 on Hs, it is required in [27, 28] for the predictable stochastic process G to take values in the space of L2(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Hs), the Hilbert-Schmidt operators from U to Hs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' We have � � t 0 GdWi, v � Hs = ∞ � k=1 � t 0 (Gek, v)HsdW i k, i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Moreover, the Burkholder-Davis-Gundy inequality E � sup t∈[0,T ] ���� � t 0 GdWi ���� p Hs � ≤ C(p, s)E � � T 0 ∥G∥2 L2(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='Hs)ds � p 2 , p ≥ 1, i = 1, 2 2 holds for some constant C(p, s) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' In this paper, we are interested in stochastic variants of the MCH2 system to model energy con- suming/exchanging mechanisms in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='1) that are driven by external stochastic influences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Adding mul- tiplicative noise has also been connected to the prevailing hypotheses that the onset of turbulence in fluid models involves randomness [29, 30, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Precisely, we consider stochastic modified two-component Camassa-Holm (SMCH2) system \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 (u − uxx)t + 3uux − 2uxuxx − uuxxx + ρρx = (1 − ∂2 x)h1(t, u, ρ) ˙ W1, t > 0, x ∈ R, ρt + (ρu)x = (1 − ∂2 x)h2(t, u, ρ) ˙ W2, t > 0, x ∈ R, u(0, x) = u0(x), x ∈ R, ρ(0, x) = ρ0(x), x ∈ R, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='2) where h1(t, u, ρ), h2(t, u, ρ) are typically nonlinear functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Let γ = ¯ρ − ¯ρ0, then (1 − ∂2 x)−1ρ = γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Notice that the deterministic MCH2 type equations with the weakly dissipative term λ2(1 − ∂2 x)h1(t, u, ρ), λ2(1 − ∂2 x)h2(t, u, ρ) have been introduced and studied by many scholars [32, 33, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' In order to model more general random energy exchange, we consider the possibly nonlinear noise term (1 − ∂2 x)h1(t, u, ρ) ˙ W1, (1 − ∂2 x)h2(t, u, ρ) ˙ W2 in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='2), which will be used to compare with deterministic weakly dissipative MCH2 type equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' In (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='2), the operator (1 − ∂2 x)−1 can be expressed by it’s associated Green’s function G(x) = e−|x|/2 with [(1 − ∂2 x)−1f](x) = [G ∗ f](x) = 1 2 � R e−|x−y|f(y)dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' So the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='2) is equivalent to tha following one \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 du + [uux + F1(u, γ)]dt = h1(t, u, γ)dW1, t > 0, x ∈ R, dγ + [uγx + F2(u, γ)]dt = h2(t, u, γ)dW2, t > 0, x ∈ R, u(0, x) = u0(x), x ∈ R, γ(0, x) = γ0(x), x ∈ R, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='3) where F1(u, γ) = ∂x(1 − ∂2 x)−1(u2 + 1 2u2 x + 1 2γ2 − 1 2γ2 x), F2(u, γ) = (1 − ∂2 x)−1((uxγx)x + uxγ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' The purpose of this paper is as follows: The first goal of the present paper is to analyze the existence and uniqueness of pathwise solutions and to determine possible blow-up criterion for the Cauchy problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Under generic assumptions on h1(t, u, γ), h2(t, u, γ), we will show that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='3) has a local unique pathwise solution(see Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='5 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' The second goal of this work is to study the case of strong nonlinear noise and consider its effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' As we will see in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='3) below, for the solution to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='3), its Hs × Hs-norm blows up if and only if its W 1,∞×W 1,∞-norm blows up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' This suggests choosing a noise coefficient involving the W 1,∞×W 1,∞-norm of (u, γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Therefore in this work we consider the case that h1(t, u, γ)dW1 = h1(t, u, γ)d(�∞ i=1 Wi(t)ei) = a(t)(1+∥u∥W 1,∞ +∥γ∥W 1,∞)θudW1, where {ei}i∈N∗ denote an orthonormal basis, {Wi}i∈N∗ is a family of independent standard real-valued Wiener processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Similarly, we consider the case that h2(t, u, γ)dW2 = a(t)(1 + ∥u∥W 1,∞ + ∥γ∥W 1,∞)θγdW1, where θ > 0, 0 < a∗ ≤ a2(t) ≤ a∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' To simplify the model, we write W1 as W, and we will try to determine the range of θ and a∗, a∗ such that the solution exists globally in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' The third goal of this paper is to consider weak linear noise effects associated with the phenomenon of wave breaking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Due to Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='6 below, we see that if wave breaking occurs, the noise term does not grow fast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Hence we consider θ = 0 in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='3), namely a non-autonomous pre-factor depending on time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Precisely, we consider the MCH2 equation with linear multiplicative noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' We will study the conditions that lead to the global existence and the blow-up in finite time of the solution, and then analyze the associated probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' 3 2 Notation and preliminaries In this section, we begin by introducing some notations and recall some elementary results, for complete- ness, we list the lemmas and skip their some proofs for conciseness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Let L2 be the usual space of square-integrable functions on R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' For any real number s ∈ R, Ds = (1 − ∂2 x)s/2 is defined by � Dsf(x) = (1 + x2) s 2 �f(x), where �f is the Fourier transform of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' The Sobolev space Hs is defined as Hs ≜ {f ∈ L2(R) : ∥f∥2 Hs = � R (1 + x2)s| �f(x)|2dx < ∞}, and the inner product (f, g)Hs := � R (1 + x2)s �f(x)�g(x)dx = ( � Dsf, � Dsg)L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' In addition, x ≲ y, x, y ∈ R means that there exists C > 0, which may vary from line to line and depend on various parameters, such that x ≤ Cy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Hereafter, C denotes a positive constant, whose value may change from one place to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Firstly, we summarize some auxiliary results, which will be used to prove our main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Define the regularizing operator Tǫ on R as Tǫf(x) := (1 − ǫ2∆)−1f(x) = � R eiξx ˆf(ξ) 1 + ǫ2|ξ|2 dξ, ǫ ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='1) Since Tǫ can be characterized by its Fourier multipliers, see [35], it is easy to see that [Ds, Tǫ] = 0, (Tǫf, g)L2 = (f, Tǫg)L2, ∥Tǫu∥Hs ≤ ∥u∥Hs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='2) Where [Ds, Tǫ] = DsTǫ − TǫDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' We therefore have the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='1 [35] Let f, g : R → R such that g ∈ W 1,∞ and f ∈ L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Then for some C > 0, ∥[Tǫ, (g · ∇)]f∥L2 ≤ C∥g∥W 1,∞∥f∥L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Furthermore, we also need to recall some useful commutator estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='2 [36] If r > 0, then Hr � L∞ is an algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Moreover, ∥uv∥Hr ≲ ∥u∥L∞∥v∥Hr+∥u∥Hr∥v∥L∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='3 [36] Let r > 0, if u ∈ Hr � W 1,∞ and v ∈ Hr−1 � L∞, then ∥[Dr, u]v∥L2 ≲ ∥∂xu∥L∞∥Dr−1v∥L2 + ∥Dru∥L2∥v∥L∞, where [Dr, u]v = Druv − uDrv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' A direct application of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='2-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='3 gives the following estimates and we omit the proof here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='4 For the F1, F2 defined in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='3) and for any u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' γ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' u1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' u2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' γ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' γ2 ∈ Hs with s > 1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' we have ∥F1(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' γ)∥Hs ≲(∥u∥W 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='∞ + ∥γ∥W 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='∞)(∥u∥Hs + ∥γ∥Hs),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' s > 3/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' ∥F2(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' γ)∥Hs ≲(∥u∥W 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='∞ + ∥γ∥W 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='∞)(∥u∥Hs + ∥γ∥Hs),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' s > 3/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' ∥F1(u1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' γ1) − F1(u2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' γ2)∥Hs ≲(∥u1∥Hs + ∥u2∥Hs + ∥γ1∥Hs + ∥γ2∥Hs) (∥u1 − u2∥Hs + ∥γ1 − γ2∥Hs),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' s > 3/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' ∥F1(u1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' γ1) − F1(u2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' γ2)∥Hs ≲(∥u1∥Hs+1 + ∥u2∥Hs+1 + ∥γ1∥Hs+1 + ∥γ2∥Hs+1) 4 (∥u1 − u2∥Hs + ∥γ1 − γ2∥Hs),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' 1/2 < s < 3/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' ∥F2(u1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' γ1) − F2(u2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' γ2)∥Hs ≲(∥u1∥Hs + ∥γ2∥Hs)(∥u1 − u2∥Hs + ∥γ1 − γ2∥Hs),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' s > 3/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' ∥F2(u1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' γ1) − F2(u2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' γ2)∥Hs ≲(∥u1∥Hs+1 + ∥γ2∥Hs+1)(∥u1 − u2∥Hs + ∥γ1 − γ2∥Hs),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' 1/2 < s < 3/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' In addition, we provide the following algebraic inequality, which will be used in the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='5 Let c, M1, M2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Assume a, b∗, b∗ > 0, either η > 1, 0 < � b∗ < b(t) < √ b∗ and 2b∗ > b∗ or η = 1, 0 < � b∗ < b(t) < √ b∗ and 2b∗ > a + b∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Then there is a constant C > 0 such that for all 0 ≤ x1 ≤ M1y1 < ∞, 0 ≤ x2 ≤ M2y2 < ∞, a(x1 + x2)(y2 1 + y2 2) + b(t)(1 + x1 + x2)η(y2 1 + y2 2) 1 + y2 1 + y2 2 −2b(t)(1 + x1 + x2)η(y2 1 + y2 2)2 (1 + y2 1 + y2 2)2 + cb(t)(1 + x1 + x2)η(y2 1 + y2 2)2 (1 + y2 1 + y2 2)2(1 + log(1 + y2 1 + y2 2)) ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Proof: Since 0 ≤ x1 M1 ≤ y1 < ∞, 0 ≤ x2 M2 ≤ y2 < ∞, we obtain a(x1 + x2)(y2 1 + y2 2) + b(t)(1 + x1 + x2)η(y2 1 + y2 2) 1 + y2 1 + y2 2 − 2b(t)(1 + x1 + x2)η(y2 1 + y2 2)2 (1 + y2 1 + y2 2)2 + cb(t)(1 + x1 + x2)η(y2 1 + y2 2)2 (1 + y2 1 + y2 2)2(1 + log(1 + y2 1 + y2 2)) ≤a(x1 + x2) + b∗(1 + x1 + x2)η − 2b∗(1 + x1 + x2)η (y2 1 + y2 2)2 (1 + y2 1 + y2 2)2 + cb∗(1 + x1 + x2)η 1 + log(1 + ( x1 M1 )2 + ( x2 M2 )2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' When η > 1 and 2b∗ > b∗ or η = 1 and 2b∗ > a + b∗, we find that the inequality will tend to −∞ for x1 → ∞ or x2 → ∞ (namely y1 → ∞ or y2 → ∞), which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' ✷ Finally, we present the following lemma to establish the Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='7 on the global existence solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='6 Let α(t) be a deterministic and locally bounded function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Suppose that λ > 0, and x(t) satisfies x(t) = e � t 0 α(t ′ )dWt′ −� t 0 λα2(t ′ )dt ′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' For R > 1 define τR = inf{t ≥ 0 : x(t) > R}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Then we have P{τR = ∞} ≥ 1 − R−2λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Proof: Noting that x(t)2λ = e � t 0 2λα(t ′ )dWt′ − 1 2 � t 0 (2λ)2α2(t ′ )dt ′ is an exponential martingale, by the martingale stopping theorem, we can derive Ex(t ∧ τR)2λ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' This yields P(τR = ∞) = lim n→∞ P(τR > n) = lim n→∞ P(x(n ∧ τR)2λ < R2λ) ≥ lim n→∞ � 1 − Ex(n ∧ τR)2λ R2λ � = 1 − R−2λ, which finishes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' ✷ 5 Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='7 Let s > 3/2, F1, F2 and Tǫ be given in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='3) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='1) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Then there is a constant K = K(s) > 0 such that for all ǫ > 0, |(Tǫ[uux], Tǫu)Hs| + |(TǫF1(u, γ), Tǫu)Hs| + |(Tǫ[uγx], Tǫγ)Hs| + |(TǫF2(u, γ), Tǫγ)Hs| ≤ K(∥u∥W 1,∞ + ∥γ∥W 1,∞)(∥u∥2 Hs + ∥γ∥2 Hs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Proof: According to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='2), we derive (Tǫ[uux], Tǫu)Hs = (DsTǫ[uux], DsTǫu)L2 = ([Ds, u]ux, DsT 2 ǫ u)L2 + ([Tǫ, u]Dsux, DsTǫu)L2 + (uDsTǫux, DsTǫu)L2, (Tǫ[uγx], Tǫγ)Hs = (DsTǫ[uγx], DsTǫγ)L2 = ([Ds, u]γx, DsT 2 ǫ γ)L2 + ([Tǫ, u]Dsγx, DsTǫγ)L2 + (uDsTǫγx, DsTǫγ)L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Then by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='1, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='2, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='2) and Sobolev embedding Theorem, we have |(Tǫ[uux], Tǫu)Hs| + |(Tǫ[uγx], Tǫγ)Hs| ≲ (∥u∥W 1,∞ + ∥γ∥W 1,∞)(∥u∥2 Hs + ∥γ∥2 Hs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' In addition, using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='4 and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='2), we obtain that |(TǫF1(u, γ), Tǫu)Hs| + |(TǫF2(u, γ), Tǫγ)Hs| ≲ (∥u∥W 1,∞ + ∥γ∥W 1,∞)(∥u∥2 Hs + ∥γ∥2 Hs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Combining the above two inequalities, we complete the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' ✷ 3 Local existence and uniqueness for SMCH2 In this section, we consider the following stochastic system with multiplicative noise (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='3) in Hs(R) × Hs(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='1 Assumptions For the main results in this paper, we rely on the following different assumptions concerning random per- turbation term in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' We assume that (h1, h2) : [0, ∞)×(Hs×Hs) ∋ (t, u, γ) → (h1(t, u, γ), h2(t, u, γ)) ∈ L2(U, Hs × Hs) are continuous in (t, u, γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Moreover, we assume Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='1 (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='1) There exists some non-decreasing function f : [0, ∞) → [0, ∞) with f(0) = 0 such that for all (u, γ) ∈ Hs × Hs, s > 1/2, 2 � i=1 ∥hi(t, u, γ)∥L2(U,Hs) ≤ f(∥u∥W 1,∞ + ∥γ∥W 1,∞)(1 + ∥u∥Hs + ∥γ∥Hs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='1) (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='2) There exists some non-decreasing function g : [0, ∞) → [0, ∞) such that for all (u1, γ1), (u2, γ2) ∈ Hs × Hs, s > 1/2, sup ∥u1∥Hs ,∥γ1∥Hs,∥u2∥Hs ,∥γ2∥Hs≤N 2 � i=1 ∥hi(t, u1, γ1) − hi(t, u2, γ2)∥L2(U,Hs) ≤ g(N) · (∥u1 − u2∥Hs + ∥γ1 − γ2∥Hs), N ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='2 h1(t, u, γ)dW1 = a(t)(1+∥u∥W 1,∞+∥γ∥W 1,∞)θudW, h2(t, u, γ)dW2 = a(t)(1+∥u∥W 1,∞+ ∥γ∥W 1,∞)θγdW for a standard 1-D Brownian motion W and θ > 0, 0 < a∗ ≤ a2(t) ≤ a∗ for all t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='3 h1(u, γ)dW1 = b(t)udW, h2(u, γ)dW2 = b(t)γdW for a standard 1-D Brownian mo- tion W, and there are constants b∗, b∗ > 0 such that 0 < b∗ ≤ b2(t) ≤ b∗ for all t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' 6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='2 Definitions of the solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Next, we give the definition of pathwise solution to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='4 (Pathwise solutions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Let S = (Ω, F, P, {Ft}t≥0, W1, W2) be a fixed stochastic basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Let s > 3/2 and z0 = (u0, γ0) be an Hs × Hs-valued F0-measurable random variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='A local pathwise solution to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='3) is a pair (z, τ), where τ ≥ 0 is a stopping time satisfying P{τ > 0} = 1 and z = (u, γ) : Ω × [0, τ) → Hs × Hs is an Ft-adapted Hs × Hs-valued process satisfying P − a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' z ∈ C([0, τ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Hs × Hs), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='2) and P − a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=', u(t) − u(0) + � t 0 [uux + F1(u, γ)]dt ′ = � t 0 h1(t ′, u, γ)dW1, γ(t) − γ(0) + � t 0 [uγx + F2(u, γ)]dt ′ = � t 0 h2(t ′, u, γ)dW2, t ∈ [0, τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Local pathwise uniqueness: if given any two local pathwise solutions (z1, τ1) and (z2, τ2) with P{z1(0) = z2(0)} = 1, we have P{z1(t) = z2(t), t ∈ [0, τ1 ∧ τ2)} = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='Additionally, (z, τ ∗) is called a maximal pathwise solution to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='3) if τ∗ > 0 almost surely and there is an increasing sequence τn → τ ∗ such that for any n ∈ N, (z, τn) is a pathwise solution to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='3) and on the set {τ ∗ < ∞}, sup t∈[0,τn] (∥u∥Hs + ∥γ∥Hs) ≥ n, n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' If (z, τ ∗) is a maximal pathwise solution and τ ∗ = ∞ almost surely, then we call that the pathwise solution exists globally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='3 Main results and remarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Now, we summarize our major contributions, such as existence of pathwise solutions, global well-posedness of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='3) and the blow-up results, and the concrete proofs will be provided later in the remainder of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='5 (Maximal solutions) Let s > 3/2, and h1(t, u, γ), h2(t, u, γ) satisfy Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' For a given stochastic basis S = (Ω, F, P, {Ft}t≥0, W1, W2), if (u0, γ0) is an Hs × Hs-valued F0-measurable random variable, then there is a local unique pathwise solution (z, τ) to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='3) in the sense of Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='4 with z ∈ C([0, τ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Hs × Hs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Moreover, (z, τ) can be extended to a unique maximal pathwise solution (z, τ ∗) and the following blow up scenario satisfies P − a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' on the set {τ∗ < ∞}, 1{lim sup t→τ∗(∥u(t)∥Hs+∥γ(t)∥Hs)=∞} = 1{lim sup t→τ∗(∥u(t)∥W 1,∞+∥γ(t)∥W 1,∞)=∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='3) Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='1 The proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='5 combines the techniques as used in the papers [35, 37, 38, 39, 40, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' By constructing the approximate sequence of the truncation problem of W 1,∞ × W 1,∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Such a cut-off means linear growth of u and γ, and guarantees the global existence of an approximate solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' 7 Turning to noise-driven regularization effects, the blow-up scenario (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='3) suggests relating the noise coefficient to the W 1,∞ × W 1,∞ of (u, γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Therefore we consider scalable noise impact, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' we assume h1(t, u, γ)dW1 = a(t)(1+∥u∥W 1,∞ +∥γ∥W 1,∞)θudW, h2(t, u, γ)dW2 = a(t)(1+∥u∥W 1,∞ +∥γ∥W 1,∞)θγdW for a standard 1-D Brownian motion W, some θ > 0, 0 < a∗ ≤ a2(t) ≤ a∗ for all t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' When a∗, a∗ and θ satisfy certain stronger conditions, the noise term remove the formation of singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='6 (Global existence for strong nonlinear noise).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Let Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='2 hold and assume that S = (Ω, F, P, {Ft}t≥0, W) is a fixed stochastic basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Let s > 5/2, (u0, γ0) ∈ Hs×Hs be an Hs×Hs-valued F0-measurable random variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Assume that θ and a∗, a∗(0 < a∗ ≤ a2(t) ≤ a∗) satisfy either 2a∗ > a∗, θ > 1/2 or 2a∗ > K + a∗, θ = 1/2, where K = K(s) is the constant introduced in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Then the corresponding maximal solution (z, τ ∗) to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='3) satisfies P{τ ∗ = ∞} = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='2 Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='6 means that blow-up of pathwise solutions might only be observed if the noise is weak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' According to Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='6, we can see that if wave breaking occurs, the noise term will not bring rapid growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Therefore, we consider θ = 0 but a non-autonomous pre-factor dependent on time t is introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' To detect such noise, we analyze the simpler form h1(t, u, γ)dW1 = b(t)udW, h2(t, u, γ)dW2 = b(t)γdW, W is a standard 1-D Brownian motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Even in this linear noise case the situation is quite interesting allowing for global existence as well as blow-up of solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' For global existence, we can identify two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='2-Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='4 and the integration by parts, we conclude that there is a C = C(s) > 1 such that − � R Dsv1Ds(v1v1x)dx − � R Dsv1DsF1(v1, v2)dx − � R Dsv2Ds(v1v2x)dx − � R Dsv2DsF2(v1, v2)dx ≤ 1 2C(∥v1∥W 1,∞ + ∥v2∥W 1,∞)(∥v1∥2 Hs + ∥v2∥2 Hs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='4) Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='7 (Global existence for weak noise I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Let s > 3/2, Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='3 be verified and S = (Ω, F, P, {Ft}t≥0, W) be a fixed stochastic basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Assume (u0, γ0) is an Hs×Hs-valued F0-measurable ran- dom variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Let Q = Q(s) > 0 be the constant such that the embedding ∥u∥W 1,∞ < Q∥u∥Hs, ∥γ∥W 1,∞ < Q∥γ∥Hs holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Let C = C(s) > 1 be in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' If there is a R > 1 and λ1 > 1 satisfying P − a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' ∥u0∥2 Hs + ∥γ0∥2 Hs < b2 ∗ 4C2Q2λ2 1R, then (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='3) has a maximal solution (z, τ ∗) satisfying for any 0 < λ2 < λ1−1 λ1 the estimate P � ∥u(t)∥2 Hs + ∥γ(t)∥2 Hs < b2 ∗ C2Q2λ2 1 for all t > 0 � ≥ 1 − R−2λ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='3 Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='7 presents a global existence solution with bounded initial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' This result can not be observed in the deterministic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='8 (Global existence for weak noise II) Let s > 5/2, Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='3 be verified and S = (Ω, F, P, {Ft}t≥0, W) be a fixed stochastic basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' (u0, γ0) is an Hs × Hs-valued F0-measurable random variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' If P{(1 − ∂2 x)u0(x) > 0, ∀x ∈ R} = p, P{(1 − ∂2 x)u0(x) < 0, ∀x ∈ R} = q, 8 and there exists some x0 ∈ R such that P{(1 − ∂2 x)u0(x) ≤ 0, x ≤ x0 and (1 − ∂2 x)u0(x) ≥ 0, x ≥ x0} = m, for some p, q, m ∈ [0, 1], then the corresponding maximal solution (z, τ ∗) to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='3) satisfies P{τ ∗ = ∞} ≥ p + q + m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='4 The proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='8 depends on the analysis of a PDE with random coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' When b(t) = 0 and taking (p, q, m) = (1, 0, 0), (p, q, m) = (0, 1, 0) or (p, q, m) = (0, 0, 1) in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='8, we obtain the global existence for the deterministic MCH2 system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Therefore, in this sense, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='8 covers the deterministic result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='9 (Wave breaking criterion for weak noise I) Let S = (Ω, F, P, {Ft}t≥0, W) be a fixed stochastic basis and s > 5/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Let Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='3 be verified and (u0, γ0) be an Hs × Hs-valued F0- measurable random variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' If for some c ∈ (0, 1) and x0 ∈ R, u0x(x0) < −1 2 � (b∗)2 c2 + 4(∥u0∥2 H1 + ∥γ0∥2 H1) − b∗ 2c P − a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=', (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='5) then the maximal solution (z, τ ∗) to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='3) satisfies P{τ ∗ < ∞} ≥ P � e � t 0 b(t ′ )dWt′ + � t 0 b∗−b2(t ′ ) 2 dt ′ ≥ c for all t � > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='5 Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='9 detects the solution singularities in finite time under certain initial data, while Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='7 provides a global existence result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' We stress that these two results do not contain each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' In Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='7, assuming ∥u0∥2 Hs + ∥γ0∥2 Hs < b2 ∗ 2C2Q2λ2 1R, then z globally exists with probability greater than 1 − R−2λ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' In Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='9, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='5) implies that ∥u0∥2 Hs > 1 Q2 ∥u0∥2 W 1,∞ > (b∗)2 c2Q2 > b2 ∗ 2C2Q2λ2 1R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='10 (Wave breaking criterion for weak noise II) Let S = (Ω, F, P, {Ft}t≥0, W) be a fixed stochastic basis and s > 5/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Let Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='3 be verified and (u0, γ0) be an Hs × Hs-valued F0- measurable random variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' If for some c ∈ (0, 1), � R u3 0x(x)dx < − � (b∗)2 4c2 (∥u0∥2 H1 + ∥γ0∥2 H1)2 + 15 8 (∥u0∥2 H1 + ∥γ0∥2 H1)3 − b∗ 2c(∥u0∥2 H1 + ∥γ0∥2 H1) P − a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=', (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='6) then the maximal solution (z, τ ∗) to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='3) satisfies P{τ ∗ < ∞} ≥ P � e � t 0 b(t ′ )dWt′ +� t 0 b∗−b2(t′ ) 2 dt ′ ≥ c for all t � > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='6 Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='10 detects the solution singularities in finite time under certain initial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='6) implies that � R u3 0x(x)dx < − b∗ c (∥u0∥2 H1 + ∥γ0∥2 H1), which combined with minx∈Ru0x(x)(∥u0∥2 H1 + ∥γ0∥2 H1) ≤ � R u3 0x(x)dx derives ∥u0∥2 Hs > 1 Q2 ∥u0∥2 W 1,∞ > (b∗)2 c2Q2 > b2 ∗ 2C2Q2λ2 1R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' So, the initial value condi- tions here and those given in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='7 do not contain each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' 9 4 Sketch of the Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='5 We consider the initial value problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' The proof of existence and uniqueness of pathwise solutions can be carried out by standard procedures used in many works, see [35, 37, 39, 40, 42, 43] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Therefore we only give a sketch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' (Approximation scheme) The first step is to construct a suitable approximation scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' For any R > 1, we let χR(x) : [0, ∞) → [0, 1] be a C∞ 0 function such that χR(x) = 1 for x ∈ [0, R] and χR(x) = 0 for x > 2R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Then we consider the following cut-off problem on R, \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 du + χR(∥u∥W 1,∞ + ∥γ∥W 1,∞)[uux + F1(u, γ)]dt = χR(∥u∥W 1,∞ + ∥γ∥W 1,∞)h1(t, u, γ)dW1, t > 0, dγ + χR(∥u∥W 1,∞ + ∥γ∥W 1,∞)[uγx + F2(u, γ)]dt = χR(∥u∥W 1,∞ + ∥γ∥W 1,∞)h2(t, u, γ)dW2, t > 0, u(ω, 0, x) = u0(ω, x), γ(ω, 0, x) = γ0(ω, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='1) From (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='4), we observe that the nonlinear term F1(u, γ), F2(u, γ), preserves the Hs × Hs-regularity of (u, γ) for any s > 3/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' However, in order to apply the stochastic differential equation (SDE) theory in Hilbert space to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='1), we will mollify the transport term uux, uγx since the products uux and uγx lose one regularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' For this reason, we consider the following approximation scheme: \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 du + G1,ǫ(u, γ)dt = χR(∥u∥W 1,∞ + ∥γ∥W 1,∞)h1(t, u, γ)dW1, t > 0, x ∈ R, dγ + G2,ǫ(u, γ)dt = χR(∥u∥W 1,∞ + ∥γ∥W 1,∞)h2(t, u, γ)dW2, t > 0, x ∈ R, G1,ǫ(u, γ) = χR(∥u∥W 1,∞ + ∥γ∥W 1,∞)[Jǫ((Jǫu)(Jǫu)x) + F1(u, γ)], G2,ǫ(u, γ) = χR(∥u∥W 1,∞ + ∥γ∥W 1,∞)[Jǫ((Jǫu)(Jǫγ)x) + F2(u, γ)], u(0, x) = u0(x) ∈ Hs, γ(0, x) = γ0(x) ∈ Hs, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='2) where Jǫ is the Friedrichs mollifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' According to the theory of SDE in Hilbert space (see for example [28, 44]), for a fixed stochastic basis S = (Ω, F, P, {Ft}t≥0, W1, W2) and for (u0, γ0) ∈ Hs × Hs with s > 5/2, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='2) admits a unique solution (uǫ, γǫ) ∈ C([0, Tǫ), Hs × Hs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' In addition, the uniform L∞(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' W 1,∞ × W 1,∞) condition provided by the cut-off function χR enables us to split the expectation E(∥uǫ∥2 Hs∥uǫ∥W 1,∞|F0), E(∥uǫ∥2 Hs∥γǫ∥W 1,∞|F0) to close a priori L2(Ω, Hs × Hs, P(·|F0)) estimate for uǫ, γǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Then we can go along the lines as we prove Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='1 to find that for each fixed ǫ, if Tǫ < ∞, then lim supt→Tǫ(∥uǫ∥W 1,∞ + ∥γǫ∥W 1,∞) = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Due to the cut-off in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='2) for P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' ω ∈ Ω, ∥uǫ∥W 1,∞, ∥γǫ∥W 1,∞ are always bounded and hence (uǫ, γǫ) is actually a global in time solution, that is (uǫ, γǫ) ∈ C([0, ∞), Hs × Hs) P − a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' (Pathwise solution to the cut-off problem in Hs × Hs with s > 5/2) By applying the stochastic com- pactness arguments from Prokhorov’s and Skorokhod’s Theorem, we obtain the almost sure convergence for a new approximation solution (( ˜uǫ, ˜γǫ), ˜ W1ǫ, ˜ W2ǫ) defined on a new probability space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' By virtue of a refined martingale representation Theorem [45, Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='1], we may set ǫ → 0 in (( ˜uǫ, ˜γǫ), ˜ W1ǫ, ˜ W2ǫ) to obtain a martingale solution in Hs × Hs with s > 5/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Here, the Gy¨ongy-Krylov characterization [46] of the convergence in probability can be used here to prove the convergence of the original approximation solutions, and one can refer to [35, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='7] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Finally, since F1(u, γ), F2(u, γ) satisfy the estimates as in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='4 and h1(t, u, γ), h2(t, u, γ) satisfies Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='1, we conclude that G1,ǫ, G2,ǫ, χR(∥u∥W 1,∞ + ∥γ∥W 1,∞)h1, χR(∥u∥W 1,∞ + ∥γ∥W 1,∞)h2 are Lipschitz continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' So, one can obtain the pathwise uniqueness easily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Then by the Yamada-Watanabe principle, we derive the existence and uniqueness of the pathwise solution to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='2) denoted by zR = (uR, γR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' (Remove the cut-off and extend the range of s to s > 3/2) Let τR := R ∧ inf{t ≥ 0 : ∥uR(t)∥W 1,∞ + ∥γR(t)∥W 1,∞ > R}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' By the pathwise uniqueness to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='2), we conclude that zR(t) = z ¯R(t), t ∈ [0, τ R ∧ τ ¯ R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' In particular, τR is increasing in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Let τ ∗ = limR→∞ τR and define z = ∞ � R=1 1[τR−1,τR)zR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Then (z, τ ∗) is the unique pathwise solution to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='3) for s > 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' 10 Next, we extend the range of s to s > 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' When z0 ∈ L∞(Ω, Hs × Hs) with s > 3/2, by mollifying the initial data, we obtain a sequence of regular solutions {zn, ζn}n∈N to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Motivated by [42], one can prove that there is some stopping time τ with P(τ > 0) = 1, a subsequence (still denoted by zn) and some process z such that P − a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' lim n→∞ sup t∈[0,s] (∥zn − z∥Hs) = 0 , s < τ and sup t∈[0,s] ∥z∥Hs×Hs ≤ ∥z0∥Hs×Hs + 2, s < τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='3) Then we can let n → ∞ to prove that (z, τ) is a solution to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Besides, a cutting argument as in [37, 39, 42] enables us to remove the L∞(Ω, Hs × Hs) assumption on (u0, γ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' More precisely, consider the decomposition Ωm = {m − 1 ≤ ∥u0∥Hs + ∥γ0∥Hs < m}, m ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' We conclude �∞ m=1 P(Ωm) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Therefore we have P − a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' z0(ω, x) = � m≥1 zm 0 (ω, x) := � m≥1 z0(ω, x)1Ωm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' For each initial value zm 0 , we let (zm, ζm) be the pathwise unique solution to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='3) satisfying (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Moreover, as Ωm ∩ Ωm′ = ∅, m ̸= m ′, F1(0, 0) = 0, F2(0, 0) = 0 and h1(t, 0, 0) = 0, h2(t, 0, 0) = 0 (see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='1)), it follows that z := � m≥1 zm1Ωm, ζ = � m≥1 ζm1Ωm is the unique pathwise solution to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='3) with corresponding initial condition z0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Since (zm, ζm) satisfies (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='3), we have P − a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' sup t∈[0,s] (∥u∥2 Hs + ∥γ∥2 Hs) = ∞ � m=1 1Ωm sup t∈[0,s] (∥um∥2 Hs + ∥γm∥2 Hs) ≤ C ∞ � m=1 1Ωm(4 + ∥um 0 ∥2 Hs + ∥γm 0 ∥2 Hs) = C(4 + ∥u0∥2 Hs + ∥γ0∥2 Hs), s < ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' So, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='2) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Since the passage from (z, ζ) to a unique maximal pathwise solution (z, τ ∗) in the sense of Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='4 can be carried out as in [37, 42, 47], we omit the details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' To finish the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='5, we only need to prove the blow-up scenario (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Motivated by [43, 48], we consider the relationship between the explosion time of ∥z(t)∥Hs×Hs and the explosion time of ∥z(t)∥W 1,∞×W 1,∞ in the next lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='1 (Blow-up scenario 1) Let (z, τ ∗) be the unique maximal solution to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Then the real- valued stochastic processes ∥z(t)∥W 1,∞×W 1,∞, ∥z(t)∥Hs×Hs are also Ft-adapted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Besides, for any m, n ∈ Z+, define τ1,m = inf{t ≥ 0 : ∥u(t)∥Hs + ∥γ(t)∥Hs ≥ m}, τ2,n = inf{t ≥ 0 : ∥u(t)∥W 1,∞ + ∥γ(t)∥W 1,∞ ≥ n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' For τ1 := τ ∗ = limm→∞ τ1,m and τ2 = limn→∞ τ2,n, we have τ1 = τ2 P − a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' 11 Consequently, 1{limt→τ∗ ∥u(t)∥W 1,∞+∥γ(t)∥W 1,∞=∞} = 1{τ ∗<∞} P − a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='. Proof: Since z ∈ C([0, τ ∗);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Hs × Hs) almost surely, by the continuous embedding Hs × Hs ֒→ W 1,∞ × W 1,∞ for s > 3/2, we conclude that ∥z(t)∥W 1,∞×W 1,∞ is Ft-adapted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Moreover, the embedding Hs × Hs ֒→ W 1,∞ × W 1,∞ for s > 3/2 means that P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' τ1 ≤ τ2 P − a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Now we only need to prove P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' τ2 ≤ τ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' We cannot directly apply the Itˆo formula for ∥u(t)∥2 Hs + ∥γ(t)∥2 Hs since we only have u, γ ∈ Hs and uux, uγx ∈ Hs−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Therefore, the Itˆo formula in Hilbert space cannot be applied directly, see ([27], Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='32) or ([49], Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Instead, we will use the mollifier operator Tǫ defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='2) to overcome this difficulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' We apply Tǫ to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='3), and then use the Itˆo formula for ∥Tǫu∥2 Hs, ∥Tǫγ∥2 Hs to derive d∥Tǫu(t)∥2 Hs = 2(Tǫh1(u, γ)dW1, Tǫu)Hs − 2(DsTǫ[uux], DsTǫu)L2dt − 2(DsTǫF1(u, γ), DsTǫu)L2dt + ∥Tǫh1(u, γ)∥2 L2(U,Hs)dt, d∥Tǫγ(t)∥2 Hs = 2(Tǫh2(u, γ)dW2, Tǫγ)Hs − 2(DsTǫ[uγx], DsTǫγ)L2dt − 2(DsTǫF2(u, γ), DsTǫγ)L2dt + ∥Tǫh2(u, γ)∥2 L2(U,Hs)dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Therefore for any n1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' m ≥ 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' r ≥ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' and t ∈ [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' τ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='n1 ∧ r ∧ τ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='m],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' ∥Tǫu(t)∥2 Hs + ∥Tǫγ(t)∥2 Hs − ∥Tǫu(0)∥2 Hs − ∥Tǫγ(0)∥2 Hs =2 ∞ � j=1 � t 0 (DsTǫh1(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' γ)ej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' DsTǫu)L2dW 1 j + 2 ∞ � i=1 � t 0 (DsTǫh2(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' γ)ei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' DsTǫγ)L2dW 2 i − 2 � t 0 (DsTǫ[uux],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' DsTǫu)L2dt ′ − 2 � t 0 (DsTǫF1(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' γ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' DsTǫu)L2dt ′ + � t 0 ∥Tǫh1(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' γ)∥2 L2(U,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='Hs)dt ′ − 2 � t 0 (DsTǫ[uγx],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' DsTǫγ)L2dt ′ − 2 � t 0 (DsTǫF2(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' γ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' DsTǫγ)L2dt ′ + � t 0 ∥Tǫh2(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' γ)∥2 L2(U,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='Hs)dt ′ =: � t 0 ∞ � j=1 L1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='jdW 1 j + � t 0 ∞ � i=1 L2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='idW 2 i + 8 � j=3 � t 0 Ljdt ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' where {ek} is the complete orthonormal basis of U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' On account of the Burkholder-Davis-Gundy inequality and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' we obtain that E � sup t∈[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='τ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='n1∧r∧τ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='m] ������ � t 0 ∞ � j=1 L1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='jdW 1 j ������ ����F0 � ≤ CE �� ∞ � j=1 � τ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='n1∧r∧τ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='m 0 |L1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='j|2dt � 1 2 ����F0 � ≤ 1 2E � sup t∈[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='τ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='n1∧r∧∧τ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='m] ∥Tǫu∥2 Hs ����F0 � + Cf 2(n1) � r 0 E � sup t′ ∈[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='τ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='n1∧t∧τ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='m] (1 + ∥u(t ′)∥2 Hs + ∥γ(t ′)∥2 Hs) ����F0 � dt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' E � sup t∈[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='τ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='n1∧r∧τ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='m] ����� � t 0 ∞ � i=1 L2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='idW 2 i ����� ����F0 � ≤ CE �� ∞ � i=1 � τ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='n1∧r∧τ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='m 0 |L1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='i|2dt � 1 2 ����F0 � ≤ 1 2E � sup t∈[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='τ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='n1∧r∧τ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='m] ∥Tǫγ∥2 Hs ����F0 � + Cf 2(n1) � r 0 E � sup t′ ∈[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='τ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='n1∧t∧τ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='m] (1 + ∥u(t ′)∥2 Hs + ∥γ(t ′)∥2 Hs) ����F0 � dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' For L3, L6, using integration by part, Sobolev’s inequality and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='3, we have (DsTǫ[uux], DsTǫu)L2 = ([Ds, u]ux, DsT 2 ǫ u)L2 + ([Tǫ, u]Dsux, DsTǫu)L2 + (uDsTǫux, DsTǫu)L2 ≤ C∥u∥W 1,∞∥u∥2 Hs, (DsTǫ[uγx], DsTǫγ)L2 = ([Ds, u]γx, DsT 2 ǫ γ)L2 + ([Tǫ, u]Dsγx, DsTǫγ)L2 + (uDsTǫγx, DsTǫγ)L2 12 ≤ C(∥u∥W 1,∞ + ∥γ∥W 1,∞)(∥u∥2 Hs + ∥γ∥2 Hs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' For L4, L7, we derive from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='4 that (DsTǫF1(u, γ), DsTǫu)L2 ≤ C(∥u∥W 1,∞ + ∥γ∥W 1,∞)(∥u∥2 Hs + ∥γ∥2 Hs), (DsTǫF2(u, γ), DsTǫγ)L2 ≤ C(∥u∥W 1,∞ + ∥γ∥W 1,∞)(∥u∥2 Hs + ∥γ∥2 Hs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' For L5, L8, it follows from the Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='1 that E � � τ2,n1∧r∧τ1,m 0 ∥Tǫh1(u, γ)∥2 L2(U,Hs)dt ′����F0 � ≤ Cf 2(n1) � r 0 E � sup t′∈[0,τ2,n1∧t∧τ1,m] (1 + ∥u(t ′)∥2 Hs + ∥γ(t ′)∥2 Hs) ����F0 � dt, E � � τ2,n1∧r∧τ1,m 0 ∥Tǫh2(u, γ)∥2 L2(U,Hs)dt ′����F0 � ≤ Cf 2(n1) � r 0 E � sup t′∈[0,τ2,n1∧t∧τ1,m] (1 + ∥u(t ′)∥2 Hs + ∥γ(t ′)∥2 Hs) ����F0 � dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Therefore combining the above estimates, we have E � sup t∈[0,τ2,n1∧r∧τ1,m] (∥Tǫu(t)∥2 Hs + ∥Tǫγ(t)∥2 Hs) ����F0 � ≤ C[∥u(0)∥2 Hs + ∥γ(0)∥2 Hs] + C � r 0 E � 1 + sup t′ ∈[0,τ2,n1∧t∧τ1,m] (∥u(t ′)∥2 Hs + ∥γ(t ′)∥2 Hs) ����F0 � dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Since the right hand side of the above estimate does not depend on ǫ, and (Tǫu, Tǫγ) tends to (u, γ) in C([0, τ1,m ∧ r], Hs × Hs) almost surely as ǫ → 0, by Fatou’s Lemma, one can send ǫ → 0 to obtain E � sup t∈[0,τ2,n1∧r∧τ1,m] (∥u(t)∥2 Hs + ∥γ(t)∥2 Hs) ����F0 � ≤ C[∥u(0)∥2 Hs + ∥γ(0)∥2 Hs] + C � r 0 E � 1 + sup t′ ∈[0,τ2,n1∧t∧τ1,m] (∥u(t ′)∥2 Hs + ∥γ(t ′)∥2 Hs)|F0 � dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Then Gronwall’s inequality shows that for each n1 ∈ Z+, r ∈ R+, there is a constant C = C(n1, r, u0, γ0) > 0 such that E � sup t∈[0,τ2,n1∧r∧τ1,m] (∥u(t)∥2 Hs + ∥γ(t)∥2 Hs) ����F0 � < C(n1, r, u0, γ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' So, it follows from Chebyshev’s inequality and Fatou’s lemma that P(τ1 ≤ τ2,n1 ∧ r|F0) ≤ lim m→∞ P(τ1,m ≤ τ2,n1 ∧ r|F0) ≤ lim m→∞ P � sup t∈[0,τ2,n1∧r∧τ1,m] (∥u(t)∥Hs + ∥γ(t)∥Hs) ≥ m ����F0 � ≤ lim m→∞ E[supt∈[0,τ2,n1∧r∧τ1,m](2∥u(t)∥2 Hs + 2∥γ(t)∥2 Hs)|F0] m2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Letting r → ∞ first and then n1 → ∞, Fatou’s lemma yields that P(τ1 ≤ τ2|F0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' 13 Therefore, we conclude P(τ1 ≥ τ2) ≥ 1 − P(τ1 ≤ τ2) = 1 − P[P(τ1 ≤ τ2|F0)] = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' We finish the section with the proof of the first blow-up scenario (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' ✷ 5 Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='6 Assume s > 5/2 and let (u0, γ0) be Hs × Hs-valued F0-measurable random variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Let h1(t, u, γ) = a(t)(1 + ∥u∥W 1,∞ + ∥γ∥W 1,∞)θu, h2(t, u, γ) = a(t)(1 + ∥u∥W 1,∞ + ∥γ∥W 1,∞)θγ with θ ≥ 1/2 and a(t) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' For s > 3/2, the embedding Hs × Hs ֒→ W 1,∞ × W 1,∞ implies that sup ∥u1∥Hs,∥γ1∥Hs,∥u2∥Hs,∥γ2∥Hs≤N 2 � i=1 (∥hi(t, u1, γ1) − hi(t, u2, γ2)∥Hs ≤ g(N)(∥u1 − u2∥Hs + ∥γ1 − γ2∥Hs), N ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Hence, by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='5, we conclude that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='3) admits a unique pathwise solution z = (u, γ) in Hs × Hs with s > 5/2 and maximal existence time τ ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Define τm = inf{t ≥ 0 : ∥u∥2 Hs + ∥γ∥2 Hs ≥ m}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Applying the Itˆo formula to ∥Tǫu∥2 Hs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' ∥Tǫγ∥2 Hs gives d∥Tǫu∥2 Hs =2a(t)(1 + ∥u∥W 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='∞ + ∥γ∥W 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='∞)θ(Tǫu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Tǫu)HsdW − 2(Tǫ[uux],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Tǫu)Hsdt − 2(TǫF1(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' γ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Tǫu)Hsdt + a2(t)(1 + ∥u∥W 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='∞ + ∥γ∥W 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='∞)2θ(Tǫu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Tǫu)Hsdt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' d∥Tǫγ∥2 Hs =2a(t)(1 + ∥u∥W 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='∞ + ∥γ∥W 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='∞)θ(Tǫγ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Tǫγ)HsdW − 2(Tǫ[uγx],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Tǫγ)Hsdt − 2(TǫF2(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' γ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Tǫγ)Hsdt + a2(t)(1 + ∥u∥W 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='∞ + ∥γ∥W 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='∞)2θ(Tǫγ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Tǫγ)Hsdt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Again,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' using Itˆo’s formula to log(1 + ∥Tǫu∥2 Hs + ∥Tǫγ∥2 Hs) yields d log(1 + ∥Tǫu∥2 Hs + ∥Tǫγ∥2 Hs) = 2a(t)(1 + ∥u∥W 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='∞ + ∥γ∥W 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='∞)θ(Tǫu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Tǫu)Hs 1 + ∥Tǫu∥2 Hs + ∥Tǫγ∥2 Hs dW − 2(Tǫ[uux],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Tǫu)Hs 1 + ∥Tǫu∥2 Hs + ∥Tǫγ∥2 Hs dt − 2(TǫF1(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' γ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Tǫu)Hs 1 + ∥Tǫu∥2 Hs + ∥Tǫγ∥2 Hs dt + a2(t)(1 + ∥u∥W 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='∞ + ∥γ∥W 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='∞)2θ(Tǫu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Tǫu)Hs 1 + ∥Tǫu∥2 Hs + ∥Tǫγ∥2 Hs dt + 2a(t)(1 + ∥u∥W 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='∞ + ∥γ∥W 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='∞)θ(Tǫγ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Tǫγ)Hs 1 + ∥Tǫu∥2 Hs + ∥Tǫγ∥2 Hs dW − 2(Tǫ[uγx],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Tǫγ)Hs 1 + ∥Tǫu∥2 Hs + ∥Tǫγ∥2 Hs dt − 2(TǫF2(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' γ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Tǫγ)Hs 1 + ∥Tǫu∥2 Hs + ∥Tǫγ∥2 Hs dt + a2(t)(1 + ∥u∥W 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='∞ + ∥γ∥W 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='∞)2θ(Tǫγ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Tǫγ)Hs 1 + ∥Tǫu∥2 Hs + ∥Tǫγ∥2 Hs dt − 2a2(t)(1 + ∥u∥W 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='∞ + ∥γ∥W 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='∞)2θ × [(Tǫu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Tǫu)2 Hs + (Tǫγ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Tǫγ)2 Hs + 2(Tǫu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Tǫu)Hs(Tǫγ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Tǫγ)Hs] (1 + ∥Tǫu∥2 Hs + ∥Tǫγ∥2 Hs)2 dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='1-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='4 and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' it follows that E[log(1 + ∥Tǫu(t ∧ τm)∥2 Hs + ∥Tǫγ(t ∧ τm)∥2 Hs)|F0] − log(1 + ∥Tǫu0∥2 Hs + ∥Tǫγ0∥2 Hs) 14 = − 2E � � t∧τm 0 (Tǫ[uux],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Tǫu)Hs 1 + ∥Tǫu∥2 Hs + ∥Tǫγ∥2 Hs dt ′����F0 � − 2E � � t∧τm 0 (TǫF1(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' γ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Tǫu)Hs 1 + ∥Tǫu∥2 Hs + ∥Tǫγ∥2 Hs dt ′����F0 � + E � � t∧τm 0 a2(t ′)(1 + ∥u∥W 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='∞ + ∥γ∥W 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='∞)2θ(Tǫu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Tǫu)Hs 1 + ∥Tǫu∥2 Hs + ∥Tǫγ∥2 Hs dt ′����F0 � − 2E � � t∧τm 0 (Tǫ[uγx],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Tǫγ)Hs 1 + ∥Tǫu∥2 Hs + ∥Tǫγ∥2 Hs dt ′����F0 � − 2E � � t∧τm 0 (TǫF2(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' γ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Tǫγ)Hs 1 + ∥Tǫu∥2 Hs + ∥Tǫγ∥2 Hs dt ′����F0 � + E � � t∧τm 0 a2(t ′)(1 + ∥u∥W 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='∞ + ∥γ∥W 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='∞)2θ(Tǫγ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Tǫγ)Hs 1 + ∥Tǫu∥2 Hs + ∥Tǫγ∥2 Hs dt ′����F0 � − 2E � � t∧τm 0 a2(t ′)(1 + ∥u∥W 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='∞ + ∥γ∥W 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='∞)2θ × [(Tǫu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Tǫu)2 Hs + (Tǫγ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Tǫγ)2 Hs + +2(Tǫu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Tǫu)Hs(Tǫγ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Tǫγ)Hs] (1 + ∥Tǫu∥2 Hs + ∥Tǫγ∥2 Hs)2 dt ′����F0 � ≤E � � t∧τm 0 K(∥u∥W 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='∞ + ∥γ∥W 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='∞)(∥u∥2 Hs + ∥γ∥2 Hs) 1 + ∥Tǫu∥2 Hs + ∥Tǫγ∥2 Hs dt ′����F0 � + E � � t∧τm 0 a2(t ′)(1 + ∥u∥W 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='∞ + ∥γ∥W 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='∞)2θ(∥Tǫu∥2 Hs + ∥Tǫγ∥2 Hs) 1 + ∥Tǫu∥2 Hs + ∥Tǫγ∥2 Hs dt ′����F0 � − 2E � � t∧τm 0 a2(t ′)(1 + ∥u∥W 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='∞ + ∥γ∥W 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='∞)2θ (∥Tǫu∥4 Hs + ∥Tǫγ∥4 Hs + 2∥Tǫu∥2 Hs∥Tǫγ∥2 Hs) (1 + ∥Tǫu∥2 Hs + ∥Tǫγ∥2 Hs)2 dt ′����F0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Let Iǫ 1(t′) = K(∥u∥W 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='∞ + ∥γ∥W 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='∞)(∥u∥2 Hs + ∥γ∥2 Hs) 1 + ∥Tǫu∥2 Hs + ∥Tǫγ∥2 Hs − K(∥u∥W 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='∞ + ∥γ∥W 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='∞)(∥u∥2 Hs + ∥γ∥2 Hs) 1 + ∥u∥2 Hs + ∥γ∥2 Hs ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Iǫ 2(t′) = a2(t ′)(1 + ∥u∥W 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='∞ + ∥γ∥W 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='∞)2θ(∥Tǫu∥2 Hs + ∥Tǫγ∥2 Hs) 1 + ∥Tǫu∥2 Hs + ∥Tǫγ∥2 Hs − a2(t ′)(1 + ∥u∥W 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='∞ + ∥γ∥W 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='∞)2θ(∥u∥2 Hs + ∥γ∥2 Hs) 1 + ∥u∥2 Hs + ∥γ∥2 Hs ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Iǫ 3(t′) = a2(t ′)(1 + ∥u∥W 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='∞ + ∥γ∥W 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='∞)2θ(∥Tǫu∥4 Hs + ∥Tǫγ∥4 Hs + 2∥Tǫu∥2 Hs∥Tǫγ∥2 Hs) (1 + ∥Tǫu∥2 Hs + ∥Tǫγ∥2 Hs)2 − a2(t ′)(1 + ∥u∥W 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='∞ + ∥γ∥W 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='∞)2θ(∥u∥4 Hs + ∥γ∥4 Hs + 2∥u∥2 Hs∥γ∥2 Hs) (1 + ∥u∥2 Hs + ∥γ∥2 Hs)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='1) Notice that for any T > 0, (Tǫu, Tǫγ) tends to (u, γ) in C([0, τm ∧ t], Hs × Hs) almost surely as ǫ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' It follows from the dominated convergence theorem that lim ǫ→0 E �� t∧τm 0 [|Iǫ 1(t′)| + |Iǫ 2(t′)| + |Iǫ 3(t′)|]dt′ ����F0 � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Then, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='2) and the dominated convergence Theorem,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' it holds E[log(1 + ∥u(t ∧ τm)∥2 Hs + ∥γ(t ∧ τm)∥2 Hs)|F0] − log(1 + ∥u0∥2 Hs + ∥γ0∥2 Hs) ≤E � � t∧τm 0 K(∥u∥W 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='∞ + ∥γ∥W 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='∞)(∥u∥2 Hs + ∥γ∥2 Hs) 1 + ∥u∥2 Hs + ∥γ∥2 Hs dt ′����F0 � + E � � t∧τm 0 a2(t ′)(1 + ∥u∥W 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='∞ + ∥γ∥W 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='∞)2θ(∥u∥2 Hs + ∥γ∥2 Hs) 1 + ∥u∥2 Hs + ∥γ∥2 Hs dt ′����F0 � 15 − 2E � � t∧τm 0 a2(t ′)(1 + ∥u∥W 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='∞ + ∥γ∥W 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='∞)2θ(∥u∥4 Hs + ∥γ∥4 Hs + 2∥u∥2 Hs∥γ∥2 Hs) (1 + ∥u∥2 Hs + ∥γ∥2 Hs)2 dt ′����F0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='5 with x1 = ∥u∥W 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='∞,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' x2 = ∥γ∥W 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='∞,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' y1 = ∥u∥Hs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' y2 = ∥γ∥Hs immediately shows that there are constants C1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' C2 > 0 such that E[log(1 + ∥u(t ∧ τm)∥2 Hs + ∥γ(t ∧ τm)∥2 Hs)|F0] − log(1 + ∥u0∥2 Hs + ∥γ0∥2 Hs) ≤ E � � t∧τm 0 C1 − C2 a2(t ′)(1 + ∥u∥W 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='∞ + ∥γ∥W 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='∞)2θ(∥u∥2 Hs + ∥γ∥2 Hs)2 (1 + ∥u∥2 Hs + ∥γ∥2 Hs)2(1 + log(1 + ∥u∥2 Hs + ∥γ∥2 Hs))dt ′����F0 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' which means that for some C(u0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' γ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' C1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' C2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' t) > 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' E � � t∧τm 0 a2(t ′)(1 + ∥u∥W 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='∞ + ∥γ∥W 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='∞)2θ(∥u∥2 Hs + ∥γ∥2 Hs)2 (1 + ∥u∥2 Hs + ∥γ∥2 Hs)2(1 + log(1 + ∥u∥2 Hs + ∥γ∥2 Hs))dt ′����F0 � ≤ C(u0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' γ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' C1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' C2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='2) Therefore, for any T > 0, by using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='5 and the Burkholder-Davis-Gundy inequality,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' we find that E � sup t∈[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='T ∧τm] log(1 + ∥Tǫu∥2 Hs + ∥Tǫγ∥2 Hs) ����F0 � − log(1 + ∥Tǫu0∥2 Hs + ∥Tǫγ0∥2 Hs) ≤C � E � � T ∧τm 0 a2(t)(1 + ∥u∥W 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='∞ + ∥γ∥W 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='∞)2θ(∥Tǫu∥2 Hs + ∥Tǫγ∥2 Hs)2 (1 + ∥Tǫu∥2 Hs + ∥Tǫγ∥2 Hs)2 dt ����F0 �� 1 2 + E � � T ∧τm 0 ����C1 − C2 a2(t)(1 + ∥u∥W 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='∞ + ∥γ∥W 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='∞)2θ(∥u∥2 Hs + ∥γ∥2 Hs)2 (1 + ∥u∥2 Hs + ∥γ∥2 Hs)2(1 + log(1 + ∥u∥2 Hs + ∥γ∥2 Hs)) ���� dt ����F0 � + E �� T ∧τm 0 [|Iǫ 1(t)| + |Iǫ 2(t)| + |Iǫ 3(t)|]dt ����F0 � ≤1 2E � sup t∈[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='T ∧τm] (1 + log(1 + ∥u∥2 Hs + ∥γ∥2 Hs)) ����F0 � + CE � � T ∧τm 0 a2(t)(1 + ∥u∥W 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='∞ + ∥γ∥W 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='∞)2θ(∥Tǫu∥2 Hs + ∥Tǫγ∥2 Hs)2 (1 + ∥Tǫu∥2 Hs + ∥Tǫγ∥2 Hs)2(1 + log(1 + ∥u∥2 Hs + ∥γ∥2 Hs))dt ����F0 � + C1T + C2E � � T ∧τm 0 a2(t)(1 + ∥u∥W 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='∞ + ∥γ∥W 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='∞)2θ(∥u∥2 Hs + ∥γ∥2 Hs)2 (1 + ∥u∥2 Hs + ∥γ∥2 Hs)2(1 + log(1 + ∥u∥2 Hs + ∥γ∥2 Hs))dt ����F0 � + E �� T ∧τm 0 [|Iǫ 1(t)| + |Iǫ 2(t)| + |Iǫ 3(t)|]dt ����F0 � ≤1 2E � sup t∈[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='T ∧τm] (1 + log(1 + ∥u∥2 Hs + ∥γ∥2 Hs)) ����F0 � + E �� T ∧τm 0 [|Iǫ 1(t)| + |Iǫ 2(t)| + |Iǫ 3(t)|]dt ����F0 � + CE � � T ∧τm 0 a2(t)(1 + ∥u∥W 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='∞ + ∥γ∥W 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='∞)2θ(∥Tǫu∥2 Hs + ∥Tǫγ∥2 Hs)2 (1 + ∥Tǫu∥2 Hs + ∥Tǫγ∥2 Hs)2(1 + log(1 + ∥u∥2 Hs + ∥γ∥2 Hs))dt ����F0 � + C(u0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' γ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' C1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' C2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' T ) + C1T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Thus, we use the dominated convergence Theorem, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='2) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='1) to obtain E � sup t∈[0,T ∧τm] log(1 + ∥u∥2 Hs + ∥γ∥2 Hs) ����F0 � ≤ C(u0, γ0, C1, C2, T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Since log(1 + x) is increasing for x > 0, we have that for any m ≥ 1, P{τm < T |F0} ≤ P � sup t∈[0,T ∧τm] log(1 + ∥u∥2 Hs + ∥γ∥2 Hs) ≥ log(1 + m) ����F0 � ≤ C(u0, γ0, C1, C2, T ) log(1 + m) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Letting m → ∞ forces P{τ ∗ < T |F0} = 0 for any T > 0, which means P{τ ∗ = ∞} = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' 16 6 Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='7 In this section, we study (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='3) with linear noise satisfying Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Depending on the strength of the noise in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='3), we provide the global existence of pathwise solutions for the maximal pathwise solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Motivated by [35, 37, 47], we introduce β(ω, t) = e � t 0 b(t ′ )dWt′ − � t 0 b2(t′ ) 2 dt ′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='1 Let s > 3/2 and h1(t, u, γ) = b(t)u, h2(t, u, γ) = b(t)γ such that b(t) satisfies As- sumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Let (u0, γ0) be an Hs × Hs-valued F0-measurable random variable and (z, τ ∗) be the corresponding unique maximal solution to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Let v1 = β−1u, v2 = β−1γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Then for t ∈ [0, τ ∗), the processes v1, v2 solve the following problem \uf8f1 \uf8f2 \uf8f3 ∂tv1 + βv1v1x + β(1 − ∂2 x)−1∂x(v2 1 + 1 2v2 1x + 1 2v2 2 − 1 2v2 2x) = 0, ∂tv2 + βv1v2x + β(1 − ∂2 x)−1((v1xv2x)x + v1xv2) = 0, v1(ω, 0, x) = u0(ω, x), v2(ω, 0, x) = γ0(ω, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='1) Moreover, we have P − a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' (v1, v2) ∈ C([0, τ ∗);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Hs × Hs) ∩ C1([0, τ ∗);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Hs−1 × Hs−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' In addition, if s > 5/2, then it holds P{∥v1∥H1 + ∥v2∥H1 = ∥u0∥H1 + ∥γ0∥H1 for all t < τ∗} = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='2) Proof: Since b(t) satisfies Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='3, h1(t, u, γ) = b(t)u, h2(t, u, γ) = b(t)γ satisfy Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='1, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='5 implies that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='3) has a unique maximal solution (z, τ ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' A direct computation with the Itˆo formula yields dβ−1 = −b(t)β−1dW + b2(t)β−1dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Therefore we have dv1 = β−1du + udβ−1 + dβ−1du = β−1 � −uux − (1 − ∂2 x)−1∂x � u2 + 1 2u2 x + 1 2γ2 − 1 2γ2 x �� dt + b(t)β−1udW +u[−b(t)β−1dW + b2(t)β−1dt] − b2(t)β−1udt = −βv1v1x − β(1 − ∂2 x)−1∂x � v2 1 + 1 2v2 1x + 1 2v2 2 − 1 2v2 2x � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' dv2 = β−1dγ + γdβ−1 + dβ−1dγ = β−1[−uγx − (1 − ∂2 x)−1((uxγx)x + uxγ)]dt + b(t)β−1γdW +γ[−b(t)β−1dW + b2(t)β−1dt] − b2(t)β−1γdt = −βv1v2x − β(1 − ∂2 x)−1((v1xv2x)x + v1xv2) and since v1(ω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' x) = u0(ω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' v2(ω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' x) = γ0(ω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' we see that (v1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' v2) satisfies (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Moreover, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='5 implies (u, γ) ∈ C([0, τ ∗), Hs × Hs) P − a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=', so is (v1, v2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' From Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='4 and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='1), we see that for P − a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=', v1t = −βv1v1x − β(1 − ∂2 x)−1∂x(v2 1 + 1 2v2 1x + 1 2v2 2 − 1 2v2 2x), v2t = −βv1v2x − β(1 − ∂2 x)−1((v1xv2x)x + v1xv2), (v1t, v2t) ∈ C([0, τ ∗), Hs−1 × Hs−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Hence, it holds P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' (v1, v2) ∈ C1([0, τ ∗), Hs−1 × Hs−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' In addition, the first two equations of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='1) are equivalent to v1t − v1xxt + 3βv1v1x − 2βv1xv1xx − βv1v1xxx + β(v2 − v2xx)v2x = 0, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='3) v2t − v2xxt + β(v1v2x + v1xv2) − β(v1v2xxx + v1xv2xx) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='4) 17 Multiplying both sides of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='3) by v1 and multiplying both sides of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='4) by v2, then integrating the equation on x ∈ R, and finally adding the two derived equations, we arrive at P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' d dt � R (v2 1 + v2 2 + v2 1x + v2 2x)dx = 0, t < τ ∗, which implies (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' ✷ Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' To begin with, we apply the operator Ds to (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='3) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='4) , multiply both sides of the resulting equation by Dsv1, Dsv2 respectively, and then integrate on R to obtain P − a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' 1 2 d dt(∥v1∥2 Hs + ∥v2∥2 Hs) = −β(ω, t) � R Dsv1Ds(v1v1x)dx − β(ω, t) � R Dsv1DsF1(v1, v2)dx −β(ω, t) � R Dsv2Ds(v1v2x)dx − β(ω, t) � R Dsv2DsF2(v1, v2)dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='4), we conclude that P − a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' d dt(∥v1∥2 Hs + ∥v2∥2 Hs) ≤ Cβ(ω, t)(∥v1∥W 1,∞ + ∥v2∥W 1,∞)(∥v1∥2 Hs + ∥v2∥2 Hs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='5) Letting w1 = e− � t 0 b(t ′ )dWt′ u = e− � t 0 b2(t′ ) 2 dt ′ v1, w2 = e− � t 0 b(t ′ )dWt′ γ = e− � t 0 b2(t′ ) 2 dt ′ v2 and α(ω, t) = e � t 0 b(t ′ )dWt′ , we obtain d dt(∥w1∥2 Hs + ∥w2∥2 Hs) + b2(t)(∥w1∥2 Hs + ∥w2∥2 Hs) ≤ Cα(ω, t)(∥w1∥W 1,∞ + ∥w2∥W 1,∞)(∥w1∥2 Hs + ∥w2∥2 Hs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Assume ∥u0∥2 Hs + ∥γ0∥2 Hs < b2 ∗ 2C2Q2λ2 1R < b2 ∗ C2Q2λ2 1 and define τ1 = inf � t < τ∗ : α(ω, t)(∥w1∥W 1,∞ + ∥w2∥W 1,∞) = (∥u∥W 1,∞ + ∥γ∥W 1,∞) > b(t)2 Cλ1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='6) Then it follows from the embedding (∥u0∥W 1,∞ + ∥γ0∥W 1,∞) ≤ Q(∥u0∥Hs + ∥γ0∥Hs) that P{τ1 > 0} = 1, and it holds d dt(∥w1∥2 Hs + ∥w2∥2 Hs) + (λ1 − 1)b2(t) λ1 (∥w1∥2 Hs + ∥w2∥2 Hs) ≤ 0, t ∈ [0, τ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' This implies that for any 0 < λ2 < λ1−1 λ1 , P − a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' ∥u(t)∥2 Hs + ∥γ(t)∥2 Hs ≤ (∥u0∥2 Hs + ∥γ0∥2 Hs)e � t 0 b(t ′ )dWt′ − � t 0 (λ1−1)b2(t ′ ) λ1 dt ′ = (∥u0∥2 Hs + ∥γ0∥2 Hs)e � t 0 b(t ′ )dWt′ −λ2 � t 0 b2(t ′ )dt ′ e− � t 0 (λ1−1)−λ1λ2 λ1 b2(t ′ )dt ′ , t ∈ [0, τ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='7) Define τ2 = inf{t > 0 : e � t 0 b(t ′ )dWt′ −λ2 � t 0 b2(t ′ )dt ′ > R}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Notice that P{τ2 > 0} = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' From (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='7), we have 2∥u(t)∥2 Hs + 2∥γ(t)∥2 Hs ≤ 2b2 ∗ 2C2Q2λ2 1R × R × e− � t 0 (λ1−1)−λ1λ2 λ1 b2(t ′ )dt ′ 18 = b2 ∗ C2Q2λ2 1 e− � t 0 (λ1−1)−λ1λ2 λ1 b2(t ′)dt ′ , t ∈ [0, τ1 ∧ τ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='8) By Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='3, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='8) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='6), we find that on [0, τ1 ∧ τ2), P − a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' (∥u∥W 1,∞ + ∥γ∥W 1,∞) ≤ Q(∥u∥Hs + ∥γ∥Hs) ≤ b∗ Cλ1 e− � t 0 (λ1−1)−λ1λ2 2λ1 b2(t ′)dt ′ ≤ b2(t) Cλ1 e− � t 0 (λ1−1)−λ1λ2 2λ1 b2(t ′)dt ′ , which together with λ2 < λ1−1 λ1 and b2 > 0 derives P{τ1 ≥ τ2} = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Therefore it follows from (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='8) and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='6 that P{∥u(t)∥2 Hs + ∥γ(t)∥2 Hs ≤ b2 ∗ 2C2Q2λ2 1 for all t > 0} ≥ P{τ2 = ∞} ≥ 1 − R−2λ2, which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' 7 Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='8 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='1 Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' By proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='1, we can proceed to prove Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Since Hs ֒→ C2 for s > 5/2, we have v1, v1x, v2, v2x ∈ C1([0, τ ∗) × R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Then for x ∈ R and P − a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' ω ∈ Ω, the problem � dq(ω,t,x) dt = β(ω, t)v1(ω, t, q(ω, t, x)), t ∈ [0, τ ∗), q(ω, 0, x) = x, x ∈ R (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='1) has a unique solution q(ω, t, x) such that q(ω, t, x) ∈ C1([0, τ ∗) × R) for P − a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' ω ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Moreover, differentiating (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='1) with respect to x yields that for P − a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' ω ∈ Ω, � dqx(ω,t,x) dt = β(ω, t)v1x(ω, t, q(ω, t, x))qx, t ∈ [0, τ ∗), qx(ω, 0, x) = 1, x ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' For P − a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' ω ∈ Ω, we solve the above equation to obtain qx(ω, t, x) = exp � � t 0 β(ω, t ′)v1x(ω, t ′, q(ω, t ′, x))dt ′� , t ∈ (0, τ ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Thus for P − a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' ω ∈ Ω, qx(ω, t, x) > 0, (t, x) ∈ [0, τ ∗) × R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Then the momentum variable V1 = (1 − ∂2 x)v1, V2 = (1 − ∂2 x)v2 satisfy P − a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' V1t + βv1V1x + 2βv1xV1 + βv2xV2 = 0, V2t + +β(v1V2)x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='2) Applying particle trajectory method (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='1) and the first equation of (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='2), we obtain d dt � e � t 0 β(ω,s)V2v2x(ω,s,x) V1(ω,s,x) dsV1(ω, t, q(ω, t, x))q2 x(ω, t, x) � =e � t 0 β(ω,s)V2v2x(ω,s,x) V1(ω,s,x) dsβ(ω, s)q2 xV2v2x(ω, s, x) + e � t 0 β(ω,s)V2v2x(ω,s,x) V1(ω,s,x) dsq2 x[V1t + βv1V1x + 2βv1xV1] =e � t 0 β(ω,s)V2v2x(ω,s,x) V1(ω,s,x) dsq2 x[V1t + βv1V1x + 2βv1xV1 + βv2xV2] 19 =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' This and qx(ω, 0, x) = 1 imply that e � t 0 β(ω,s)V2v2x(ω,s,x) V1(ω,s,x) dsV1(ω, t, q(ω, t, x))q2 x(ω, t, x) = V1(ω, 0, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='3) Consequently, we have sign(V1(ω, t, x))=sign(V1(ω, 0, x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' The next step, we give the following useful lemma that will be used in the sequel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='1 (Blow-up scenario 2) Let s > 3/2 and (u0, γ0) be an Hs ×Hs-valued F0-measurable random variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Assume that (z, τ ∗) is the corresponding maximal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Then z as a W 1,∞ × W 1,∞-valued process is Ft-adapted for t < τ∗ and P − a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' on the set {τ ∗ < ∞} 1{lim sup t→τ∗(∥u(t)∥Hs+∥γ(t)∥Hs)=∞} = 1{lim sup t→τ∗ ∥u(t)∥W 1,∞=∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='4) Proof: It is clear that {lim sup t→τ ∗ ∥u(t)∥W 1,∞ = ∞} ⊂ {lim sup t→τ ∗(∥u(t)∥Hs + ∥γ(t)∥Hs) = ∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' It is sufficient to prove {lim sup t→τ ∗ ∥u(t)∥W 1,∞ = ∞}C ⊂ {lim sup t→τ ∗(∥u(t)∥Hs + ∥γ(t)∥Hs) = ∞}C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Notice that {lim sup t→τ ∗ ∥u(ω, t)∥W 1,∞ = ∞}C = {∃M(ω) > 0, s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' ∥u(ω, t)∥W 1,∞ ≤ M(ω), ∀t < τ ∗}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='5) By the equation (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='4) and using the identity ∂2 xG ∗ f = ∂2 x(1 − ∂2 x)−1f = (1 − ∂2 x)−1f − f, we have ���� dv2x(ω, t, q(ω, t, x)) dt ���� = |v2tx(t, q) + v2xx(t, q)βv1| = | − βv1xv2x − β∂2 x(1 − ∂2 x)−1(v1xv2x) − β∂x(1 − ∂2 x)−1(v1xv2)| = |β(1 − ∂2 x)−1(v1xv2x) − β∂x(1 − ∂2 x)−1(v1xv2)| ≤ β∥G∥L∞∥v1xv2x∥L1 + β∥∂xG∥L∞∥v1xv2∥L1 ≤ Cβ(2∥v1x∥L2 + ∥v2x∥L2 + ∥v2∥L2) ≤ Cβ(∥v1(0)∥H1 + ∥v2(0)∥H1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='6) For m ≥ 1, define τm = inf{t < τ ∗ : ∥u(t)∥Hs + ∥γ(t)∥Hs ≥ m}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' By (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='6), Sobolev’s embedding and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='2), we have ∥v2(ω, t, q(ω, t, ·))∥W 1,∞ ≤ C � t 0 β(ω, t ′)dt ′(∥u0∥H1 + ∥γ0∥H1) + ∥γ0∥W 1,∞, t ≤ τm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='7) In addition, we derive from (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='5) that d dt(∥v1∥2 Hs + ∥v2∥2 Hs) ≤ C(∥u∥W 1,∞ + β(ω, t)∥v2∥W 1,∞)(∥v1∥2 Hs + ∥v2∥2 Hs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' By means of Gronwall’s inequality and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='5), for any ω ∈ {lim sup t→τ ∗ ∥u(ω, t)∥W 1,∞ = ∞}C, we obtain ∥v1(T ∧ τm)∥2 Hs + ∥v2(T ∧ τm)∥2 Hs ≤ (∥u0∥2 Hs + ∥γ0∥2 Hs) exp �� T ∧τm 0 C(∥u∥W 1,∞ + β(ω, t)∥v2∥W 1,∞)dt � , ≤ (∥u0∥2 Hs + ∥γ0∥2 Hs) × exp � C � M(T ∧ τm) + � T ∧τm 0 β(ω, t) � � t 0 β(ω, t ′)dt ′(∥u0∥H1 + ∥γ0∥H1) + ∥γ0∥W 1,∞ � dt �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' 20 This implies on the set {τ ∗ < ∞} ∩ {lim sup t→τ ∗ ∥u(ω, t)∥W 1,∞ = ∞}C, ∥u(T ∧ τm)∥2 Hs + ∥γ(T ∧ τm)∥2 Hs ≤(∥u0∥2 Hs + ∥γ0∥2 Hs)β(ω, T ∧ τm) × exp � C � Mτm + � T ∧τm 0 � β(ω, t) � t 0 β(ω, t ′)dt ′(∥u0∥H1 + ∥γ0∥H1) + ∥γ0∥W 1,∞ � dt �� <∞, where we used supt>0 β(ω, t) < ∞ due to supt>0 Eβ(ω, t) = 1 and Doob’s L1-inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Hence we can see that on the set {τ ∗ < ∞}, {lim sup t→τ ∗ ∥u(t)∥W 1,∞ = ∞}C ⊂ {lim sup t→τ ∗(∥u(t)∥Hs + ∥γ(t)∥Hs) = ∞}C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' So, we finish the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' ✷ Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='2 (Blow-up scenario 3) Let s > 3/2 and z0 be an Hs × Hs-valued F0-measurable random variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Assume that (z, τ ∗) is the corresponding maximal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Then z as a W 1,∞ × W 1,∞-valued process is Ft-adapted for t < τ∗ and P − a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' on the set {τ ∗ < ∞}, 1{lim sup t→τ∗(∥u(t)∥Hs +∥γ(t)∥Hs)=∞} = 1{lim inft→τ∗ minx∈R{ux(ω,t,x)}=−∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='8) Proof: It is clear that {lim inft→τ ∗ minx∈R{ux(ω, t, x)} = −∞} ⊂ {lim sup t→τ ∗(∥u(t)∥Hs +∥γ(t)∥Hs) = ∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' The rest of proof is similar to that of Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='1 by replacing equation (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='5) with {lim inf t→τ ∗ min x∈R{ux(ω, t, x)} = −∞}C = {∃M(ω) > 0, s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' ux(ω, t, x) > −M(ω), ∀t < τ ∗}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='9) Without loss of generality, we only need to show that this Lemma holds for s = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Multiplying the first equation in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='2) by V1 = (1 − ∂2 x)v1 and integrating by parts, we get d dt � R V 2 1 dx = −2β(w, t) � R v1V1V1xdx − 4β(w, t) � R V 2 1 v1xdx − 2β(w, t) � R V1V2v2xdx = −3β(w, t) � R V 2 1 v1xdx − 2β(w, t) � R V1V2v2xdx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='10) Multiplying the second equation in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='2) by V2 = (1 − ∂2 x)v2 and integrating by parts, we obtain d dt � R V 2 2 dx = −β(w, t) � R v1xV 2 2 dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='11) Thus, in view of (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='9), (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='10), (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='11) and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='7),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' for any ω ∈ {lim inft→τ ∗ minx∈R{ux(ω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' x)} = −∞}C,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' we obtain d dt � R (V 2 1 + V 2 2 )dx = −3β(w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' t) � R V 2 1 v1xdx − β(w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' t) � R v1xV 2 2 dx − 2β(w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' t) � R V1V2v2xdx ≤ 3M � R (V 2 1 + V 2 2 )dx + β(ω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' t) � � t 0 β(ω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' t ′)dt ′(∥u0∥H1 + ∥γ0∥H1) + ∥γ0∥W 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='∞ � � R (V 2 1 + V 2 2 )dx By means of Gronwall’s inequality,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' we arrive at ∥v1(T ∧ τm)∥H2 + ∥v2(T ∧ τm)∥H2 = ∥V1(T ∧ τm)∥L2 + ∥V2(T ∧ τm)∥L2 21 ≤ (∥V1(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' ·)∥L2 + ∥V2(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' ·)∥L2) × exp � � T ∧τm 0 � 3M + β(w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' t) � � t 0 β(ω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' t ′)dt ′(∥u0∥H1 + ∥γ0∥H1) + ∥γ0∥W 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='∞ �� dt � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Then on the set {τ ∗ < ∞} ∩ {lim inft→τ ∗ minx∈R{ux(ω, t, x)} = −∞}C, ∥u(T ∧ τm)∥H2 + ∥γ(T ∧ τm)∥H2 ≤ β(w, T ∧ τm)(∥V1(0, ·)∥L2 + ∥V2(0, ·)∥L2) × exp � � T ∧τm 0 � 3M + β(w, t) � � t 0 β(ω, t ′)dt ′(∥u0∥H1 + ∥γ0∥H1) + ∥γ0∥W 1,∞ �� dt � < ∞, where we used supt>0 β(ω, t) < ∞ due to supt>0 Eβ(ω, t) = 1 and Doob’s L1-inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Hence we can see that {lim inft→τ ∗ minx∈R{ux(ω, t, x)} = −∞}C ⊂ {lim sup t→τ ∗(∥u(t)∥Hs + ∥γ(t)∥Hs) = ∞}C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' ✷ Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='3 Let V1 = v1 − v1xx and P{V1(ω, 0, x) > 0, ∀x ∈ R} = p, P{V1(ω, 0, x) < 0, ∀x ∈ R} = q, P � V1(ω, 0, x) ≤ 0, x ≤ x0 and V1(ω, 0, x) ≥ 0, x ≥ x0 � = m for some p, q, m ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Then the maximal solution (z, τ ∗) of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='3) satisfies P � ∥ux(ω, t)∥L∞ ≤ √ 2 2 β(ω, t)(∥u0∥H1 + ∥γ0∥H1) or ux(ω, t) ≥ − √ 2 2 β(ω, t, x)(∥u0∥H1 + ∥γ0∥H1), ∀t ∈ [0, τ ∗) � ≥ p + q + m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='12) Proof: Denote Ap = {V1(ω, 0, x) > 0, ∀x ∈ R}, Aq = {V1(ω, 0, x) < 0, ∀x ∈ R}, Am = {V1(ω, 0, x) ≤ 0, x ≤ x0 and V1(ω, 0, x) ≥ 0, x ≥ x0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Using G(x) = e−|x| 2 , one can derive that for P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' ω ∈ Ω, and for all (t, x) ∈ [0, τ ∗) × R, v1(ω, t, x) = 1 2e−x � x −∞ eξV1(ω, t, ξ)dξ + 1 2ex � ∞ x e−ξV1(ω, t, ξ)dξ, v1x(ω, t, x) = −1 2e−x � x −∞ eξV1(ω, t, ξ)dξ + 1 2ex � ∞ x e−ξV1(ω, t, ξ)dξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Therefore, v1(ω, t, x) + v1x(ω, t, x) = ex � ∞ x e−ξV1(ω, t, ξ)dξ, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='13) v1(ω, t, x) − v1x(ω, t, x) = e−x � x −∞ eξV1(ω, t, ξ)dξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='14) Then one can employ (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='13), (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='14) and sign(V1)=sign(V1(ω, 0, x)), (t, x) ∈ [0, τ∗) × R to obtain that for all (t, x) ∈ [0, τ ∗) × R, � −v1(ω, t, x) ≤ v1x(ω, t, x) ≤ v1(ω, t, x), ω ∈ Ap, v1(ω, t, x) ≤ v1x(ω, t, x) ≤ −v1(ω, t, x), ω ∈ Aq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='15) 22 In addition, since q(ω, t, ·) is an increasing diffeomorphism of R with qx(ω, t, x) > 0 for all (t, x) ∈ [0, τ ∗) × R, by (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='3), it follows that for any ω ∈ Am, \uf8f1 \uf8f2 \uf8f3 V1(ω, t, x) ≤ 0 if x ≤ q(ω, t, x0), V1(ω, t, x) ≥ 0 if x ≥ q(ω, t, x0), V1(ω, t, q(ω, t, x0)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='16) Therefore, for any ω ∈ Am, when x ≤ q(ω, t, x0), by (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='14) and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='16), we have v1(ω, t, x) ≤ v1x(ω, t, x);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' when x ≥ q(ω, t, x0), by (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='13) and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='16), we have v1x(ω, t, x) ≥ −v1(ω, t, x), Therefore, it follows from (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='2) that for any ω ∈ Am, −v1x(ω, t, x) ≤ |v1(ω, t, x)| ≤ ∥v1(ω, t, x)∥L∞ ≤ √ 2 2 (∥u0∥H1 + ∥γ0∥H1), ∀(t, x) ∈ [0, τ ∗) × R (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='17) Then for any ω ∈ Am, ux(ω, t) ≥ − √ 2 2 β(ω, t, x)(∥u0∥H1 + ∥γ0∥H1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' This together with Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='2 and supt>0 β(ω, t, x) < ∞ implies that z globally exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' For any ω ∈ Ap ∪ Aq, it follows from (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='15) that |v1x(ω, t, x)| ≤ |v1(ω, t, x)|, in view of Sobolev inequality and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='2), we arrive at ∥v1x(ω, t, x)∥L∞ ≤ ∥v1(ω, t, x)∥L∞ ≤ √ 2 2 (∥u0∥H1 + ∥γ0∥H1), ∀(t, x) ∈ [0, τ ∗) × R, ω ∈ Ap ∪ Aq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='18) Combining Ap ∩ Aq ∩ Am = ∅, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='17) and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='18), we derive (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' ✷ Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Note that supt>0 Eβ(ω, t, x) = 1 and Doob’s L1-inequality implies that supt>0 β(ω, t, x) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Then we can infer from (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='4), (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='8) and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='12) that P{τ ∗ = ∞} ≥ p + q + m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='2 Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='9 The proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='9 relies on certain properties of the solution v1, v2 to the equations (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='3) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' We first prove the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='4 Let s > 5/2 and b(t) satisfy Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Assume (u0, γ0) is an Hs × Hs-valued F0- measurable random variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Let K = √ 2 2 (∥u0∥2 H1 + ∥γ0∥2 H1) 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Then for v1, v2 defined by (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='3), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='4) and any x0 ∈ R, g(ω, t) := v1x(ω, t, q(ω, t, x0)) satisfies P − a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' d dtg(t) ≤ βK2 − β 2 g2(t), t < τ ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='19) Moreover, if there exists some x0 ∈ R such that P−a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' g(0) < − √ 2K, then P−a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='. g(t) is non-increasing on [0, τ ∗) and g(t) < − √ 2K, t ∈ [0, τ ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='20) Proof: For any v1, v2 ∈ H1, by the representation of G ∗ f = (1 − ∂2 x)−1f, we have G ∗ � v2 1 + 1 2v2 1x � (x) = 1 2 � x −∞ e−x+y � v2 1 + 1 2v2 1x � (y)dy + 1 2 � ∞ x ex−y � v2 1 + 1 2v2 1x � (y)dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='21) The following inequality � x −∞ ey � v2 1 + v2 1x � (y)dy ≥ 2 � x −∞ eyv1v1x(y)dy = exv2 1(x) − � x −∞ eyv2 1dy 23 implies that 1 2 � x −∞ e−x+y � v2 1 + 1 2v2 1x � (y)dy ≥ 1 4v2 1(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='22) Similarly, we get the estimate of the second term in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='21) as 1 2 � ∞ x ex−y � v2 1 + 1 2v2 1x � (y)dy ≥ 1 4v2 1(x), (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='23) Combining (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='21), (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='22) and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='23), we deduce G ∗ (v2 1 + 1 2v2 1x)(x) ≥ 1 2v2 1(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' In addition, ∥G ∗ v2 2x∥L∞ ≤ ∥G∥L∞∥v2 2x∥L1 = 1 2∥v2 2x∥L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='24) Differentiating the first equation of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='1) with respect to x, and using (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='2) and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='24), we have d dtv1x(ω, t, q(t, ω, x)) = v1xt + v1xxβ(ω, t, x)v1(ω, t, q(ω, t, x)) = −βv2 1x − β∂2 x(1 − ∂2 x)−1 � v2 1 + 1 2v2 1x + 1 2v2 2 − 1 2v2 2x � ≤ −1 2βv2 1x + 1 2βv2 1 + 1 4βv2 2 + 3 4βG ∗ (v2 2x) ≤ −1 2βv2 1x + β 2 (∥u0∥2 H1 + ∥γ0∥2 H1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' In view of the assumptions of Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='4, we have P − a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' d dtg(t) ≤ −β 2 g2(t) + βK2, t < τ ∗, which is (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' In order to prove (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='20), define ζ(w) := inf � t ∈ [0, τ ∗) : g(w, t) > − √ 2K � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' If g(0) < − √ 2K, then P{ζ > 0} = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' From the definition of ζ(ω), we find that ζ(ω) ≤ τ ∗, for P − a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' w ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' From (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='19), we have that g(ω, t) is nonincreasing for t ∈ [0, ζ(ω)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Hence by the continuity of the path of g(ω, t), we obtain that g(ω, t) ≤ g(0) < − √ 2K, t ∈ [0, ζ(ω)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' In view of the time continuity of g(ω, t) again, we find that P{ζ = τ ∗} = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Hence (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='20) is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' ✷ Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' From Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='4 and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='5), we rewrite (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='19) as d dtg(t) ≤ −β(t) 2 � 1 − 2K2 g2(0) � g2(t) − � g2(t) g2(0) − 1 � β(t)K2 ≤ −β(t) 2 � 1 − 2K2 g2(0) � g2(t), t ∈ [0, τ ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Integrating on both sides leads to P − a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' 1 g(t) − 1 g(0) ≥ � 1 − 2K2 g2(0) � � t 0 β(t ′) 2 dt ′, t ∈ [0, τ ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Assuming Ω ′ = {ω : β(t, ω) ≥ ce− b∗ 2 t for all t}, g(t) ≤ − √ 2K means that P − a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' ω ∈ Ω ′ − 1 g(0) ≥ �1 2 − K2 g2(0) � � τ ∗ 0 β(t ′)dt ′ ≥ �1 2 − K2 g2(0) ��2c b∗ − 2c b∗ e− b∗ 2 τ ∗� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' 24 If g(0) < − 1 2 � (b∗)2 c2 + 8K2 − b∗ 2c, we obtain on Ω′ �1 2 − K2 g2(0) �2c b∗ e− b∗ 2 τ ∗ ≥ 2c b∗ �1 2 − K2 g2(0) � + 1 g(0) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Therefore we have τ ∗ < ∞ P − a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' on Ω ′, which implies that P{τ ∗ < ∞} ≥ P{β(t) ≥ ce− b∗ 2 t for all t} = P � e � t 0 b(t ′ )dWt′ + � t 0 b∗−b2(t ′ ) 2 dt ′ ≥ c for all t � > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' We finish the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='3 Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='10 The proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='10 is similar to that of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' We first prove the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='5 Let s > 5/2 and b(t) satisfy Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Assume (u0, γ0) is an Hs × Hs-valued F0- measurable random variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Let K = √ 2 2 (∥u0∥2 H1 + ∥γ0∥2 H1) 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Then for v1, v2 defined by (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='3), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='4), N(ω, t) := � R v3 1x(ω, t, q(ω, t, x))dx satisfies P − a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' d dtN(t) ≤ 15β 4 K4 − β 4K2 N 2(t), t < τ ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='25) Moreover, if P − a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' N(0) < − √ 15K3, then P − a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='. N(t) is non-increasing on [0, τ ∗) and N(t) < − √ 15K3, t ∈ [0, τ ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='26) Proof: Differentiating the first equation of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='1) with respect to x, and using the ∂2 x(1 − ∂2 x)−1f = ∂2 xG ∗ f = G ∗ f − f, we have v1xt + β 2 v2 1x + βv1v1xx + βG ∗ � v2 1 + 1 2v2 1x + 1 2v2 2 − 1 2v2 2x � − β � v2 1 + 1 2v2 2 − 1 2v2 2x � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='27) Let N(t) := � R v3 1x(ω, t, x)dx, t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Multiplying (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='27) with v2 1x and integrating by parts subsequently, by G ∗ (v2 1 + 1 2v2 1x)(x) ≥ 1 2v2 1(x), we get 1 3 dN(t) dt = − β 6 � R v4 1xdx − β � R v2 1xG ∗ (v2 1 + 1 2v2 1x + 1 2v2 2 − 1 2v2 2x)dx + β � R v2 1x(v2 1 + 1 2v2 2 − 1 2v2 2x)dx ≤ − β 6 � R v4 1xdx + β 2 � R v2 1v2 1xdx + β 2 � R v2 1xG ∗ v2 2xdx + β 2 � R v2 1xv2 2dx ≤ − β 6 � R v4 1xdx + β 2 � R v2 1x(v2 1 + v2 2)dx + β 4 ∥v2 2x∥L1 � R v2 1xdx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' In view of Sobolev’s embedding and the invariant property of ∥v1(t)∥2 H1 + ∥v2(t)∥2 H1 = ∥u0∥2 H1 + ∥γ0∥2 H1, we find that 3 2 � R v2 1x(v2 1 + v2 2)dx + 3 4∥v2 2x∥L1 � R v2 1xdx ≤ 3 4(∥u0∥2 H1 + ∥γ0∥2 H1)2 + 3 16(∥u0∥2 H1 + ∥γ0∥2 H1)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' 25 On the other hand, the Cauchy-Schwarz inequality implies that ���� � R v3 1xdx ���� ≤ � � R v4 1xdx � 1 2 � � R v2 1xdx � 1 2 , hence, � R v4 1xdx ≥ 1 ∥u0∥2 H1 + ∥γ0∥2 H1 � � R v3 1xdx �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' As defined in Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='5, K = √ 2 2 (∥u0∥2 H1 + ∥γ0∥2 H1) 1 2 , we have the similar Riccati type equation dN(t) dt ≤ − β 4K2 N 2(t) + 15β 4 K4, which is (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' In order to prove (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='26), define stopping time χ(w) := inf � t ∈ [0, τ ∗) : N(w, t) > − √ 15K3 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' If N(0) < − √ 15K3, then P{χ(ω) > 0} = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' From the definition of χ(ω), we find that w ∈ Ω, χ(ω) ≤ τ ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' From (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='25), we conclude that N(ω, t) is nonincreasing for t ∈ [0, χ(ω)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Hence by the continuity of the path of N(ω, t), we obtain that N(ω, t) ≤ N(0) < − √ 15K3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' In view of the time continuity of N(ω, t) again, we find that P{χ = τ ∗} = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Therefore, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='26) is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' ✷ Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' From Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='5 and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='6), we rewrite (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='25) as d dtN(t) ≤ − β(t) 4K2 � 1 − 15K6 N 2(0) � N 2(t) − � N 2(t) N 2(0) − 1 � 15β(t) 4 K4 ≤ − β(t) 4K2 � 1 − 15K6 N 2(0) � N 2(t), t ∈ [0, τ ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Integrating on both sides leads to P − a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' 1 N(t) − 1 N(0) ≥ � 1 − 15K6 N 2(0) � � t 0 β(t ′) 4K2 dt ′, t ∈ [0, τ ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Assuming Ω ′ = {ω : β(t, ω) ≥ ce− b∗ 2 t for all t}, also due to N(t) < − √ 15K3, we get P − a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' ω ∈ Ω ′ − 1 N(0) ≥ � 1 4K2 − 15K4 4N 2(0) � � τ ∗ 0 β(t ′)dt ′ ≥ � 1 4K2 − 15K4 4N 2(0) ��2c b∗ − 2c b∗ e− b∗ 2 τ ∗� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' If N(0) < − � (b∗)2K4 c2 + 15K6 − b∗K2 c , we obtain on Ω′ � 1 4K2 − 15K4 4N 2(0) �2c b∗ e− b∗ 2 τ ∗ ≥ 2c b∗ � 1 4K2 − 15K4 4N 2(0) � + 1 N(0) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Therefore we obtain τ∗ < ∞ P − a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' on Ω ′, which means that P{τ ∗ < ∞} ≥ P{β(t) ≥ ce− b∗ 2 t for all t} = P � e � t 0 b(t ′ )dWt′ +� t 0 b∗−b2(t′ ) 2 dt ′ ≥ c for all t � > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' So, the proof is finished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' 26 8 Acknowledgments This paper is supported by Fundamental Research Funds for the Central Universities (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' 22D110913).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' References [1] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Olver, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Rosenau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' E 53 (1996) 1900-1906.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' [2] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Constantin, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Ivanov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' On an integrable two-component Camassa-Holm shallow water system, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' A 372 (2008) 7129-7132.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' [3] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Chen, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Liu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' A two-component generalization of the Camassa-Holm equation and its solutions, Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' 75 (2006) 1-15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' [4] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Falqui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' On a Camassa-Holm type equation with two dependent variables, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' A 39 (2006) 327-342.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' [5] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Escher, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Lechtenfeld, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Yin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Well-posedness and blow-up phenomena for the 2-component Camassa-Holm equation, Discrete Contin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Dyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' 19 (2007) 493-513.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' [6] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Gui, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Liu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' On the global existence and wave-breaking criteria for the two-component Camassa- Holm system, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Funct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' 258 (2010) 4251-4278.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' [7] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Guo, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Zhou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' On solutions to a two-component generalized Camassa-Holm equation, Stud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' 124 (2010) 307-322.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' [8] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Guo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Blow-up and global solutions to a new integrable model with two components, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' 372 (2010) 316-327.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' [9] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Camassa, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Holm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' An integrable shallow water equation with peaked solitons, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' 71 (1993) 1661-1664.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' [10] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Fuchssteiner, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Fokas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Symplectic structures, their Bcklund transformations and hereditary symmetries, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' 4 (1)(1981/1982) 4766.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' [11] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Constantin, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' McKean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' A shallow water equation on the circle, Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Pure Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' 52 (1999) 949-982.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' [12] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Constantin, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Gerdjikov, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Ivanov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Inverse scattering transform for the Camassa-Holm equation, Inverse Problems 22(2006) 2197-2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' [13] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Lenells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Conservation laws of the Camassa-Holm equation, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' A 38 (2005) 869-880.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' [14] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Constantin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Escher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Well-posedness, global existence and blow-up phenomenon for a periodic quasi-linear hyperbolic equation, Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Pure Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' 51 (1998) 475-504.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' [15] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Li, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Olver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Well-posedness and blow-up solutions for an integrable nonlinear dispersive model wave equation, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Differ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' 162 (2000) 27-63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' [16] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Constantin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Escher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Wave breaking for nonlinear nonlocal shallow water equations, Acta Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' 181 (1998) 229-243.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' [17] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' McKean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Breakdown of a shallow water equation, Asian J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' 2 (1998) 767-774.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' [18] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Zhou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Wave breaking for a shallow water equation, Nonlinear Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' 57 (2004) 137-152.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' 27 [19] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Zhou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Wave breaking for a periodic shallow water equation, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' 290 (2004) 591-604.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' [20] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Constantin, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Strauss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Stability of peakons, Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Pure Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' 53 (2000) 603-610.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' [21] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Holm, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Naraigh, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Tronci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Singular solution of a modified two-component Camassa-Holm equation, Phy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' 79 (2009) 1-13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' [22] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Guan, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Karlsen, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Yin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Well-posedness and blow-up phenomena for a modified tow- component Camassa-Holm equation, Contemp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' 526 (2010)199-220.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' [23] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Guo, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Zhu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Wave breaking for a modified two-component Camassa-Holm system, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Differ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' 252 (2012) 2759-2770.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' [24] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Liu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Yin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' On the blow-up phenomena for a modified periodic two-component Camassa-Holm equation, IMA J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' 77 (2012) 563-577.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' [25] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Guo, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Zhu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Blow-up criteria of solutions to a modified two-component Camassa-Holm system, Nonlinear Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Real World Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' 12 (2011) 3531-3540.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' [26] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Tan, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Yin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Global periodic conservative solutions of a periodic modified two-component Camassa-Holm equation, J Funct Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' 261 (2011)1204-1226.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' [27] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Prato, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Zabczyk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Stochastic Equations in Infinite Dimensions, Encyclopedia of Mathematics and its Applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' 152 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' [28] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Prevot, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Rockner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' A Concise Course on Stochastic Partial Differential Equations, Lecture Notes in Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Berlin (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' [29] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Brzezniak, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Capinski, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Flandoli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Stochastic partial differential equations and turbulence, Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Models Methods Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' 1 (1991) 41-59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' [30] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Kuksin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Shirikyan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Mathematics of two-dimensional turbulence, Cambridge Tracts in Mathe- matics, 194 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' [31] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Weinan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Stochastic hydrodynamics, In Current developments in mathematics, (2000) 109-147.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' [32] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Lenells, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Wunsch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' On the weakly dissipative Camassa-Holm, Degasperis-Procesi, and Novikov equations, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Differ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' 255 (2013)441-448.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' [33] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Wu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Yin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Blow-up and decay of the solution of the weakly dissipative Degasperis-Procesi equation, SIAM J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' 40 (2008) 475-490.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' [34] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Wu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Yin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Global existence and blow-up phenomena for the weakly dissipative Camassa-Holm equation, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Differ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' 246 (2009) 4309-4321.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' [35] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Tang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' On the pathwise solutions to the Camassa-Holm equation with multiplicative noise, SIAM J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' 50 (2018) 1322-1366.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' [36] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Kato, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Ponce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Commutator estimates and the Euler and Navier-Stokes equations, Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Pure Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' 41 (1988) 891-907.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' [37] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Glatt, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Vicol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Local and global existence of smooth solutions for the stochastic Euler equations with multiplicative noise, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Probab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' 42 (2014) 80-145.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' [38] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Debussche, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Glatt, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Temam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Local martingale and pathwise solutions for an abstract fluids model, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' 240 (2011) 1123-1144.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' 28 [39] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Breit, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Feireisl, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Hofmanova.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Local strong solutions to the stochastic compressible Navier- Stokes system, Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Partial Differential Equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' 43 (2018) 313-345.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' [40] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Breit, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Feireisl, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Hofmanova.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Stochastically forced compressible fluid flows, De Gruyter Series in Applied and Numerical Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' 3 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' [41] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Breit, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Hofmanova.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Stochastic Navier-Stokes equations for compressible fluids, Indiana Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' 65 (2016) 1183-1250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' [42] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Glatt, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Ziane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Strong pathwise solutions of the stochastic Navier-Stokes system, Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Differ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Equ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' 14 (2009) 567-600.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' [43] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Rohde, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Tang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' On a stochastic camassa-holm type equation with higher order nonlinearities, Journal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Dyn Differ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Equ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' 4(2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' [44] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Kallianpur, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Xiong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Stochastic differential equations in infinite-dimensional spaces, Expanded version of the lectures delivered as part of the 1993 Barrett Lectures at the University of Tennessee, Knoxville, TN, March (1995) 25-27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' [45] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Hofmanova.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Degenerate parabolic stochastic partial differential equations, Stochastic Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' 123 (2013) 4294-4336.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' [46] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Gyongy, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Krylov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Existence of strong solutions for Itˆo stochastic equations via approximations, Probab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Theory Related Fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' 105 (1996) 143-158.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' [47] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Rockner, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Zhu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Zhu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Local existence and non-explosion of solutions for stochastic fractional partial differential equations driven by multiplicative noise, Stoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' 124 (2014) 1974-2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' [48] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Crisan, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Flandoli, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Holm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Solution Properties of a 3D Stochastic Euler Fluid Equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Nonlinear Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' 29 (2019) 813-870.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' [49] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Gawarecki, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Mandrekar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Stochastic differential equations in infinite dimensions with applications to stochastic partial differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Probability and its Applications (NewYork).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' Springer, Heidelberg (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} +page_content=' 29' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE2T4oBgHgl3EQfqAhW/content/2301.04034v1.pdf'} diff --git a/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf b/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..86c8482cbbe7e5ac033d533a9d729ca935cbfdcc --- /dev/null +++ b/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:626da0bf32ebe9aaf761036bc30a31c51d5c2970101915bd06bb128fbe65287f +size 496277 diff --git a/LtE4T4oBgHgl3EQfJwyq/vector_store/index.faiss b/LtE4T4oBgHgl3EQfJwyq/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..0a99114792789659c8d414eb2562041543880196 --- /dev/null +++ b/LtE4T4oBgHgl3EQfJwyq/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:ce8b4e1f0b749b5854aa7c8263b42d364aff24a59c6046561beb4bf2961cc8aa +size 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C. Witten +,1 Nicolas Laporte +,2, 3 and Harley Katz4 +1Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK +2Kavli Institute for Cosmology, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK +3Cavendish Laboratory, University of Cambridge, 19 JJ Thomson Avenue, Cambridge CB3 0HE, UK +4Sub-department of Astrophysics, University of Oxford, Keble Road, Oxford OX1 3RH, UK +Accepted for publication in ApJ +ABSTRACT +Although low-mass star-forming galaxies are the leading candidates of the reionisation process, we +cannot conclusively rule out high-mass star-forming galaxies as candidates. While most simulations +indicate the former is the best candidate some models suggest that at z ≥ 6 massive, UV-bright galaxies +- “oligarchs” - account for at least 80% of the ionising budget. To test this hypothesis we target massive +(log10(M⋆[M⊙]) > 10), UV-bright (MUV ∼ −22) Lyα emitters at z > 7 in archival data, observed with +similar resolution spectrographs (VLT/X-shooter and Keck/MOSFIRE). To increase the reliability of +our conclusions we stack all spectra and obtain a deep-stacked spectrum of 24.75hrs. The stacked Ly-α +profile displays a clear asymmetric red peak and an absence of a blue peak. We additionally estimate +the intrinsic stacked Lyα profile of our targets by correcting for IGM transmission using a range of +neutral hydrogen fractions, finding no significant change in the profile. We measure a velocity offset +Vred > 300 km/s and an asymmetry in our red peak A ∼ 3. Using various models and estimators +such as the peak separation, the asymmetry of the red peak, the ratio between Lyα and Hβ and the +β slope, we conclude that the escape fraction in these three UV bright, massive (∼ 1010M⊙), z ≥ 7 +galaxies is fesc(LyC) ≤ 10%. +1. INTRODUCTION +Cosmic reionisation is the processes by which the neu- +tral hydrogen in the universe, formed after the recom- +bination phase (380 000 years after the Big-Bang), un- +derwent a phase change - to almost complete ionisation +by z = 5.5 (Bosman et al. 2022, Kulkarni et al. 2019, +Robertson et al. 2015). Although the period over which +reionisation occurs is well constrained, the sources of the +ionising photons are not. The current debate is regard- +ing whether or not the faint galaxies which represent +the bulk of galaxies at z ≥ 6 (e.g. Bouwens et al. 2015, +Finkelstein et al. 2015, Atek et al. 2018) are the main +drivers (Ocvirk et al. 2020, Trebitsch et al. 2022) or if the +most massive objects are the major contributors (Naidu +et al. 2020; Sharma et al. 2017) to reionisation. +For +example, Naidu et al. (2020) claim that bright galaxies +may produce > 80% of the reionisation budget in order +Corresponding author: Callum E. C. Witten +cw795@cam.ac.uk +to match the observed rapid fall in neutral hydrogen +fraction at z < 8. Directly measuring the Lyman con- +tinuum radiation escaping a source and hence ionising +the universe is however not possible due to the neutral- +ity of the IGM during the epoch of reionisation (Inoue +et al. 2014, eg.). Instead the Lyman-α profile is heavily +affected by neutral hydrogen and hence it can be used +as a probe of the amount of ionising photons escaping +the galaxy. +Given the potential Lyα emission has as a probe of +neutral hydrogen, research into the properties of the line +emerging from a source through surrounding neutral hy- +drogen were conducted by Harrington (1973); Neufeld +(1990). The absorption of Lyα at line centre by neu- +tral hydrogen results in the main escape mechanism for +photons being through diffuse in frequency producing +a double-peaked spectrum. The wings of the Lyα line, +that form the consequent two peaks are defined by the +density of the scattering neutral hydrogen. As such the +separation between the two peaks (Vsep) has been pro- +posed as a key diagnostic of the Lyman-continuum es- +arXiv:2301.03599v1 [astro-ph.GA] 9 Jan 2023 + +ID2 +Witten, Laporte and Katz +cape fraction fesc(LyC) due to their tight dependence +(Izotov et al. 2017; Verhamme et al. 2015). +While this double-peaked spectrum has now been ob- +served (eg. Yee & De Robertis 1991; Venemans et al. +2005; Vanzella et al. 2008), especially at high-redshift, +we often see reduced or a complete absence of the blue +peak (Verhamme et al. 2015). This process is now well +understood as a result of inter-galactic medium (IGM) +radiative transfer (RT) effects (Gunn & Peterson 1965) +at high redshifts where we do not expect to find strong +outflows (Vito et al. 2022, Murray et al. 2005). Pho- +tons are redshifted as they travel through the expanding +universe, and hence blue peak photons get shifted into +resonance. If this occurs in the vicinity of neutral hydro- +gen, we observe absorption, and hence any intrinsic blue +peak is not observed for high neutral hydrogen column +densities as predicted by Garel et al. (2021); Laursen +et al. (2011). +Emission from high redshift objects typically travels +through high neutral hydrogen patches in the IGM and +as such, we rarely observe blue Lyα peaks. There have +however been instances of double-peaked high redshift +galaxies such as that discussed in Meyer et al. (2021) +and Matthee et al. (2018). These galaxies typically live +within large ionised bubbles resulting in a largely re- +duced neutral hydrogen column density, allowing the +presence of a blue-peak. +In this letter, we aim to measure the mean LyC escape +fraction of bright z ≥7 Lyman Alpha Emitters (LAEs) +based on their Lyα profile and to give new insights on +their contribution to the reionisation. In section 2, we +describe the sample of LAEs we identified at z ≥7. We +then describe the stacking method we use to improve the +signal-to-noise ratio on our Lyα line in section 3 and we +discuss the results and their implications in sections 4 +and 5. +2. TARGETS SELECTION +Our primary goal is to infer the mean escape frac- +tion of bright, z ≥ 7 galaxies by constraining the shape +and properties of Lyα, 1215.67˚A, redshifted into the +near-infrared. We therefore search in the literature for +all z ≥7 spectroscopically confirmed galaxies that have +been observed by high-resolution NIR spectrographs, to +guarantee the ability to resolve any double-peak in the +Lyα profile. Moreover, to obtain a good estimate of the +UV beta slope, which can be used to measure the es- +cape fraction at high-redshifts (Zackrisson et al. 2017), +we require that our selected galaxies have at least 3 con- +straints on their SEDs. 11 objects with a redshift rang- +ing from 7.15 to 9.11 were selected with Lyα luminosities +ranging from ∼0.05 to 2×1043erg/s. +Additionally, Schenker et al. (2012) reports the detec- +tion of A1703-zd6, a redshift z = 7.045 galaxy that satis- +fies all of our selection criteria, except one - it is a lensed, +low-mass galaxy (Stark et al. 2015 - log(M⋆) ∼ 8.7). +We therefore do not include this target within our stack, +but we note the measured Lyα red-peak separation from +line-center of ∼ 60 km/s. This is consistent with a high +Lyman-continuum escape fraction as expected if faint +galaxies are responsible for re-ionization (Gazagnes et al. +2020; Izotov et al. 2018a). +Because the observed shape of the Lyα line profile +is a result of propagation through neutral gas, it does +not trace the precise redshift of a galaxy. In order to +produce a proper stack of this emission line, we also +require that the galaxies used in our study have a sys- +temic redshift measurement through the observation of +another emission line. Among the 11 galaxies mentioned +above, only 3 (described in table 1) satisfy this crite- +ria, returning 4 datasets combining 24 hours and 45 +minutes on-source exposure time. +For MOSFIRE, we +make use of the following programs : C228M (PI: A. +Zitrin), Y288M (PI: Moncheva) and N190 (PI: Malho- +tra) ; and for XSHOOTER : 097.A-0043(A) (PI: Ellis). +The three aforementioned galaxies have UV luminosi- +ties MUV ∼ −22, placing them on the extreme end of +galaxies luminosities at z = 7 − 8 (Bowler et al. 2014, +2020). +The Spectral Energy Distribution (SED) of the 3 +galaxies have been extracted from 3DHST catalogues +(Brammer et al. 2012, Skelton et al. 2014). The physi- +cal properties of each individual galaxy have been esti- +mated by SED-fitting using BAGPIPES (Carnall et al. +2018). One of the main advantages of this code is to +allow the user to choose between several Star Formation +Histories (SFHs). In this work, we run BAGPIPES with +4 different SFHs, namely a burst, a constant, a delayed +and a combination of burst+constant. The best SED- +fit is obtained for the SFH that minimised the BIC (see +Laporte et al. 2021 for more details). BAGPIPES being +a parametric code, we used the following ranges for the +parameters of the SFH : +• ionisation parameter : log U ∈ [-3.0,-1.0] +• dust attenuation (assuming a Calzetti law) : +Av[mag] ∈ [0.0, 1.0] +• age of the stellar population : Age[Gyr] ∈ [0.0,1.0] +• mass formed : M⋆[M⊙] ∈ [106, 1012] +• metallicity : Z [Z⊙] ∈ [0.0, 1.5] +The redshift was fixed to the spectroscopic redshift +of each galaxy, and the IMF used in BAGPIPES is a + +Low fesc(LyC) in three Massive, High-z Galaxies +3 +Table 1. The properties of the archival observations of our three target galaxies. zsyst indicates the systemic redshift of the +target galaxy that we use for our analysis and the stellar mass is determined by the BAGPIPES fitting discussed in Section 2. +Name +zsyst +log(M⋆) +Exposure time +Lyα luminosity +Telescope +Reference +[M⊙] +[1043 erg/s] +EGSY-8p68 +8.671 +10.1+0.1 +−0.2 +4 hrs. 45 min. +1.95±0.49 +MOSFIRE +Zitrin et al. (2015) +EGS-zs8-1 +7.721 +10.2+0.2 +−0.1 +4 hrs. +1.2±0.1 +MOSFIRE +Tilvi et al. (2020) +4 hrs. +1.2±0.2 +MOSFIRE +Oesch et al. (2015) +COSY +7.142 +10.2+0.1 +−0.8 +12 hrs. +1.43±0.19 +X-Shooter +Laporte et al. (2017) +Kroupa IMF (Kroupa 2001). As expected by the selec- +tion function of our sample, our galaxies are good ex- +amples of the most massive galaxies at z ≥7 with stellar +masses ranging from 1.36×1010 to 1.75×1010 M⊙, plac- +ing them on the extreme end of the galactic stellar mass +function at z = 7−8 (see Table 1). The properties of our +stacked spectrum obtained using BAGPIPES are found +in Table 1. +The detection of Lyα at z ≥7 implies the formation of +an ionized bubble around the galaxies. (e.g. Castellano +et al. 2022; Roberts-Borsani et al. 2022). The origin of +the ionising photons that produce these ionised bubbles +is still highly debated and could be either due to the +intrinsic nature of the object (e.g. star formation or an +active galactic nucleus) or the over-dense environment +near the most massive galaxies formed at high-redshift +(Leonova et al. 2021; Laporte et al. 2022). Indeed previ- +ous research into these galaxies indicates that they reside +within ionised bubbles, large enough that any blue-peak +escaping the host galaxy would be redshifted past Lyα +line-centre before leaving the ionized bubble and hence +should be unaffected by IGM absorption. +Based on a relation between Lyα luminosity and ion- +ized bubble size derived from theoretical models, Tilvi +et al. (2020) estimate that EGS-zs8-1 sits in a common +bubble with at least 3 neighbouring galaxies, with a +size of 1.02 pMpc. More recently, Leonova et al. (2021) +present analysis of Hubble Space Telescope (HST) imag- +ing that supports the conclusion of a bubble surrounding +EGS-zs8-1 given it resides in an overdensity. They addi- +tionally find that EGSY-z8p68 is found with an overden- +sity and again comparing to simulations find an expected +bubble radius of ∼ 1 pMpc. Laporte et al. (2017) de- +tect Lyα, HeII and NV, as well as upper bounds on the +flux of CIII] and CIV in the spectrum of COSY allowing +for the determination of the source of its radiation field. +The most likely hypothesis is that the radiation field of +COSY is inconsistent with that from star-forming galax- +ies, instead it likely is produced by an active galactic +nucleus (AGN) (Laporte et al. 2017; Costa et al. 2014) +and additionally its high UV luminosity makes it likely +to trace an overdense region (eg. Barkana & Loeb 2004; +Furlanetto et al. 2004). Therefore it is likely COSY ad- +ditionally resides within a large, ionized bubble. +3. METHOD +Data reduction of archival Keck MOSFIRE and VLT +XSHOOTER data was performed through the stan- +dard MOSFIRE data reduction pipeline1 and EsoRe- +flex2 which include standard data reduction procedures +such as flat-fielding, wavelength calibration and back- +ground subtraction. The flux calibration was performed +using a bright photometric standard observed during +each night. +The systemic redshift is obtained from the additional +emission lines present in each galaxy’s spectrum (EGSY- +8p69: N V, Mainali et al. (2018); EGS-zs8-1: C III], +Stark et al. (2017); COSY: [C II], Pentericci et al. +(2016)). We then shift each spectrum into the rest frame +wavelength based on the systemic redshift reported in +Table 1. Following this we define a new wavelength ba- +sis, spanning the range of wavelengths of interest with +a wavelength bin equal to that of our lowest resolution +spectra (∼ 0.15 ˚A). The total flux of each object within +each wavelength bin is calculated, converted into a lu- +minosity, and the median of these luminosities is then +taken as the value for our stacked spectrum. We then +return this median spectrum to units of flux by dividing +through by the luminosity distance of the median red- +shift in our sample. We take the standard deviation of +the fluxes in each bin to obtain the error in our stacked +spectrum. +We then use a Monte Carlo (MC) error propagation +method to estimate the uncertainties on the observed +red-peak velocity offset and asymmetry. We use the er- +ror in our spectrum, described above, to redraw each +spectral bin’s flux from a Gaussian centred on the me- +dian stack value with a standard deviation equal to the +aforementioned error. We then repeat this process, mea- +suring the red-peak offset and asymmetry, one hundred +thousand times in order to understand the uncertainty +1 https://keck-datareductionpipelines.github.io/MosfireDRP/ +2 https://www.eso.org/sci/software/esoreflex/ + +4 +Witten, Laporte and Katz +Table 2. The values for various parameters used for a range of diagnostics, and the fesc(Lyα) that these diagnostics predict. +Diagnostic +Value +fesc(Lyα) +Spitzer: flux (Hβ + [OIII]) +7.13 × 10−17erg/s/cm2/˚A +[0.02:0.32] +BAGPIPES: flux (Hβ) +1.1+0.9 +−0.5 × 10−17erg/s/cm2/˚A +∼ [0.09:0.18] +Spitzer: UV slope β and log10(EW(Hβ + [OIII])) +[-0.6, -2.35], [2.0 ˚A, 3.1 ˚A ] +∼ 0 +BAGPIPES: UV slope β and log10(EW(Hβ)) +−1.89+0.09 +−0.11, 2.67 ± 0.25 +∼ 0 +Lyα profile: Asymmetry and Peak separation +A = 3.3+1.4 +−0.8, Vredpeak = 330+190 +−70 km/s +< 0.15 +in our measurements driven by the error associated with +each flux measurement. We take the red-peak offset and +asymmetry associated with our stack and measure the +standard deviation of those values above and below the +median for our upper and lower bound uncertainties on +this measurement respectively. +We have additionally evaluated the results when nor- +malising all of our spectra by dividing through by their +peak Lyα flux, as not to weight our stack based on the +most luminous galaxy. Given that every galaxy has a +similar luminosity, we see no change in our Lyα profile +regardless of normalisation. A median stacking method +is chosen as it acts to reduce the affect of sky-lines, +however, in the case of COSY, significant contamina- +tion from a sky-line in a region of the spectrum where +we expect to observe the tail of our red-peak profile did +act to pollute our stacked spectrum and it was there- +fore masked. +Adopting a mean stacking method and +additionally changing the position and widths of bins +provides a very similar stacked spectrum thus indicat- +ing the robustness of our stack. +4. RESULTS +4.1. Constraints from SED +The Lyα escape fraction is the ratio of the Lyα flux +escaping a galaxy over the intrinsic Lyα flux of the ob- +ject. In order to ascertain limits on the potential intrin- +sic flux of Lyα we use recombination lines with known +relations to Lyα. Unfortunately no direct observations +of recombination lines are made in any of our target +spectra. Instead we make use of two independent meth- +ods to estimate the equivalent width (EW) of the Hβ +emission line: (i) using the flux ratio between the 4.5µm +and 3.6µm assuming that the 3.6µm flux is the stellar +continuum and the 4.5µm is the sum of the stellar con- +tinuum with a contamination of OIII+Hβ and (ii) using +BAGPIPES to directly predict the flux of Hβ (see Ta- +ble 2 for the results of both methods). The first method +returns the flux of Hβ and OIII in combination and thus +by assuming no contribution from OIII supplies us with +a lower bound estimate of the flux of the recombina- +tion line Hβ. We assume a maximum contribution of +log10([OIII]/Hβ) = 1 - based on the assumption our +stack lies in the extreme AGN region of the Baldwin, +Philips and Terlevich (BPT) diagram (Baldwin et al. +1981; Veilleux & Osterbrock 1987). This conclusion is +in itself unlikely as NV emission lines are either weak or +not present in our individual galaxy spectra but allows +us to obtain an absolute upper bound on the Hβ flux. +The second method is based on SED-fitting and depends +on the best fit parameters such as the stellar mass, the +reddening, the age, the metallicity, etc. To verify that +the Hβ flux estimated by BAGPIPES is not strongly de- +pendant on other parameters, we study the evolution of +Hβ flux as a function of the metallicity and stellar mass +and as a function of the metallicity and reddening. The +variation in the EW is estimated as ∆ log EW < 0.5. +We assume, following Gazagnes et al. (2020), that +LLyα = 8.7LHα, and in turn that LHα = 2.85LHβ, how- +ever Lyα emission can include a large contribution from +collisional excitation (Mitchell et al. 2021; Smith et al. +2021) thus increasing the intrinsic LLyα further decreas- +ing the measured escape fraction. +However, given we observe no Lyα blue-peak we must +assume the Lyα profile has undergone some absorption +due to IGM attenuation. Given many Lyα profiles of +low redshift LAEs, that have not travelled through a +high neutral hydrogen density IGM, exhibit often equal +or less than equal blue-to-red peak flux ratios (eg. Izo- +tov et al. 2018a,b) we assume that the upper bound of +the LLyα escaping the galaxy is double the LLyα that +we observe, while the LLyα that we observe represents +a lower bound. Taking the bounds of LHβ that we ob- +tain from Spitzer provides us with a potential range of +0.02 < fesc(Lyα) < 0.32, while the best-fit LHβ from +BAGPIPES returns fesc(Lyα) = 0.09+0.07 +−0.04, while tak- +ing the bounds on LLyα the best-fit value ranges from +0.09 < fesc(LyC) < 0.18. +The bounds of the Hβ flux as well as the best-fit value +and the associated escape fraction for the stacked spec- +trum can be found in Table 2. +Using relations from +Maji et al. (2022) we can obtain the escape fraction of +the Lyman-continuum (LyC) which is found to be lower +than the Lyα escape fraction, hence we take the bounds +on fesc(LyC) to be the same as those on fesc(Lyα) (also +reported in Table 2). +We additionally consider the relation between the UV +slope β and the EW(Hβ) first determined by Zackrisson + +Low fesc(LyC) in three Massive, High-z Galaxies +5 +Figure 1. Line profiles of Lyα emission in our target galaxies (top) and the stacked spectrum (bottom). Line centre is denoted +by a black dashed line, while the red-peak flux is indicated by a red dashed line. The velocity offset of the red-peak from line +centre is additionally found within each panel. (Top:) From left to right: EGS-zs8-1, EGSY-8p68, COSY. The resolution of +the spectrum of COSY has been reduced to the resolution of the two MOSFIRE spectra (0.14 ˚A). The hatched box indicates a +region of the spectrum that has been removed due to significant pollution by a sky line. (Bottom:) The grey region indicates +the 1-sigma error obtained by taking the standard deviation of the constituent galaxies of the stack. Additionally, red peak +velocity offset and the asymmetry of the red-peak as defined in Section 4 are found within the panel. +et al. (2013) at redshifts z > 6 and then at z ≈ 7 − 9 by +Zackrisson et al. (2017) by studying the evolution in syn- +thetic galaxy spectra with changing fesc(LyC). The re- +sults of this analysis can be seen in figure 2 where galax- +ies with log(EW(Hβ)) ≳ 2 exclusively have fesc(LyC) += 0. As is clear in figure 2 both the best-fit EW(Hβ) +and UV slope β (corrected for dust extinction following +Meurer et al. 1999) returned from BAGPIPES and the +range in EW(Hβ) and UV slope β (determined follow- +ing Bouwens et al. 2014) that can be estimated from +Spitzer data (the values of which are reported in Ta- +ble 2) constrain our stack to a region of the figure that +is not compatible with an fesc(LyC) ≫ 0. +We do note that this diagnostic is potentially limited +in its ability to diagnose high escape fractions given ex- +amples of low redshift galaxies that have high fesc(LyC) +and high EW(Hβ) (eg. Izotov et al. 2018b). +The di- +agnostic requires many assumptions in order to esti- +mate fesc(LyC), notably in the stellar models employed. +Changing these stellar models can significantly affect the +result of the diagnostic, as seen when binary evolution is +considered (see figure 6 in Zackrisson et al. 2017). How- +ever, we are aware of the main outcome of Zackrisson +et al. (2017) - that galaxies with fesc(LyC)>0.5 should +have EW(Hβ) < 30 ˚A. Therefore, we instead consider +a high EW(Hβ) to be a necessity for low fesc(LyC), al- +though perhaps not sufficient. +Given the inclusion of +multiple diagnostics all indicating low fesc(LyC) we con- +sider the potential uncertainty surrounding this diagnos- +tic not to be a significant issue. +We additionally note that Zackrisson et al. (2017) +provides this diagnostic for a range of different dust +attenuation laws. +While the panel in figure 2 as- +sumes a Calzetti attenuation law with E(B − V )stars = +E(B−V )neb, we find that the conclusion, that our stack +lies within a region of the diagram corresponding to +fesc(LyC) = 0, is consistent regardless of the dust at- +tenuation law used in Zackrisson et al. (2017). +4.2. Lyα profile +Figure 1 clearly indicates that for all of our targets, we +find the red-peak of the Lyα profile to be offset from line +centre by ∼ 340 km/s, such a large separation is indica- +tive of a low Lyman-continuum escape fraction (Izotov +et al. 2018b; Gazagnes et al. 2020; Kakiichi & Gronke +2021). The large offset of the red-peak is additionally +present in the stacked spectrum, as well as a clear asym- +metry. This asymmetry, A, is the ratio of the blue-to-red + +EGS-zs8-1 +EGSY-8p68 +COSY +2.0 +Vsep = 342 km/s +Vsep = 341 km/s +Vsep = 344 km/s +1.5 +1.0 +A +0.5 +0.0 +-0.5 +S +1212 +1214 +1216 +1218 +1212 +1214 +1216 +1218 +1212 +1214 +1216 +1218 +erg +2.0 +-17 +(10- +1.5 +-70 +A = 3.3±1:4 +1.0 +.0.8 +Flux +0.5 +0.0 +-0.5 +1212 +1213 +1214 +1215 +1216 +1217 +1218 +1219 +Wavelength (A)6 +Witten, Laporte and Katz +Figure 2. The indirect diagnostics of fesc(LyC), with the region which our stack resides indicated with a grey hatched box. +(Left) The EW(Hβ) of simulated z = 7 − 9 galaxies against their UV slope β, assuming a Calzetti attenuation law with +E(B − V )stars = E(B − V )neb (from Zackrisson et al. 2017). Their Lyman-continuum escape fraction is denoted by their colour, +fesc(LyC) = 0.0, 0.5, 0.7, 0.9 correspond to red, orange, green and blue respectively. +The grey hatched region indicates the +location of our stack using Spitzer data, while the black data point indicates the position using the BAGPIPIES best-fit on the +SED. (Centre) The peak separation of Lyα profiles against their LyC escape fraction (from Kakiichi & Gronke 2021). The dots +denote simulated results from Kakiichi & Gronke (2021), while crosses indicate the results for z ∼ 0.3 LyC-detected galaxies +from Izotov et al. (2016, 2018a,b). The coloured regions indicate the three regimes of LyC escape - leakage by full break, through +holes and small leakage with few or no holes indicated by blue, red and grey respectively. (Right) The red peak asymmetry of +Lyα profiles against their LyC escape fraction (from Kakiichi & Gronke 2021). The markers and shading are the same as the +central panel. +flux of the red peak (as defined in Kakiichi & Gronke +2021). We find that the two targets for which we are +able to observe the shape of the red-peak profile, we +observe clear asymmetry. +5. DISCUSSION +The limits placed on fesc(Lyα) from photometry dis- +cussed in Section 4.1 are already low enough to rule out +the possibility of these three massive, bright galaxies +currently being significant contributors to re-ionization. +However, we wish to use multiple diagnostics in order to +confirm these findings. The results from our Lyα stack +are therefore crucial to further constrain the escape frac- +tion. +5.1. Interpretation of the velocity offset +While the velocity offset of Lyα from the systemic +redshift initially appears as though it may be primarily +driven by outflows in these massive galaxies, we believe +this to be unlikely. Neufeld (1990) and Michel-Dansac +et al. (2020), using a static medium with large neutral +hydrogen column densities, find velocity offsets in their +simulations that are comparable to those that we ob- +serve indicating these velocity separations are achievable +within simulations without modelling for outflows. +Any such shift in the Lyα profile due to the expansion +velocity, vexp, of neutral gas would still result in the +expected double-peaked profile of Lyα centred on the +systemic redshift. Results from Verhamme et al. (2015) +indicate that increasing vexp has the effect of reducing +the peak separation, such that when vexp > 300 km/s, +they cannot recreate the red peak to line-centre separa- +tion that we observe. These results constrain the neu- +tral gas column density in our stack to be greater than +1020cm−2 and any outflow velocity vexp < 300 km/s. +The conclusion that any expansion velocity will act to +reduce peak separation thus allows us to conclude that +our red-peak offset is a minimum separation. Addition- +ally, most diagnostics of escape fraction that use the sep- +aration between the peaks of the Lyα profile are based +on observations of lower mass galaxies than our targets +and therefore using any such diagnostic is challenging +(Izotov et al. 2018b; Gazagnes et al. 2020). Instead we +choose to use the relation determined by simulations +from Kakiichi & Gronke (2021), allowing us to avoid +mass biases. The observed red-peak-offset in Figure 1 +can be used as a lowest bound for the blue-red peak sep- +aration given that we know the blue peak lies on the blue +side of line-centre. As such, we expect the blue-red peak +separation to far exceed the red-peak offset of ∼ 300 +km/s and comparing this to the relation from Kakiichi +& Gronke (2021) we find that fesc(Lyα)≲ 10% and Lyα +photons escape through an optically-thick medium with +few or no holes. +5.2. Interpretation of the red-peak asymmetry + +700 +-1.00 +fesc = 0.0 +=0.5 +600 +1.25 += 0.9 +Red peak asymmetry +1.50 +Peak separation [km +500 +Slope +x +1.75 +400 +3 +2.00 +300 +Xx +2.25 +200 +x +2.50 +100 +3.0 +2.5 +2.0 +1.5 +1.0 +0.5 +0.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +log10(EW(Hβ))(A) +fesc(LyC) +fesc(LyC)Low fesc(LyC) in three Massive, High-z Galaxies +7 +Figure 3. The potential intrinsic stacked spectrum created by dividing the observed spectrum by the IGM transmission for a +range of different volumetric neutral fractions indicated in the top left of each panel. The spectrum normalised by the red peak +flux is indicated by the solid black line, while the IGM attenuation curve associated with the volumetric neutral fraction, taken +from Garel et al. (2021), is indicated by the red dashed line. +This asymmetry allows us to quantify the amount Lyα +photons have to scatter, in doing so creating a broad +wing component of the emission line, in order to escape +the galaxy. A high asymmetry (A > 3) is hence indica- +tive of Lyα photons having multiple routes to escape and +hence scatter significant amounts in order to find low- +density channels to escape the galaxy (leakage through +holes), while a low asymmetry (A < 3) is indicative of +Lyα photons only having one method of escape possible +either through predominantly optically-thin (leakage by +full break) or optically thick (small leakage due to few +or no holes) media Kakiichi & Gronke (2021). There- +fore we can use the asymmetry to attempt to diagnose +the properties of the medium through which the Lyα +photons have traversed. +The asymmetry that we observe, in Figure 1, is an +upper bound on the asymmetry of the intrinsic spec- +trum. This is due to IGM attenuation reducing the flux +close to line-centre hence reducing the flux between the +red-peak and line-centre relative to the flux on the red +side of the red-peak, therefore the observed asymmetry +is greater than the intrinsic asymmetry. +As such we +find the asymmetry, A < 3, results in the interpreta- +tion that Lyα photons have either escaped through a +full break environment or by leakage through few or no +holes. Given the aforementioned limits on the escape +fraction (fesc(Lyα)≲ 10%) we can constrain ourselves +to small leakage without the presence of optically-thin +channels (Kakiichi & Gronke 2021). +5.3. Effects of IGM attenuation +Neutral hydrogen in the IGM causes attenuation of +Lyα close to line-centre (see Garel et al. 2021), therefore +in order to confirm that we do not misidentify the loca- +tion of the red-peak flux, we divide our observed stacked +spectrum through by attenuation curves for varying vol- +umetric neutral fraction, taken from Garel et al. (2021). +Given our target galaxies all likely reside within large +ionized bubbles we do not expect significant IGM ab- +sorption. Figure 3 indicates the effect of correcting for +the different IGM transmission curves and we see no no- +table difference in our observed spectra even at the most +extreme volumetric neutral fraction, indicating our ob- +served red peak separation likely trace that intrinsic to +the galaxy before IGM absorption of Lyα. While we do +see a notable decrease in the asymmetry, as predicted +in Section 5.2, we have already consider the asymmetry +to be a lower bound and as such this does not affect the +interpretation of the result. +We do note a significant increase in the flux at line- +centre, in Figure 3, due to the effectively zero transmis- +sion through the IGM at that wavelength. This is merely +an artefact of noise being divided through by a number +tending to zero rather than any physical intrinsic prop- +erty of the galaxy. +We know this to be true as Lyα +emission will immediately be absorbed at line-centre by +any neutral hydrogen within the host galaxy and as such +we must observe negligible flux at line-centre escaping +the host galaxy. +Finally, in order to confirm that we are not being af- +fected by high escape fraction interlopers that due to +IGM transmission, spectral resolution and noise are be- +ing interpreted as having a low escape fraction, we at- +tempt to recreate high redshift observations of galax- +ies with fesc(LyC) greater than our sample (fesc(LyC)> +0.1). We use the spectra of galaxies from Izotov et al. + +1.0 +XHI = 0.000057 +XHI = 0.0065 +XHI = 0.35 +XHI = 0.64 +1.0 +0.8 +0.8 +flux +lission +0.6. +0.6 +Transmi +0.4 +JON +0.4 +IGM +0.2 +0.2 +0.0 +12161217 +12181219 +12181219 +12181219 +0.0 +1216 +1216 +1217 +1216 +1218 +1219 +Wavelength[A] +Wavelength [A] +Wavelength[Ai +Wavelength[A]8 +Witten, Laporte and Katz +Figure 4. The simulated spectra of high redshift, high fesc(LyC) galaxies for varying assumed neutral hydrogen fractions. +Each column uses an increasing neutral hydrogen fraction from left to right, that in turn dictates which IGM attenuation curve, +taken from Garel et al. (2021), is applied to the original spectrum. Each panel includes the same 4 galaxies from Izotov et al. +(2018a,b) (clockwise from top left sub-panel: J1011+1947, J1256+4509, J1243+4646, J1154+2443), whose Lyman continuum +escape fraction are indicated at the top of each sub-panel. The simulated observed spectrum normalised by the red peak flux is +indicated by the solid black line, while the original spectrum (with resolution degraded) is indicated in grey. +(2018a,b) as examples of Lyα profiles associated with an +fesc(LyC) greater than our sample up to a value of 72% +at low redshifts (z ∼ 0.3). We apply the IGM attenua- +tion curves from Garel et al. (2021) to the Lyα profile, +we then reduce the resolution of these spectra down to +the resolution of our stacked Lyα profile and finally we +use the MC error propagation described in Section 3 to +estimate uncertainties on the Vred and asymmetry of +each galaxy given a noise level similar to our stacked +spectrum (by assuming the peak flux to be at SN = 5). +Figure 4 shows our simulated high redshift, high +fesc(LyC) spectra with various IGM transmission curves +applied for differing neutral hydrogen fractions. We find +that applying the IGM transmission and degrading the +resolution of the spectra result in velocity offsets that +are consistent with the original spectra, even for the +most extreme neutral hydrogen density, to within 40 +km/s. This peak separation when considered in the con- +text of the Kakiichi & Gronke (2021) diagnostic appears +to indicate these galaxies are likely high fesc(LyC). All +of these galaxies exhibit red peak offsets ≪ 300 km/s +thus allowing us to conclude that the observation of a +peak offset of ∼ 300 km/s is not only indicative of a +low fesc(LyC) but also that we are likely not being af- +fected by high fesc(LyC) interlopers. +We do however +note that the asymmetry is more challenging to under- +stand as a function of the neutral fraction. It is clear +that for a sharp Lyα red-peak, increasing the neutral +density will act to increase the flux on the red side of +this peak hence increasing the asymmetry. When the +red-peak is more broad, increasing the neutral density +can push the peak of the Lyα profile red-ward and hence +act to increase the amount of flux on the blue side of the +peak. This complicated interplay of effects leads to a +highly uncertain asymmetry in some of our Lyα profiles. +Therefore, the use of asymmetry to diagnose fesc(LyC) +alone at high redshifts, where neutral hydrogen in the +IGM causes large uncertainties on the intrinsic asymme- +try, should be avoided. However, the asymmetry of our +stacked spectrum appears relatively well defined with a +sharp drop in flux blue-ward of the peak flux. There- +fore, we conclude that our asymmetry is most likely an +upper-bound asymmetry, where the relative boosting of +flux to the red side of the Lyα peak due to IGM attenua- +tion, is the most likely of the two aforementioned effects +at play. Given the relatively small uncertainty on the +intrinsic Lyα asymmetry of our stacked spectrum we +believe that, in combination with multiple other diag- + +XHl 0.0065 +XHl- 0.35 +XHl 0.64 +1 +fesc = 0.11 +fesc = 0.38 +1 +fesc = 0.11 +fesc = 0.38 +1 +fesc = 0.11 +fesc = 0.38 +Vsep = 14028 +Vsep = 1408 +Vsep = 1408 +Vsep = 1408 +Vsep = 1408 +Vsep = 180±28 +A= 2.4: +A= 3.13 +A= 4.08 +A= 5.12 +A= 6.98 +A= 3.2±3 +Observed +Intrinsic +0.5 +0.5 +0.5 +flux +flux +flux +lised +lised +lised +0 +Q +Q +Normal +1 +fesc = 0.46 +fesc = 0.73 +Normal +1 +fesc = 0.46 +fesc = 0.73 +Normal +1 +fesc = 0.46 + fesc= 0.73 +Vsep = 110+28 +Vsep =14020 +Vsep = 140±28 +Vsep = 1800 +Vsep = 140+8 +Vsep =1800 +A= 3.8±:9 +A= 3.5±:8 +A= 2.8±:6 +A= 3.1±3:3 +A= 5.1: +A元5.6.3 +0.5 +0.5 +0.5 +Q - +0 +1216 1217.5 1219 +1216 1217.5 1219 +1216 1217.5 1219 +1216 1217.5 1219 +1216 1217.5 1219 +1216 1217.5 1219 +wavelength (A) +Wavelength (A) +Wavelength (A)Low fesc(LyC) in three Massive, High-z Galaxies +9 +nostics, the observed asymmetry can be used to infer +the ability of Lyα photons to escape their host galaxy, +with the caveat that this diagnostic should not be used +on high redshift LAEs without the use of other supple- +mentary diagnostics. +6. SUMMARY +In order to probe the potential ionising properties +of the most massive (log10(M⋆[M⊙]) > 10), UV-bright +(MUV ∼ −22), high redshift (z > 7) galaxies, we target +all archival data on telescopes with resolution R ≳ 3000, +allowing us to obtain a resolved Lyα profile. We find +a total of four observations of three satisfactory galax- +ies with Lyα emission, totalling an exposure of 24 hours +and 45 minutes. Using a median stacking method we ob- +tain a deep stacked spectrum representing massive, UV- +bright, high redshift Lyα leaking galaxies. Through the +analysis of the stacked Lyα profile, using the red-peak +velocity offset from line-centre and the red-peak asym- +metry, we deduce the Lyman-continuum escape fraction +to be less than 10% and that the few Lyman-continuum +photons that do escape, escape through an optically- +thick medium with few or no holes. Through the use +of Spitzer observations of our target galaxies, stack- +ing these and SED-fitting using BAGPIPES we obtain +bounds on the recombination line Hβ. Given this we +constrain the escape fraction to 9% < fesc(Lyα)< 18% +in strong agreement with the results of our stacked Lyα +profile. We additionally confirm that neither IGM at- +tenuation or a significant outflow velocity could affect +our conclusion regarding a low fesc(Lyα) for massive, +UV-bright, high redshift galaxies. +Our study shows +that despite the fact the 3 galaxies analysed lie within +ionised bubbles, they are not capable themselves of ion- +ising their own bubbles. +However, we emphasize that our result is obtained +using only 4 datasets of 3 different galaxies at z ≥7 +– the only observations currently available in telescope +archives. Increasing the number of Lyα detections at +z ≥7 with high-resolution spectrographs is therefore +crucial to confirm our conclusions. +Furthermore, the +high-fraction of neutral gas underlying galaxies within +the epoch of reionisation limits the detection of Lyα +to galaxies in overdense regions. Spectrographs with a +large field-of-view will therefore be ideal instruments to +push forward this project. MOONS, a 3rd generation +instrument at the Very Large Telescope, will be one of +those. +It combines high-resolution (R>4000), a large +field of view (∼500 arcmin2) and a huge number of fi- +bres (∼1000). +7. ACKNOWLEDGEMENTS +We thank the anonymous referee for providing help- +ful comments which improved the quality of this paper. +CW and NL acknowledge advice and comments from +Debora Sijacki, Martin Haehnelt, Roberto Maiolino, +Sergio Martin-Alvarez and Yuxuan Yuan that helped +to direct our analysis and the diagnostics used. +CW +acknowledges support from the Science and Technol- +ogy Facilities Council (STFC) for a Ph.D. studentship. +NL acknowledges support from the Kavli foundation. +This research has made use of the Keck Observatory +Archive (KOA), which is operated by the W. M. Keck +Observatory and the NASA Exoplanet Science Institute +(NExScI), under contract with the National Aeronau- +tics and Space Administration. Based on observations +collected at the European Southern Observatory under +ESO programme 097.A-0043(A). This work is based on +observations taken by the 3D-HST Treasury Program +(GO 12177 and 12328) with the NASA/ESA HST, which +is operated by the Association of Universities for Re- +search in Astronomy, Inc., under NASA contract NAS5- +26555 +8. DATA AVAILABILITY +The data underlying this article will be shared on rea- +sonable request to the corresponding author. +REFERENCES +Atek, H., Richard, J., Kneib, J.-P., & Schaerer, D. 2018, +MNRAS, 479, 5184, doi: 10.1093/mnras/sty1820 +Baldwin, J. A., Phillips, M. M., & Terlevich, R. 1981, +PASP, 93, 5, doi: 10.1086/130766 +Barkana, R., & Loeb, A. 2004, ApJ, 609, 474, +doi: 10.1086/421079 +Bosman, S. E. I., Davies, F. B., Becker, G. D., et al. 2022, +MNRAS, 514, 55, doi: 10.1093/mnras/stac1046 +Bouwens, R. J., Illingworth, G. D., Oesch, P. A., et al. +2014, ApJ, 793, 115, doi: 10.1088/0004-637X/793/2/115 +—. 2015, ApJ, 803, 34, doi: 10.1088/0004-637X/803/1/34 +Bowler, R. A. A., Jarvis, M. J., Dunlop, J. S., et al. 2020, +MNRAS, 493, 2059, doi: 10.1093/mnras/staa313 +Bowler, R. A. A., Dunlop, J. S., McLure, R. J., et al. 2014, +MNRAS, 440, 2810, doi: 10.1093/mnras/stu449 +Brammer, G. B., van Dokkum, P. G., Franx, M., et al. +2012, ApJS, 200, 13, doi: 10.1088/0067-0049/200/2/13 +Carnall, A. C., McLure, R. J., Dunlop, J. S., & Dav´e, R. +2018, MNRAS, 480, 4379, doi: 10.1093/mnras/sty2169 + +10 +Witten, Laporte and Katz +Castellano, M., Pentericci, L., Cupani, G., et al. 2022, +arXiv e-prints, arXiv:2203.04976. +https://arxiv.org/abs/2203.04976 +Costa, T., Sijacki, D., Trenti, M., & Haehnelt, M. G. 2014, +MNRAS, 439, 2146, doi: 10.1093/mnras/stu101 +Finkelstein, S. L., Ryan, Russell E., J., Papovich, C., et al. +2015, ApJ, 810, 71, doi: 10.1088/0004-637X/810/1/71 +Furlanetto, S. R., Zaldarriaga, M., & Hernquist, L. 2004, +ApJ, 613, 1, doi: 10.1086/423025 +Garel, T., Blaizot, J., Rosdahl, J., et al. 2021, MNRAS, +504, 1902, doi: 10.1093/mnras/stab990 +Gazagnes, S., Chisholm, J., Schaerer, D., Verhamme, A., & +Izotov, Y. 2020, A&A, 639, A85, +doi: 10.1051/0004-6361/202038096 +Gunn, J. E., & Peterson, B. A. 1965, ApJ, 142, 1633, +doi: 10.1086/148444 +Harrington, J. P. 1973, MNRAS, 162, 43, +doi: 10.1093/mnras/162.1.43 +Inoue, A. K., Shimizu, I., Iwata, I., & Tanaka, M. 2014, +MNRAS, 442, 1805, doi: 10.1093/mnras/stu936 +Izotov, Y. I., Schaerer, D., Thuan, T. X., et al. 2016, +MNRAS, 461, 3683, doi: 10.1093/mnras/stw1205 +Izotov, Y. I., Schaerer, D., Worseck, G., et al. 2018a, +MNRAS, 474, 4514, doi: 10.1093/mnras/stx3115 +Izotov, Y. I., Thuan, T. X., & Guseva, N. G. 2017, +MNRAS, 471, 548, doi: 10.1093/mnras/stx1629 +Izotov, Y. I., Worseck, G., Schaerer, D., et al. 2018b, +MNRAS, 478, 4851, doi: 10.1093/mnras/sty1378 +Kakiichi, K., & Gronke, M. 2021, ApJ, 908, 30, +doi: 10.3847/1538-4357/abc2d9 +Kroupa, P. 2001, MNRAS, 322, 231, +doi: 10.1046/j.1365-8711.2001.04022.x +Kulkarni, G., Keating, L. C., Haehnelt, M. G., et al. 2019, +MNRAS, 485, L24, doi: 10.1093/mnrasl/slz025 +Laporte, N., Meyer, R. A., Ellis, R. S., et al. 2021, +MNRAS, 505, 3336, doi: 10.1093/mnras/stab1239 +Laporte, N., Nakajima, K., Ellis, R. S., et al. 2017, ApJ, +851, 40, doi: 10.3847/1538-4357/aa96a8 +Laporte, N., Zitrin, A., Dole, H., et al. 2022, arXiv e-prints, +arXiv:2208.04930. https://arxiv.org/abs/2208.04930 +Laursen, P., Sommer-Larsen, J., & Razoumov, A. O. 2011, +ApJ, 728, 52, doi: 10.1088/0004-637X/728/1/52 +Leonova, E., Oesch, P. A., Qin, Y., et al. 2021, arXiv +e-prints, arXiv:2112.07675. +https://arxiv.org/abs/2112.07675 +Mainali, R., Zitrin, A., Stark, D. P., et al. 2018, MNRAS, +479, 1180, doi: 10.1093/mnras/sty1640 +Maji, M., Verhamme, A., Rosdahl, J., et al. 2022, arXiv +e-prints, arXiv:2204.02440. +https://arxiv.org/abs/2204.02440 +Matthee, J., Sobral, D., Gronke, M., et al. 2018, A&A, 619, +A136, doi: 10.1051/0004-6361/201833528 +Meurer, G. R., Heckman, T. M., & Calzetti, D. 1999, ApJ, +521, 64, doi: 10.1086/307523 +Meyer, R. A., Laporte, N., Ellis, R. S., Verhamme, A., & +Garel, T. 2021, MNRAS, 500, 558, +doi: 10.1093/mnras/staa3216 +Michel-Dansac, L., Blaizot, J., Garel, T., et al. 2020, A&A, +635, A154, doi: 10.1051/0004-6361/201834961 +Mitchell, P. D., Blaizot, J., Cadiou, C., et al. 2021, +MNRAS, 501, 5757, doi: 10.1093/mnras/stab035 +Murray, N., Quataert, E., & Thompson, T. A. 2005, ApJ, +618, 569, doi: 10.1086/426067 +Naidu, R. P., Tacchella, S., Mason, C. A., et al. 2020, ApJ, +892, 109, doi: 10.3847/1538-4357/ab7cc9 +Neufeld, D. A. 1990, ApJ, 350, 216, doi: 10.1086/168375 +Ocvirk, P., Aubert, D., Sorce, J. G., et al. 2020, MNRAS, +496, 4087, doi: 10.1093/mnras/staa1266 +Oesch, P. A., van Dokkum, P. G., Illingworth, G. D., et al. +2015, ApJL, 804, L30, doi: 10.1088/2041-8205/804/2/L30 +Pentericci, L., Carniani, S., Castellano, M., et al. 2016, +ApJL, 829, L11, doi: 10.3847/2041-8205/829/1/L11 +Roberts-Borsani, G., Treu, T., Mason, C., et al. 2022, arXiv +e-prints, arXiv:2207.01629. +https://arxiv.org/abs/2207.01629 +Robertson, B. E., Ellis, R. S., Furlanetto, S. R., & Dunlop, +J. S. 2015, ApJL, 802, L19, +doi: 10.1088/2041-8205/802/2/L19 +Schenker, M. A., Stark, D. P., Ellis, R. S., et al. 2012, ApJ, +744, 179, doi: 10.1088/0004-637X/744/2/179 +Sharma, M., Theuns, T., Frenk, C., et al. 2017, MNRAS, +468, 2176, doi: 10.1093/mnras/stx578 +Skelton, R. E., Whitaker, K. E., Momcheva, I. G., et al. +2014, ApJS, 214, 24, doi: 10.1088/0067-0049/214/2/24 +Smith, A., Kannan, R., Tacchella, S., et al. 2021, arXiv +e-prints, arXiv:2111.13721. +https://arxiv.org/abs/2111.13721 +Stark, D. P., Walth, G., Charlot, S., et al. 2015, MNRAS, +454, 1393, doi: 10.1093/mnras/stv1907 +Stark, D. P., Ellis, R. S., Charlot, S., et al. 2017, MNRAS, +464, 469, doi: 10.1093/mnras/stw2233 +Tilvi, V., Malhotra, S., Rhoads, J. E., et al. 2020, ApJL, +891, L10, doi: 10.3847/2041-8213/ab75ec +Trebitsch, M., Hutter, A., Dayal, P., et al. 2022, arXiv +e-prints, arXiv:2202.02337. +https://arxiv.org/abs/2202.02337 +Vanzella, E., Cristiani, S., Dickinson, M., et al. 2008, A&A, +478, 83, doi: 10.1051/0004-6361:20078332 +Veilleux, S., & Osterbrock, D. E. 1987, ApJS, 63, 295, +doi: 10.1086/191166 + +Low fesc(LyC) in three Massive, High-z Galaxies +11 +Venemans, B. P., R¨ottgering, H. J. A., Miley, G. K., et al. +2005, A&A, 431, 793, doi: 10.1051/0004-6361:20042038 +Verhamme, A., Orlitov´a, I., Schaerer, D., & Hayes, M. +2015, A&A, 578, A7, doi: 10.1051/0004-6361/201423978 +Vito, F., Di Mascia, F., Gallerani, S., et al. 2022, MNRAS, +514, 1672, doi: 10.1093/mnras/stac1422 +Yee, H. K. C., & De Robertis, M. M. 1991, ApJ, 381, 386, +doi: 10.1086/170661 +Zackrisson, E., Inoue, A. K., & Jensen, H. 2013, ApJ, 777, +39, doi: 10.1088/0004-637X/777/1/39 +Zackrisson, E., Binggeli, C., Finlator, K., et al. 2017, ApJ, +836, 78, doi: 10.3847/1538-4357/836/1/78 +Zitrin, A., Labb´e, I., Belli, S., et al. 2015, ApJL, 810, L12, +doi: 10.1088/2041-8205/810/1/L12 + diff --git a/MNE2T4oBgHgl3EQfBQYB/content/tmp_files/load_file.txt b/MNE2T4oBgHgl3EQfBQYB/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..32309fa6482e67cc49c67c4b558fd49ad0af7811 --- /dev/null +++ b/MNE2T4oBgHgl3EQfBQYB/content/tmp_files/load_file.txt @@ -0,0 +1,959 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf,len=958 +page_content='Draft version January 11, 2023 Typeset using LATEX twocolumn style in AASTeX631 Evidence for a low Lyman Continuum Escape fraction in three Massive, UV-bright galaxies at z > 7 Callum E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' Witten ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='1 Nicolas Laporte ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 3 and Harley Katz4 1Institute of Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' University of Cambridge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' Madingley Road,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' Cambridge CB3 0HA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' UK 2Kavli Institute for Cosmology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' University of Cambridge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' Madingley Road,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' Cambridge CB3 0HA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' UK 3Cavendish Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' University of Cambridge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 19 JJ Thomson Avenue,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' Cambridge CB3 0HE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' UK 4Sub-department of Astrophysics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' University of Oxford,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' Keble Road,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' Oxford OX1 3RH,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' UK Accepted for publication in ApJ ABSTRACT Although low-mass star-forming galaxies are the leading candidates of the reionisation process,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' we cannot conclusively rule out high-mass star-forming galaxies as candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' While most simulations indicate the former is the best candidate some models suggest that at z ≥ 6 massive, UV-bright galaxies “oligarchs” - account for at least 80% of the ionising budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' To test this hypothesis we target massive (log10(M⋆[M⊙]) > 10), UV-bright (MUV ∼ −22) Lyα emitters at z > 7 in archival data, observed with similar resolution spectrographs (VLT/X-shooter and Keck/MOSFIRE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' To increase the reliability of our conclusions we stack all spectra and obtain a deep-stacked spectrum of 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='75hrs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' The stacked Ly-α profile displays a clear asymmetric red peak and an absence of a blue peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' We additionally estimate the intrinsic stacked Lyα profile of our targets by correcting for IGM transmission using a range of neutral hydrogen fractions, finding no significant change in the profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' We measure a velocity offset Vred > 300 km/s and an asymmetry in our red peak A ∼ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' Using various models and estimators such as the peak separation, the asymmetry of the red peak, the ratio between Lyα and Hβ and the β slope, we conclude that the escape fraction in these three UV bright, massive (∼ 1010M⊙), z ≥ 7 galaxies is fesc(LyC) ≤ 10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' INTRODUCTION Cosmic reionisation is the processes by which the neu- tral hydrogen in the universe, formed after the recom- bination phase (380 000 years after the Big-Bang), un- derwent a phase change - to almost complete ionisation by z = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='5 (Bosman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2022, Kulkarni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2019, Robertson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' Although the period over which reionisation occurs is well constrained, the sources of the ionising photons are not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' The current debate is regard- ing whether or not the faint galaxies which represent the bulk of galaxies at z ≥ 6 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' Bouwens et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2015, Finkelstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2015, Atek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2018) are the main drivers (Ocvirk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2020, Trebitsch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2022) or if the most massive objects are the major contributors (Naidu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' Sharma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2017) to reionisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' For example, Naidu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' (2020) claim that bright galaxies may produce > 80% of the reionisation budget in order Corresponding author: Callum E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' Witten cw795@cam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='uk to match the observed rapid fall in neutral hydrogen fraction at z < 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' Directly measuring the Lyman con- tinuum radiation escaping a source and hence ionising the universe is however not possible due to the neutral- ity of the IGM during the epoch of reionisation (Inoue et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2014, eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' Instead the Lyman-α profile is heavily affected by neutral hydrogen and hence it can be used as a probe of the amount of ionising photons escaping the galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' Given the potential Lyα emission has as a probe of neutral hydrogen, research into the properties of the line emerging from a source through surrounding neutral hy- drogen were conducted by Harrington (1973);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' Neufeld (1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' The absorption of Lyα at line centre by neu- tral hydrogen results in the main escape mechanism for photons being through diffuse in frequency producing a double-peaked spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' The wings of the Lyα line, that form the consequent two peaks are defined by the density of the scattering neutral hydrogen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' As such the separation between the two peaks (Vsep) has been pro- posed as a key diagnostic of the Lyman-continuum es- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='03599v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='GA] 9 Jan 2023 ID2 Witten, Laporte and Katz cape fraction fesc(LyC) due to their tight dependence (Izotov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' Verhamme et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' While this double-peaked spectrum has now been ob- served (eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' Yee & De Robertis 1991;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' Venemans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' Vanzella et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2008), especially at high-redshift, we often see reduced or a complete absence of the blue peak (Verhamme et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' This process is now well understood as a result of inter-galactic medium (IGM) radiative transfer (RT) effects (Gunn & Peterson 1965) at high redshifts where we do not expect to find strong outflows (Vito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2022, Murray et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' Pho- tons are redshifted as they travel through the expanding universe, and hence blue peak photons get shifted into resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' If this occurs in the vicinity of neutral hydro- gen, we observe absorption, and hence any intrinsic blue peak is not observed for high neutral hydrogen column densities as predicted by Garel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' Laursen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' Emission from high redshift objects typically travels through high neutral hydrogen patches in the IGM and as such, we rarely observe blue Lyα peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' There have however been instances of double-peaked high redshift galaxies such as that discussed in Meyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' (2021) and Matthee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' These galaxies typically live within large ionised bubbles resulting in a largely re- duced neutral hydrogen column density, allowing the presence of a blue-peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' In this letter, we aim to measure the mean LyC escape fraction of bright z ≥7 Lyman Alpha Emitters (LAEs) based on their Lyα profile and to give new insights on their contribution to the reionisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' In section 2, we describe the sample of LAEs we identified at z ≥7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' We then describe the stacking method we use to improve the signal-to-noise ratio on our Lyα line in section 3 and we discuss the results and their implications in sections 4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' TARGETS SELECTION Our primary goal is to infer the mean escape frac- tion of bright, z ≥ 7 galaxies by constraining the shape and properties of Lyα, 1215.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='67˚A, redshifted into the near-infrared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' We therefore search in the literature for all z ≥7 spectroscopically confirmed galaxies that have been observed by high-resolution NIR spectrographs, to guarantee the ability to resolve any double-peak in the Lyα profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' Moreover, to obtain a good estimate of the UV beta slope, which can be used to measure the es- cape fraction at high-redshifts (Zackrisson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2017), we require that our selected galaxies have at least 3 con- straints on their SEDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 11 objects with a redshift rang- ing from 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='15 to 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='11 were selected with Lyα luminosities ranging from ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='05 to 2×1043erg/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' Additionally, Schenker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' (2012) reports the detec- tion of A1703-zd6, a redshift z = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='045 galaxy that satis- fies all of our selection criteria, except one - it is a lensed, low-mass galaxy (Stark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2015 - log(M⋆) ∼ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' We therefore do not include this target within our stack, but we note the measured Lyα red-peak separation from line-center of ∼ 60 km/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' This is consistent with a high Lyman-continuum escape fraction as expected if faint galaxies are responsible for re-ionization (Gazagnes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' Izotov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2018a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' Because the observed shape of the Lyα line profile is a result of propagation through neutral gas, it does not trace the precise redshift of a galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' In order to produce a proper stack of this emission line, we also require that the galaxies used in our study have a sys- temic redshift measurement through the observation of another emission line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' Among the 11 galaxies mentioned above, only 3 (described in table 1) satisfy this crite- ria, returning 4 datasets combining 24 hours and 45 minutes on-source exposure time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' For MOSFIRE, we make use of the following programs : C228M (PI: A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' Zitrin), Y288M (PI: Moncheva) and N190 (PI: Malho- tra) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' and for XSHOOTER : 097.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='A-0043(A) (PI: Ellis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' The three aforementioned galaxies have UV luminosi- ties MUV ∼ −22, placing them on the extreme end of galaxies luminosities at z = 7 − 8 (Bowler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2014, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' The Spectral Energy Distribution (SED) of the 3 galaxies have been extracted from 3DHST catalogues (Brammer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2012, Skelton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' The physi- cal properties of each individual galaxy have been esti- mated by SED-fitting using BAGPIPES (Carnall et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' One of the main advantages of this code is to allow the user to choose between several Star Formation Histories (SFHs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' In this work, we run BAGPIPES with 4 different SFHs, namely a burst, a constant, a delayed and a combination of burst+constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' The best SED- fit is obtained for the SFH that minimised the BIC (see Laporte et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2021 for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' BAGPIPES being a parametric code, we used the following ranges for the parameters of the SFH : ionisation parameter : log U ∈ [-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='0,-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='0] dust attenuation (assuming a Calzetti law) : Av[mag] ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='0] age of the stellar population : Age[Gyr] ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='0,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='0] mass formed : M⋆[M⊙] ∈ [106, 1012] metallicity : Z [Z⊙] ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='5] The redshift was fixed to the spectroscopic redshift of each galaxy, and the IMF used in BAGPIPES is a Low fesc(LyC) in three Massive, High-z Galaxies 3 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' The properties of the archival observations of our three target galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' zsyst indicates the systemic redshift of the target galaxy that we use for our analysis and the stellar mass is determined by the BAGPIPES fitting discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' Name zsyst log(M⋆) Exposure time Lyα luminosity Telescope Reference [M⊙] [1043 erg/s] EGSY-8p68 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='671 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='1+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='2 4 hrs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 45 min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='95±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='49 MOSFIRE Zitrin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' (2015) EGS-zs8-1 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='721 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='2+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='1 4 hrs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='2±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='1 MOSFIRE Tilvi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' (2020) 4 hrs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='2±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='2 MOSFIRE Oesch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' (2015) COSY 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='142 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='2+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='8 12 hrs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='43±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='19 X-Shooter Laporte et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' (2017) Kroupa IMF (Kroupa 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' As expected by the selec- tion function of our sample, our galaxies are good ex- amples of the most massive galaxies at z ≥7 with stellar masses ranging from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='36×1010 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='75×1010 M⊙, plac- ing them on the extreme end of the galactic stellar mass function at z = 7−8 (see Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' The properties of our stacked spectrum obtained using BAGPIPES are found in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' The detection of Lyα at z ≥7 implies the formation of an ionized bubble around the galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' Castellano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' Roberts-Borsani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' The origin of the ionising photons that produce these ionised bubbles is still highly debated and could be either due to the intrinsic nature of the object (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' star formation or an active galactic nucleus) or the over-dense environment near the most massive galaxies formed at high-redshift (Leonova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' Laporte et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' Indeed previ- ous research into these galaxies indicates that they reside within ionised bubbles, large enough that any blue-peak escaping the host galaxy would be redshifted past Lyα line-centre before leaving the ionized bubble and hence should be unaffected by IGM absorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' Based on a relation between Lyα luminosity and ion- ized bubble size derived from theoretical models, Tilvi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' (2020) estimate that EGS-zs8-1 sits in a common bubble with at least 3 neighbouring galaxies, with a size of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='02 pMpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' More recently, Leonova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' (2021) present analysis of Hubble Space Telescope (HST) imag- ing that supports the conclusion of a bubble surrounding EGS-zs8-1 given it resides in an overdensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' They addi- tionally find that EGSY-z8p68 is found with an overden- sity and again comparing to simulations find an expected bubble radius of ∼ 1 pMpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' Laporte et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' (2017) de- tect Lyα, HeII and NV, as well as upper bounds on the flux of CIII] and CIV in the spectrum of COSY allowing for the determination of the source of its radiation field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' The most likely hypothesis is that the radiation field of COSY is inconsistent with that from star-forming galax- ies, instead it likely is produced by an active galactic nucleus (AGN) (Laporte et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' Costa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2014) and additionally its high UV luminosity makes it likely to trace an overdense region (eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' Barkana & Loeb 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' Furlanetto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' Therefore it is likely COSY ad- ditionally resides within a large, ionized bubble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' METHOD Data reduction of archival Keck MOSFIRE and VLT XSHOOTER data was performed through the stan- dard MOSFIRE data reduction pipeline1 and EsoRe- flex2 which include standard data reduction procedures such as flat-fielding, wavelength calibration and back- ground subtraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' The flux calibration was performed using a bright photometric standard observed during each night.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' The systemic redshift is obtained from the additional emission lines present in each galaxy’s spectrum (EGSY- 8p69: N V, Mainali et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' EGS-zs8-1: C III], Stark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' COSY: [C II], Pentericci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' (2016)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' We then shift each spectrum into the rest frame wavelength based on the systemic redshift reported in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' Following this we define a new wavelength ba- sis, spanning the range of wavelengths of interest with a wavelength bin equal to that of our lowest resolution spectra (∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='15 ˚A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' The total flux of each object within each wavelength bin is calculated, converted into a lu- minosity, and the median of these luminosities is then taken as the value for our stacked spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' We then return this median spectrum to units of flux by dividing through by the luminosity distance of the median red- shift in our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' We take the standard deviation of the fluxes in each bin to obtain the error in our stacked spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' We then use a Monte Carlo (MC) error propagation method to estimate the uncertainties on the observed red-peak velocity offset and asymmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' We use the er- ror in our spectrum, described above, to redraw each spectral bin’s flux from a Gaussian centred on the me- dian stack value with a standard deviation equal to the aforementioned error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' We then repeat this process, mea- suring the red-peak offset and asymmetry, one hundred thousand times in order to understand the uncertainty 1 https://keck-datareductionpipelines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='io/MosfireDRP/ 2 https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='eso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='org/sci/software/esoreflex/ 4 Witten, Laporte and Katz Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' The values for various parameters used for a range of diagnostics, and the fesc(Lyα) that these diagnostics predict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' Diagnostic Value fesc(Lyα) Spitzer: flux (Hβ + [OIII]) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='13 × 10−17erg/s/cm2/˚A [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='02:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='32] BAGPIPES: flux (Hβ) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='1+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='9 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='5 × 10−17erg/s/cm2/˚A ∼ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='09:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='18] Spitzer: UV slope β and log10(EW(Hβ + [OIII])) [-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='6, -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='35], [2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='0 ˚A, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='1 ˚A ] ∼ 0 BAGPIPES: UV slope β and log10(EW(Hβ)) −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='89+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='09 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='11, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='67 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='25 ∼ 0 Lyα profile: Asymmetry and Peak separation A = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='3+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='8, Vredpeak = 330+190 −70 km/s < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='15 in our measurements driven by the error associated with each flux measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' We take the red-peak offset and asymmetry associated with our stack and measure the standard deviation of those values above and below the median for our upper and lower bound uncertainties on this measurement respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' We have additionally evaluated the results when nor- malising all of our spectra by dividing through by their peak Lyα flux, as not to weight our stack based on the most luminous galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' Given that every galaxy has a similar luminosity, we see no change in our Lyα profile regardless of normalisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' A median stacking method is chosen as it acts to reduce the affect of sky-lines, however, in the case of COSY, significant contamina- tion from a sky-line in a region of the spectrum where we expect to observe the tail of our red-peak profile did act to pollute our stacked spectrum and it was there- fore masked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' Adopting a mean stacking method and additionally changing the position and widths of bins provides a very similar stacked spectrum thus indicat- ing the robustness of our stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' RESULTS 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' Constraints from SED The Lyα escape fraction is the ratio of the Lyα flux escaping a galaxy over the intrinsic Lyα flux of the ob- ject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' In order to ascertain limits on the potential intrin- sic flux of Lyα we use recombination lines with known relations to Lyα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' Unfortunately no direct observations of recombination lines are made in any of our target spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' Instead we make use of two independent meth- ods to estimate the equivalent width (EW) of the Hβ emission line: (i) using the flux ratio between the 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='5µm and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='6µm assuming that the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='6µm flux is the stellar continuum and the 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='5µm is the sum of the stellar con- tinuum with a contamination of OIII+Hβ and (ii) using BAGPIPES to directly predict the flux of Hβ (see Ta- ble 2 for the results of both methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' The first method returns the flux of Hβ and OIII in combination and thus by assuming no contribution from OIII supplies us with a lower bound estimate of the flux of the recombina- tion line Hβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' We assume a maximum contribution of log10([OIII]/Hβ) = 1 - based on the assumption our stack lies in the extreme AGN region of the Baldwin, Philips and Terlevich (BPT) diagram (Baldwin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 1981;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' Veilleux & Osterbrock 1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' This conclusion is in itself unlikely as NV emission lines are either weak or not present in our individual galaxy spectra but allows us to obtain an absolute upper bound on the Hβ flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' The second method is based on SED-fitting and depends on the best fit parameters such as the stellar mass, the reddening, the age, the metallicity, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' To verify that the Hβ flux estimated by BAGPIPES is not strongly de- pendant on other parameters, we study the evolution of Hβ flux as a function of the metallicity and stellar mass and as a function of the metallicity and reddening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' The variation in the EW is estimated as ∆ log EW < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' We assume, following Gazagnes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' (2020), that LLyα = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='7LHα, and in turn that LHα = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='85LHβ, how- ever Lyα emission can include a large contribution from collisional excitation (Mitchell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2021) thus increasing the intrinsic LLyα further decreas- ing the measured escape fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' However, given we observe no Lyα blue-peak we must assume the Lyα profile has undergone some absorption due to IGM attenuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' Given many Lyα profiles of low redshift LAEs, that have not travelled through a high neutral hydrogen density IGM, exhibit often equal or less than equal blue-to-red peak flux ratios (eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' Izo- tov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2018a,b) we assume that the upper bound of the LLyα escaping the galaxy is double the LLyα that we observe, while the LLyα that we observe represents a lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' Taking the bounds of LHβ that we ob- tain from Spitzer provides us with a potential range of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='02 < fesc(Lyα) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='32, while the best-fit LHβ from BAGPIPES returns fesc(Lyα) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='09+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='07 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='04, while tak- ing the bounds on LLyα the best-fit value ranges from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='09 < fesc(LyC) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' The bounds of the Hβ flux as well as the best-fit value and the associated escape fraction for the stacked spec- trum can be found in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' Using relations from Maji et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' (2022) we can obtain the escape fraction of the Lyman-continuum (LyC) which is found to be lower than the Lyα escape fraction, hence we take the bounds on fesc(LyC) to be the same as those on fesc(Lyα) (also reported in Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' We additionally consider the relation between the UV slope β and the EW(Hβ) first determined by Zackrisson Low fesc(LyC) in three Massive, High-z Galaxies 5 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' Line profiles of Lyα emission in our target galaxies (top) and the stacked spectrum (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' Line centre is denoted by a black dashed line, while the red-peak flux is indicated by a red dashed line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' The velocity offset of the red-peak from line centre is additionally found within each panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' (Top:) From left to right: EGS-zs8-1, EGSY-8p68, COSY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' The resolution of the spectrum of COSY has been reduced to the resolution of the two MOSFIRE spectra (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='14 ˚A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' The hatched box indicates a region of the spectrum that has been removed due to significant pollution by a sky line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' (Bottom:) The grey region indicates the 1-sigma error obtained by taking the standard deviation of the constituent galaxies of the stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' Additionally, red peak velocity offset and the asymmetry of the red-peak as defined in Section 4 are found within the panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' (2013) at redshifts z > 6 and then at z ≈ 7 − 9 by Zackrisson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' (2017) by studying the evolution in syn- thetic galaxy spectra with changing fesc(LyC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' The re- sults of this analysis can be seen in figure 2 where galax- ies with log(EW(Hβ)) ≳ 2 exclusively have fesc(LyC) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' As is clear in figure 2 both the best-fit EW(Hβ) and UV slope β (corrected for dust extinction following Meurer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 1999) returned from BAGPIPES and the range in EW(Hβ) and UV slope β (determined follow- ing Bouwens et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2014) that can be estimated from Spitzer data (the values of which are reported in Ta- ble 2) constrain our stack to a region of the figure that is not compatible with an fesc(LyC) ≫ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' We do note that this diagnostic is potentially limited in its ability to diagnose high escape fractions given ex- amples of low redshift galaxies that have high fesc(LyC) and high EW(Hβ) (eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' Izotov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2018b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' The di- agnostic requires many assumptions in order to esti- mate fesc(LyC), notably in the stellar models employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' Changing these stellar models can significantly affect the result of the diagnostic, as seen when binary evolution is considered (see figure 6 in Zackrisson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' How- ever, we are aware of the main outcome of Zackrisson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' (2017) - that galaxies with fesc(LyC)>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='5 should have EW(Hβ) < 30 ˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' Therefore, we instead consider a high EW(Hβ) to be a necessity for low fesc(LyC), al- though perhaps not sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' Given the inclusion of multiple diagnostics all indicating low fesc(LyC) we con- sider the potential uncertainty surrounding this diagnos- tic not to be a significant issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' We additionally note that Zackrisson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' (2017) provides this diagnostic for a range of different dust attenuation laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' While the panel in figure 2 as- sumes a Calzetti attenuation law with E(B − V )stars = E(B−V )neb, we find that the conclusion, that our stack lies within a region of the diagram corresponding to fesc(LyC) = 0, is consistent regardless of the dust at- tenuation law used in Zackrisson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' Lyα profile Figure 1 clearly indicates that for all of our targets, we find the red-peak of the Lyα profile to be offset from line centre by ∼ 340 km/s, such a large separation is indica- tive of a low Lyman-continuum escape fraction (Izotov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2018b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' Gazagnes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' Kakiichi & Gronke 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' The large offset of the red-peak is additionally present in the stacked spectrum, as well as a clear asym- metry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' This asymmetry, A, is the ratio of the blue-to-red EGS-zs8-1 EGSY-8p68 COSY 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='0 Vsep = 342 km/s Vsep = 341 km/s Vsep = 344 km/s 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='0 A 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='5 S 1212 1214 1216 1218 1212 1214 1216 1218 1212 1214 1216 1218 erg 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='0 17 (10- 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='5 70 A = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='3±1:4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='8 Flux 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='5 1212 1213 1214 1215 1216 1217 1218 1219 Wavelength (A)6 Witten, Laporte and Katz Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' The indirect diagnostics of fesc(LyC), with the region which our stack resides indicated with a grey hatched box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' (Left) The EW(Hβ) of simulated z = 7 − 9 galaxies against their UV slope β, assuming a Calzetti attenuation law with E(B − V )stars = E(B − V )neb (from Zackrisson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' Their Lyman-continuum escape fraction is denoted by their colour, fesc(LyC) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='7, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='9 correspond to red, orange, green and blue respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' The grey hatched region indicates the location of our stack using Spitzer data, while the black data point indicates the position using the BAGPIPIES best-fit on the SED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' (Centre) The peak separation of Lyα profiles against their LyC escape fraction (from Kakiichi & Gronke 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' The dots denote simulated results from Kakiichi & Gronke (2021), while crosses indicate the results for z ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='3 LyC-detected galaxies from Izotov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' (2016, 2018a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' The coloured regions indicate the three regimes of LyC escape - leakage by full break, through holes and small leakage with few or no holes indicated by blue, red and grey respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' (Right) The red peak asymmetry of Lyα profiles against their LyC escape fraction (from Kakiichi & Gronke 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' The markers and shading are the same as the central panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' flux of the red peak (as defined in Kakiichi & Gronke 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' We find that the two targets for which we are able to observe the shape of the red-peak profile, we observe clear asymmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' DISCUSSION The limits placed on fesc(Lyα) from photometry dis- cussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='1 are already low enough to rule out the possibility of these three massive, bright galaxies currently being significant contributors to re-ionization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' However, we wish to use multiple diagnostics in order to confirm these findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' The results from our Lyα stack are therefore crucial to further constrain the escape frac- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' Interpretation of the velocity offset While the velocity offset of Lyα from the systemic redshift initially appears as though it may be primarily driven by outflows in these massive galaxies, we believe this to be unlikely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' Neufeld (1990) and Michel-Dansac et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' (2020), using a static medium with large neutral hydrogen column densities, find velocity offsets in their simulations that are comparable to those that we ob- serve indicating these velocity separations are achievable within simulations without modelling for outflows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' Any such shift in the Lyα profile due to the expansion velocity, vexp, of neutral gas would still result in the expected double-peaked profile of Lyα centred on the systemic redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' Results from Verhamme et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' (2015) indicate that increasing vexp has the effect of reducing the peak separation, such that when vexp > 300 km/s, they cannot recreate the red peak to line-centre separa- tion that we observe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' These results constrain the neu- tral gas column density in our stack to be greater than 1020cm−2 and any outflow velocity vexp < 300 km/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' The conclusion that any expansion velocity will act to reduce peak separation thus allows us to conclude that our red-peak offset is a minimum separation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' Addition- ally, most diagnostics of escape fraction that use the sep- aration between the peaks of the Lyα profile are based on observations of lower mass galaxies than our targets and therefore using any such diagnostic is challenging (Izotov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2018b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' Gazagnes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' Instead we choose to use the relation determined by simulations from Kakiichi & Gronke (2021), allowing us to avoid mass biases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' The observed red-peak-offset in Figure 1 can be used as a lowest bound for the blue-red peak sep- aration given that we know the blue peak lies on the blue side of line-centre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' As such, we expect the blue-red peak separation to far exceed the red-peak offset of ∼ 300 km/s and comparing this to the relation from Kakiichi & Gronke (2021) we find that fesc(Lyα)≲ 10% and Lyα photons escape through an optically-thick medium with few or no holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' Interpretation of the red-peak asymmetry 700 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='00 fesc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='0 =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='5 600 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='25 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='9 Red peak asymmetry 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='50 Peak separation [km 500 Slope x 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='75 400 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='00 300 Xx 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='25 200 x 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='50 100 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='0 log10(EW(Hβ))(A) fesc(LyC) fesc(LyC)Low fesc(LyC) in three Massive, High-z Galaxies 7 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' The potential intrinsic stacked spectrum created by dividing the observed spectrum by the IGM transmission for a range of different volumetric neutral fractions indicated in the top left of each panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' The spectrum normalised by the red peak flux is indicated by the solid black line, while the IGM attenuation curve associated with the volumetric neutral fraction, taken from Garel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' (2021), is indicated by the red dashed line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' This asymmetry allows us to quantify the amount Lyα photons have to scatter, in doing so creating a broad wing component of the emission line, in order to escape the galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' A high asymmetry (A > 3) is hence indica- tive of Lyα photons having multiple routes to escape and hence scatter significant amounts in order to find low- density channels to escape the galaxy (leakage through holes), while a low asymmetry (A < 3) is indicative of Lyα photons only having one method of escape possible either through predominantly optically-thin (leakage by full break) or optically thick (small leakage due to few or no holes) media Kakiichi & Gronke (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' There- fore we can use the asymmetry to attempt to diagnose the properties of the medium through which the Lyα photons have traversed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' The asymmetry that we observe, in Figure 1, is an upper bound on the asymmetry of the intrinsic spec- trum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' This is due to IGM attenuation reducing the flux close to line-centre hence reducing the flux between the red-peak and line-centre relative to the flux on the red side of the red-peak, therefore the observed asymmetry is greater than the intrinsic asymmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' As such we find the asymmetry, A < 3, results in the interpreta- tion that Lyα photons have either escaped through a full break environment or by leakage through few or no holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' Given the aforementioned limits on the escape fraction (fesc(Lyα)≲ 10%) we can constrain ourselves to small leakage without the presence of optically-thin channels (Kakiichi & Gronke 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' Effects of IGM attenuation Neutral hydrogen in the IGM causes attenuation of Lyα close to line-centre (see Garel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2021), therefore in order to confirm that we do not misidentify the loca- tion of the red-peak flux, we divide our observed stacked spectrum through by attenuation curves for varying vol- umetric neutral fraction, taken from Garel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' Given our target galaxies all likely reside within large ionized bubbles we do not expect significant IGM ab- sorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' Figure 3 indicates the effect of correcting for the different IGM transmission curves and we see no no- table difference in our observed spectra even at the most extreme volumetric neutral fraction, indicating our ob- served red peak separation likely trace that intrinsic to the galaxy before IGM absorption of Lyα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' While we do see a notable decrease in the asymmetry, as predicted in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='2, we have already consider the asymmetry to be a lower bound and as such this does not affect the interpretation of the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' We do note a significant increase in the flux at line- centre, in Figure 3, due to the effectively zero transmis- sion through the IGM at that wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' This is merely an artefact of noise being divided through by a number tending to zero rather than any physical intrinsic prop- erty of the galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' We know this to be true as Lyα emission will immediately be absorbed at line-centre by any neutral hydrogen within the host galaxy and as such we must observe negligible flux at line-centre escaping the host galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' Finally, in order to confirm that we are not being af- fected by high escape fraction interlopers that due to IGM transmission, spectral resolution and noise are be- ing interpreted as having a low escape fraction, we at- tempt to recreate high redshift observations of galax- ies with fesc(LyC) greater than our sample (fesc(LyC)> 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' We use the spectra of galaxies from Izotov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='0 XHI = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='000057 XHI = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='0065 XHI = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='35 XHI = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='64 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='8 flux lission 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='6 Transmi 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='4 JON 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='4 IGM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='0 12161217 12181219 12181219 12181219 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='0 1216 1216 1217 1216 1218 1219 Wavelength[A] Wavelength [A] Wavelength[Ai Wavelength[A]8 Witten, Laporte and Katz Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' The simulated spectra of high redshift, high fesc(LyC) galaxies for varying assumed neutral hydrogen fractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' Each column uses an increasing neutral hydrogen fraction from left to right, that in turn dictates which IGM attenuation curve, taken from Garel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' (2021), is applied to the original spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' Each panel includes the same 4 galaxies from Izotov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' (2018a,b) (clockwise from top left sub-panel: J1011+1947, J1256+4509, J1243+4646, J1154+2443), whose Lyman continuum escape fraction are indicated at the top of each sub-panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' The simulated observed spectrum normalised by the red peak flux is indicated by the solid black line, while the original spectrum (with resolution degraded) is indicated in grey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' (2018a,b) as examples of Lyα profiles associated with an fesc(LyC) greater than our sample up to a value of 72% at low redshifts (z ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' We apply the IGM attenua- tion curves from Garel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' (2021) to the Lyα profile, we then reduce the resolution of these spectra down to the resolution of our stacked Lyα profile and finally we use the MC error propagation described in Section 3 to estimate uncertainties on the Vred and asymmetry of each galaxy given a noise level similar to our stacked spectrum (by assuming the peak flux to be at SN = 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' Figure 4 shows our simulated high redshift, high fesc(LyC) spectra with various IGM transmission curves applied for differing neutral hydrogen fractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' We find that applying the IGM transmission and degrading the resolution of the spectra result in velocity offsets that are consistent with the original spectra, even for the most extreme neutral hydrogen density, to within 40 km/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' This peak separation when considered in the con- text of the Kakiichi & Gronke (2021) diagnostic appears to indicate these galaxies are likely high fesc(LyC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' All of these galaxies exhibit red peak offsets ≪ 300 km/s thus allowing us to conclude that the observation of a peak offset of ∼ 300 km/s is not only indicative of a low fesc(LyC) but also that we are likely not being af- fected by high fesc(LyC) interlopers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' We do however note that the asymmetry is more challenging to under- stand as a function of the neutral fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' It is clear that for a sharp Lyα red-peak, increasing the neutral density will act to increase the flux on the red side of this peak hence increasing the asymmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' When the red-peak is more broad, increasing the neutral density can push the peak of the Lyα profile red-ward and hence act to increase the amount of flux on the blue side of the peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' This complicated interplay of effects leads to a highly uncertain asymmetry in some of our Lyα profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' Therefore, the use of asymmetry to diagnose fesc(LyC) alone at high redshifts, where neutral hydrogen in the IGM causes large uncertainties on the intrinsic asymme- try, should be avoided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' However, the asymmetry of our stacked spectrum appears relatively well defined with a sharp drop in flux blue-ward of the peak flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' There- fore, we conclude that our asymmetry is most likely an upper-bound asymmetry, where the relative boosting of flux to the red side of the Lyα peak due to IGM attenua- tion, is the most likely of the two aforementioned effects at play.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' Given the relatively small uncertainty on the intrinsic Lyα asymmetry of our stacked spectrum we believe that, in combination with multiple other diag- XHl 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='0065 XHl- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='35 XHl 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='64 1 fesc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='11 fesc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='38 1 fesc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='11 fesc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='38 1 fesc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='11 fesc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='38 Vsep = 14028 Vsep = 1408 Vsep = 1408 Vsep = 1408 Vsep = 1408 Vsep = 180±28 A= 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='4: A= 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='13 A= 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='08 A= 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='12 A= 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='98 A= 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='2±3 Observed Intrinsic 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='5 flux flux flux lised lised lised 0 Q Q Normal 1 fesc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='46 fesc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='73 Normal 1 fesc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='46 fesc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='73 Normal 1 fesc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='46 fesc= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='73 Vsep = 110+28 Vsep =14020 Vsep = 140±28 Vsep = 1800 Vsep = 140+8 Vsep =1800 A= 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='8±:9 A= 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='5±:8 A= 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='8±:6 A= 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='1±3:3 A= 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='1: A元5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='5 Q - 0 1216 1217.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='5 1219 1216 1217.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='5 1219 1216 1217.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='5 1219 1216 1217.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='5 1219 1216 1217.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='5 1219 1216 1217.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='5 1219 wavelength (A) Wavelength (A) Wavelength (A)Low fesc(LyC) in three Massive, High-z Galaxies 9 nostics, the observed asymmetry can be used to infer the ability of Lyα photons to escape their host galaxy, with the caveat that this diagnostic should not be used on high redshift LAEs without the use of other supple- mentary diagnostics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' SUMMARY In order to probe the potential ionising properties of the most massive (log10(M⋆[M⊙]) > 10), UV-bright (MUV ∼ −22), high redshift (z > 7) galaxies, we target all archival data on telescopes with resolution R ≳ 3000, allowing us to obtain a resolved Lyα profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' We find a total of four observations of three satisfactory galax- ies with Lyα emission, totalling an exposure of 24 hours and 45 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' Using a median stacking method we ob- tain a deep stacked spectrum representing massive, UV- bright, high redshift Lyα leaking galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' Through the analysis of the stacked Lyα profile, using the red-peak velocity offset from line-centre and the red-peak asym- metry, we deduce the Lyman-continuum escape fraction to be less than 10% and that the few Lyman-continuum photons that do escape, escape through an optically- thick medium with few or no holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' Through the use of Spitzer observations of our target galaxies, stack- ing these and SED-fitting using BAGPIPES we obtain bounds on the recombination line Hβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' Given this we constrain the escape fraction to 9% < fesc(Lyα)< 18% in strong agreement with the results of our stacked Lyα profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' We additionally confirm that neither IGM at- tenuation or a significant outflow velocity could affect our conclusion regarding a low fesc(Lyα) for massive, UV-bright, high redshift galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' Our study shows that despite the fact the 3 galaxies analysed lie within ionised bubbles, they are not capable themselves of ion- ising their own bubbles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' However, we emphasize that our result is obtained using only 4 datasets of 3 different galaxies at z ≥7 – the only observations currently available in telescope archives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' Increasing the number of Lyα detections at z ≥7 with high-resolution spectrographs is therefore crucial to confirm our conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' Furthermore, the high-fraction of neutral gas underlying galaxies within the epoch of reionisation limits the detection of Lyα to galaxies in overdense regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' Spectrographs with a large field-of-view will therefore be ideal instruments to push forward this project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' MOONS, a 3rd generation instrument at the Very Large Telescope, will be one of those.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' It combines high-resolution (R>4000), a large field of view (∼500 arcmin2) and a huge number of fi- bres (∼1000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' ACKNOWLEDGEMENTS We thank the anonymous referee for providing help- ful comments which improved the quality of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' CW and NL acknowledge advice and comments from Debora Sijacki, Martin Haehnelt, Roberto Maiolino, Sergio Martin-Alvarez and Yuxuan Yuan that helped to direct our analysis and the diagnostics used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' CW acknowledges support from the Science and Technol- ogy Facilities Council (STFC) for a Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' studentship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' NL acknowledges support from the Kavli foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' This research has made use of the Keck Observatory Archive (KOA), which is operated by the W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' Keck Observatory and the NASA Exoplanet Science Institute (NExScI), under contract with the National Aeronau- tics and Space Administration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' Based on observations collected at the European Southern Observatory under ESO programme 097.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='A-0043(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' This work is based on observations taken by the 3D-HST Treasury Program (GO 12177 and 12328) with the NASA/ESA HST, which is operated by the Association of Universities for Re- search in Astronomy, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', under NASA contract NAS5- 26555 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' DATA AVAILABILITY The data underlying this article will be shared on rea- sonable request to the corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' REFERENCES Atek, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', Richard, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', Kneib, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', & Schaerer, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2018, MNRAS, 479, 5184, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='1093/mnras/sty1820 Baldwin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', Phillips, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', & Terlevich, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 1981, PASP, 93, 5, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='1086/130766 Barkana, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', & Loeb, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2004, ApJ, 609, 474, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='1086/421079 Bosman, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', Davies, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', Becker, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2022, MNRAS, 514, 55, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='1093/mnras/stac1046 Bouwens, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', Illingworth, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', Oesch, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2014, ApJ, 793, 115, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='1088/0004-637X/793/2/115 —.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2015, ApJ, 803, 34, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='1088/0004-637X/803/1/34 Bowler, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', Jarvis, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', Dunlop, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2020, MNRAS, 493, 2059, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='1093/mnras/staa313 Bowler, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', Dunlop, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', McLure, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2014, MNRAS, 440, 2810, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='1093/mnras/stu449 Brammer, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', van Dokkum, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', Franx, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2012, ApJS, 200, 13, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='1088/0067-0049/200/2/13 Carnall, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', McLure, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', Dunlop, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', & Dav´e, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2018, MNRAS, 480, 4379, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='1093/mnras/sty2169 10 Witten, Laporte and Katz Castellano, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', Pentericci, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', Cupani, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2022, arXiv e-prints, arXiv:2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='04976.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='org/abs/2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='04976 Costa, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', Sijacki, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', Trenti, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', & Haehnelt, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2014, MNRAS, 439, 2146, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='1093/mnras/stu101 Finkelstein, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', Ryan, Russell E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', Papovich, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2015, ApJ, 810, 71, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='1088/0004-637X/810/1/71 Furlanetto, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', Zaldarriaga, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', & Hernquist, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2004, ApJ, 613, 1, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='1086/423025 Garel, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', Blaizot, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', Rosdahl, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2021, MNRAS, 504, 1902, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='1093/mnras/stab990 Gazagnes, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', Chisholm, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', Schaerer, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', Verhamme, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', & Izotov, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2020, A&A, 639, A85, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='1051/0004-6361/202038096 Gunn, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', & Peterson, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 1965, ApJ, 142, 1633, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='1086/148444 Harrington, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 1973, MNRAS, 162, 43, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='1093/mnras/162.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='43 Inoue, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', Shimizu, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', Iwata, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', & Tanaka, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2014, MNRAS, 442, 1805, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='1093/mnras/stu936 Izotov, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', Schaerer, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', Thuan, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2016, MNRAS, 461, 3683, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='1093/mnras/stw1205 Izotov, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', Schaerer, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', Worseck, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2018a, MNRAS, 474, 4514, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='1093/mnras/stx3115 Izotov, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', Thuan, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', & Guseva, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2017, MNRAS, 471, 548, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='1093/mnras/stx1629 Izotov, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', Worseck, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', Schaerer, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2018b, MNRAS, 478, 4851, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='1093/mnras/sty1378 Kakiichi, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', & Gronke, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2021, ApJ, 908, 30, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='3847/1538-4357/abc2d9 Kroupa, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2001, MNRAS, 322, 231, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='1046/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='1365-8711.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='04022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='x Kulkarni, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', Keating, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', Haehnelt, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2019, MNRAS, 485, L24, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='1093/mnrasl/slz025 Laporte, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', Meyer, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', Ellis, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2021, MNRAS, 505, 3336, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='1093/mnras/stab1239 Laporte, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', Nakajima, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', Ellis, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2017, ApJ, 851, 40, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='3847/1538-4357/aa96a8 Laporte, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', Zitrin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', Dole, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2022, arXiv e-prints, arXiv:2208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='04930.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='org/abs/2208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='04930 Laursen, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', Sommer-Larsen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', & Razoumov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2011, ApJ, 728, 52, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='1088/0004-637X/728/1/52 Leonova, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', Oesch, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', Qin, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2021, arXiv e-prints, arXiv:2112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='07675.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='org/abs/2112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='07675 Mainali, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', Zitrin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', Stark, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2018, MNRAS, 479, 1180, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='1093/mnras/sty1640 Maji, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', Verhamme, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', Rosdahl, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2022, arXiv e-prints, arXiv:2204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='02440.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='org/abs/2204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='02440 Matthee, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', Sobral, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', Gronke, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2018, A&A, 619, A136, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='1051/0004-6361/201833528 Meurer, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', Heckman, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', & Calzetti, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 1999, ApJ, 521, 64, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='1086/307523 Meyer, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', Laporte, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', Ellis, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', Verhamme, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', & Garel, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2021, MNRAS, 500, 558, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='1093/mnras/staa3216 Michel-Dansac, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', Blaizot, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', Garel, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2020, A&A, 635, A154, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='1051/0004-6361/201834961 Mitchell, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', Blaizot, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', Cadiou, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2021, MNRAS, 501, 5757, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='1093/mnras/stab035 Murray, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', Quataert, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', & Thompson, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2005, ApJ, 618, 569, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='1086/426067 Naidu, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', Tacchella, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', Mason, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2020, ApJ, 892, 109, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='3847/1538-4357/ab7cc9 Neufeld, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 1990, ApJ, 350, 216, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='1086/168375 Ocvirk, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', Aubert, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', Sorce, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2020, MNRAS, 496, 4087, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='1093/mnras/staa1266 Oesch, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', van Dokkum, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', Illingworth, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2015, ApJL, 804, L30, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='1088/2041-8205/804/2/L30 Pentericci, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', Carniani, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', Castellano, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2016, ApJL, 829, L11, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='3847/2041-8205/829/1/L11 Roberts-Borsani, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', Treu, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', Mason, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2022, arXiv e-prints, arXiv:2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='01629.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='org/abs/2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='01629 Robertson, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', Ellis, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', Furlanetto, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', & Dunlop, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2015, ApJL, 802, L19, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='1088/2041-8205/802/2/L19 Schenker, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', Stark, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', Ellis, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2012, ApJ, 744, 179, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='1088/0004-637X/744/2/179 Sharma, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', Theuns, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', Frenk, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2017, MNRAS, 468, 2176, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='1093/mnras/stx578 Skelton, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', Whitaker, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', Momcheva, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2014, ApJS, 214, 24, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='1088/0067-0049/214/2/24 Smith, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', Kannan, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', Tacchella, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2021, arXiv e-prints, arXiv:2111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='13721.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='org/abs/2111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='13721 Stark, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', Walth, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', Charlot, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2015, MNRAS, 454, 1393, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='1093/mnras/stv1907 Stark, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', Ellis, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', Charlot, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2017, MNRAS, 464, 469, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='1093/mnras/stw2233 Tilvi, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', Malhotra, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', Rhoads, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2020, ApJL, 891, L10, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='3847/2041-8213/ab75ec Trebitsch, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', Hutter, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', Dayal, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2022, arXiv e-prints, arXiv:2202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='02337.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='org/abs/2202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='02337 Vanzella, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', Cristiani, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', Dickinson, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2008, A&A, 478, 83, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='1051/0004-6361:20078332 Veilleux, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', & Osterbrock, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 1987, ApJS, 63, 295, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='1086/191166 Low fesc(LyC) in three Massive, High-z Galaxies 11 Venemans, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', R¨ottgering, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', Miley, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2005, A&A, 431, 793, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='1051/0004-6361:20042038 Verhamme, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', Orlitov´a, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', Schaerer, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', & Hayes, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2015, A&A, 578, A7, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='1051/0004-6361/201423978 Vito, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', Di Mascia, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', Gallerani, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2022, MNRAS, 514, 1672, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='1093/mnras/stac1422 Yee, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', & De Robertis, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 1991, ApJ, 381, 386, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='1086/170661 Zackrisson, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', Inoue, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', & Jensen, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2013, ApJ, 777, 39, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='1088/0004-637X/777/1/39 Zackrisson, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', Binggeli, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', Finlator, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2017, ApJ, 836, 78, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='3847/1538-4357/836/1/78 Zitrin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', Labb´e, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', Belli, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content=' 2015, ApJL, 810, L12, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} +page_content='1088/2041-8205/810/1/L12' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE2T4oBgHgl3EQfBQYB/content/2301.03599v1.pdf'} diff --git a/MdE1T4oBgHgl3EQfZQR8/content/tmp_files/2301.03148v1.pdf.txt b/MdE1T4oBgHgl3EQfZQR8/content/tmp_files/2301.03148v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..b52c17db0d1703cbc5c8fc0c574ef5091d1db907 --- /dev/null +++ b/MdE1T4oBgHgl3EQfZQR8/content/tmp_files/2301.03148v1.pdf.txt @@ -0,0 +1,1910 @@ +Reducing Datacenter Operational Carbon Emissions Effectively +by Cooperating with the Grid +Liuzixuan Lin +University of Chicago +Chicago, IL, USA +lzixuan@uchicago.edu +Andrew A. Chien +University of Chicago & Argonne National Lab +Chicago, IL, USA +achien@cs.uchicago.edu +ABSTRACT +Facing growing concerns about power consumption and carbon +emissions, cloud providers are adapting datacenter loads to reduce +carbon emissions. With datacenters exceeding 100MW, they can +affect grid dynamics, so doing this without damaging grid perfor- +mance is difficult. +We study power adaptation algorithms that use grid metrics +and seek to reduce datacenter (DC) operational carbon emissions. +First, we consider local, online adaptation. Second, to reduce grid +disruption that can arise from adaptation, we consider using an +external coordinator that limits aggregate impact. Finally, we study +a novel cooperative scheme, where datacenters first create a full- +day adapted power plan, based on day-ahead information, and then +share it with the power grid. This novel approach is a partner- +ship that reflects the shared responsibility between datacenter and +the grid for carbon minimization. For each, we report DC carbon +emissions reduction and grid impacts. +Results show that PlanShare, the novel cooperative scheme is +superior to all online approaches considered, achieving a net bene- +fit of 12.6% reduction in datacenter operational carbon emissions. +PlanShare achieves 26%–132% better performance than the best +online approach, and far larger for many. This benefit arises from +allowing the grid optimization with full future DC load information. +Further it enables efficient DC resource management with a 24-hour +capacity schedule. It also decreases power cost for the entire grid +(datacenters and other customers). +CCS CONCEPTS +• Applied computing → Data centers; • Hardware → Power +and energy. +KEYWORDS +Data centers, Carbon emissions, Power management, Adaptive +loads +ACM Reference Format: +Liuzixuan Lin and Andrew A. Chien. 2023. Reducing Datacenter Operational +Carbon Emissions Effectively by Cooperating with the Grid. In Proceedings +Permission to make digital or hard copies of all or part of this work for personal or +classroom use is granted without fee provided that copies are not made or distributed +for profit or commercial advantage and that copies bear this notice and the full citation +on the first page. Copyrights for components of this work owned by others than ACM +must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, +to post on servers or to redistribute to lists, requires prior specific permission and/or a +fee. Request permissions from permissions@acm.org. +Conference’17, July 2017, Washington, DC, USA +© 2023 Association for Computing Machinery. +ACM ISBN 978-x-xxxx-xxxx-x/YY/MM...$15.00 +https://doi.org/10.1145/nnnnnnn.nnnnnnn +of ACM Conference (Conference’17). ACM, New York, NY, USA, 15 pages. +https://doi.org/10.1145/nnnnnnn.nnnnnnn +1 +INTRODUCTION +With the commercial success of both internet-scale applications +and public cloud computing, cloud computing infrastructure has +continued to grow rapidly. A recent article documented the addi- +tion of over 50 datacenters a year by a single cloud provider [67]. +Rapid growth of cloud provider revenue and documented power +purchases confirm Amazon, Microsoft, and Google’s cloud growth +rates exceeding 30% annually for the past 5 years [25, 39, 63]. As +of 2021, the total power consumption of these three leading cloud +providers is over 64 TWh, equivalent to the power consumption of +approximately 6.4 million American homes. Some estimates project +datacenter power consumption growing to more than 10% of global +electric power use by 2030 [38, 46, 56]. The largest cloud datacenter +sites consist of multiple buildings with power footprints from 200 +MW to 1 GW [2, 6, 29, 59, 67]. +This rapid growth raises concerns about associated carbon emis- +sions. Cloud providers purchase renewable power (long-term pur- +chase contracts) or renewable offsets (renewable-energy credits— +RECs) to assuage concerns [16, 63]. Several have announced “24x7” +matching efforts to match their power use and renewable generation +hourly [16, 28, 30]. However, these contracts are legal, accounting +arrangements, not actual transfers of power. That is, even with +these arrangements, the cloud datacenters cause and consume large +quantities of fossil-fuel generated power. +Datacenters are major load contributors in many power grids. +For example, in Virginia, datacenters account for 12% of grid power +consumption today, and are projected to grow to 18% by 2027 and +22% by 2032 [23, 24]. The environmental impacts of datacenters +there have drawn much attention [2]. Another hotspot is Ireland +where datacenters currently account for 14% of national electricity +use [10] and could be 30% by 2029 [26]. Worried that datacenter load +growth may retard grid decarbonization, Ireland’s state grid opera- +tor is canceling datacenter projects [40]. As computing continues its +rapid growth, datacenters will exceed 10% of load in an increasing +number of grids—with the concomitant challenges [2, 35, 38, 56]. +One promising idea to reduce the carbon emissions (and also +power cost from cloud provider perspective) of datacenters is power +adaptation. This reflects the goal of aligning power consumption +with renewable generation. Over a decade, computer systems re- +searchers have proposed many creative ideas for dynamic load +shaping across Internet datacenters with the objective of reducing +carbon emissions [19, 20, 27, 31, 50, 51] or power cost [5, 43, 47, 52, +53, 65, 68, 73, 78]. These approaches seek to align power use with +1 +arXiv:2301.03148v1 [cs.DC] 9 Jan 2023 + +Conference’17, July 2017, Washington, DC, USA +Liuzixuan Lin and Andrew A. Chien +low-carbon or low-price power, using online control techniques as +illustrated in Figure 1. +Power Grid +Datacenters +Resource +Manager +Varying +Power +Load +Carbon Emissions, +Power Price… +Workload +Figure 1: Datacenters can adapt power use to align with pe- +riods of lower carbon emissions. +Many efforts to shape power consumption or shift load to re- +duce carbon emissions focus on internal challenges of datacenter +management and compute load prediction. One recently deployed +at scale creates a daily compute capacity plan to enable effective +resource management [66]. However, most ignore the negative +impacts of varying datacenter power on grid dynamics (e.g. trigger- +ing unnecessary generation starts and load shedding). With large +datacenter loads, this coupling of power adaptation and grid im- +pacts is critical: 10-20% of grid load is large enough to alter power +grid dynamics. Load control schemes thought to be productive for +small datacenter loads can fail to deliver benefits (carbon emissions, +power cost) and even damage grid performance if used with large +datacenter loads (e.g. 21.7% increase in datacenter carbon emissions +from large loads’ overshifting in Section 3). As a result, it’s impor- +tant for datacenter power adaptation to be studied with power grid +models [46]. One consequence of this new insight is that there is no +known solution to coupled management of large-scale datacenter +power adaptation and power grids. +With the goal of finding widely usable solutions to effectively +reduce datacenter operational carbon emissions, we explore grid +metrics that can guide datacenter power adaptation and different +power adaptation approaches. For each approach, we focus on +the effectiveness in datacenter carbon reduction. We also consider +datacenter capacity variation from power adaptation, and impacts +on power cost for both datacenters and other grid customers. +To achieve this goal, we propose a novel cooperative scheme, +PlanShare, where datacenters share their full-day adaptation plan +with the grid in advance. The certainty for the varied datacen- +ter power use enables the power grid to optimize generation and +transmission schedule across time periods, achieving most carbon +reduction without harming the grid. Overall, this is a widely usable +solution that satisfies both datacenter and power grid objectives. +Specific contributions of the paper include: +• We consider three metrics (average carbon intensity (ACI), +grid price (GPrice), and locational marginal price (LMPrice)), +comparing their effectiveness for online control algorithms +that adapt computing load to reduce carbon emissions. LM- +Price is the most effective by reducing 1–5% of datacenter +carbon emissions vs. ACI and 0.7–1.5% vs. GPrice. A further +advantage is that LMPrice is practically available. +• We next consider how to improve online power adaptation. +More detailed future price information (+2.3%) and step size +(+2.3%) can increase the effectiveness of local adaptation +based on LMPrice, bringing a total datacenter carbon reduc- +tion of 10%. Another approach with a coordinator to limit +aggregate load change gives a lower improvement, reducing +datacenter carbon emissions by 7.3%. +• The plan sharing approach (PlanShare) achieves the greatest +benefit, decreasing 12.6% of operational carbon emissions. +PlanShare significantly outperforms best online approach +with 26%–132% increase in carbon-emissions reductions for +datacenter. Other benefits include efficient DC resource man- +agement with a 24-hour capacity plan and power cost reduc- +tion for both the datacenters and non-datacenter customers. +The remainder of the paper includes background in Section 2. In +Section 3 we describe the challenges of effective power adaptation +for carbon reduction with datacenters as large loads in the grid, +and then in Section 4 we give an overview of the datacenter power +adaptation approaches we study addressing the challenges. Section +5 introduces the methodology of grid-coupled simulation. In Section +6 we evaluate different power adaptation approaches’ effectiveness +in datacenter carbon reduction, with other impacts on datacenter +and non-datacenter grid customers considered. Finally, in Section +7 and 8, we discuss related work, summarize the paper, and give +some future directions. +2 +BACKGROUND +2.1 +Growth of Cloud Datacenter Load in Power +Grids +Hyperscale cloud providers (e.g. Amazon, Microsoft, Google, Al- +ibaba) are building larger and more datacenters to meet the needs +of digitalization, which accelerated since the Covid-19 pandemic +[24, 62, 69]. The rapid growth of hyperscale cloud industry is re- +flected in its power consumption: 1% of worldwide power con- +sumption today, projections suggest it could exceed 10% by 2030 +[38, 46, 56]. Similar rapid growth rates are also supported by cor- +porate renewable power purchase data [39, 63]. +As a result, datacenters have become the key driver of new elec- +tric power demand, direct cause of new emissions, as well as con- +struction of power plants, transmission, and energy storage infras- +tructure [23, 60]—it’s no exaggeration to say that cloud datacenters +are shaping the future of power grids. In Virginia, US, datacenters +account for about 12% of regional power consumption as of today, +projected to be 18% in 5 years and 22% in 10 years [23, 24]. An- +other example is Ireland, where datacenters reach 14% of national +electricity use today [10] and could use 30% by 2029 [26]. Worried +that datacenter load growth may hold back grid decarbonization, +Ireland’s state grid operator is canceling datacenter projects [40], +unless datacenters are equipped with their own renewable genear- +ion or flex their power demand. Similar story is happening in more +and more parts of the world as the scale and reach of datacenters +continue to grow [2, 35, 38, 56]. In short, growing cloud datacenter +power demand is a top-shelf concern for power grids around the +world, challenging grid operation and decarbonization. +2 + +Reducing Datacenter Operational Carbon Emissions Effectively by Cooperating with the Grid +Conference’17, July 2017, Washington, DC, USA +From the cloud provider perspective, such exponential growth in +datacenter power consumption first translates into growing power +cost, as power price [3] don’t decrease at the same speed. Also, it +raises concern about carbon emissions growth. Obscured by car- +bon offset or renewable purchase, additional datacenters can still +result in more fossil fuel power plants if the load is not aligned +with renewable generation [7, 22, 26]. To improve sustainability, +Google and Microsoft seek to match their load with renewable +generation (24x7, 100/100/0) hourly [30, 39], with attempts such +as temporal load shaping [66]. While the fraction of dynamic load +and scale of load change are small (a few percent of capacity [66]), +they are expected to increase in future. Furthermore, such load +change behaviors can disturb the grid even at a small percentage +of dynamic load. There are already reports of power variations +from supercomputer loads (all on or all off) affecting grid stability +[70]. For gigawatt cloud datacenters, a 10% load change produces +a 100 MW swing, similar to the dynamic range we study; a 40% +load change, which is not unusual for some diurnal peak to trough, +would be 400 MW! +2.2 +Dynamics of a Renewable-based Power +Grid +Climate agreements seek to limit global warming to below 1.5◦C +above pre-industrial to avoid catastrophic climate-related risk. This +means the world’s carbon emissions need to be halved by 2030 and +net-zero by 2050 [57]. Power grid decarbonization is necessary for +achieving these goals. Recent years have seen the rise of renewable +sources in energy mix of many power grids across the world, such +as 34% in California (2021), 22% in China (2019), and 22% in Europe +(2020). Some regions have set more ambitious goals for this decade. +For example, California aims at 60% renewable fraction by 2030 +[18], and Germany plans to phase out coal power plants (20% of +power generation in 2020) by 2030 [74]. +Integrating intermittent renewable sources (mainly wind and +solar) is challenging for the power grid. For example, renewable +generation can be wasted due to temporal mismatch with energy +demand and transmission limits, producing “curtailment” [8, 13, 15, +33, 34, 45]. A complementary problem is generation shortage, such +as under extreme weather. One key to meet this challenge is to +increase supply or demand flexibility, and corresponding solutions +include energy storage, adaptive loads, etc [37]. +0 +6 +12 +18 +24 +Time +0 +100 +200 +300 +ACI (kg CO2/MWh) +0 +6 +12 +18 +24 +Time +0 +50 +100 +Price ($/MWh) +Figure 2: CAISO’s Daily Average Carbon Intensity and Price +Variation, 2022/05/02. Left: Average Carbon Intensity (kg +CO2/MWh), Right: Grid Price ($/MWh). Source: CAISO. +The intermittency of renewable sources reflects in grid metrics, +producing more fluctuating generation mix and associated carbon +intensity (carbon emissions per MWh energy consumption) as re- +newable fraction increases. Figure 2 (left) shows California’s typical +daily carbon intensity pattern: the average carbon intensity keeps +low at near zero during the daytime when solar generation domi- +nates, but it climbs up to about 200 kg CO2/MWh as natural gas +generators are up. In addition, as wind and solar generation comes +at zero fuel cost, the fluctuating generation mix results in time- +varying price. When there is plenty of solar generation, the price +can drop to near zero (Figure 2, right)! Adaptive loads such as data- +centers, electric vehicles, and smart appliances may use variation +in these grid metrics to reduce their carbon emissions or power +cost. +3 +PROBLEM +Cloud datacenters seek to reduce their carbon emissions by align- +ing their power use with periods of plentiful renewable generation +(low-carbon periods). However, power grid carbon is determined +by a complex set of factors including load, generation available, +transmission topology, and ramping rate limits. In contrast to the +complex grid dynamics, the availability of important grid metrics +(e.g. locational marginal generation) that can guide power adapta- +tion is limited. Therefore, it’s challenging for cloud datacenters of +realistic-scale to use dynamic power control to reduce carbon emis- +sions. Attempts can achieve only a fraction of anticipated benefits +or worse harm the grid, themselves, and other customers. +One example is a phenomenon called “overshifting” illustrated +in Figure 3. Simulating a Spring day in California, datacenter loads +that adapt power based on average carbon intensity (1st row) in- +crease the average carbon intensity (2nd row), producing increase +in grid carbon emissions equivalent to 21.7% of datacenter carbon +emissions in a day. Tremendous harm! This is because the data- +centers act in unison, increasing or decreasing load at the same +time (3rd row). Large aggregate load changes produce competition +for power and oversubscribe the opportunity, causing dispatch of +additional fossil-fuel generation and increasing carbon emissions. +So the key question for datacenters is: How to adapt data- +center power to reduce operational carbon-emissions effec- +tively? That is, how to control their power load to better align +with low-carbon opportunity— and to do so without disturbing +grid dispatch (which generators). To achieve this, datacenters must +solve several challenges: +(1) Identify opportunities: datacenters need to find the right +times to increase and decrease load to reduce their carbon +emissions. +(2) Avoid contention or overshifting: to realize the goal datacen- +ters must avoid oversubscribing the opportunity. +(3) Avoid harming power grid: ensure the datacenter power load +adaptation does not harm power grid participants (datacen- +ters, other customers, generators, grid) by increasing prices +or overall carbon emissions. +4 +APPROACH +Overall, to meet the challenges faced by datacenters, we explore +different datacenter power adaptation approaches and couple them +to grid simulations. We first consider three grid metrics (average +carbon intensity, grid price, and locational marginal price) which +3 + +Conference’17, July 2017, Washington, DC, USA +Liuzixuan Lin and Andrew A. Chien +Figure 3: Overshifting: local, online adaptation using aver- +age carbon intensity (ACI) increases datacenter carbon emis- +sions by 21.7% because they all make the same load adapta- +tion. +datacneters can identify opportunities to reduce carbon emissions +from and adapt power load to. Then we compare three approaches +(Figure 4) to adapt datacenter load for operational carbon reduction. +Grid +Day +Ahead +Hourly +Data- +centers +Day +Ahead +Hourly +Coordinator +Day +Ahead +Hourly +Data- +centers +Day +Ahead Plan +DCs adapt based +on Grid Metric +DCs adapt, limited +by Coordinator +DCs share Plan, +Grid adapts to it +Data- +centers +Figure 4: Different Approaches for Datacenters to Adapt +Their Power Loads to Reduce Carbon-emissions. +Representing a classes of control techniques in previous work, +local online adaptation (Figure 4, left) makes hourly decisions +using real-time and future metrics from the grid. Our studies of +local adaptation provide insights into the limits of these approaches +to reduce carbon emissions, showing the improvements from more +future grid information and smoothed adapted load step size. +At times, datacenters adapting load can harm their efforts to +reduce carbon emissions (e.g. overshifting shown in Figure 3). We +consider use of an external coordinator to eliminate such harm +by limiting the total power change of a set of datacenters (Figure 4, +middle). Each DC makes a request to adapt power level hourly, but +the external coordinator limits their collective (total) power level +change. +Finally, we consider a new approach that combines adaptation +based on day-ahead grid information and sharing of the resulting +adapted plan with the power grid (Figure 4, right). This approach +requires each DC to plan ahead—making binding choices 24-hours +in advance. The fixed plan gives the grid certainty for the datacenter +power load, enabling it to optimize generation and transmission +scheduling under dynamic constraints that bind decisions across +time periods. +We introduce the algorithms of each approach and evaluate +them in Section 6. We focus on the datacenter carbon reduction +achieved, and also consider impacts on datacenter operation and +power price impacts for both datacenters and other grid customers. +The simulations are with realistic settings: datacenter sizes (200 +MW), power grids with wind penetration levels for today and the +coming decade (15%–60%, 2015 levels–2050 target), and 10 to 40 +datacenters, which match current and near-future (2027) loads in +Northern Virginia and Ireland power grids. For clarity of exposition, +we focus on a 30-DC scenario, representing ≈10% grid load, below +current levels in NoVA’s (12%) and Ireland’s (14%) power grids. +5 +GRID-COUPLED SIMULATION +METHODOLOGY +This section sets up the framework under which we evaluate the dat- +acenter power adaptation approaches. Adaptation adapts power to +grid metrics (Section 5.1), respecting the load flexibility constraints +(Section 5.2). Resulting time-varying loads affect the dynamics (pric- +ing, generation, etc.) in the power grid (Section 5.3). +5.1 +Grid Metrics for Carbon Optimization +Power grids have complex dynamics, so the “best” grid metric for +reducing carbon emissions is an open research question [17, 48]. A +good metric enables carbon reduction and is available in most or +all power grids. We consider several candidates: +• Average carbon intensity or ACI (kg CO2/MWh) is the +carbon emissions per MWh energy consumption in the grid. +Derived from fuel mix, ACI is only recently available from +unvalidated 3rd parties (e.g. Electricity Maps [55]) in a frac- +tion of the world’s grids. +• Grid price or GPrice ($/MWh) is power price in a grid or +region (e.g. “hub price”). Renewable generation often bids +low causing low power price to be correlated with carbon +intensity (Figure 2). Price information is widely available in +day-ahead and real-time markets. +• Locational marginal price or LMPrice ($/MWh) is price +at a specific node in the power grid. LMPrice reflects local +properties such as nearby renewables and grid transmission +constraints. LMPrice is widely-available. +Recently, researchers and companies [49, 75] propose marginal +carbon intensity as a metric. Though an interesting direction, mar- +ginal carbon intensity is not broadly available, so we do not consider +it. +5.2 +Modeling Datacenter Load Flexibility +Datacenter load flexibility is the basis of power adaptation. We +follow the datacenter load flexibility model in [46] wherein every +datacenters adjust load within a dynamic range and can defer +4 + +200 +ACI +100 +0 +5 +10 +15 +20 +25 +FixedLoad +200 +ACI +Adaptive Load +100 +0 +5 +10 +15 +20 +25 +Change +50 +Load +0 +-50 +0 +5 +10 +15 +20 +25 +TimeReducing Datacenter Operational Carbon Emissions Effectively by Cooperating with the Grid +Conference’17, July 2017, Washington, DC, USA +workload (backlog), but must catch up within a 24-hour day (Figure +5). +2 +Average +Load +Load +Time of Day +t t+1 +Step Size: max. load change +between two hours +Backlog +(deferred work) +Catchup +Dynamic Range +Figure 5: Datacenter Load Flexibility Model. +Formally, +𝑙𝑜𝑎𝑑𝑚𝑖𝑛 ≤ 𝑙𝑖,𝑡 ≤ 𝑙𝑜𝑎𝑑𝑚𝑎𝑥, ∀𝑖,𝑡 +(1) +𝑏𝑎𝑐𝑘𝑙𝑜𝑔𝑖,𝑡 = 𝑏𝑎𝑐𝑘𝑙𝑜𝑔𝑖,𝑡−1 + (𝑎𝑣𝑔𝐿𝑜𝑎𝑑 − 𝑙𝑖,𝑡) +(2) +𝑏𝑎𝑐𝑘𝑙𝑜𝑔𝑖,24 = 0, ∀𝑖 +(3) +Datacenter capacity variation can harm a datacenter’s compu- +tation performance [79] and harm power markets. We consider a +step size limit that bounds datacenter power change in one-hour: +|𝑙𝑖,𝑡 − 𝑙𝑖,𝑡−1| ≤ 𝑠𝑡𝑒𝑝𝑆𝑖𝑧𝑒, ∀𝑖,𝑡 +(4) +The configurations for the attributes above are shown in Table +1. These resource utilization and capacity assumptions are typical +of hyperscale datacenters [2, 29, 59, 72]. We focus on the [0.4, 1.0] +dynamic range where the potential benefit is large but overshifting +problem is more prominent. This level of flexibility (30% flexible +workload) is realistic according to Google’s BorgTNG trace [72] +and growing use of batch workloads such as ML training. +Table 1: Configurations of Datacenter Attributes +Attribute +Configuration +Maximum Capacity +200 MW +Average Utilization Level +70% +Average Power Load (𝑎𝑣𝑔𝐿𝑜𝑎𝑑) +140 MW +Dynamic Range +[0.6, 0.8] (small) +[0.4, 1.0] (large) +Step Size +10, 20, 40, 80, 120 +MW/h +5.3 +Power Grid Model, Base Load and +Generation Profiles +We evaluate the coupling approaches under a realistic grid model. +Grid operation is simulated by solving the direct-current optimal +power flow (DC-OPF) problem in [41, 46] and Appendix A, which +minimizes the grid dispatch cost in one-day time horizon with +hourly intervals, subject to typical grid constraints. The grid metrics +for carbon optimization (ACI, GPrice, LMPrice) are derived from the +OPF solutions. Reflecting reality, we assume renewable generation +sources have lower generation costs and curtailment penalties that +encourage use, producing low prices when renewable generation is +dispatched at the margin. This makes price highly correlated with +carbon metrics, as what happens in the real world (Figure 2). +The grid model is a reduced California power system (CAISO) +consisting of 225 buses, 375 transmission lines, 130 thermal genera- +tors (31.2 GW total capacity), 11 non-wind renewable power plants, +5 wind power plants, and 40 loads. Power can also be imported at 5 +boundary buses. The thermal power plants’ ramp rates are scaled +up by 4-fold to reflect current ramping capability of CAISO as in +[46]. This model is originally from [61] and has been used to assess +the impact of dynamic datacenter load management in [41, 46]. +There are 8 base load (DCs excluded), imports, and non-wind +renewable generation profiles that cover the four seasons (Spring, +Summer, Fall, Winter) and weekday/weekend (WD/WE). Figure 6 +shows how each load profile varies in a day, with the average span- +ning from 23,780 MW (WinterWE) to 31,089 MW (SummerWD). +Comparing the weekday and weekend in a season, there is a clear +weekly pattern that the two profiles are in similar shape but week- +end’s load is lower. For each season, we use 100 wind generation +profiles (shown in Figure 7), and a season’s weekday and week- +end share these profiles. The wind generation presents intra-day +variation—higher in the late night and early morning, which is a +misalignment with load and becomes more evident as wind pen- +etration increases. To model higher wind penetration, the wind +generation are scaled up equally at current sites, assuming those +sites can be expanded or replace old wind turbines with higher +capacity turbines [64]. +0 +10 +20 +Hour +0 +5 +10 +15 +20 +25 +30 +35 +40 +Base Load (GW) +WD +WE +(a) Spring +0 +10 +20 +Hour +0 +5 +10 +15 +20 +25 +30 +35 +40 +(b) Summer +0 +10 +20 +Hour +0 +5 +10 +15 +20 +25 +30 +35 +40 +(c) Fall +0 +10 +20 +Hour +0 +5 +10 +15 +20 +25 +30 +35 +40 +(d) Winter +Figure 6: Grid Base (Non-DC) Load Profiles. +0 +10 +20 +Hour +0 +2 +4 +6 +8 +10 +12 +14 +Wind Gen. (GW) +(a) Spring +0 +10 +20 +Hour +0 +2 +4 +6 +8 +10 +12 +14 +(b) Summer +0 +10 +20 +Hour +0 +2 +4 +6 +8 +10 +12 +14 +(c) Fall +0 +10 +20 +Hour +0 +2 +4 +6 +8 +10 +12 +14 +(d) Winter +Figure 7: Used Wind Scenarios (15% Penetration Level). Red +lines represent average wind scenarios for each season. +Datacenters are added to random buses in the grid, which reflects +the fact that cloud company site selection is more based on business +5 + +Conference’17, July 2017, Washington, DC, USA +Liuzixuan Lin and Andrew A. Chien +considerations external to the power grid (e.g. tax breaks, jobs, +internet hookups, etc.). +5.4 +Evaluation Metrics +The metrics used to evaluate effects of datacenter power adaptation +in achieving datacenter goals, and impact on other grid customers, +include: +• Datacenter Carbon Reduction. Because datacenter load +adaptation accounts for the grid carbon emissions change, +we report the percentage reduction in datacenter operational +carbon emissions as: +(𝑔𝑟𝑖𝑑𝐶𝑎𝑟𝑏𝑜𝑛𝑓 𝑖𝑥𝑒𝑑−𝑙𝑜𝑎𝑑 − 𝑔𝑟𝑖𝑑𝐶𝑎𝑟𝑏𝑜𝑛𝑎𝑑𝑎𝑝𝑡𝑎𝑡𝑖𝑜𝑛) +𝑑𝑎𝑡𝑎𝑐𝑒𝑛𝑡𝑒𝑟𝐶𝑎𝑟𝑏𝑜𝑛𝑓 𝑖𝑥𝑒𝑑−𝑙𝑜𝑎𝑑 +∗ 100% +and +𝑔𝑟𝑖𝑑𝐶𝑎𝑟𝑏𝑜𝑛 = +24 +∑︁ +𝑡=1 +𝑔𝑒𝑛𝑓 ,𝑡 ∗ 𝑒𝑚𝑖𝑠𝑠𝑖𝑜𝑛𝑅𝑎𝑡𝑒𝑓 +where 𝑔𝑒𝑛𝑓 ,𝑡 is generation from fuel 𝑓 in the 𝑡-th hour. Fuel +emission rates are from US EPA eGrid database [4, 36] and +listed in Appendix B. 𝑑𝑎𝑡𝑎𝑐𝑒𝑛𝑡𝑒𝑟𝐶𝑎𝑟𝑏𝑜𝑛 in the fixed DC load +scenario is calculated using the grid emission rate. +• Datacenter Average Power Price ($/MWh) is how much +the datacenter would pay for power considering the location +of datacenter and the time when power is consumed. For +multiple datacenters, we compute the average across them. +• Datacenter Average Capacity Variation (MW/h) is de- +fined as the average change in power level (load) of a dat- +acenter between adjacent one-hour periods, the lower the +better. More formally: +1 +23 +24 +∑︁ +𝑡=2 +|𝑙𝑖,𝑡 − 𝑙𝑖,𝑡−1| +where 𝑙𝑖,𝑡 denotes the power level of datacenter 𝑖 at time 𝑡. +• Non-datacenter Customer Average Power Price ($/MWh) +is the average power price across grid customers other than +datacenters, weighted by power demand. +Conventional datacenter operation has fixed capacity and power +load. We use it (“Fixed”) as the baseline to illustrate the impacts +from datacenter power adaptation. +5.5 +Experiment Setup +To reflect the impact of wind variation, we run the simulations with +100 wind scenarios for each day type (WD and WE of a season share +the same set of wind scenarios), which is equivalent to simulating +a total of 800 days [61]. For each day, we vary the wind penetration +level and simulate datacenter operation using different adaptation +models. The results reported in Section 6 are the average of 8 day +types (weighted for the number of weekdays and weekend days) +and varied wind scenarios. +We used Julia 1.5.2 with JuMP v0.21.5 [21] to implement the grid +simulation and solved grid OPF with Gurobi Optimizer v9.0 [32]. +6 +EVALUATING DATACENTER POWER +ADAPTATION APPROACHES +We explore representative datacenter power adaptation algorithms +for the three approaches outlined in Section 4. Each compared to +fixed datacenter loads, and evaluated coupled to grid dynamics. +Performance is compared based on datacenter carbon reduction, as +well as price for datacenters and other customers. +We present a subset of our results, focused on a representative +scenario with 30 datacenters and dynamic range of [0.4, 1.0]. This +corresponds to 10% of grid load, well below current leading edge +grids, and increasingly realistic for dozens of grids throughout +the world in the near future. We explore varied levels of wind +penetration, to capture future evolution as grids decarbonize. We +also plot standard deviation as “whiskers” to indicate variation +across the wind scenarios used. A specific wind scenario typically +produces correlated results for different algorithms, so these capture +variability, not uncertainty. +6.1 +Local, Online Adaptation +Datacenter-local online power adaptation approaches make real- +time load decisions based on current and future information about +grid metrics. We first consider a dynamic programming algorithm +that makes hourly decisions using current value of metric and its +daily average (Adapt-Avg). The algorithm selects amongst +{𝑎𝑣𝑔𝐿𝑜𝑎𝑑,𝑎𝑣𝑔𝐿𝑜𝑎𝑑 ± dynamic range/2} to minimize the expecta- +tion, e.g. +𝑙𝑖,𝑡 ∗ 𝑚𝑒𝑡𝑟𝑖𝑐𝑖,𝑡 + (𝑏𝑎𝑐𝑘𝑙𝑜𝑔𝑖,𝑡−1 + 𝑎𝑣𝑔𝐿𝑜𝑎𝑑 − 𝑙𝑖,𝑡) ∗ 𝑚𝑒𝑡𝑟𝑖𝑐𝑖 +which can be either carbon emissions or power cost. The 𝑏𝑎𝑐𝑘𝑙𝑜𝑔 +is then updated given the determined power level. +Datacenter load adaptation is coupled to grid dispatch (OPF +optimization), as below, where datacenters are denoted by 𝑖, and +hours by 𝑡: +(1) For all i,t: 𝑙𝑖,𝑡 = 𝑎𝑣𝑔𝐿𝑜𝑎𝑑 (neutral initial condition). +(2) Solve grid OPF with {𝑙𝑖,𝑡 }, defining 𝑚𝑒𝑡𝑟𝑖𝑐𝑖,𝑡 as day-ahead +information. +(3) At the beginning of 𝑡 = 1, ..., 24-th hour, each datacenter +decides 𝑎𝑑𝑎𝑝𝑡(𝑖,𝑡) based on the metrics. +(4) Then the grid solves OPF with updated datacenter loads +{𝑙𝑖,𝑡 } +(5) This new OPF solution redefines 𝑚𝑒𝑡𝑟𝑖𝑐𝑖,𝑡 and for the future +(e.g. [𝑡 + 1, ..., 24]). +6.1.1 +Comparing Grid Metrics. We evaluate the effectiveness of +different grid metrics (average carbon intensity (ACI), grid price +(GPrice), and locational marginal price (LMPrice)) for online adapta- +tion algorithm. LMPrice consistently outperforms ACI and GPrice +(see Figure 8). +At 15% wind, online adaptation fails to achieve carbon-emissions +reduction because the generation supply is tight, and oversubscrip- +tion happens during low-carbon periods. Higher wind penetration +provides more opportunity, enabling online adaptation using LM- +Price to reduce datacenter carbon emissions by 5.4% (60% wind), +consistently exceeding ACI (by 5.4%) and Gprice (by 1.4%), In short, +price metrics work better, and finer-grained detailed pricing is the +best amongst those. +6 + +Reducing Datacenter Operational Carbon Emissions Effectively by Cooperating with the Grid +Conference’17, July 2017, Washington, DC, USA +-6% +-4% +-2% +0% +2% +4% +6% +8% +10% +12% +15% +30% +45% +60% +DC Carbon Reduction +Wind Penetration +AdaptCarbon-Avg +AdaptGPrice-Avg +AdaptLMPrice-Avg +Figure 8: Achieved Datacenter Carbon Reduction based +on Carbon Emissions, Grid Price, and Locational-Marginal +Price. +To illustrate this, Figure 9 shows a single-day timeline, The +graphs show that ACI and GPrice as grid-wide signals, drive uni- +form datacenter behavior, maximizing their collective power swings +for the grid. In contrast, LMPrice is a local metric, and drives diverse +behavior, better reflecting grid constraints such as transmission +and ramping. +Figure 9: Global and local metrics (LMPrice) drive differ- +ent Datacenter Load Adaptation behaviors (Spring Weekday, +45% Wind Penetration). +6.1.2 +Improving Local Online Adaptation. We build on online adap- +tation with the most effective metric (LMPrice), improving these +approaches by addition of finer-resolution or better estimates of +future (forecasts) and smoothing power level changes. +Forecasts (Future Information). Hourly LMPrice is generally +available in the day-ahead market, which can be thought of as a +forecast1. We add this information—the full 24 hours of day-ahead +prices as a forecast—to enable better adaptation in AdaptLMPrice- +Hourly. These forecasts are not guaranteed to be accurate as the +final OPF that runs at the specific hour will determine grid fuel mix, +prices, etc. +Step Size. Our results show that online adaptation can cause +large datacenter load fluctuations, harming both datacenter com- +puting efficiency and grid dynamics (generation dispatch, carbon +1It’s a forecast because it’s subject to revision by other markets (eg. hourly, 15-minute, +5-minute, real-time) as the time approaches. +intensity, price). To mitigate the harm from large, frequent load +changes, we add a step size constraint that smoothes hour-to-hour +individual DC power level changes. +Using the day-ahead hourly locational prices at the DC (price +array {predLMPrice𝑖,𝑡 },𝑡 = 1, ..., 24), in the 𝑗-th hour, datacenter +𝑖 performs dynamic programming (DP) on the price array {𝑝𝑖,𝑡 } +with: +𝑝𝑖,𝑡 = +� +LMPrice𝑖,𝑗, if 𝑡 = 𝑗 and 𝑗 ≠ 1 +predLMPrice𝑖,𝑡, otherwise +, 𝑡 = 𝑗, ..., 24 +where LMPrice𝑖,𝑗 is the real-time price after {𝑙𝑖,𝑗−1} are set. The +dynamic programming algorithm produces a load array based on +the following recurrence formula: +𝑐𝑜𝑠𝑡𝑖 (𝑛,𝑡,𝑙) = 𝑙 ∗ 𝑝𝑖,𝑡 + min +𝑙′ {𝑐𝑜𝑠𝑡𝑖 (𝑛 + 𝑙 − 𝑎𝑣𝑔𝐿𝑜𝑎𝑑,𝑡 − 1,𝑙′) +| |𝑙 − 𝑙 ′| ≤ 𝑠𝑡𝑒𝑝𝑆𝑖𝑧𝑒} +(5) +where 𝑐𝑜𝑠𝑡𝑖 (𝑛,𝑡,𝑙) denotes the minimum power cost of the sub- +problem ending at 𝑡-th hour with backlog 𝑛 and power level 𝑙 in the +𝑡-th hour. The datacenter takes the first element of the load array +as 𝑙𝑖,𝑡—similar to the receding horizon control [42] but with a fixed +horizon. +We add detailed future price information, and then tune the step +size, showing the results in Figure 10. The results reflect the best +step sizes (40 MW/h for AdaptLMPrice-Hourly, and no limit for +AdaptLMPrice-Avg). +Comparing these two variants, we can see the impact of detailed +future price information and step size: AdaptLMPrice-Hourly’s +improvement is roughly 50% from hourly information and 50% +from step size. +To further study the impact of detailed future information, we +develop a variant, which at each hour is given hourly future informa- +tion for the next 6 hours but only daily average after that. The two +approaches with detailed future information are clearly superior, +with the 6-hour future price information variant (AdaptLMPrice- +6hr+Avg) already better than AdaptLMPrice-Avg, and benefit +grows with more future information in AdaptLMPrice-Hourly. +Quantitatively, at 30% wind penetration, 6 hours’ information gives +a benefit of additional 2.6% reduction and 24-hours gives a benefit of +additional 4.2% of datacenter carbon emissions. These benefits grow +to 3.5% and 4.6% at 60% wind penetration respectively, enabling +AdaptLMPrice-Hourly to reach a 10% of DC carbon reduction. +6.2 +Constraining Adaptive Datacenters with +Coordinator +Independently controlled DCs can react together, when decisions +are based on grid-wide or other strongly-correlated metrics pro- +ducing a large aggregate power change (see Section 3). We have +seen LMPrice is better, but even its correlation across sites can +produces synchronized DC load changes that are difficult for the +grid to manage. Addressing this, we add an external limiter, called +coordinator, to mitigate the harm. In AdaptLMPrice-Avg-Coord, +each datacenter runs independent online control algorithm then +submits their adaptation to a coordinator. The coordinator limits +total power change for a set of datacenter, using a quota: +(1) Generate a random permutation of datacenters. +7 + +AdaptCarbon-Avg +AdaptGPrice-Avg +AdaptLMPrice-Avg +200 + 150 +ice +0 +A +100 +-200 +-200 +0 +20 +0 +20 +0 +20 +200 +200 +200 +pe +9 150 +150 +150 +0100 +100 +100 +0 +20 +0 +20 +0 +20Conference’17, July 2017, Washington, DC, USA +Liuzixuan Lin and Andrew A. Chien +-2% +0% +2% +4% +6% +8% +10% +12% +14% +16% +30% +60% +DC Carbon Reduction +Wind Penetration +AdaptLMPrice-Avg +AdaptLMPrice-6hr+Avg +AdaptLMPrice-Hourly +Figure 10: Achieved Datacenter Carbon Reduction with Var- +ied Future LMPrice Information (daily average, 6 hours, 24 +hours). +(2) For each datacenter, if 𝑐ℎ𝑎𝑛𝑔𝑒 ≤ 𝑞𝑢𝑜𝑡𝑎 then accept, 𝑞𝑢𝑜𝑡𝑎 = +𝑞𝑢𝑜𝑡𝑎 − 𝑐ℎ𝑎𝑛𝑔𝑒; else, reject. +If local controllers were unable to get their requested change, +they must update their backlogs accordingly. +Figure 11: Achieved Datacenter Carbon Reduction vs. Coor- +dinator Quota. +Figure 11 shows DC carbon reduction with varied coordinator +quotas (3600–1200 MW/h). Each line represents a different wind +penetration level. Coordination improves performance, mitigating +overshifting harm, at 15-60% wind penetration and growing benefits +as the quota is tightened. All of points on each line reflect greater +carbon reduction than that for AdaptLMPrice-Avg (leftmost point +on each line), and steady improvement as the quota is reduced. +The mitigation of overshifting is a larger benefits for datacenters +when generation is tight (15% wind penetration), improving DC +carbon reduction by 2.3%. Benefits are smaller than those achieved +by exploiting future price information (e.g. AdaptLMPrice-Hourly), +which yields a 50% higher DC carbon reduction. +Multiple Coordinators. In many geographic areas, there are mul- +tiple cloud providers (e.g. Northern Virginia, Texas, Ireland, or +China’s Ningxia), and each cloud provider has multiple datacenter +sites in that area. As competitors, they may not be willing to share +a coordinator. To model this multi-coordinator scenario, we assume +there are 2 or 3 coordinators in the grid, each coordinating 10 or 15 +DCs. We use an overall quota of 1200 MW/h, and divide it equally +across the coordinators. +Increasing the number of coordinators further decreases DC car- +bon emissions slightly. The reason for this is narrower coordinator +scope increases the smoothness of total datacenter load, but the +improvements are small. 3 coordinators produce improvements +up to 0.7% of DC carbon emissions, achieving 7.3% of DC carbon +reduction at 60% wind penetration. +6.3 +Adapted Power Plan Sharing +Adaptive DC loads cause grid problems as their large power changes +are unpredictable and strain generator ramp constraints. Further, +unplanned adaptation can produce rapid changes in compute capac- +ity, making it difficult for cloud resource managers to be efficient. +In view of these insights, we propose a new approach—PlanShare: +datacenters create a 24-hour adapted power plan based on LMPrice +in day-ahead grid market [9] ahead of operating day, and then share +the plan with the grid , allowing it to optimize grid-wide based on +the DC information. Formally, with datacenters denoted by 𝑖 and +hours denoted by 𝑡: +(1) For all i,t: 𝑙𝑖,𝑡 = 𝑎𝑣𝑔𝐿𝑜𝑎𝑑 (neutral initial condition). +(2) Solve grid OPF with {𝑙𝑖,𝑡 }, defining initial LMPrice𝑖,𝑡 as day- +ahead information. +(3) Each datacenter makes 24-hour adaptation plan using AdaptLMPrice- +Hourly’s dynamic programming algorithm and shares it with +the grid. +(4) Solve grid OPF for [1, ..., 24] with adapted {𝑙𝑖,𝑡 } to model the +next day’s operation. +The datacenter must follow the full-day power plan it shares +with the power grid. +Figure 12 compares PlanShare and with the local, online ap- +proaches. The results show the benefits of plan sharing with the +grid. Using essentially the same adaptation algorithm, and on less +accurate information, by working with the grid, PlanShare reduces +DC carbon emissions by up to 12.6%. This is 1.26x (26%) better +than the best online adaptation result, and the advantage is even +higher at lower wind penetration as large as 2.32x (132%). By con- +tributing its adaptation plan to the grid optimization—in advance +and as a committed schedule—PlanShare dramatically improves the +datacenter carbon emissions reduction that can be achieved. +-10% +-5% +0% +5% +10% +15% +20% +15% +30% +45% +60% +DC Carbon Reduction +Wind Penetration +AdaptCarbon-Avg +AdaptLMPrice-Avg +AdaptLMPrice-Hourly +PlanShare +Figure 12: PlanShare outperforms all online adaptation ap- +proaches. +8 + +(%） +CarbonReduction +10.0 +7.5 +5.0 +60%Wind +30%Wind +2.5 +-15%Wind +0.0 +DC +35003000250020001500 +CoordinatorQuota(MW/h)Reducing Datacenter Operational Carbon Emissions Effectively by Cooperating with the Grid +Conference’17, July 2017, Washington, DC, USA +Sensitivity to Length of Shared Plan. Having demonstrated the +benefits of a practical scheme (24-hour day-ahead plans are widely +available in power grids around the world), we ask an intellec- +tual curiosity question—how much plan-ahead is really needed +by the grid? We vary the length of plan shared from 1 to 24 +hours, reporting results in Figure 13. Interestingly, a single hour +of plan-ahead enables plan sharing to match the performance of +AdaptLMPrice-Hourly, the best online approach, despite the fact +that PlanShare has no online adaptation—the full 24-hour schedule +is fixed in advance. As the length of plan is increased to 3, 6, and 12 +hours, the benefit of plan sharing increases significantly, reaching +the large benefits previously highlighted in Figure 12. +0% +2% +4% +6% +8% +10% +12% +14% +15% +30% +45% +60% +DC Carbon Reduction +Wind Penetration +1 Hour +3 Hours +6 Hours +12 Hours +24 Hours +Figure 13: Length of Plans Shared: PlanShare variants with +1-24 hours of plan shared. +6.4 +Datacenter Adaptation Impacts beyond +Carbon +Datacenter load adaption produces other impacts on datacenter +operation and other customers in the grid. We study several here. +Datacenter Capacity Variation. Capacity variation is a critical +concern for datacenter operators as it affects workload efficiency of +the available capacity. Using the average capacity variation metric +(see Section 5), Figure 14 shows DC carbon reduction on y-axis +and capacity variation on x-axis. As before, DC carbon reduction +is relative to the fixed DC load scenario, and capacity variation +(MW/h) is normalized to DC average capacity (140 MW). An ideal +adaptation approach would fall in the upper-left corner (high carbon +reduction, low capacity variation). Each line connnects results as +wind penetration increases for a given adaptation approach. +The plot clearly shows how the higher-performing adaptation +techniques exploit increased changes to track the changing grid car- +bon properties. A progression from AdaptCarbon-Avg to AdaptLMPrice- +Avg to AdaptLMPrice-Hourly shows a clear tradeoff of carbon re- +duction for online capacity variation. The PlanShare results achieve +the greatest carbon reduction, do require greater capacity variation +to do so. However, it’s worth noting that PlanShare fixes the capac- +ity plan in advance, so the resource manager has a statically known +resource schedule at the start of the day, facilitating computing +workload scheduling. Scheduling studies and other proposals argue +for the benefits of this stability [66, 79]. +Figure 14: Datacenter Carbon Reduction and Capacity Vari- +ation. +-20 +-10 +0 +10 +20 +30 +40 +50 +60 +70 +30% +60% +Change in Power Price ($/MWh) +Wind Penetration +AdaptCarbon-Avg +AdaptLMPrice-Avg +AdaptLMPrice-Hourly +PlanShare +Figure 15: Change in Average Power Price for Datacenters +Compared to Fixed. +Datacenter Power Cost. Adaptive DC loads affect power pricing +in the grid. In Figure 15, results show that local, online adaptation +can cause significant power prices increases of $20 to $50/MWh, +corresponding to 59%–490% increase in power cost. This is because +locally controlled adaptation clashes with grid constraints and dy- +namics (overshifting in Section 3). This is likely a major deterrent +for datacenter adoption. In contrast, sharing the datacenter’s load +schedule in advance as in PlanShare decreases average power price +stably, up to 30% compared with the fixed-load scenario. +Non-DC Customer Power Cost. In Figure 16, We explore how +adapting datacenter power impacts non-datacenter (non-DC) cus- +tomers. Online approaches, including AdaptCarbon-Avg and AdaptLMPrice- +Avg, significantly increase the price for non-DC customers, espe- +cially for lower wind penetration with less excess renewable gen- +eration (AdaptLMPrice-Hourly also increases price at 15% wind). +Datacenters as growing consumers of power are already subject to +growing scrutiny and negative publicity. Pricing hard to other cus- +tomers has the potential to cause a backlash, so datacenters should +be careful about deploying such local, online adaptive power-level +control. PlanShare avoids this price harm for non-DC customers at +all wind penetration levels. +9 + +AdaptCarbon-Avg +AdaptLMPrice-Avg +AdaptLMPrice-Hourly +PlanShare +15 +15% Wind +Carbon Reduction (%) +30% Wind +10 ++ +45% Wind +60% Wind +DC +0 +5.0 +7.5 +10.0 +12.5 +15.0 +17.5 +20.0 +22.5 +25.0 +%CapacityChange/hourConference’17, July 2017, Washington, DC, USA +Liuzixuan Lin and Andrew A. Chien +Table 2: Summary of Datacenter Power Adaptation Approaches +AdaptCarbon- +AdaptLMPrice- +AdaptLMPrice- +PlanShare +PlanShare +Avg +Avg +Hourly +(1 hour) +(24 hours) +DC Carbon Reduction +– +neutral ++ ++ +++ +DC Capacity Variation ++ ++ +neutral +neutral +neutral +DC Power Cost +– – +– +neutral ++ +++ +Non-DC Power Cost +– – +– +neutral ++ +++ +-20 +-10 +0 +10 +20 +30 +40 +50 +60 +70 +30% +60% +Change in Power Price ($/MWh) +Wind Penetration +AdaptCarbon-Avg +AdaptLMPrice-Avg +AdaptLMPrice-Hourly +PlanShare +Figure 16: Change in Average Power Price for Non-DC Cus- +tomers Compared to Fixed. +6.5 +Summary +Table 2 summarizes the impacts of datacenter power adaptation in +different dimensions, where “+” means advantage and “–” means +disadvantage. The weakest performance plan sharing approach— +PlanShare with 1 hour’s shared plan—matches and outperforms all +of the other approaches. And, as we can see in Figure 13, PlanShare +with 24-hour adaptation plan outperforms others significantly and +is by far the best, delivering the greatest DC carbon reduction while +enabling datacenter resource managers to have known capacity +plans. Further, PlanShare-24 produces benefits in power cost for +both datacenters and other customers in the grid. +7 +DISCUSSION AND RELATED WORK +To the best of our knowledge, this is the first paper that proposes +sharing datacenter adaptive power plan with the grid to reduce +operational carbon emissions effectively. Given the LMPrice metric +availability and industrial practice in making day-ahead adaptive +power plan [66], this approach is feasible and widely usable. A +major concern might be that cloud providers would be unwilling to +share their capacity plan a day in advance, for proprietary or com- +petitive reasons. It’s worth pointing out that the datacenter’s local +utility/grids (e.g. Dominion Energy, PGE, etc.) already know data- +centers’ historical power consumption, going back days, months, +and years. Further, if the cloud datacenters wanted to intentionally +mask their actual compute load, they could still do so with on-site +batteries or even generators. +We review related work below: +Datacenter Power Load Adaptation (Shaping). Early ideas like +“follow-the-moon”“chase-the-wind” propose to shift datacenter work- +load to a time or place with low energy prices or carbon emissions +[1], exploiting variations across geography, competitive power +load, grids, and renewables. The datacenter load adaptation ex- +plored in this paper addresses a subset of these ideas—temporal +load shaping or shifting, which has been widely studied, typically +employing sophisticated online control or optimization techniques +[20, 47, 51, 53]. An additional variant is to manage colocated energy +storage [5, 27, 68, 73, 78]. This work usually assumes datacenters +are small loads (reflected in grid trace-based studies or DC-centric +evaluation). Consequently, they don’t consider the impact of DC +dynamic load change on the grid—and further circular impacts on +carbon, prices, generation, etc. In this paper, we consider realistic +cases when a collection of large hyperscale datacenters (200 MW) +collectively comprise a significant fraction of grid load, and thus +directly affect grid dispatch. +Another dimension of “follow-the-moon” load shifting is spatial +or geographic, with similar goals [31, 43, 50, 65, 82, 83, 85, 86]. We +do not consider this direction, but it is certainly of promise. +Datacenters in Demand Response. Datacenters’ ability to delay +workload (e.g. deferring batch jobs) can be used to participate in +demand-response programs, reducing load during peak periods +according to requests from the grid. Such participation can reduce +DC power cost, and research has explored how to balance this ben- +efit while respecting service-level objectives (SLO) [44, 52]. Further +efforts in this area design sophisticated markets that incentivize +DC operators and even their colocation tenants to participate in +demand response [11, 12, 71, 80, 84]. +Demand response is designed for emergency reduction in load +to protect grid stability. As a result, actions are rare, and the power +reduction is small relative to the total power consumption [58]. +In contrast, we consider multiple datacenters’ active, continuous +power adaptation for reducing operational carbon emissions with +dynamic range up to 60% of capacity. +Grid-coupled Datacenter Adaptation. Some researchers propose +building datacenters as dispatchable loads controlled by the grid to +harness excess carbon-free power [14, 76, 77], and the grid benefits +like decreased dispatch cost and renewable curtailment are reported +in a study including a grid model [41]. Other efforts include explor- +ing the grid benefits of temporal or spatial load shifting [49, 81]. +[48] studies what grid metrics can effectively guide carbon-aware +spatial load shifting. While also showing locational marginal price +is effective, they claim locational mariginal carbon emissions is +better, which, however, is not broadly available today. +This paper builds on the insights of [46]. The authors showed +that without modeling the grid’s dynamics, the carbon emission pro- +jections could be significantly wrong. Thus, its necessary to include +grid models in studies of large-scale temporal workload shifting. +10 + +Reducing Datacenter Operational Carbon Emissions Effectively by Cooperating with the Grid +Conference’17, July 2017, Washington, DC, USA +However, that work provides no solution to coupled management +of datacenters and power grids. +Industrial Efforts to Reduce Datacenter Carbon Emissions. Hyper- +scale cloud providers have taken actions on sustainability such +as renewable power purchase agreements (PPAs) that match their +annual power consumption. Several (e.g. Google, Microsoft) are +targeting so-called “24×7” or “100/100/0”, hourly matching data- +center’s power consumption with carbon-free generation [30, 39]. +Recent efforts also propose application and resource-management +level load shifting to exploit varying power carbon-intensity. These +efforts involve small dynamic range (only a few percent of load), +and do not consider their impact on power grids [54, 66]. +Notably, Google’s effort, termed carbon-aware computing [66], +creates day-ahead virtual capacity curves (VCC) that addresses con- +cerns of online capacity variation (and its impact on ability to use +resources efficiently). However, VCC is used within datacenters to +optimize compute resource management, not to inform the power +grid of upcoming load power changes. This perhaps could be attrib- +uted to the fact that only very small percentages of load are under +control of these schemes today. +8 +SUMMARY AND FUTURE WORK +Cloud providers are adapting datacenter power to renewable gener- +ation to reduce operational carbon emissions. However, for today’s +large cloud datacenters, the numerous prior techniques that seek +datacenter benefits with independent, online control can fail to +achieve carbon emissions reduction and even harm the grid, in- +creasing carbon emissions and other customers’ power costs. +We first consider grid metrics which datacenters can adapt to +for reducing operational carbon emissions and identify locational +marginal price (LMPrice) as a widely available and the most effective +metric. Exploring improvements for online adaptation and other +new approaches, we then find that sharing adapted power plan is +critical: datacenters create an adapted power plan based on day- +ahead grid metric, and share that with the grid. This approach +can not only enable datacenters to achieve more of their goals— +more carbon reduction (12.6%), lower average power prices (-30%), +predictable capacity (and thereby better internal utilization), but +also eliminate the harm on other customers in the grid. In short, +this approach is a widely usable solution for cloud providers to +effectively reduce operational carbon emissions without harming +the power grid. +There are a number of exciting future directions for this research. +First, from datacenter resource management perspective, creating +load flexibility and responding to capacity change, while respect- +ing service-level objectives (SLO) continues to be an interesting +area. The resource managers and applications today lack a clear +cost metric, and further it’s unclear what types and extents of flex- +ibility are possible or valuable. Second, here we have focused on +temporal load adaptation, but as cloud providers run datacenters +across multiple regions, spatial shifting combined with day-ahead +capacity plan schedules is an interesting space to explore. Finally, a +more direct follow-on question is how we can balance datacenter +privacy considerations with the clear benefit of sharing capacity +plan information. +ACKNOWLEDGMENTS +This Work is supported in part by NSF Grants CMMI-1832230, OAC- +2019506, and the VMware University Research Fund. We also thank +the Zero-carbon Cloud team members! +11 + +Conference’17, July 2017, Washington, DC, USA +Liuzixuan Lin and Andrew A. Chien +A +DIRECT-CURRENT OPTIMAL POWER +FLOW FORMULATION +We model the grid operation using the direct-current optimal power +flow (DC-OPF) model. Notations in this model are listed below: +Table 3: DC-OPF Notation: Sets +Notation +Description +Notation +Description +D (D𝑛) +Demand loads (at bus 𝑛) +G (G𝑛) +Generators (at bus 𝑛) +I (I𝑛) +Import points (at bus 𝑛) +L +Transmission lines +L+𝑛/L−𝑛 +Transmission +lines +to/from bus 𝑛 +N +Buses +R (R𝑛) +Renewable generators +(at bus 𝑛) +T +Time periods +W (W𝑛) +Wind farms (at bus 𝑛) +𝐷𝐶(𝐷𝐶𝑛) +Datacenters (at bus 𝑛) +Table 4: DC-OPF Notation: Parameters +Notation +Description +Notation +Description +𝐵𝑙 +Susceptance of trans- +mission line 𝑙 +𝐶𝑖 +Generation cost of gen- +erator 𝑖 +𝐶𝑑 +𝑗 +Load-shedding penalty +at load 𝑗 +𝐶𝑤 +𝑖 +Curtailment penalty at +wind farm 𝑖 +𝐶𝑚 +𝑖 +Curtailment penalty at +import point 𝑖 +𝐶𝑟 +𝑖 +Curtailment penalty at +renewable 𝑖 +𝐷𝑗,𝑡 +Demand load of con- +sumer 𝑗 at time 𝑡 +𝐹𝑚𝑎𝑥 +𝑙 +Maximum power flow +of transmission line 𝑙 +𝑀𝑖,𝑡 +Power production of +import 𝑖 at time 𝑡 +𝑃𝑚𝑎𝑥 +𝑖 +Maximum power out- +put of generator 𝑖 +𝑅𝑖,𝑡 +Power production of re- +newable 𝑖 at time 𝑡 +𝑅𝑈𝑖 +Ramp-up limit of gener- +ator 𝑖 +𝑅𝐷𝑖 +Ramp-down limit of +generator 𝑖 +𝑊𝑤,𝑡 +Power from wind farm +𝑤 at time 𝑡 +Θ𝑚𝑖𝑛 +𝑛,𝑡 +Minimum phase angle +at bus 𝑛 at time 𝑡 +Θ𝑚𝑎𝑥 +𝑛,𝑡 +Maximum phase angle +at bus 𝑛 at time 𝑡 +Table 5: DC-OPF Notation: Decision Variables +Notation +Description +Notation +Description +𝑑𝑗,𝑡 +Load shedding at load 𝑗 +at time 𝑡 +𝑓𝑙,𝑡 +Power flow of line 𝑙 at +time 𝑡 +𝑚𝑖,𝑡 +Curtailment at import 𝑖 +at time 𝑡 +𝑝𝑖,𝑡 +Power from generator 𝑖 +at time 𝑡 +𝑟𝑖,𝑡 +Curtailment at renew- +able 𝑖 at time 𝑡 +𝑤𝑖,𝑡 +Curtailment at wind +farm 𝑖 at time 𝑡 +𝜃𝑛,𝑡 +Phase angle at bus 𝑛 at +time 𝑡 +Datacenter power levels 𝑙𝑖,𝑡 are either decision variables (grid- +controlled optimization) or external decisions (independent or co- +ordination) according to the coupling model. The power grid solves +the DC-OPF model (one-day time horizon with hourly intervals +in our studies), minimizing the dispatch cost (6a) that consists of +generation cost, load shedding penalty, and curtailment penalties: +min +∑︁ +𝑡 ∈T +�� +� +∑︁ +𝑖 ∈G +𝐶𝑖𝑝𝑖,𝑡 + +∑︁ +𝑗 ∈D +𝐶𝑑 +𝑗 𝑑𝑗,𝑡 + +∑︁ +𝑖 ∈I +𝐶𝑚 +𝑖 𝑚𝑖,𝑡 ++ +∑︁ +𝑖 ∈W +𝐶𝑤 +𝑖 𝑤𝑖,𝑡 + +∑︁ +𝑖 ∈R +𝐶𝑟 +𝑖 𝑟𝑖,𝑡 +� +(6a) +s.t. +∑︁ +𝑙 ∈L+𝑛 +𝑓𝑙,𝑡 − +∑︁ +𝑙 ∈L−𝑛 +𝑓𝑙,𝑡 + +∑︁ +𝑖 ∈G𝑛 +𝑝𝑖,𝑡 + +∑︁ +𝑖 ∈I𝑛 +(𝑀𝑖,𝑡 − 𝑚𝑖,𝑡) ++ +∑︁ +𝑖 ∈W𝑛 +(𝑊𝑖,𝑡 − 𝑤𝑖,𝑡) + +∑︁ +𝑖 ∈R𝑛 +(𝑅𝑖,𝑡 − 𝑟𝑖,𝑡) += +∑︁ +𝑗 ∈D𝑛 +(𝐷𝑗,𝑡 − 𝑑𝑗,𝑡) + +∑︁ +𝑖 ∈𝐷𝐶𝑛 +𝑙𝑖,𝑡, +∀𝑛 ∈ N,𝑡 ∈ T, +(6b) +𝑓𝑙,𝑡 = 𝐵𝑙 (𝜃𝑛,𝑡 − 𝜃𝑚,𝑡), +∀𝑙 = (𝑚,𝑛) ∈ L,𝑡 ∈ T, +(6c) +− 𝐹𝑚𝑎𝑥 +𝑙 +≤ 𝑓𝑙,𝑡 ≤ 𝐹𝑚𝑎𝑥 +𝑙 +, +∀𝑙 ∈ L,𝑡 ∈ T, +(6d) +Θ𝑚𝑖𝑛 +𝑛 +≤ 𝜃𝑛,𝑡 ≤ Θ𝑚𝑎𝑥 +𝑛 +∀𝑛 ∈ N,𝑡 ∈ T, +(6e) +− 𝑅𝐷𝑖 ≤ 𝑝𝑖,𝑡 − 𝑝𝑖,𝑡−1 ≤ 𝑅𝑈𝑖, +∀𝑖 ∈ G,𝑡 ∈ T, +(6f) +0 ≤ 𝑝𝑖,𝑡 ≤ 𝑃𝑚𝑎𝑥 +𝑖 +, +∀𝑖 ∈ G,𝑡 ∈ T, +(6g) +0 ≤ 𝑑𝑗,𝑡 ≤ 𝐷𝑗,𝑡, +∀𝑗 ∈ D,𝑡 ∈ T, +(6h) +0 ≤ 𝑚𝑖,𝑡 ≤ 𝑀𝑗,𝑡, +∀𝑖 ∈ I,𝑡 ∈ T, +(6i) +0 ≤ 𝑤𝑖,𝑡 ≤ 𝑊𝑗,𝑡, +∀𝑖 ∈ W,𝑡 ∈ T, +(6j) +0 ≤ 𝑟𝑖,𝑡 ≤ 𝑅𝑗,𝑡, +∀𝑖 ∈ R,𝑡 ∈ T . +(6k) +The generation sources include conventional thermal power +plants (e.g. gas, nuclear, coal), non-wind renewables (e.g. hydro), +imports, and wind power plants. Due to long-term commitments +for imports or goal of reducing carbon emissions with renewables, +the imports and renewables are non-dispatchable as in [41] but +can be curtailed at the cost of 𝐶𝑚 +𝑖 +and 𝐶𝑟 +𝑖 $/MWh (𝐶𝑤 +𝑖 $/MWh for +wind) respectively. Sometimes the power supply may not meet +the demand, and each unit of load shedding (not served load) is +at the cost of value of lost load (VOLL) 𝐶𝑑 +𝑗 . In this paper, the unit +generation cost is 1/2/4 $/MWh for nuclear/coal/gas. The penalties +are 500/100/1,000 $/MWh for import/wind/non-wind renewables +curtailment, 1,000 $/MWh for load shedding. +The constraints are typical for DC-OPF. Constraint 6b represents +the balance constraint at each bus, whose associated dual value is +the locational marginal price (LMP) at that bus, indicating marginal +cost of adding 1 MW load at a specific location, so the price can +go negative/high when curtailment/load shedding happens. Con- +straints 6c–6e represent how the power flow (6c) is determined +given the line capacity (6d) and phase angle (6e) limits. Constraint +6f limits conventional power plants’ rate of ramping up/down gen- +eration. Constraints 6g–6k bound the conventional generation, load +shedding, and curtailments (6i–6k) respectively. +B +FUEL CARBON EMISSION RATES +Below are the fuel carbon emission rates (carbon emissions per +MWh energy generated from that fuel) we use to calculate carbon +emissions: +12 + +Reducing Datacenter Operational Carbon Emissions Effectively by Cooperating with the Grid +Conference’17, July 2017, Washington, DC, USA +Table 6: Carbon Emission Rates of Different Fuels [4, 36] +Generation Type +Carbon Emission Rate (kg CO2/MWh) +Coal +895.2 +Natural Gas +388.9 +Oil +877.6 +Dual-fuel +633.3 +Nuclear +0 +Geothermal +107.6 +Biomass +0 +Hydro +0 +Wind +0 +Import +428 +13 + +Conference’17, July 2017, Washington, DC, USA +Liuzixuan Lin and Andrew A. Chien +REFERENCES +[1] Follow the sun, wind, moon, 2009. https://www.vertatique.com/cloud-computing- +starting-follow-sunwindmoon. +[2] Clicking Clean Virginia: The Dirty Energy Powering Data Center Alley, February +2019. https://www.greenpeace.org/usa/reports/click-clean-virginia/. +[3] U.S. Energy Information Administration. Electric power monthly, 2022. +[4] United States Environmental Protection Agency. Emissions & generation resource +integrated database, 2020. +[5] Sohaib Ahmad, Arielle Rosenthal, Mohammad H. Hajiesmaili, and Ramesh K. +Sitaraman. Learning from optimal: Energy procurement strategies for data +centers. In Proceedings of the Tenth ACM International Conference on Future Energy +Systems, e-Energy ’19, page 326–330, New York, NY, USA, 2019. Association for +Computing Machinery. +[6] Amazon Web Services (AWS). Amazon global datacenters. +[7] Noman Bashir, Tian Guo, Mohammad Hajiesmaili, David Irwin, Prashant Shenoy, +Ramesh Sitaraman, Abel Souza, and Adam Wierman. Enabling sustainable clouds: +The case for virtualizing the energy system. In Proceedings of the ACM Symposium +on Cloud Computing, pages 350–358, 2021. +[8] Lori Bird, M Milligan, and Debra Lew. Integrating Variable Renewable Energy: +Challenges and Solutions. Technical report, NREL, 2013. +[9] CAISO. Market processes and products, 2022. +[10] John Campbell. Data centres used 14% of republic of ireland’s electricity use, +2022. +[11] Niangjun Chen, Xiaoqi Ren, Shaolei Ren, and Adam Wierman. Greening multi- +tenant data center demand response. Performance Evaluation, 91:229–254, 2015. +[12] Shutong Chen, Lei Jiao, Lin Wang, and Fangming Liu. An online market mecha- +nism for edge emergency demand response via cloudlet control. In IEEE INFOCOM +2019-IEEE Conference on Computer Communications, pages 2566–2574. IEEE, 2019. +[13] Andrew A Chien. Characterizing opportunity power in the california indepen- +dent system operator (caiso) in years 2015-2017. Energy and Earth Science, 3(2), +December 2020. Also available as University of Chicago, Computer Science +TR-2018-07, https://newtraell.cs.uchicago.edu/research/publications/techreports. +[14] Andrew A Chien, Richard Wolski, and Fan Yang. The zero-carbon cloud: High- +value, dispatchable demand for renewable power generators. The Electricity +Journal, pages 110–118, 2015. +[15] Andrew A. Chien, Fan Yang, and Chaojie Zhang. Characterizing curtailed and +uneconomic renewable power in the mid-continent independent system operator. +AIMS Energy, 6(2):376–401, December 2018. +[16] Microsoft Corporation. +Microsft 2021 environmental sustainability report, +2021. Available from https://www.microsoft.com/en-us/corporate-responsibility/ +sustainability/report. +[17] Olivier Corradi. Marginal vs average: which one to use in practice?, 2022. +[18] CPUC. California Public Utilities Commission (CPUC). http://www.cpuc.ca.gov/ +PUC/energy/Renewables. +[19] Wei Deng, Fangming Liu, Hai Jin, Bo Li, and Dan Li. Harnessing renewable energy +in cloud datacenters: opportunities and challenges. iEEE Network, 28(1):48–55, +2014. +[20] Hui Dou, Yong Qi, Wei Wei, and Houbing Song. Carbon-aware electricity cost +minimization for sustainable data centers. IEEE Transactions on Sustainable +Computing, 2(2):211–223, 2017. +[21] Iain Dunning, Joey Huchette, and Miles Lubin. Jump: A modeling language for +mathematical optimization. SIAM review, 59(2):295–320, 2017. +[22] Dominion Energy. 2018 virginia integrated resource plan, 2018. +[23] Dominion Energy. 2020 virginia integrated resource plan, 2020. +[24] Dominion Energy. 2021 update to the 2020 integrated resource plan, 2021. +[25] Mary Jo Foley. Cloud revenues power microsoft’s $51.7 billion q2 in fiscal year +2022, January 2022. https://www.zdnet.com/article/microsoft-cloud-revenues- +power-microsofts-51-7-billion-second-fy22-quarter/. +[26] Robbie Galvin. Data centers are pushing ireland’s electric grid to the brink, 2021. +[27] Íñigo Goiri, William Katsak, Kien Le, Thu D Nguyen, and Ricardo Bianchini. +Parasol and greenswitch: Managing datacenters powered by renewable energy. +In ACM SIGARCH Computer Architecture News, pages 51–64. ACM, 2013. +[28] Caroline Golin and Nick Pearson. A policy roadmap for 24/7 carbon-free energy, +2022. +[29] Google. About google datacenters. https://www.google.com/about/datacenters/. +[30] Google. Moving toward 24x7 carbon-free energy at google data centers: Progress +and insights. Technical report, Google, October 2018. +[31] V. Gupta, P. Shenoy, and R. K. Sitaraman. Efficient solar provisioning for net-zero +internet-scale distributed networks. In 2018 10th International Conference on +Communication Systems Networks (COMSNETS), pages 372–379, 2018. +[32] LLC Gurobi Optimization. Gurobi optimizer reference manual, 2020. +[33] GWEC. Global wind report: Annual market update. Technical report, Global +Wind Energy Council, 2016. Documents curtailment around the world. +[34] Siqi Han. The wind is wasted in china. https://www.wilsoncenter.org/, 2015. +[35] Kevin Imboden. 2022 global data center market comparison, 2021. +[36] California ISO. Greenhouse gas emission tracking methodology, 2016. +[37] California ISO. Managing oversupply, 2022. +[38] Nicola Jones. How to stop data centres from gobbling up the world’s electricity. +Nature, September 2018. +[39] Lucas Joppa. Made to measure: Sustainability commitment progress and updates, +2021. +[40] Peter Judge. Eirgrid pulls plug on 30 irish data center projects, 2022. +[41] Kibaek Kim, Fan Yang, Victor Zavala, and Andrew A. Chien. Data centers as +dispatchable loads to harness stranded power. IEEE Transactions on Sustainable +Energy, 2016. DOI 10.1109/TSTE.2016.2593607. +[42] Wook Hyun Kwon and Soo Hee Han. Receding horizon control: model predictive +control for state models. Springer Science & Business Media, 2006. +[43] Kien Le, Ricardo Bianchini, Thu D. Nguyen, Ozlem Bilgir, and Margaret Martonosi. +Capping the brown energy consumption of internet services at low cost. In +Proceedings of the International Conference on Green Computing, GREENCOMP +’10, page 3–14, USA, 2010. IEEE Computer Society. +[44] Tan N Le, Zhenhua Liu, Yuan Chen, and Cullen Bash. Joint capacity planning +and operational management for sustainable data centers and demand response. +In Proceedings of the Seventh International Conference on Future Energy Systems, +pages 1–12, 2016. +[45] Liuzixuan Lin and Andrew A. Chien. Characterizing stranded power in the +ercot in years 2012-2019: A preliminary report. Technical Report TR-2020-06, +University of Chicago, August 2020. +[46] Liuzixuan Lin, Victor M Zavala, and Andrew A Chien. Evaluating coupling +models for cloud datacenters and power grids. In Proceedings of the Twelfth ACM +International Conference on Future Energy Systems, pages 171–184, 2021. +[47] Minghong Lin, Adam Wierman, Lachlan LH Andrew, and Eno Thereska. Dynamic +right-sizing for power-proportional data centers. IEEE/ACM Transactions on +Networking, 21(5):1378–1391, 2012. +[48] Julia Lindberg, Yasmine Abdennadher, Jiaqi Chen, Bernard C Lesieutre, and Line +Roald. A guide to reducing carbon emissions through data center geographical +load shifting. In Proceedings of the Twelfth ACM International Conference on Future +Energy Systems, pages 430–436, 2021. +[49] Julia Lindberg, Line Roald, and Bernard Lesieutre. The environmental potential +of hyper-scale data centers: Using locational marginal co2 emissions to guide geo- +graphical load shifting. In Proceedings of the 54th Hawaii International Conference +on System Sciences, page 3158, 2020. +[50] Z. Liu, M. Lin, A. Wierman, S. Low, and L. L. H. Andrew. Greening geographical +load balancing. IEEE/ACM Transactions on Networking, 23(2):657–671, 2015. +[51] Zhenhua Liu, Yuan Chen, Cullen Bash, Adam Wierman, Daniel Gmach, Zhikui +Wang, Manish Marwah, and Chris Hyser. Renewable and cooling aware work- +load management for sustainable data centers. In Proceedings of the 12th ACM +SIGMETRICS/PERFORMANCE joint international conference on Measurement and +Modeling of Computer Systems, pages 175–186, 2012. +[52] Zhenhua Liu, Adam Wierman, Yuan Chen, Benjamin Razon, and Niangjun Chen. +Data center demand response: Avoiding the coincident peak via workload shifting +and local generation. Performance Evaluation, 70(10):770–791, 2013. +[53] Jianying Luo, Lei Rao, and Xue Liu. Temporal load balancing with service delay +guarantees for data center energy cost optimization. IEEE Transactions on Parallel +and Distributed Systems, 25(3):775–784, 2013. +[54] Srilatha Manne. Architecting a sustainable planet. In MICRO 2020, https://www. +microarch.org/micro53/, October 2020. Keynote Talk. +[55] Electricity Maps. Electricity maps, 2022. +[56] Eric Masanet, Arman Shehabi, Nuoa Lei, Sarah Smith, and Jonathan Koomey. +Recalibrating global data center energy-use estimates. Science, 367(6481):984–986, +2020. +[57] Valérie Masson-Delmotte, Panmao Zhai, Hans-Otto Pörtner, Debra Roberts, Jim +Skea, Priyadarshi R Shukla, Anna Pirani, W Moufouma-Okia, C Péan, R Pidcock, +et al. Global warming of 1.5 c. An IPCC Special Report on the impacts of global +warming of, 1(5), 2018. +[58] James McAnany. 2021 demand response operations markets activity report: +January 2022, 2021. +[59] Microsoft Azure. Azure global datacenters. https://azure.microsoft.com/en- +us/global-infrastructure/. +[60] Rich Miller. Dominion energy plans more green power for virginia’s data centers, +2022. +[61] Anthony Papavasiliou and Shmuel S Oren. Multiarea stochastic unit commitment +for high wind penetration in a transmission constrained network. Operations +Research, 61(3):578–592, 2013. +[62] Brian Peccarelli. Three ways covid-19 is accelerating digital transformation in +professional services. June 2020. https://bit.ly/34Qitb5, 37% growth. +[63] Sundar Pichai. Our biggest renewable energy purchase ever, 2019. +[64] SC Pryor, RJ Barthelmie, and TJ Shepherd. 20% of us electricity from wind will +have limited impacts on system efficiency and regional climate. Scientific reports, +10(1):1–14, 2020. +[65] Asfandyar Qureshi, Rick Weber, Hari Balakrishnan, John Guttag, and Bruce +Maggs. Cutting the electric bill for internet-scale systems. In Proceedings of the +ACM SIGCOMM 2009 Conference on Data Communication, SIGCOMM ’09, page +123–134, New York, NY, USA, 2009. Association for Computing Machinery. +14 + +Reducing Datacenter Operational Carbon Emissions Effectively by Cooperating with the Grid +Conference’17, July 2017, Washington, DC, USA +[66] Ana Radovanovic, Ross Koningstein, Ian Schneider, Bokan Chen, Alexandre +Duarte, Binz Roy, Diyue Xiao, Maya Haridasan, Patrick Hung, Nick Care, et al. +Carbon-aware computing for datacenters. arXiv preprint arXiv:2106.11750, 2021. +[67] John Roach. +Microsoft’s virtual datacenter grounds ‘the cloud’ in re- +ality, April 2021. +Microsoft to build 50 to 100 datacenters per year, +https://news.microsoft.com/innovation-stories/microsofts-virtual-datacenter- +grounds-the-cloud-in-reality/. +[68] Yuanyuan Shi, Bolun Xu, Baosen Zhang, and Di Wang. Leveraging energy storage +to optimize data center electricity cost in emerging power markets. In Proceedings +of the Seventh International Conference on Future Energy Systems, pages 1–13, +2016. +[69] Staff. Covid-19 accelerates cloud adoption, market to reach $1 trillion, idc. Equip- +ment FA News, October 2020. https://www.equipmentfa.com/news/31459/covid- +19-accelerates-cloud-adoption-market-to-reach-1t-idc. +[70] Grant L. Stewart, Gregory A. Koenig, Jingjing Liu, Anders Clausen, Sonja Klingert, +and Natalie Bates. Grid accommodation of dynamic hpc demand. In Proceedings +of the 48th International Conference on Parallel Processing: Workshops, ICPP 2019, +New York, NY, USA, 2019. Association for Computing Machinery. +[71] Qihang Sun, Shaolei Ren, Chuan Wu, and Zongpeng Li. An online incentive +mechanism for emergency demand response in geo-distributed colocation data +centers. In Proceedings of the seventh international conference on future energy +systems, pages 1–13, 2016. +[72] Muhammad Tirmazi, Adam Barker, Nan Deng, Md E. Haque, Zhijing Gene Qin, +Steven Hand, Mor Harchol-Balter, and John Wilkes. Borg: The next generation. +In Proceedings of the Fifteenth European Conference on Computer Systems, EuroSys +’20, New York, NY, USA, 2020. Association for Computing Machinery. +[73] Rahul Urgaonkar, Bhuvan Urgaonkar, Michael J Neely, and Anand Sivasubrama- +niam. Optimal power cost management using stored energy in data centers. In +Proceedings of the ACM SIGMETRICS joint international conference on Measurement +and modeling of computer systems, pages 221–232, 2011. +[74] Markus Wacket. German parties agree on 2030 coal phase-out in coalition talks, +2021. +[75] WattTime. Automated emissions reduction, 2022. +[76] Fan Yang and Andrew A Chien. Zccloud: Exploring wasted green power for +high-performance computing. In 2016 IEEE International Parallel and Distributed +Processing Symposium (IPDPS), pages 1051–1060. IEEE, 2016. +[77] Fan Yang and Andrew A. Chien. Large-scale and extreme-scale computing with +stranded green power: Opportunities and costs. IEEE Transactions on Parallel +and Distributed Systems, 29(5), December 2017. +[78] Lin Yang, Mohammad Hassan Hajiesmaili, Ramesh K. Sitaraman, Enrique Mallada, +Wing Shing Wong, and Adam Wierman. Online inventory management with +application to energy procurement in data centers. CoRR, abs/1901.04372, 2019. +[79] Chaojie Zhang and Andrew A Chien. Scheduling challenges for variable capacity +resources. In Workshop on Job Scheduling for Parallel Processing (JSSPP), 2021. +[80] Linquan Zhang, Shaolei Ren, Chuan Wu, and Zongpeng Li. A truthful incentive +mechanism for emergency demand response in colocation data centers. In 2015 +IEEE Conference on Computer Communications (INFOCOM), pages 2632–2640. +IEEE, 2015. +[81] Weiqi Zhang, Line A Roald, Andrew A Chien, John R Birge, and Victor M Zavala. +Flexibility from networks of data centers: A market clearing formulation with +virtual links. Electric Power Systems Research, 189:106723, 2020. +[82] Yanwei Zhang, Yefu Wang, and Xiaorui Wang. Electricity bill capping for cloud- +scale data centers that impact the power markets. In 2012 41st International +Conference on Parallel Processing, pages 440–449. IEEE, 2012. +[83] Jiajia Zheng, Andrew A. Chien, and Sangwon Suh. Mitigating curtailment and +carbon emissions through load migration between data centers. Joule, October +2020. +[84] Zhi Zhou, Fangming Liu, Shutong Chen, and Zongpeng Li. A truthful and +efficient incentive mechanism for demand response in green datacenters. IEEE +Transactions on Parallel and Distributed Systems, 31(1):1–15, 2018. +[85] Zhi Zhou, Fangming Liu, Bo Li, Baochun Li, Hai Jin, Ruolan Zou, and Zhiyong +Liu. Fuel cell generation in geo-distributed cloud services: A quantitative study. +In 2014 IEEE 34th International Conference on Distributed Computing Systems, +pages 52–61. IEEE, 2014. +[86] Zhi Zhou, Fangming Liu, Ruolan Zou, Jiangchuan Liu, Hong Xu, and Hai Jin. +Carbon-aware online control of geo-distributed cloud services. IEEE Transactions +on Parallel and Distributed Systems, 27(9):2506–2519, 2016. +15 + diff --git a/MdE1T4oBgHgl3EQfZQR8/content/tmp_files/load_file.txt b/MdE1T4oBgHgl3EQfZQR8/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..92c492fd074cc277bf1c188fdce54ae351639c7d --- /dev/null +++ b/MdE1T4oBgHgl3EQfZQR8/content/tmp_files/load_file.txt @@ -0,0 +1,1036 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf,len=1035 +page_content='Reducing Datacenter Operational Carbon Emissions Effectively by Cooperating with the Grid Liuzixuan Lin University of Chicago Chicago, IL, USA lzixuan@uchicago.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='edu Andrew A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Chien University of Chicago & Argonne National Lab Chicago, IL, USA achien@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='uchicago.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='edu ABSTRACT Facing growing concerns about power consumption and carbon emissions, cloud providers are adapting datacenter loads to reduce carbon emissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' With datacenters exceeding 100MW, they can affect grid dynamics, so doing this without damaging grid perfor- mance is difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' We study power adaptation algorithms that use grid metrics and seek to reduce datacenter (DC) operational carbon emissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' First, we consider local, online adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Second, to reduce grid disruption that can arise from adaptation, we consider using an external coordinator that limits aggregate impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Finally, we study a novel cooperative scheme, where datacenters first create a full- day adapted power plan, based on day-ahead information, and then share it with the power grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' This novel approach is a partner- ship that reflects the shared responsibility between datacenter and the grid for carbon minimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' For each, we report DC carbon emissions reduction and grid impacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Results show that PlanShare, the novel cooperative scheme is superior to all online approaches considered, achieving a net bene- fit of 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='6% reduction in datacenter operational carbon emissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' PlanShare achieves 26%–132% better performance than the best online approach, and far larger for many.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' This benefit arises from allowing the grid optimization with full future DC load information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Further it enables efficient DC resource management with a 24-hour capacity schedule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' It also decreases power cost for the entire grid (datacenters and other customers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' CCS CONCEPTS Applied computing → Data centers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' • Hardware → Power and energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' KEYWORDS Data centers, Carbon emissions, Power management, Adaptive loads ACM Reference Format: Liuzixuan Lin and Andrew A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Chien.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Reducing Datacenter Operational Carbon Emissions Effectively by Cooperating with the Grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' In Proceedings Permission to make digital or hard copies of all or part of this work for personal or 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='1145/nnnnnnn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='nnnnnnn 1 INTRODUCTION With the commercial success of both internet-scale applications and public cloud computing, cloud computing infrastructure has continued to grow rapidly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' A recent article documented the addi- tion of over 50 datacenters a year by a single cloud provider [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Rapid growth of cloud provider revenue and documented power purchases confirm Amazon, Microsoft, and Google’s cloud growth rates exceeding 30% annually for the past 5 years [25, 39, 63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' As of 2021, the total power consumption of these three leading cloud providers is over 64 TWh, equivalent to the power consumption of approximately 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='4 million American homes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Some estimates project datacenter power consumption growing to more than 10% of global electric power use by 2030 [38, 46, 56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' The largest cloud datacenter sites consist of multiple buildings with power footprints from 200 MW to 1 GW [2, 6, 29, 59, 67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' This rapid growth raises concerns about associated carbon emis- sions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Cloud providers purchase renewable power (long-term pur- chase contracts) or renewable offsets (renewable-energy credits— RECs) to assuage concerns [16, 63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Several have announced “24x7” matching efforts to match their power use and renewable generation hourly [16, 28, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' However, these contracts are legal, accounting arrangements, not actual transfers of power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' That is, even with these arrangements, the cloud datacenters cause and consume large quantities of fossil-fuel generated power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Datacenters are major load contributors in many power grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' For example, in Virginia, datacenters account for 12% of grid power consumption today, and are projected to grow to 18% by 2027 and 22% by 2032 [23, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' The environmental impacts of datacenters there have drawn much attention [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Another hotspot is Ireland where datacenters currently account for 14% of national electricity use [10] and could be 30% by 2029 [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Worried that datacenter load growth may retard grid decarbonization, Ireland’s state grid opera- tor is canceling datacenter projects [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' As computing continues its rapid growth, datacenters will exceed 10% of load in an increasing number of grids—with the concomitant challenges [2, 35, 38, 56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' One promising idea to reduce the carbon emissions (and also power cost from cloud provider perspective) of datacenters is power adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' This reflects the goal of aligning power consumption with renewable generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Over a decade, computer systems re- searchers have proposed many creative ideas for dynamic load shaping across Internet datacenters with the objective of reducing carbon emissions [19, 20, 27, 31, 50, 51] or power cost [5, 43, 47, 52, 53, 65, 68, 73, 78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' These approaches seek to align power use with 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='03148v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='DC] 9 Jan 2023 Conference’17, July 2017, Washington, DC, USA Liuzixuan Lin and Andrew A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Chien low-carbon or low-price power, using online control techniques as illustrated in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Power Grid Datacenters Resource Manager Varying Power Load Carbon Emissions, Power Price… Workload Figure 1: Datacenters can adapt power use to align with pe- riods of lower carbon emissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Many efforts to shape power consumption or shift load to re- duce carbon emissions focus on internal challenges of datacenter management and compute load prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' One recently deployed at scale creates a daily compute capacity plan to enable effective resource management [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' However, most ignore the negative impacts of varying datacenter power on grid dynamics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' trigger- ing unnecessary generation starts and load shedding).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' With large datacenter loads, this coupling of power adaptation and grid im- pacts is critical: 10-20% of grid load is large enough to alter power grid dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Load control schemes thought to be productive for small datacenter loads can fail to deliver benefits (carbon emissions, power cost) and even damage grid performance if used with large datacenter loads (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='7% increase in datacenter carbon emissions from large loads’ overshifting in Section 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' As a result, it’s impor- tant for datacenter power adaptation to be studied with power grid models [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' One consequence of this new insight is that there is no known solution to coupled management of large-scale datacenter power adaptation and power grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' With the goal of finding widely usable solutions to effectively reduce datacenter operational carbon emissions, we explore grid metrics that can guide datacenter power adaptation and different power adaptation approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' For each approach, we focus on the effectiveness in datacenter carbon reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' We also consider datacenter capacity variation from power adaptation, and impacts on power cost for both datacenters and other grid customers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' To achieve this goal, we propose a novel cooperative scheme, PlanShare, where datacenters share their full-day adaptation plan with the grid in advance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' The certainty for the varied datacen- ter power use enables the power grid to optimize generation and transmission schedule across time periods, achieving most carbon reduction without harming the grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Overall, this is a widely usable solution that satisfies both datacenter and power grid objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Specific contributions of the paper include: We consider three metrics (average carbon intensity (ACI), grid price (GPrice), and locational marginal price (LMPrice)), comparing their effectiveness for online control algorithms that adapt computing load to reduce carbon emissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' LM- Price is the most effective by reducing 1–5% of datacenter carbon emissions vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' ACI and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='7–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='5% vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' GPrice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' A further advantage is that LMPrice is practically available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' We next consider how to improve online power adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' More detailed future price information (+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='3%) and step size (+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='3%) can increase the effectiveness of local adaptation based on LMPrice, bringing a total datacenter carbon reduc- tion of 10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Another approach with a coordinator to limit aggregate load change gives a lower improvement, reducing datacenter carbon emissions by 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='3%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' The plan sharing approach (PlanShare) achieves the greatest benefit, decreasing 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='6% of operational carbon emissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' PlanShare significantly outperforms best online approach with 26%–132% increase in carbon-emissions reductions for datacenter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Other benefits include efficient DC resource man- agement with a 24-hour capacity plan and power cost reduc- tion for both the datacenters and non-datacenter customers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' The remainder of the paper includes background in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' In Section 3 we describe the challenges of effective power adaptation for carbon reduction with datacenters as large loads in the grid, and then in Section 4 we give an overview of the datacenter power adaptation approaches we study addressing the challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Section 5 introduces the methodology of grid-coupled simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' In Section 6 we evaluate different power adaptation approaches’ effectiveness in datacenter carbon reduction, with other impacts on datacenter and non-datacenter grid customers considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Finally, in Section 7 and 8, we discuss related work, summarize the paper, and give some future directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' 2 BACKGROUND 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='1 Growth of Cloud Datacenter Load in Power Grids Hyperscale cloud providers (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Amazon, Microsoft, Google, Al- ibaba) are building larger and more datacenters to meet the needs of digitalization, which accelerated since the Covid-19 pandemic [24, 62, 69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' The rapid growth of hyperscale cloud industry is re- flected in its power consumption: 1% of worldwide power con- sumption today, projections suggest it could exceed 10% by 2030 [38, 46, 56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Similar rapid growth rates are also supported by cor- porate renewable power purchase data [39, 63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' As a result, datacenters have become the key driver of new elec- tric power demand, direct cause of new emissions, as well as con- struction of power plants, transmission, and energy storage infras- tructure [23, 60]—it’s no exaggeration to say that cloud datacenters are shaping the future of power grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' In Virginia, US, datacenters account for about 12% of regional power consumption as of today, projected to be 18% in 5 years and 22% in 10 years [23, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' An- other example is Ireland, where datacenters reach 14% of national electricity use today [10] and could use 30% by 2029 [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Worried that datacenter load growth may hold back grid decarbonization, Ireland’s state grid operator is canceling datacenter projects [40], unless datacenters are equipped with their own renewable genear- ion or flex their power demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Similar story is happening in more and more parts of the world as the scale and reach of datacenters continue to grow [2, 35, 38, 56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' In short, growing cloud datacenter power demand is a top-shelf concern for power grids around the world, challenging grid operation and decarbonization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' 2 Reducing Datacenter Operational Carbon Emissions Effectively by Cooperating with the Grid Conference’17, July 2017, Washington, DC, USA From the cloud provider perspective, such exponential growth in datacenter power consumption first translates into growing power cost, as power price [3] don’t decrease at the same speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Also, it raises concern about carbon emissions growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Obscured by car- bon offset or renewable purchase, additional datacenters can still result in more fossil fuel power plants if the load is not aligned with renewable generation [7, 22, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' To improve sustainability, Google and Microsoft seek to match their load with renewable generation (24x7, 100/100/0) hourly [30, 39], with attempts such as temporal load shaping [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' While the fraction of dynamic load and scale of load change are small (a few percent of capacity [66]), they are expected to increase in future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Furthermore, such load change behaviors can disturb the grid even at a small percentage of dynamic load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' There are already reports of power variations from supercomputer loads (all on or all off) affecting grid stability [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' For gigawatt cloud datacenters, a 10% load change produces a 100 MW swing, similar to the dynamic range we study;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' a 40% load change, which is not unusual for some diurnal peak to trough, would be 400 MW!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='2 Dynamics of a Renewable-based Power Grid Climate agreements seek to limit global warming to below 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='5◦C above pre-industrial to avoid catastrophic climate-related risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' This means the world’s carbon emissions need to be halved by 2030 and net-zero by 2050 [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Power grid decarbonization is necessary for achieving these goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Recent years have seen the rise of renewable sources in energy mix of many power grids across the world, such as 34% in California (2021), 22% in China (2019), and 22% in Europe (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Some regions have set more ambitious goals for this decade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' For example, California aims at 60% renewable fraction by 2030 [18], and Germany plans to phase out coal power plants (20% of power generation in 2020) by 2030 [74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Integrating intermittent renewable sources (mainly wind and solar) is challenging for the power grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' For example, renewable generation can be wasted due to temporal mismatch with energy demand and transmission limits, producing “curtailment” [8, 13, 15, 33, 34, 45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' A complementary problem is generation shortage, such as under extreme weather.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' One key to meet this challenge is to increase supply or demand flexibility, and corresponding solutions include energy storage, adaptive loads, etc [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' 0 6 12 18 24 Time 0 100 200 300 ACI (kg CO2/MWh) 0 6 12 18 24 Time 0 50 100 Price ($/MWh) Figure 2: CAISO’s Daily Average Carbon Intensity and Price Variation, 2022/05/02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Left: Average Carbon Intensity (kg CO2/MWh), Right: Grid Price ($/MWh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Source: CAISO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' The intermittency of renewable sources reflects in grid metrics, producing more fluctuating generation mix and associated carbon intensity (carbon emissions per MWh energy consumption) as re- newable fraction increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Figure 2 (left) shows California’s typical daily carbon intensity pattern: the average carbon intensity keeps low at near zero during the daytime when solar generation domi- nates, but it climbs up to about 200 kg CO2/MWh as natural gas generators are up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' In addition, as wind and solar generation comes at zero fuel cost, the fluctuating generation mix results in time- varying price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' When there is plenty of solar generation, the price can drop to near zero (Figure 2, right)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Adaptive loads such as data- centers, electric vehicles, and smart appliances may use variation in these grid metrics to reduce their carbon emissions or power cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' 3 PROBLEM Cloud datacenters seek to reduce their carbon emissions by align- ing their power use with periods of plentiful renewable generation (low-carbon periods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' However, power grid carbon is determined by a complex set of factors including load, generation available, transmission topology, and ramping rate limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' In contrast to the complex grid dynamics, the availability of important grid metrics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' locational marginal generation) that can guide power adapta- tion is limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Therefore, it’s challenging for cloud datacenters of realistic-scale to use dynamic power control to reduce carbon emis- sions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Attempts can achieve only a fraction of anticipated benefits or worse harm the grid, themselves, and other customers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' One example is a phenomenon called “overshifting” illustrated in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Simulating a Spring day in California, datacenter loads that adapt power based on average carbon intensity (1st row) in- crease the average carbon intensity (2nd row), producing increase in grid carbon emissions equivalent to 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='7% of datacenter carbon emissions in a day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Tremendous harm!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' This is because the data- centers act in unison, increasing or decreasing load at the same time (3rd row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Large aggregate load changes produce competition for power and oversubscribe the opportunity, causing dispatch of additional fossil-fuel generation and increasing carbon emissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' So the key question for datacenters is: How to adapt data- center power to reduce operational carbon-emissions effec- tively?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' That is, how to control their power load to better align with low-carbon opportunity— and to do so without disturbing grid dispatch (which generators).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' To achieve this, datacenters must solve several challenges: (1) Identify opportunities: datacenters need to find the right times to increase and decrease load to reduce their carbon emissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' (2) Avoid contention or overshifting: to realize the goal datacen- ters must avoid oversubscribing the opportunity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' (3) Avoid harming power grid: ensure the datacenter power load adaptation does not harm power grid participants (datacen- ters, other customers, generators, grid) by increasing prices or overall carbon emissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' 4 APPROACH Overall, to meet the challenges faced by datacenters, we explore different datacenter power adaptation approaches and couple them to grid simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' We first consider three grid metrics (average carbon intensity, grid price, and locational marginal price) which 3 Conference’17, July 2017, Washington, DC, USA Liuzixuan Lin and Andrew A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Chien Figure 3: Overshifting: local, online adaptation using aver- age carbon intensity (ACI) increases datacenter carbon emis- sions by 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='7% because they all make the same load adapta- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' datacneters can identify opportunities to reduce carbon emissions from and adapt power load to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Then we compare three approaches (Figure 4) to adapt datacenter load for operational carbon reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Grid Day Ahead Hourly Data- centers Day Ahead Hourly Coordinator Day Ahead Hourly Data- centers Day Ahead Plan DCs adapt based on Grid Metric DCs adapt, limited by Coordinator DCs share Plan, Grid adapts to it Data- centers Figure 4: Different Approaches for Datacenters to Adapt Their Power Loads to Reduce Carbon-emissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Representing a classes of control techniques in previous work, local online adaptation (Figure 4, left) makes hourly decisions using real-time and future metrics from the grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Our studies of local adaptation provide insights into the limits of these approaches to reduce carbon emissions, showing the improvements from more future grid information and smoothed adapted load step size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' At times, datacenters adapting load can harm their efforts to reduce carbon emissions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' overshifting shown in Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' We consider use of an external coordinator to eliminate such harm by limiting the total power change of a set of datacenters (Figure 4, middle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Each DC makes a request to adapt power level hourly, but the external coordinator limits their collective (total) power level change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Finally, we consider a new approach that combines adaptation based on day-ahead grid information and sharing of the resulting adapted plan with the power grid (Figure 4, right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' This approach requires each DC to plan ahead—making binding choices 24-hours in advance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' The fixed plan gives the grid certainty for the datacenter power load, enabling it to optimize generation and transmission scheduling under dynamic constraints that bind decisions across time periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' We introduce the algorithms of each approach and evaluate them in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' We focus on the datacenter carbon reduction achieved, and also consider impacts on datacenter operation and power price impacts for both datacenters and other grid customers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' The simulations are with realistic settings: datacenter sizes (200 MW), power grids with wind penetration levels for today and the coming decade (15%–60%, 2015 levels–2050 target), and 10 to 40 datacenters, which match current and near-future (2027) loads in Northern Virginia and Ireland power grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' For clarity of exposition, we focus on a 30-DC scenario, representing ≈10% grid load, below current levels in NoVA’s (12%) and Ireland’s (14%) power grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' 5 GRID-COUPLED SIMULATION METHODOLOGY This section sets up the framework under which we evaluate the dat- acenter power adaptation approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Adaptation adapts power to grid metrics (Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='1), respecting the load flexibility constraints (Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Resulting time-varying loads affect the dynamics (pric- ing, generation, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=') in the power grid (Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='1 Grid Metrics for Carbon Optimization Power grids have complex dynamics, so the “best” grid metric for reducing carbon emissions is an open research question [17, 48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' A good metric enables carbon reduction and is available in most or all power grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' We consider several candidates: Average carbon intensity or ACI (kg CO2/MWh) is the carbon emissions per MWh energy consumption in the grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Derived from fuel mix, ACI is only recently available from unvalidated 3rd parties (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Electricity Maps [55]) in a frac- tion of the world’s grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Grid price or GPrice ($/MWh) is power price in a grid or region (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' “hub price”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Renewable generation often bids low causing low power price to be correlated with carbon intensity (Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Price information is widely available in day-ahead and real-time markets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Locational marginal price or LMPrice ($/MWh) is price at a specific node in the power grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' LMPrice reflects local properties such as nearby renewables and grid transmission constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' LMPrice is widely-available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Recently, researchers and companies [49, 75] propose marginal carbon intensity as a metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Though an interesting direction, mar- ginal carbon intensity is not broadly available, so we do not consider it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='2 Modeling Datacenter Load Flexibility Datacenter load flexibility is the basis of power adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' We follow the datacenter load flexibility model in [46] wherein every datacenters adjust load within a dynamic range and can defer 4 200 ACI 100 0 5 10 15 20 25 FixedLoad 200 ACI Adaptive Load 100 0 5 10 15 20 25 Change 50 Load 0 50 0 5 10 15 20 25 TimeReducing Datacenter Operational Carbon Emissions Effectively by Cooperating with the Grid Conference’17, July 2017, Washington, DC, USA workload (backlog), but must catch up within a 24-hour day (Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' 2 Average Load Load Time of Day t t+1 Step Size: max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' load change between two hours Backlog (deferred work) Catchup Dynamic Range Figure 5: Datacenter Load Flexibility Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Formally, 𝑙𝑜𝑎𝑑𝑚𝑖𝑛 ≤ 𝑙𝑖,𝑡 ≤ 𝑙𝑜𝑎𝑑𝑚𝑎𝑥, ∀𝑖,𝑡 (1) 𝑏𝑎𝑐𝑘𝑙𝑜𝑔𝑖,𝑡 = 𝑏𝑎𝑐𝑘𝑙𝑜𝑔𝑖,𝑡−1 + (𝑎𝑣𝑔𝐿𝑜𝑎𝑑 − 𝑙𝑖,𝑡) (2) 𝑏𝑎𝑐𝑘𝑙𝑜𝑔𝑖,24 = 0, ∀𝑖 (3) Datacenter capacity variation can harm a datacenter’s compu- tation performance [79] and harm power markets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' We consider a step size limit that bounds datacenter power change in one-hour: |𝑙𝑖,𝑡 − 𝑙𝑖,𝑡−1| ≤ 𝑠𝑡𝑒𝑝𝑆𝑖𝑧𝑒, ∀𝑖,𝑡 (4) The configurations for the attributes above are shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' These resource utilization and capacity assumptions are typical of hyperscale datacenters [2, 29, 59, 72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' We focus on the [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='4, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='0] dynamic range where the potential benefit is large but overshifting problem is more prominent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' This level of flexibility (30% flexible workload) is realistic according to Google’s BorgTNG trace [72] and growing use of batch workloads such as ML training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Table 1: Configurations of Datacenter Attributes Attribute Configuration Maximum Capacity 200 MW Average Utilization Level 70% Average Power Load (𝑎𝑣𝑔𝐿𝑜𝑎𝑑) 140 MW Dynamic Range [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='6, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='8] (small) [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='4, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='0] (large) Step Size 10, 20, 40, 80, 120 MW/h 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='3 Power Grid Model, Base Load and Generation Profiles We evaluate the coupling approaches under a realistic grid model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Grid operation is simulated by solving the direct-current optimal power flow (DC-OPF) problem in [41, 46] and Appendix A, which minimizes the grid dispatch cost in one-day time horizon with hourly intervals, subject to typical grid constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' The grid metrics for carbon optimization (ACI, GPrice, LMPrice) are derived from the OPF solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Reflecting reality, we assume renewable generation sources have lower generation costs and curtailment penalties that encourage use, producing low prices when renewable generation is dispatched at the margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' This makes price highly correlated with carbon metrics, as what happens in the real world (Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' The grid model is a reduced California power system (CAISO) consisting of 225 buses, 375 transmission lines, 130 thermal genera- tors (31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='2 GW total capacity), 11 non-wind renewable power plants, 5 wind power plants, and 40 loads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Power can also be imported at 5 boundary buses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' The thermal power plants’ ramp rates are scaled up by 4-fold to reflect current ramping capability of CAISO as in [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' This model is originally from [61] and has been used to assess the impact of dynamic datacenter load management in [41, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' There are 8 base load (DCs excluded), imports, and non-wind renewable generation profiles that cover the four seasons (Spring, Summer, Fall, Winter) and weekday/weekend (WD/WE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Figure 6 shows how each load profile varies in a day, with the average span- ning from 23,780 MW (WinterWE) to 31,089 MW (SummerWD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Comparing the weekday and weekend in a season, there is a clear weekly pattern that the two profiles are in similar shape but week- end’s load is lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' For each season, we use 100 wind generation profiles (shown in Figure 7), and a season’s weekday and week- end share these profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' The wind generation presents intra-day variation—higher in the late night and early morning, which is a misalignment with load and becomes more evident as wind pen- etration increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' To model higher wind penetration, the wind generation are scaled up equally at current sites, assuming those sites can be expanded or replace old wind turbines with higher capacity turbines [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' 0 10 20 Hour 0 5 10 15 20 25 30 35 40 Base Load (GW) WD WE (a) Spring 0 10 20 Hour 0 5 10 15 20 25 30 35 40 (b) Summer 0 10 20 Hour 0 5 10 15 20 25 30 35 40 (c) Fall 0 10 20 Hour 0 5 10 15 20 25 30 35 40 (d) Winter Figure 6: Grid Base (Non-DC) Load Profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' 0 10 20 Hour 0 2 4 6 8 10 12 14 Wind Gen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' (GW) (a) Spring 0 10 20 Hour 0 2 4 6 8 10 12 14 (b) Summer 0 10 20 Hour 0 2 4 6 8 10 12 14 (c) Fall 0 10 20 Hour 0 2 4 6 8 10 12 14 (d) Winter Figure 7: Used Wind Scenarios (15% Penetration Level).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Red lines represent average wind scenarios for each season.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Datacenters are added to random buses in the grid, which reflects the fact that cloud company site selection is more based on business 5 Conference’17, July 2017, Washington, DC, USA Liuzixuan Lin and Andrew A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Chien considerations external to the power grid (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' tax breaks, jobs, internet hookups, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='4 Evaluation Metrics The metrics used to evaluate effects of datacenter power adaptation in achieving datacenter goals, and impact on other grid customers, include: Datacenter Carbon Reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Because datacenter load adaptation accounts for the grid carbon emissions change, we report the percentage reduction in datacenter operational carbon emissions as: (𝑔𝑟𝑖𝑑𝐶𝑎𝑟𝑏𝑜𝑛𝑓 𝑖𝑥𝑒𝑑−𝑙𝑜𝑎𝑑 − 𝑔𝑟𝑖𝑑𝐶𝑎𝑟𝑏𝑜𝑛𝑎𝑑𝑎𝑝𝑡𝑎𝑡𝑖𝑜𝑛) 𝑑𝑎𝑡𝑎𝑐𝑒𝑛𝑡𝑒𝑟𝐶𝑎𝑟𝑏𝑜𝑛𝑓 𝑖𝑥𝑒𝑑−𝑙𝑜𝑎𝑑 ∗ 100% and 𝑔𝑟𝑖𝑑𝐶𝑎𝑟𝑏𝑜𝑛 = 24 ∑︁ 𝑡=1 𝑔𝑒𝑛𝑓 ,𝑡 ∗ 𝑒𝑚𝑖𝑠𝑠𝑖𝑜𝑛𝑅𝑎𝑡𝑒𝑓 where 𝑔𝑒𝑛𝑓 ,𝑡 is generation from fuel 𝑓 in the 𝑡-th hour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Fuel emission rates are from US EPA eGrid database [4, 36] and listed in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' 𝑑𝑎𝑡𝑎𝑐𝑒𝑛𝑡𝑒𝑟𝐶𝑎𝑟𝑏𝑜𝑛 in the fixed DC load scenario is calculated using the grid emission rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Datacenter Average Power Price ($/MWh) is how much the datacenter would pay for power considering the location of datacenter and the time when power is consumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' For multiple datacenters, we compute the average across them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Datacenter Average Capacity Variation (MW/h) is de- fined as the average change in power level (load) of a dat- acenter between adjacent one-hour periods, the lower the better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' More formally: 1 23 24 ∑︁ 𝑡=2 |𝑙𝑖,𝑡 − 𝑙𝑖,𝑡−1| where 𝑙𝑖,𝑡 denotes the power level of datacenter 𝑖 at time 𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Non-datacenter Customer Average Power Price ($/MWh) is the average power price across grid customers other than datacenters, weighted by power demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Conventional datacenter operation has fixed capacity and power load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' We use it (“Fixed”) as the baseline to illustrate the impacts from datacenter power adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='5 Experiment Setup To reflect the impact of wind variation, we run the simulations with 100 wind scenarios for each day type (WD and WE of a season share the same set of wind scenarios), which is equivalent to simulating a total of 800 days [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' For each day, we vary the wind penetration level and simulate datacenter operation using different adaptation models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' The results reported in Section 6 are the average of 8 day types (weighted for the number of weekdays and weekend days) and varied wind scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' We used Julia 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='2 with JuMP v0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='5 [21] to implement the grid simulation and solved grid OPF with Gurobi Optimizer v9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='0 [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' 6 EVALUATING DATACENTER POWER ADAPTATION APPROACHES We explore representative datacenter power adaptation algorithms for the three approaches outlined in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Each compared to fixed datacenter loads, and evaluated coupled to grid dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Performance is compared based on datacenter carbon reduction, as well as price for datacenters and other customers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' We present a subset of our results, focused on a representative scenario with 30 datacenters and dynamic range of [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='4, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' This corresponds to 10% of grid load, well below current leading edge grids, and increasingly realistic for dozens of grids throughout the world in the near future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' We explore varied levels of wind penetration, to capture future evolution as grids decarbonize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' We also plot standard deviation as “whiskers” to indicate variation across the wind scenarios used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' A specific wind scenario typically produces correlated results for different algorithms, so these capture variability, not uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='1 Local, Online Adaptation Datacenter-local online power adaptation approaches make real- time load decisions based on current and future information about grid metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' We first consider a dynamic programming algorithm that makes hourly decisions using current value of metric and its daily average (Adapt-Avg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' The algorithm selects amongst {𝑎𝑣𝑔𝐿𝑜𝑎𝑑,𝑎𝑣𝑔𝐿𝑜𝑎𝑑 ± dynamic range/2} to minimize the expecta- tion, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' 𝑙𝑖,𝑡 ∗ 𝑚𝑒𝑡𝑟𝑖𝑐𝑖,𝑡 + (𝑏𝑎𝑐𝑘𝑙𝑜𝑔𝑖,𝑡−1 + 𝑎𝑣𝑔𝐿𝑜𝑎𝑑 − 𝑙𝑖,𝑡) ∗ 𝑚𝑒𝑡𝑟𝑖𝑐𝑖 which can be either carbon emissions or power cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' The 𝑏𝑎𝑐𝑘𝑙𝑜𝑔 is then updated given the determined power level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Datacenter load adaptation is coupled to grid dispatch (OPF optimization), as below, where datacenters are denoted by 𝑖, and hours by 𝑡: (1) For all i,t: 𝑙𝑖,𝑡 = 𝑎𝑣𝑔𝐿𝑜𝑎𝑑 (neutral initial condition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' (2) Solve grid OPF with {𝑙𝑖,𝑡 }, defining 𝑚𝑒𝑡𝑟𝑖𝑐𝑖,𝑡 as day-ahead information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' (3) At the beginning of 𝑡 = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=', 24-th hour, each datacenter decides 𝑎𝑑𝑎𝑝𝑡(𝑖,𝑡) based on the metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' (4) Then the grid solves OPF with updated datacenter loads {𝑙𝑖,𝑡 } (5) This new OPF solution redefines 𝑚𝑒𝑡𝑟𝑖𝑐𝑖,𝑡 and for the future (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' [𝑡 + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=', 24]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='1 Comparing Grid Metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' We evaluate the effectiveness of different grid metrics (average carbon intensity (ACI), grid price (GPrice), and locational marginal price (LMPrice)) for online adapta- tion algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' LMPrice consistently outperforms ACI and GPrice (see Figure 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' At 15% wind, online adaptation fails to achieve carbon-emissions reduction because the generation supply is tight, and oversubscrip- tion happens during low-carbon periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Higher wind penetration provides more opportunity, enabling online adaptation using LM- Price to reduce datacenter carbon emissions by 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='4% (60% wind), consistently exceeding ACI (by 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='4%) and Gprice (by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='4%), In short, price metrics work better, and finer-grained detailed pricing is the best amongst those.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' 6 Reducing Datacenter Operational Carbon Emissions Effectively by Cooperating with the Grid Conference’17, July 2017, Washington, DC, USA 6% 4% 2% 0% 2% 4% 6% 8% 10% 12% 15% 30% 45% 60% DC Carbon Reduction Wind Penetration AdaptCarbon-Avg AdaptGPrice-Avg AdaptLMPrice-Avg Figure 8: Achieved Datacenter Carbon Reduction based on Carbon Emissions, Grid Price, and Locational-Marginal Price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' To illustrate this, Figure 9 shows a single-day timeline, The graphs show that ACI and GPrice as grid-wide signals, drive uni- form datacenter behavior, maximizing their collective power swings for the grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' In contrast, LMPrice is a local metric, and drives diverse behavior, better reflecting grid constraints such as transmission and ramping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Figure 9: Global and local metrics (LMPrice) drive differ- ent Datacenter Load Adaptation behaviors (Spring Weekday, 45% Wind Penetration).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='2 Improving Local Online Adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' We build on online adap- tation with the most effective metric (LMPrice), improving these approaches by addition of finer-resolution or better estimates of future (forecasts) and smoothing power level changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Forecasts (Future Information).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Hourly LMPrice is generally available in the day-ahead market, which can be thought of as a forecast1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' We add this information—the full 24 hours of day-ahead prices as a forecast—to enable better adaptation in AdaptLMPrice- Hourly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' These forecasts are not guaranteed to be accurate as the final OPF that runs at the specific hour will determine grid fuel mix, prices, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Step Size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Our results show that online adaptation can cause large datacenter load fluctuations, harming both datacenter com- puting efficiency and grid dynamics (generation dispatch, carbon 1It’s a forecast because it’s subject to revision by other markets (eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' hourly, 15-minute, 5-minute, real-time) as the time approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' intensity, price).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' To mitigate the harm from large, frequent load changes, we add a step size constraint that smoothes hour-to-hour individual DC power level changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Using the day-ahead hourly locational prices at the DC (price array {predLMPrice𝑖,𝑡 },𝑡 = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=', 24), in the 𝑗-th hour, datacenter 𝑖 performs dynamic programming (DP) on the price array {𝑝𝑖,𝑡 } with: 𝑝𝑖,𝑡 = � LMPrice𝑖,𝑗, if 𝑡 = 𝑗 and 𝑗 ≠ 1 predLMPrice𝑖,𝑡, otherwise , 𝑡 = 𝑗, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=', 24 where LMPrice𝑖,𝑗 is the real-time price after {𝑙𝑖,𝑗−1} are set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' The dynamic programming algorithm produces a load array based on the following recurrence formula: 𝑐𝑜𝑠𝑡𝑖 (𝑛,𝑡,𝑙) = 𝑙 ∗ 𝑝𝑖,𝑡 + min 𝑙′ {𝑐𝑜𝑠𝑡𝑖 (𝑛 + 𝑙 − 𝑎𝑣𝑔𝐿𝑜𝑎𝑑,𝑡 − 1,𝑙′) | |𝑙 − 𝑙 ′| ≤ 𝑠𝑡𝑒𝑝𝑆𝑖𝑧𝑒} (5) where 𝑐𝑜𝑠𝑡𝑖 (𝑛,𝑡,𝑙) denotes the minimum power cost of the sub- problem ending at 𝑡-th hour with backlog 𝑛 and power level 𝑙 in the 𝑡-th hour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' The datacenter takes the first element of the load array as 𝑙𝑖,𝑡—similar to the receding horizon control [42] but with a fixed horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' We add detailed future price information, and then tune the step size, showing the results in Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' The results reflect the best step sizes (40 MW/h for AdaptLMPrice-Hourly, and no limit for AdaptLMPrice-Avg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Comparing these two variants, we can see the impact of detailed future price information and step size: AdaptLMPrice-Hourly’s improvement is roughly 50% from hourly information and 50% from step size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' To further study the impact of detailed future information, we develop a variant, which at each hour is given hourly future informa- tion for the next 6 hours but only daily average after that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' The two approaches with detailed future information are clearly superior, with the 6-hour future price information variant (AdaptLMPrice- 6hr+Avg) already better than AdaptLMPrice-Avg, and benefit grows with more future information in AdaptLMPrice-Hourly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Quantitatively, at 30% wind penetration, 6 hours’ information gives a benefit of additional 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='6% reduction and 24-hours gives a benefit of additional 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='2% of datacenter carbon emissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' These benefits grow to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='5% and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='6% at 60% wind penetration respectively, enabling AdaptLMPrice-Hourly to reach a 10% of DC carbon reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='2 Constraining Adaptive Datacenters with Coordinator Independently controlled DCs can react together, when decisions are based on grid-wide or other strongly-correlated metrics pro- ducing a large aggregate power change (see Section 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' We have seen LMPrice is better, but even its correlation across sites can produces synchronized DC load changes that are difficult for the grid to manage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Addressing this, we add an external limiter, called coordinator, to mitigate the harm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' In AdaptLMPrice-Avg-Coord, each datacenter runs independent online control algorithm then submits their adaptation to a coordinator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' The coordinator limits total power change for a set of datacenter, using a quota: (1) Generate a random permutation of datacenters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' 7 AdaptCarbon-Avg AdaptGPrice-Avg AdaptLMPrice-Avg 200 150 ice 0 A 100 200 200 0 20 0 20 0 20 200 200 200 pe 9 150 150 150 0100 100 100 0 20 0 20 0 20Conference’17, July 2017, Washington, DC, USA Liuzixuan Lin and Andrew A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Chien 2% 0% 2% 4% 6% 8% 10% 12% 14% 16% 30% 60% DC Carbon Reduction Wind Penetration AdaptLMPrice-Avg AdaptLMPrice-6hr+Avg AdaptLMPrice-Hourly Figure 10: Achieved Datacenter Carbon Reduction with Var- ied Future LMPrice Information (daily average, 6 hours, 24 hours).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' (2) For each datacenter, if 𝑐ℎ𝑎𝑛𝑔𝑒 ≤ 𝑞𝑢𝑜𝑡𝑎 then accept, 𝑞𝑢𝑜𝑡𝑎 = 𝑞𝑢𝑜𝑡𝑎 − 𝑐ℎ𝑎𝑛𝑔𝑒;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' else, reject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' If local controllers were unable to get their requested change, they must update their backlogs accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Figure 11: Achieved Datacenter Carbon Reduction vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Coor- dinator Quota.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Figure 11 shows DC carbon reduction with varied coordinator quotas (3600–1200 MW/h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Each line represents a different wind penetration level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Coordination improves performance, mitigating overshifting harm, at 15-60% wind penetration and growing benefits as the quota is tightened.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' All of points on each line reflect greater carbon reduction than that for AdaptLMPrice-Avg (leftmost point on each line), and steady improvement as the quota is reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' The mitigation of overshifting is a larger benefits for datacenters when generation is tight (15% wind penetration), improving DC carbon reduction by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='3%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Benefits are smaller than those achieved by exploiting future price information (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' AdaptLMPrice-Hourly), which yields a 50% higher DC carbon reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Multiple Coordinators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' In many geographic areas, there are mul- tiple cloud providers (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Northern Virginia, Texas, Ireland, or China’s Ningxia), and each cloud provider has multiple datacenter sites in that area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' As competitors, they may not be willing to share a coordinator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' To model this multi-coordinator scenario, we assume there are 2 or 3 coordinators in the grid, each coordinating 10 or 15 DCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' We use an overall quota of 1200 MW/h, and divide it equally across the coordinators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Increasing the number of coordinators further decreases DC car- bon emissions slightly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' The reason for this is narrower coordinator scope increases the smoothness of total datacenter load, but the improvements are small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' 3 coordinators produce improvements up to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='7% of DC carbon emissions, achieving 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='3% of DC carbon reduction at 60% wind penetration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='3 Adapted Power Plan Sharing Adaptive DC loads cause grid problems as their large power changes are unpredictable and strain generator ramp constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Further, unplanned adaptation can produce rapid changes in compute capac- ity, making it difficult for cloud resource managers to be efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' In view of these insights, we propose a new approach—PlanShare: datacenters create a 24-hour adapted power plan based on LMPrice in day-ahead grid market [9] ahead of operating day, and then share the plan with the grid , allowing it to optimize grid-wide based on the DC information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Formally, with datacenters denoted by 𝑖 and hours denoted by 𝑡: (1) For all i,t: 𝑙𝑖,𝑡 = 𝑎𝑣𝑔𝐿𝑜𝑎𝑑 (neutral initial condition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' (2) Solve grid OPF with {𝑙𝑖,𝑡 }, defining initial LMPrice𝑖,𝑡 as day- ahead information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' (3) Each datacenter makes 24-hour adaptation plan using AdaptLMPrice- Hourly’s dynamic programming algorithm and shares it with the grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' (4) Solve grid OPF for [1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=', 24] with adapted {𝑙𝑖,𝑡 } to model the next day’s operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' The datacenter must follow the full-day power plan it shares with the power grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Figure 12 compares PlanShare and with the local, online ap- proaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' The results show the benefits of plan sharing with the grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Using essentially the same adaptation algorithm, and on less accurate information, by working with the grid, PlanShare reduces DC carbon emissions by up to 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='6%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' This is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='26x (26%) better than the best online adaptation result, and the advantage is even higher at lower wind penetration as large as 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='32x (132%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' By con- tributing its adaptation plan to the grid optimization—in advance and as a committed schedule—PlanShare dramatically improves the datacenter carbon emissions reduction that can be achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' 10% 5% 0% 5% 10% 15% 20% 15% 30% 45% 60% DC Carbon Reduction Wind Penetration AdaptCarbon-Avg AdaptLMPrice-Avg AdaptLMPrice-Hourly PlanShare Figure 12: PlanShare outperforms all online adaptation ap- proaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' 8 (%） CarbonReduction 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='0 60%Wind 30%Wind 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='5 15%Wind 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='0 DC 35003000250020001500 CoordinatorQuota(MW/h)Reducing Datacenter Operational Carbon Emissions Effectively by Cooperating with the Grid Conference’17, July 2017, Washington, DC, USA Sensitivity to Length of Shared Plan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Having demonstrated the benefits of a practical scheme (24-hour day-ahead plans are widely available in power grids around the world), we ask an intellec- tual curiosity question—how much plan-ahead is really needed by the grid?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' We vary the length of plan shared from 1 to 24 hours, reporting results in Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Interestingly, a single hour of plan-ahead enables plan sharing to match the performance of AdaptLMPrice-Hourly, the best online approach, despite the fact that PlanShare has no online adaptation—the full 24-hour schedule is fixed in advance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' As the length of plan is increased to 3, 6, and 12 hours, the benefit of plan sharing increases significantly, reaching the large benefits previously highlighted in Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' 0% 2% 4% 6% 8% 10% 12% 14% 15% 30% 45% 60% DC Carbon Reduction Wind Penetration 1 Hour 3 Hours 6 Hours 12 Hours 24 Hours Figure 13: Length of Plans Shared: PlanShare variants with 1-24 hours of plan shared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='4 Datacenter Adaptation Impacts beyond Carbon Datacenter load adaption produces other impacts on datacenter operation and other customers in the grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' We study several here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Datacenter Capacity Variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Capacity variation is a critical concern for datacenter operators as it affects workload efficiency of the available capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Using the average capacity variation metric (see Section 5), Figure 14 shows DC carbon reduction on y-axis and capacity variation on x-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' As before, DC carbon reduction is relative to the fixed DC load scenario, and capacity variation (MW/h) is normalized to DC average capacity (140 MW).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' An ideal adaptation approach would fall in the upper-left corner (high carbon reduction, low capacity variation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Each line connnects results as wind penetration increases for a given adaptation approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' The plot clearly shows how the higher-performing adaptation techniques exploit increased changes to track the changing grid car- bon properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' A progression from AdaptCarbon-Avg to AdaptLMPrice- Avg to AdaptLMPrice-Hourly shows a clear tradeoff of carbon re- duction for online capacity variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' The PlanShare results achieve the greatest carbon reduction, do require greater capacity variation to do so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' However, it’s worth noting that PlanShare fixes the capac- ity plan in advance, so the resource manager has a statically known resource schedule at the start of the day, facilitating computing workload scheduling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Scheduling studies and other proposals argue for the benefits of this stability [66, 79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Figure 14: Datacenter Carbon Reduction and Capacity Vari- ation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' 20 10 0 10 20 30 40 50 60 70 30% 60% Change in Power Price ($/MWh) Wind Penetration AdaptCarbon-Avg AdaptLMPrice-Avg AdaptLMPrice-Hourly PlanShare Figure 15: Change in Average Power Price for Datacenters Compared to Fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Datacenter Power Cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Adaptive DC loads affect power pricing in the grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' In Figure 15, results show that local, online adaptation can cause significant power prices increases of $20 to $50/MWh, corresponding to 59%–490% increase in power cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' This is because locally controlled adaptation clashes with grid constraints and dy- namics (overshifting in Section 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' This is likely a major deterrent for datacenter adoption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' In contrast, sharing the datacenter’s load schedule in advance as in PlanShare decreases average power price stably, up to 30% compared with the fixed-load scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Non-DC Customer Power Cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' In Figure 16, We explore how adapting datacenter power impacts non-datacenter (non-DC) cus- tomers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Online approaches, including AdaptCarbon-Avg and AdaptLMPrice- Avg, significantly increase the price for non-DC customers, espe- cially for lower wind penetration with less excess renewable gen- eration (AdaptLMPrice-Hourly also increases price at 15% wind).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Datacenters as growing consumers of power are already subject to growing scrutiny and negative publicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Pricing hard to other cus- tomers has the potential to cause a backlash, so datacenters should be careful about deploying such local, online adaptive power-level control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' PlanShare avoids this price harm for non-DC customers at all wind penetration levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' 9 AdaptCarbon-Avg AdaptLMPrice-Avg AdaptLMPrice-Hourly PlanShare 15 15% Wind Carbon Reduction (%) 30% Wind 10 + 45% Wind 60% Wind DC 0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='0 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='5 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='0 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='5 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='0 %CapacityChange/hourConference’17, July 2017, Washington, DC, USA Liuzixuan Lin and Andrew A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Chien ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='Table 2: Summary of Datacenter Power Adaptation Approaches ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='AdaptCarbon- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='AdaptLMPrice- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='AdaptLMPrice- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='PlanShare ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='PlanShare ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='Avg ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='Avg ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='Hourly ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='(1 hour) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='(24 hours) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='DC Carbon Reduction ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='– ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='neutral ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='++ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='DC Capacity Variation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='neutral ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='neutral ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='neutral ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='DC Power Cost ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='– – ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='– ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='neutral ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='++ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='Non-DC Power Cost ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='– – ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='– ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='neutral ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='++ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='30% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='60% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='Change in Power Price ($/MWh) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='Wind Penetration ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='AdaptCarbon-Avg ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='AdaptLMPrice-Avg ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='AdaptLMPrice-Hourly ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='PlanShare ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='Figure 16: Change in Average Power Price for Non-DC Cus- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='tomers Compared to Fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='5 Summary Table 2 summarizes the impacts of datacenter power adaptation in different dimensions, where “+” means advantage and “–” means disadvantage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' The weakest performance plan sharing approach— PlanShare with 1 hour’s shared plan—matches and outperforms all of the other approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' And, as we can see in Figure 13, PlanShare with 24-hour adaptation plan outperforms others significantly and is by far the best, delivering the greatest DC carbon reduction while enabling datacenter resource managers to have known capacity plans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Further, PlanShare-24 produces benefits in power cost for both datacenters and other customers in the grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' 7 DISCUSSION AND RELATED WORK To the best of our knowledge, this is the first paper that proposes sharing datacenter adaptive power plan with the grid to reduce operational carbon emissions effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Given the LMPrice metric availability and industrial practice in making day-ahead adaptive power plan [66], this approach is feasible and widely usable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' A major concern might be that cloud providers would be unwilling to share their capacity plan a day in advance, for proprietary or com- petitive reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' It’s worth pointing out that the datacenter’s local utility/grids (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Dominion Energy, PGE, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=') already know data- centers’ historical power consumption, going back days, months, and years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Further, if the cloud datacenters wanted to intentionally mask their actual compute load, they could still do so with on-site batteries or even generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' We review related work below: Datacenter Power Load Adaptation (Shaping).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Early ideas like “follow-the-moon”“chase-the-wind” propose to shift datacenter work- load to a time or place with low energy prices or carbon emissions [1], exploiting variations across geography, competitive power load, grids, and renewables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' The datacenter load adaptation ex- plored in this paper addresses a subset of these ideas—temporal load shaping or shifting, which has been widely studied, typically employing sophisticated online control or optimization techniques [20, 47, 51, 53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' An additional variant is to manage colocated energy storage [5, 27, 68, 73, 78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' This work usually assumes datacenters are small loads (reflected in grid trace-based studies or DC-centric evaluation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Consequently, they don’t consider the impact of DC dynamic load change on the grid—and further circular impacts on carbon, prices, generation, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' In this paper, we consider realistic cases when a collection of large hyperscale datacenters (200 MW) collectively comprise a significant fraction of grid load, and thus directly affect grid dispatch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Another dimension of “follow-the-moon” load shifting is spatial or geographic, with similar goals [31, 43, 50, 65, 82, 83, 85, 86].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' We do not consider this direction, but it is certainly of promise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Datacenters in Demand Response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Datacenters’ ability to delay workload (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' deferring batch jobs) can be used to participate in demand-response programs, reducing load during peak periods according to requests from the grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Such participation can reduce DC power cost, and research has explored how to balance this ben- efit while respecting service-level objectives (SLO) [44, 52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Further efforts in this area design sophisticated markets that incentivize DC operators and even their colocation tenants to participate in demand response [11, 12, 71, 80, 84].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Demand response is designed for emergency reduction in load to protect grid stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' As a result, actions are rare, and the power reduction is small relative to the total power consumption [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' In contrast, we consider multiple datacenters’ active, continuous power adaptation for reducing operational carbon emissions with dynamic range up to 60% of capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Grid-coupled Datacenter Adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Some researchers propose building datacenters as dispatchable loads controlled by the grid to harness excess carbon-free power [14, 76, 77], and the grid benefits like decreased dispatch cost and renewable curtailment are reported in a study including a grid model [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Other efforts include explor- ing the grid benefits of temporal or spatial load shifting [49, 81].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' [48] studies what grid metrics can effectively guide carbon-aware spatial load shifting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' While also showing locational marginal price is effective, they claim locational mariginal carbon emissions is better, which, however, is not broadly available today.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' This paper builds on the insights of [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' The authors showed that without modeling the grid’s dynamics, the carbon emission pro- jections could be significantly wrong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Thus, its necessary to include grid models in studies of large-scale temporal workload shifting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' 10 Reducing Datacenter Operational Carbon Emissions Effectively by Cooperating with the Grid Conference’17, July 2017, Washington, DC, USA However, that work provides no solution to coupled management of datacenters and power grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Industrial Efforts to Reduce Datacenter Carbon Emissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Hyper- scale cloud providers have taken actions on sustainability such as renewable power purchase agreements (PPAs) that match their annual power consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Several (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Google, Microsoft) are targeting so-called “24×7” or “100/100/0”, hourly matching data- center’s power consumption with carbon-free generation [30, 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Recent efforts also propose application and resource-management level load shifting to exploit varying power carbon-intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' These efforts involve small dynamic range (only a few percent of load), and do not consider their impact on power grids [54, 66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Notably, Google’s effort, termed carbon-aware computing [66], creates day-ahead virtual capacity curves (VCC) that addresses con- cerns of online capacity variation (and its impact on ability to use resources efficiently).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' However, VCC is used within datacenters to optimize compute resource management, not to inform the power grid of upcoming load power changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' This perhaps could be attrib- uted to the fact that only very small percentages of load are under control of these schemes today.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' 8 SUMMARY AND FUTURE WORK Cloud providers are adapting datacenter power to renewable gener- ation to reduce operational carbon emissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' However, for today’s large cloud datacenters, the numerous prior techniques that seek datacenter benefits with independent, online control can fail to achieve carbon emissions reduction and even harm the grid, in- creasing carbon emissions and other customers’ power costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' We first consider grid metrics which datacenters can adapt to for reducing operational carbon emissions and identify locational marginal price (LMPrice) as a widely available and the most effective metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Exploring improvements for online adaptation and other new approaches, we then find that sharing adapted power plan is critical: datacenters create an adapted power plan based on day- ahead grid metric, and share that with the grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' This approach can not only enable datacenters to achieve more of their goals— more carbon reduction (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='6%), lower average power prices (-30%), predictable capacity (and thereby better internal utilization), but also eliminate the harm on other customers in the grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' In short, this approach is a widely usable solution for cloud providers to effectively reduce operational carbon emissions without harming the power grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' There are a number of exciting future directions for this research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' First, from datacenter resource management perspective, creating load flexibility and responding to capacity change, while respect- ing service-level objectives (SLO) continues to be an interesting area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' The resource managers and applications today lack a clear cost metric, and further it’s unclear what types and extents of flex- ibility are possible or valuable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Second, here we have focused on temporal load adaptation, but as cloud providers run datacenters across multiple regions, spatial shifting combined with day-ahead capacity plan schedules is an interesting space to explore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Finally, a more direct follow-on question is how we can balance datacenter privacy considerations with the clear benefit of sharing capacity plan information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' ACKNOWLEDGMENTS This Work is supported in part by NSF Grants CMMI-1832230, OAC- 2019506, and the VMware University Research Fund.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' We also thank the Zero-carbon Cloud team members!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' 11 Conference’17, July 2017, Washington, DC, USA Liuzixuan Lin and Andrew A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Chien A DIRECT-CURRENT OPTIMAL POWER FLOW FORMULATION We model the grid operation using the direct-current optimal power flow (DC-OPF) model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Notations in this model are listed below: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='Table 3: DC-OPF Notation: Sets ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='Notation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='Description ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='Notation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='Description ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='D (D𝑛) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='Demand loads (at bus 𝑛) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='G (G𝑛) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='Generators (at bus 𝑛) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='I (I𝑛) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='Import points (at bus 𝑛) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='L ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='Transmission lines ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='L+𝑛/L−𝑛 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='Transmission ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='lines ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='to/from bus 𝑛 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='N ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='Buses ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='R (R𝑛) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='Renewable generators ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='(at bus 𝑛) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='Time periods ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='W (W𝑛) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='Wind farms (at bus 𝑛) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='𝐷𝐶(𝐷𝐶𝑛) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='Datacenters (at bus 𝑛) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='Table 4: DC-OPF Notation: Parameters ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='Notation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='Description ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='Notation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='Description ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='𝐵𝑙 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='Susceptance of trans- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='mission line 𝑙 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='𝐶𝑖 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='Generation cost of gen- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='erator 𝑖 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='𝐶𝑑 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='𝑗 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='Load-shedding penalty ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='at load 𝑗 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='𝐶𝑤 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='𝑖 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='Curtailment penalty at ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='wind farm 𝑖 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='𝐶𝑚 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='𝑖 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='Curtailment penalty at ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='import point 𝑖 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='𝐶𝑟 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='𝑖 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='Curtailment penalty at ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='renewable 𝑖 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='𝐷𝑗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='𝑡 Demand load of con- sumer 𝑗 at time 𝑡 𝐹𝑚𝑎𝑥 𝑙 Maximum power flow of transmission line 𝑙 𝑀𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='𝑡 Power production of import 𝑖 at time 𝑡 𝑃𝑚𝑎𝑥 𝑖 Maximum power out- put of generator 𝑖 𝑅𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='𝑡 Power production of re- newable 𝑖 at time 𝑡 𝑅𝑈𝑖 Ramp-up limit of gener- ator 𝑖 𝑅𝐷𝑖 Ramp-down limit of generator 𝑖 𝑊𝑤,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='𝑡 Power from wind farm 𝑤 at time 𝑡 Θ𝑚𝑖𝑛 𝑛,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='𝑡 Minimum phase angle at bus 𝑛 at time 𝑡 Θ𝑚𝑎𝑥 𝑛,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='𝑡 Maximum phase angle at bus 𝑛 at time 𝑡 Table 5: DC-OPF Notation: Decision Variables Notation Description Notation Description 𝑑𝑗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='𝑡 Load shedding at load 𝑗 at time 𝑡 𝑓𝑙,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='𝑡 Power flow of line 𝑙 at time 𝑡 𝑚𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='𝑡 Curtailment at import 𝑖 at time 𝑡 𝑝𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='𝑡 Power from generator 𝑖 at time 𝑡 𝑟𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='𝑡 Curtailment at renew- able 𝑖 at time 𝑡 𝑤𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='𝑡 Curtailment at wind farm 𝑖 at time 𝑡 𝜃𝑛,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='𝑡 Phase angle at bus 𝑛 at time 𝑡 Datacenter power levels 𝑙𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='𝑡 are either decision variables (grid- controlled optimization) or external decisions (independent or co- ordination) according to the coupling model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' The power grid solves the DC-OPF model (one-day time horizon with hourly intervals in our studies), minimizing the dispatch cost (6a) that consists of generation cost, load shedding penalty, and curtailment penalties: min ∑︁ 𝑡 ∈T �� � ∑︁ 𝑖 ∈G 𝐶𝑖𝑝𝑖,𝑡 + ∑︁ 𝑗 ∈D 𝐶𝑑 𝑗 𝑑𝑗,𝑡 + ∑︁ 𝑖 ∈I 𝐶𝑚 𝑖 𝑚𝑖,𝑡 + ∑︁ 𝑖 ∈W 𝐶𝑤 𝑖 𝑤𝑖,𝑡 + ∑︁ 𝑖 ∈R 𝐶𝑟 𝑖 𝑟𝑖,𝑡 � (6a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' ∑︁ 𝑙 ∈L+𝑛 𝑓𝑙,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='𝑡 − ∑︁ 𝑙 ∈L−𝑛 𝑓𝑙,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='𝑡 + ∑︁ 𝑖 ∈G𝑛 𝑝𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='𝑡 + ∑︁ 𝑖 ∈I𝑛 (𝑀𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='𝑡 − 𝑚𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='𝑡) + ∑︁ 𝑖 ∈W𝑛 (𝑊𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='𝑡 − 𝑤𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='𝑡) + ∑︁ 𝑖 ∈R𝑛 (𝑅𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='𝑡 − 𝑟𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='𝑡) = ∑︁ 𝑗 ∈D𝑛 (𝐷𝑗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='𝑡 − 𝑑𝑗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='𝑡) + ∑︁ 𝑖 ∈𝐷𝐶𝑛 𝑙𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='𝑡,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' ∀𝑛 ∈ N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='𝑡 ∈ T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' (6b) 𝑓𝑙,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='𝑡 = 𝐵𝑙 (𝜃𝑛,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='𝑡 − 𝜃𝑚,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='𝑡),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' ∀𝑙 = (𝑚,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='𝑛) ∈ L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='𝑡 ∈ T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' (6c) − 𝐹𝑚𝑎𝑥 𝑙 ≤ 𝑓𝑙,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='𝑡 ≤ 𝐹𝑚𝑎𝑥 𝑙 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' ∀𝑙 ∈ L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='𝑡 ∈ T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' (6d) Θ𝑚𝑖𝑛 𝑛 ≤ 𝜃𝑛,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='𝑡 ≤ Θ𝑚𝑎𝑥 𝑛 ∀𝑛 ∈ N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='𝑡 ∈ T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' (6e) − 𝑅𝐷𝑖 ≤ 𝑝𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='𝑡 − 𝑝𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='𝑡−1 ≤ 𝑅𝑈𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' ∀𝑖 ∈ G,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='𝑡 ∈ T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' (6f) 0 ≤ 𝑝𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='𝑡 ≤ 𝑃𝑚𝑎𝑥 𝑖 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' ∀𝑖 ∈ G,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='𝑡 ∈ T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' (6g) 0 ≤ 𝑑𝑗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='𝑡 ≤ 𝐷𝑗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='𝑡,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' ∀𝑗 ∈ D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='𝑡 ∈ T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' (6h) 0 ≤ 𝑚𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='𝑡 ≤ 𝑀𝑗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='𝑡,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' ∀𝑖 ∈ I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='𝑡 ∈ T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' (6i) 0 ≤ 𝑤𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='𝑡 ≤ 𝑊𝑗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='𝑡,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' ∀𝑖 ∈ W,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='𝑡 ∈ T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' (6j) 0 ≤ 𝑟𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='𝑡 ≤ 𝑅𝑗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='𝑡,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' ∀𝑖 ∈ R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='𝑡 ∈ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' (6k) The generation sources include conventional thermal power plants (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' gas, nuclear, coal), non-wind renewables (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' hydro), imports, and wind power plants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Due to long-term commitments for imports or goal of reducing carbon emissions with renewables, the imports and renewables are non-dispatchable as in [41] but can be curtailed at the cost of 𝐶𝑚 𝑖 and 𝐶𝑟 𝑖 $/MWh (𝐶𝑤 𝑖 $/MWh for wind) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Sometimes the power supply may not meet the demand, and each unit of load shedding (not served load) is at the cost of value of lost load (VOLL) 𝐶𝑑 𝑗 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' In this paper, the unit generation cost is 1/2/4 $/MWh for nuclear/coal/gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' The penalties are 500/100/1,000 $/MWh for import/wind/non-wind renewables curtailment, 1,000 $/MWh for load shedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' The constraints are typical for DC-OPF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Constraint 6b represents the balance constraint at each bus, whose associated dual value is the locational marginal price (LMP) at that bus, indicating marginal cost of adding 1 MW load at a specific location, so the price can go negative/high when curtailment/load shedding happens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Con- straints 6c–6e represent how the power flow (6c) is determined given the line capacity (6d) and phase angle (6e) limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Constraint 6f limits conventional power plants’ rate of ramping up/down gen- eration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Constraints 6g–6k bound the conventional generation, load shedding, and curtailments (6i–6k) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' B FUEL CARBON EMISSION RATES Below are the fuel carbon emission rates (carbon emissions per MWh energy generated from that fuel) we use to calculate carbon emissions: 12 Reducing Datacenter Operational Carbon Emissions Effectively by Cooperating with the Grid Conference’17, July 2017, Washington, DC, USA Table 6: Carbon Emission Rates of Different Fuels [4, 36] Generation Type Carbon Emission Rate (kg CO2/MWh) Coal 895.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='2 Natural Gas 388.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='9 Oil 877.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='6 Dual-fuel 633.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='3 Nuclear 0 Geothermal 107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='6 Biomass 0 Hydro 0 Wind 0 Import 428 13 Conference’17, July 2017, Washington, DC, USA Liuzixuan Lin and Andrew A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Chien REFERENCES [1] Follow the sun, wind, moon, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='vertatique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='com/cloud-computing- starting-follow-sunwindmoon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' [2] Clicking Clean Virginia: The Dirty Energy Powering Data Center Alley, February 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='greenpeace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='org/usa/reports/click-clean-virginia/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' [3] U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Energy Information Administration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Electric power monthly, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' [4] United States Environmental Protection Agency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Emissions & generation resource integrated database, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' [5] Sohaib Ahmad, Arielle Rosenthal, Mohammad H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Hajiesmaili, and Ramesh K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Sitaraman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Learning from optimal: Energy procurement strategies for data centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' In Proceedings of the Tenth ACM International Conference on Future Energy Systems, e-Energy ’19, page 326–330, New York, NY, USA, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Association for Computing Machinery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' [6] Amazon Web Services (AWS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Amazon global datacenters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' [7] Noman Bashir, Tian Guo, Mohammad Hajiesmaili, David Irwin, Prashant Shenoy, Ramesh Sitaraman, Abel Souza, and Adam Wierman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Enabling sustainable clouds: The case for virtualizing the energy system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' In Proceedings of the ACM Symposium on Cloud Computing, pages 350–358, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' [8] Lori Bird, M Milligan, and Debra Lew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Integrating Variable Renewable Energy: Challenges and Solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Technical report, NREL, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' [9] CAISO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Market processes and products, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' [10] John Campbell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Data centres used 14% of republic of ireland’s electricity use, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' [11] Niangjun Chen, Xiaoqi Ren, Shaolei Ren, and Adam Wierman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Greening multi- tenant data center demand response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Performance Evaluation, 91:229–254, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' [12] Shutong Chen, Lei Jiao, Lin Wang, and Fangming Liu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' An online market mecha- nism for edge emergency demand response via cloudlet control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' In IEEE INFOCOM 2019-IEEE Conference on Computer Communications, pages 2566–2574.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' IEEE, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' [13] Andrew A Chien.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Characterizing opportunity power in the california indepen- dent system operator (caiso) in years 2015-2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Energy and Earth Science, 3(2), December 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Also available as University of Chicago, Computer Science TR-2018-07, https://newtraell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='uchicago.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='edu/research/publications/techreports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' [14] Andrew A Chien, Richard Wolski, and Fan Yang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' The zero-carbon cloud: High- value, dispatchable demand for renewable power generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' The Electricity Journal, pages 110–118, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' [15] Andrew A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Chien, Fan Yang, and Chaojie Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Characterizing curtailed and uneconomic renewable power in the mid-continent independent system operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' AIMS Energy, 6(2):376–401, December 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' [16] Microsoft Corporation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Microsft 2021 environmental sustainability report, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Available from https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='microsoft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='com/en-us/corporate-responsibility/ sustainability/report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' [17] Olivier Corradi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Marginal vs average: which one to use in practice?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=', 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' [18] CPUC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' California Public Utilities Commission (CPUC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='cpuc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='gov/ PUC/energy/Renewables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' [19] Wei Deng, Fangming Liu, Hai Jin, Bo Li, and Dan Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Harnessing renewable energy in cloud datacenters: opportunities and challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' iEEE Network, 28(1):48–55, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' [20] Hui Dou, Yong Qi, Wei Wei, and Houbing Song.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Carbon-aware electricity cost minimization for sustainable data centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' IEEE Transactions on Sustainable Computing, 2(2):211–223, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' [21] Iain Dunning, Joey Huchette, and Miles Lubin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Jump: A modeling language for mathematical optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' SIAM review, 59(2):295–320, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' [22] Dominion Energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' 2018 virginia integrated resource plan, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' [23] Dominion Energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' 2020 virginia integrated resource plan, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' [24] Dominion Energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' 2021 update to the 2020 integrated resource plan, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' [25] Mary Jo Foley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Cloud revenues power microsoft’s $51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='7 billion q2 in fiscal year 2022, January 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='zdnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='com/article/microsoft-cloud-revenues- power-microsofts-51-7-billion-second-fy22-quarter/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' [26] Robbie Galvin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Data centers are pushing ireland’s electric grid to the brink, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' [27] Íñigo Goiri, William Katsak, Kien Le, Thu D Nguyen, and Ricardo Bianchini.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Parasol and greenswitch: Managing datacenters powered by renewable energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' In ACM SIGARCH Computer Architecture News, pages 51–64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' ACM, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' [28] Caroline Golin and Nick Pearson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' A policy roadmap for 24/7 carbon-free energy, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' [29] Google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' About google datacenters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='com/about/datacenters/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' [30] Google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Moving toward 24x7 carbon-free energy at google data centers: Progress and insights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Technical report, Google, October 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' [31] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Gupta, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Shenoy, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Sitaraman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Efficient solar provisioning for net-zero internet-scale distributed networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' In 2018 10th International Conference on Communication Systems Networks (COMSNETS), pages 372–379, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' [32] LLC Gurobi Optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Gurobi optimizer reference manual, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' [33] GWEC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Global wind report: Annual market update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Technical report, Global Wind Energy Council, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Documents curtailment around the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' [34] Siqi Han.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' The wind is wasted in china.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='wilsoncenter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='org/, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' [35] Kevin Imboden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' 2022 global data center market comparison, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' [36] California ISO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Greenhouse gas emission tracking methodology, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' [37] California ISO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Managing oversupply, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' [38] Nicola Jones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' How to stop data centres from gobbling up the world’s electricity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Nature, September 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' [39] Lucas Joppa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Made to measure: Sustainability commitment progress and updates, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' [40] Peter Judge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Eirgrid pulls plug on 30 irish data center projects, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' [41] Kibaek Kim, Fan Yang, Victor Zavala, and Andrew A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Chien.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Data centers as dispatchable loads to harness stranded power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' IEEE Transactions on Sustainable Energy, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='1109/TSTE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='2593607.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' [42] Wook Hyun Kwon and Soo Hee Han.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Receding horizon control: model predictive control for state models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Springer Science & Business Media, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' [43] Kien Le, Ricardo Bianchini, Thu D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Nguyen, Ozlem Bilgir, and Margaret Martonosi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Capping the brown energy consumption of internet services at low cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' In Proceedings of the International Conference on Green Computing, GREENCOMP ’10, page 3–14, USA, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' IEEE Computer Society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' [44] Tan N Le, Zhenhua Liu, Yuan Chen, and Cullen Bash.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Joint capacity planning and operational management for sustainable data centers and demand response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' In Proceedings of the Seventh International Conference on Future Energy Systems, pages 1–12, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' [45] Liuzixuan Lin and Andrew A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Chien.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Characterizing stranded power in the ercot in years 2012-2019: A preliminary report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Technical Report TR-2020-06, University of Chicago, August 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' [46] Liuzixuan Lin, Victor M Zavala, and Andrew A Chien.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Evaluating coupling models for cloud datacenters and power grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' In Proceedings of the Twelfth ACM International Conference on Future Energy Systems, pages 171–184, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' [47] Minghong Lin, Adam Wierman, Lachlan LH Andrew, and Eno Thereska.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Dynamic right-sizing for power-proportional data centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' IEEE/ACM Transactions on Networking, 21(5):1378–1391, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' [48] Julia Lindberg, Yasmine Abdennadher, Jiaqi Chen, Bernard C Lesieutre, and Line Roald.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' A guide to reducing carbon emissions through data center geographical load shifting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' In Proceedings of the Twelfth ACM International Conference on Future Energy Systems, pages 430–436, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' [49] Julia Lindberg, Line Roald, and Bernard Lesieutre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' The environmental potential of hyper-scale data centers: Using locational marginal co2 emissions to guide geo- graphical load shifting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' In Proceedings of the 54th Hawaii International Conference on System Sciences, page 3158, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' [50] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Liu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Lin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Wierman, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Low, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Andrew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Greening geographical load balancing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' IEEE/ACM Transactions on Networking, 23(2):657–671, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' [51] Zhenhua Liu, Yuan Chen, Cullen Bash, Adam Wierman, Daniel Gmach, Zhikui Wang, Manish Marwah, and Chris Hyser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Renewable and cooling aware work- load management for sustainable data centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' In Proceedings of the 12th ACM SIGMETRICS/PERFORMANCE joint international conference on Measurement and Modeling of Computer Systems, pages 175–186, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' [52] Zhenhua Liu, Adam Wierman, Yuan Chen, Benjamin Razon, and Niangjun Chen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Data center demand response: Avoiding the coincident peak via workload shifting and local generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Performance Evaluation, 70(10):770–791, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' [53] Jianying Luo, Lei Rao, and Xue Liu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Temporal load balancing with service delay guarantees for data center energy cost optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' IEEE Transactions on Parallel and Distributed Systems, 25(3):775–784, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' [54] Srilatha Manne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Architecting a sustainable planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' In MICRO 2020, https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' microarch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='org/micro53/, October 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Keynote Talk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' [55] Electricity Maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Electricity maps, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' [56] Eric Masanet, Arman Shehabi, Nuoa Lei, Sarah Smith, and Jonathan Koomey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Recalibrating global data center energy-use estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Science, 367(6481):984–986, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' [57] Valérie Masson-Delmotte, Panmao Zhai, Hans-Otto Pörtner, Debra Roberts, Jim Skea, Priyadarshi R Shukla, Anna Pirani, W Moufouma-Okia, C Péan, R Pidcock, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Global warming of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='5 c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' An IPCC Special Report on the impacts of global warming of, 1(5), 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' [58] James McAnany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' 2021 demand response operations markets activity report: January 2022, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' [59] Microsoft Azure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Azure global datacenters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' https://azure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='microsoft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='com/en- us/global-infrastructure/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' [60] Rich Miller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Dominion energy plans more green power for virginia’s data centers, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' [61] Anthony Papavasiliou and Shmuel S Oren.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Multiarea stochastic unit commitment for high wind penetration in a transmission constrained network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Operations Research, 61(3):578–592, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' [62] Brian Peccarelli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Three ways covid-19 is accelerating digital transformation in professional services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' June 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' https://bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='ly/34Qitb5, 37% growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' [63] Sundar Pichai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Our biggest renewable energy purchase ever, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' [64] SC Pryor, RJ Barthelmie, and TJ Shepherd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' 20% of us electricity from wind will have limited impacts on system efficiency and regional climate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Scientific reports, 10(1):1–14, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' [65] Asfandyar Qureshi, Rick Weber, Hari Balakrishnan, John Guttag, and Bruce Maggs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Cutting the electric bill for internet-scale systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' In Proceedings of the ACM SIGCOMM 2009 Conference on Data Communication, SIGCOMM ’09, page 123–134, New York, NY, USA, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Association for Computing Machinery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' 14 Reducing Datacenter Operational Carbon Emissions Effectively by Cooperating with the Grid Conference’17, July 2017, Washington, DC, USA [66] Ana Radovanovic, Ross Koningstein, Ian Schneider, Bokan Chen, Alexandre Duarte, Binz Roy, Diyue Xiao, Maya Haridasan, Patrick Hung, Nick Care, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Carbon-aware computing for datacenters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' arXiv preprint arXiv:2106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='11750, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' [67] John Roach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Microsoft’s virtual datacenter grounds ‘the cloud’ in re- ality, April 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Microsoft to build 50 to 100 datacenters per year, https://news.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='microsoft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='com/innovation-stories/microsofts-virtual-datacenter- grounds-the-cloud-in-reality/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' [68] Yuanyuan Shi, Bolun Xu, Baosen Zhang, and Di Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Leveraging energy storage to optimize data center electricity cost in emerging power markets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' In Proceedings of the Seventh International Conference on Future Energy Systems, pages 1–13, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' [69] Staff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Covid-19 accelerates cloud adoption, market to reach $1 trillion, idc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Equip- ment FA News, October 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='equipmentfa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='com/news/31459/covid- 19-accelerates-cloud-adoption-market-to-reach-1t-idc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' [70] Grant L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Stewart, Gregory A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Koenig, Jingjing Liu, Anders Clausen, Sonja Klingert, and Natalie Bates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Grid accommodation of dynamic hpc demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' In Proceedings of the 48th International Conference on Parallel Processing: Workshops, ICPP 2019, New York, NY, USA, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Association for Computing Machinery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' [71] Qihang Sun, Shaolei Ren, Chuan Wu, and Zongpeng Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' An online incentive mechanism for emergency demand response in geo-distributed colocation data centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' In Proceedings of the seventh international conference on future energy systems, pages 1–13, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' [72] Muhammad Tirmazi, Adam Barker, Nan Deng, Md E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Haque, Zhijing Gene Qin, Steven Hand, Mor Harchol-Balter, and John Wilkes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Borg: The next generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' In Proceedings of the Fifteenth European Conference on Computer Systems, EuroSys ’20, New York, NY, USA, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Association for Computing Machinery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' [73] Rahul Urgaonkar, Bhuvan Urgaonkar, Michael J Neely, and Anand Sivasubrama- niam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Optimal power cost management using stored energy in data centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' In Proceedings of the ACM SIGMETRICS joint international conference on Measurement and modeling of computer systems, pages 221–232, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' [74] Markus Wacket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' German parties agree on 2030 coal phase-out in coalition talks, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' [75] WattTime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Automated emissions reduction, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' [76] Fan Yang and Andrew A Chien.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Zccloud: Exploring wasted green power for high-performance computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' In 2016 IEEE International Parallel and Distributed Processing Symposium (IPDPS), pages 1051–1060.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' IEEE, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' [77] Fan Yang and Andrew A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Chien.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Large-scale and extreme-scale computing with stranded green power: Opportunities and costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' IEEE Transactions on Parallel and Distributed Systems, 29(5), December 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' [78] Lin Yang, Mohammad Hassan Hajiesmaili, Ramesh K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Sitaraman, Enrique Mallada, Wing Shing Wong, and Adam Wierman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Online inventory management with application to energy procurement in data centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' CoRR, abs/1901.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content='04372, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' [79] Chaojie Zhang and Andrew A Chien.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Scheduling challenges for variable capacity resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' In Workshop on Job Scheduling for Parallel Processing (JSSPP), 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' [80] Linquan Zhang, Shaolei Ren, Chuan Wu, and Zongpeng Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' A truthful incentive mechanism for emergency demand response in colocation data centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' In 2015 IEEE Conference on Computer Communications (INFOCOM), pages 2632–2640.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' IEEE, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' [81] Weiqi Zhang, Line A Roald, Andrew A Chien, John R Birge, and Victor M Zavala.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Flexibility from networks of data centers: A market clearing formulation with virtual links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Electric Power Systems Research, 189:106723, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' [82] Yanwei Zhang, Yefu Wang, and Xiaorui Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Electricity bill capping for cloud- scale data centers that impact the power markets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' In 2012 41st International Conference on Parallel Processing, pages 440–449.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' IEEE, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' [83] Jiajia Zheng, Andrew A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Chien, and Sangwon Suh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Mitigating curtailment and carbon emissions through load migration between data centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Joule, October 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' [84] Zhi Zhou, Fangming Liu, Shutong Chen, and Zongpeng Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' A truthful and efficient incentive mechanism for demand response in green datacenters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' IEEE Transactions on Parallel and Distributed Systems, 31(1):1–15, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' [85] Zhi Zhou, Fangming Liu, Bo Li, Baochun Li, Hai Jin, Ruolan Zou, and Zhiyong Liu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Fuel cell generation in geo-distributed cloud services: A quantitative study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' In 2014 IEEE 34th International Conference on Distributed Computing Systems, pages 52–61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' IEEE, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' [86] Zhi Zhou, Fangming Liu, Ruolan Zou, Jiangchuan Liu, Hong Xu, and Hai Jin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' Carbon-aware online control of geo-distributed cloud services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' IEEE Transactions on Parallel and Distributed Systems, 27(9):2506–2519, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} +page_content=' 15' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE1T4oBgHgl3EQfZQR8/content/2301.03148v1.pdf'} diff --git a/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf b/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..ca816f074ab02daafb05f0a66414f33cc5b398d8 --- /dev/null +++ b/N9AzT4oBgHgl3EQfzP7F/content/2301.01767v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:d783fa20d4f483eff4579105f1999949488c84fa34d64887a59a40ef07058199 +size 1666786 diff --git a/O9FOT4oBgHgl3EQf4DQT/content/tmp_files/2301.12948v1.pdf.txt b/O9FOT4oBgHgl3EQf4DQT/content/tmp_files/2301.12948v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..1e88062a36eccbd186a7afcb9efbe47eab0a17c9 --- /dev/null +++ b/O9FOT4oBgHgl3EQf4DQT/content/tmp_files/2301.12948v1.pdf.txt @@ -0,0 +1,185 @@ +IL NUOVO CIMENTO +Vol. ?, N. ? +? +Measurement of the cosmic ray flux by an ArduSiPM-based muon +telescope in the framework of the Lab2Go project. +V. Agostini(1), B. Arcese(1), N. Ascani(1), P. Astone(3), V. Bocci(3), S. Caperna(1), +F. Casaburo(2)(3)(∗)(∗∗), A. Cerica(1), C. DAuria(1), G. De Bonis(3), D. Deda(1), +F. Di Mauro(1), A. Di Vico(1), R. Faccini(2)(3), L. Frasca(1), G. Galuppi(1), +G. Giovannetti(1), F. Iacoangeli(3), G. Ludovici(1), L. Martone(1), B. Marucci(1), +L. Mizzoni(1), A. Moriconi(1), G. Organtini(2)(3), F. Piacentini(2)(3), A. Pietrobono(1), +F. Pongelli(1), F. Severa(1) and D. Vona(1) +(1) Liceo Scientifico ”L. Pietrobono” - Alatri, Italy +(2) Dipartimento di Fisica, Universit`a di Roma - Roma, Italy +(3) INFN, Sezione di Roma - Roma, Italy +Summary. — Whitin Istituto Nazionale di Fisica Nucleare (INFN) outreach ac- +tivities, the Lab2Go project is of great significance. Its goal is involving high school +teachers and students in several laboratory activities, aiming at increasing the weight +of experimental contents in teaching and learning. In this article we present the mea- +surement, carried out in the framework of the Lab2Go project, of the cosmic muon +flux made by an ArduSiPM-based muon telescope. +1. – Introduction +At the end of the XVIII century, it has been observed the spontaneous discharge of +electroscopes [1]. After R¨ontgen’s X-rays discovery [2], it has been proposed that the +leakage of electric charge was due to the ionization caused by radioactivity coming from +the underground. This hypothesis was dismissed by Pacini, who measured the variations +of the electroscope’s discharge at sea level and under the sea level. Results showed that +the radiation under the sea was significantly lower than at the surface [3]. In 1912, during +a balloon flight up to an altitude of 5200 m, Hess observed an increasing of the radiation +with the altitude [4], proving that it comes from the space. Later, Millikan named this +radiation Cosmic Rays (CRs) [5]. +(∗) Corresponding author. +(∗∗) Currently at Dipartimento di Fisica, Universit`a di Genova and INFN- Sezione di Genova +© Societ`a Italiana di Fisica +1 +arXiv:2301.12948v1 [physics.ed-ph] 30 Jan 2023 + +2V. AGOSTINI, B. ARCESE, N. ASCANI, P. ASTONE, V. BOCCI, S. CAPERNA, F. CASABURO, A. CERICA, C. DAURIA, G. DE BONIS, D. DEDA, F. DI MAURO, A. DI VICO, R. FACCINI, L. FRASCA, G. GALUPPI, G. GIOVANNETTI, F. IACOANGELI, G. LUDOVICI, L. MARTONE, B. MARUCCI, L. MIZZONI, A. MORICONI, G. ORGANTINI, F. PIACENTINI, A. PIETROBONO, F. PONGELLI, F. SEVERA and D. VONA +When CRs enter the atmosphere, they interact producing a particle showers called +Extensive Air Showers (EAS), that can be divided in three components: hadronic, elec- +tromagnetic and muonic [6]. The latter, due to the low interaction between muons and +the atmosphere, can be detected at sea level. Their flux depends on the Zenith angle θ +and, for θ < 75◦, it is: +F (θ) = F0 cos2 θ +(1) +being F0 the flux at θ = 0 [7]. +Experimentally, we can measure it rotating the +detector, and thus verifying the expected formal relation between measured flux and +angle. In this article, we present the measurement of the cosmic muon flux made by an +ArduSiPM-based muon telescope in the framework of the Lab2Go project [8] at Liceo +”L. Pietrobono” in Alatri (Italy). +2. – Experimental setup and procedure +The experimental setup (Fig. +1a) consists of a muon telescope made of two Ar- +duSiPMs. ArduSiPM is a transportable particle detector [9] created and developed by +INFN Roma, constituted by an electronic shield connected to an Arduino DUE [10], a +scintillator and a Silicon PhotoMultiplier (SiPM). To build up the muon telescope, the +two ArduSiPMs are placed on a mechanical structure that hosts the scinitllators as well, +and is free to rotate. It is connected to an accelerometer to measure the angle. The +number of events measured by each one of the ArduSiPMs and by both of them, the +acquisition time and the angle are read by an M5Stack (Fig. 1b) [11]. +(a) +(b) +Fig. 1.: (a) Muon telescope during the measurement at Liceo ”L. Pietrobono”. (b) Data +read by the M5Stack. +3. – Data analysis and results +By the number of time coincidence events N (θ) and the acquisition time T, we have +measured the flux as F (θ) = N(θ) +T +. Due to the low rate of muons, long acquisition time +was needed for each angle, from approximately 1h at θ = 0 up to approximately 2h at +θ = 60◦. The uncertainty on N (θ) has been evaluated assuming a Gaussian uncertainty +when counts were > 30, and a Poisson uncertainty otherwise. +Values of F (θ) for several angles have been interpolated (Fig. 2) with the function +Eq. 1, resulting in F0esti = (6.22 ± 0.43)·10−1 min−1 as derived from the fit. Despite the + +Muon telescopeTime +Angle +900 +Coinoiclenza! +Events +Coincidenze totali->4 +Total time coincidence events +Last coincidenceMEASUREMENT OF THE COSMIC RAY FLUX BY AN ARDUSIPM-BASED MUON TELESCOPE IN THE FRAMEWORK OF THE LAB2GO PROJECT.3 +large uncertainties due to poor statistics, the relation in Eq. 1 is qualitatively verified as +shown in Fig. 2. +Fig. 2.: Flux interpolation. +In addition, to further stress out the need for a quantitative validation and to allow +the students to acquire the concept of agreement between measurements, we evaluated +a test function. In particular, we started assuming no-correlation between the estimated +F0esti and measured F0meas = (6.5 ± 1.1) · 10−1 min−1 flux parameters (zero-hypothesis) +and we consider t = +|F0esti−F0meas| +� +σ2 +F0esti ++σ2 +F0meas +, whose result (t = 0.24) has been used to estimate +a one-sided gaussian p-value. We got p-value=0.81 (> 0.05, then verified at 95% C.L.) +and an agreement within 1σ. +4. – Students statisfaction +At the end of the Lab2Go course, an anonymous satisfaction questionnaire has been +proposed to the students, in order to evaluate their level of satisfaction concerning the +lecture topics, their engagement, the quality of teaching and the overall project. For each +of the 10 questions, students could express their impression, from a minimum of 1 (un- +satisfactory) to a maximum of 4 (very satisfactory). The grades have been summarised +in an histogram (Fig. 3) resulting in a mean grade µ = 3.435 ± 0.048. +Fig. 3.: Results of satisfaction questionnaires. + +0.6 +0.5 +0.4 +0.3 +0.2 +0. +0 +0.2 +0.4 +0.6 +0.8 +Angle (rad)Counts +90 +80 +70 +60 +50 +40 +30 +20 +10 +0 +2 +3 +4 +Grade4V. AGOSTINI, B. ARCESE, N. ASCANI, P. ASTONE, V. BOCCI, S. CAPERNA, F. CASABURO, A. CERICA, C. DAURIA, G. DE BONIS, D. DEDA, F. DI MAURO, A. DI VICO, R. FACCINI, L. FRASCA, G. GALUPPI, G. GIOVANNETTI, F. IACOANGELI, G. LUDOVICI, L. MARTONE, B. MARUCCI, L. MIZZONI, A. MORICONI, G. ORGANTINI, F. PIACENTINI, A. PIETROBONO, F. PONGELLI, F. SEVERA and D. VONA +5. – Conclusions +During the school year 2021/22, in the framework of Lab2Go, the measurement of +the cosmic muon flux by an ArduSiPM-based muon telescope has been proposed to high +school students at Liceo ”L. Pietrobono” in Alatri. Thanks to this experiment we had the +possibility to introduce many topics, as CRs, particle physics, muons, antimatter, special +relativity, particle detectors and time coincidence, going far beyond the topics commonly +taught in physics lectures at high schools. +Moreover, since students had to produce +a laboratory report, they learned how to process, interpolate and show data, taking +into account the uncertainties and estimating the agreement between measurements. +A satisfaction questionnaire proposed to students at the end of the course reported a +positive evaluation of the proposed activity and, in general, of Lab2Go overall. +∗ ∗ ∗ +The authors acknowledge Mauro Mancini, Francesco Safai Therani, and Comitato di +Coordinamento III missione (CC3M)-INFN. +REFERENCES +[1] P. Kir´aly, “Two centenaries: The discovery of cosmic rays and the birth of Lajos J´anossy,” +Journal of Physics: Conference Series, vol. 409, p. 012001, feb 2013. +[2] R. I. Frankel, “Centennial of r¨ontgen’s discovery of x-rays,” The Western journal of +medicine, vol. 164, pp. 497–501, Jun 1996. 8764624[pmid]. +[3] D. +Pacini, +“Penetrating +radiation +at +the +surface +of +and +in +water.” +https://arxiv.org/abs/1002.1810, 2010. +[4] V. Hess, “On the observations of the penetrating radiation during seven balloon flights.” +https://arxiv.org/abs/1808.02927, 2018. +[5] L. Bonolis, “Walther Bothe and Bruno Rossi: The birth and development of coincidence +methods in cosmic-ray physics,” American Journal of Physics, vol. 79, pp. 1133–1150, nov +2011. +[6] A. Haungs, H. Rebel, and M. Roth, “Energy spectrum and mass composition of high-energy +cosmic rays,” Reports on Progress in Physics, vol. 66, p. 1145, jun 2003. +[7] M. Bektasoglu and H. Arslan, “Investigation of the zenith angle dependence of cosmic-ray +muons at sea level,” Pramana, vol. 80, pp. 837–846, 2013. +[8] M. Andreotti and et al., “Il progetto Lab2Go per la diffusione della pratica laboratoriale +nelle scuole,” La Fisica nella Scuola, vol. 3-4, 2020. +[9] V. Bocci, G. Chiodi, F. Iacoangeli, M. Nuccetelli, and L. Recchia, “The ArduSiPM a +compact trasportable software/hardware data acquisition system for sipm detector,” in +2014 IEEE Nuclear Science Symposium and Medical Imaging Conference (NSS/MIC), +pp. 1–5, 2014. +[10] Arduino collaboration. https://www.arduino.cc/. +[11] M5Stack collaboration. https://m5stack.com. + diff --git a/O9FOT4oBgHgl3EQf4DQT/content/tmp_files/load_file.txt b/O9FOT4oBgHgl3EQf4DQT/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3bf67ddad27c72e187ab744a6f0f4b93b230e80c --- /dev/null +++ b/O9FOT4oBgHgl3EQf4DQT/content/tmp_files/load_file.txt @@ -0,0 +1,233 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf,len=232 +page_content='IL NUOVO CIMENTO Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=', N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' Measurement of the cosmic ray flux by an ArduSiPM-based muon telescope in the framework of the Lab2Go project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' Agostini(1), B.' metadata={'source': 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+page_content=' Cerica(1), C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' DAuria(1), G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' De Bonis(3), D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' Deda(1), F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' Di Mauro(1), A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' Di Vico(1), R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' Faccini(2)(3), L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' Frasca(1), G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' Galuppi(1), G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' Giovannetti(1), F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' Iacoangeli(3), G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' Ludovici(1), L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' Martone(1), B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' Marucci(1), L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' Mizzoni(1), A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' Moriconi(1), G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' Organtini(2)(3), F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' Piacentini(2)(3), A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' Pietrobono(1), F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' Pongelli(1), F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' Severa(1) and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' Vona(1) (1) Liceo Scientifico ”L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' Pietrobono” - Alatri, Italy (2) Dipartimento di Fisica, Universit`a di Roma - Roma, Italy (3) INFN, Sezione di Roma - Roma, Italy Summary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' — Whitin Istituto Nazionale di Fisica Nucleare (INFN) outreach ac- tivities, the Lab2Go project is of great significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' Its goal is involving high school teachers and students in several laboratory activities, aiming at increasing the weight of experimental contents in teaching and learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' In this article we present the mea- surement, carried out in the framework of the Lab2Go project, of the cosmic muon flux made by an ArduSiPM-based muon telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' – Introduction At the end of the XVIII century, it has been observed the spontaneous discharge of electroscopes [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' After R¨ontgen’s X-rays discovery [2], it has been proposed that the leakage of electric charge was due to the ionization caused by radioactivity coming from the underground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' This hypothesis was dismissed by Pacini, who measured the variations of the electroscope’s discharge at sea level and under the sea level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' Results showed that the radiation under the sea was significantly lower than at the surface [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' In 1912, during a balloon flight up to an altitude of 5200 m, Hess observed an increasing of the radiation with the altitude [4], proving that it comes from the space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' Later, Millikan named this radiation Cosmic Rays (CRs) [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' (∗) Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' (∗∗) Currently at Dipartimento di Fisica, Universit`a di Genova and INFN- Sezione di Genova © Societ`a Italiana di Fisica 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content='12948v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content='ed-ph] 30 Jan 2023 2V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' AGOSTINI, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' ARCESE, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' ASCANI, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' ASTONE, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' BOCCI, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' CAPERNA, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' CASABURO, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' CERICA, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' DAURIA, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' DE BONIS, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' DEDA, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' DI MAURO, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' DI VICO, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' FACCINI, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' FRASCA, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' GALUPPI, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' GIOVANNETTI, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' IACOANGELI, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' LUDOVICI, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' MARTONE, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' MARUCCI, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' MIZZONI, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' MORICONI, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' ORGANTINI, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' PIACENTINI, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' PIETROBONO, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' PONGELLI, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' SEVERA and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' VONA When CRs enter the atmosphere, they interact producing a particle showers called Extensive Air Showers (EAS), that can be divided in three components: hadronic, elec- tromagnetic and muonic [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' The latter, due to the low interaction between muons and the atmosphere, can be detected at sea level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' Their flux depends on the Zenith angle θ and, for θ < 75◦, it is: F (θ) = F0 cos2 θ (1) being F0 the flux at θ = 0 [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' Experimentally, we can measure it rotating the detector, and thus verifying the expected formal relation between measured flux and angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' In this article, we present the measurement of the cosmic muon flux made by an ArduSiPM-based muon telescope in the framework of the Lab2Go project [8] at Liceo ”L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' Pietrobono” in Alatri (Italy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' – Experimental setup and procedure The experimental setup (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' 1a) consists of a muon telescope made of two Ar- duSiPMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' ArduSiPM is a transportable particle detector [9] created and developed by INFN Roma, constituted by an electronic shield connected to an Arduino DUE [10], a scintillator and a Silicon PhotoMultiplier (SiPM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' To build up the muon telescope, the two ArduSiPMs are placed on a mechanical structure that hosts the scinitllators as well, and is free to rotate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' It is connected to an accelerometer to measure the angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' The number of events measured by each one of the ArduSiPMs and by both of them, the acquisition time and the angle are read by an M5Stack (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' 1b) [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' (a) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=': (a) Muon telescope during the measurement at Liceo ”L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' Pietrobono”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' (b) Data read by the M5Stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' – Data analysis and results By the number of time coincidence events N (θ) and the acquisition time T, we have measured the flux as F (θ) = N(θ) T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' Due to the low rate of muons, long acquisition time was needed for each angle, from approximately 1h at θ = 0 up to approximately 2h at θ = 60◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' The uncertainty on N (θ) has been evaluated assuming a Gaussian uncertainty when counts were > 30, and a Poisson uncertainty otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' Values of F (θ) for several angles have been interpolated (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' 2) with the function Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' 1, resulting in F0esti = (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content='22 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content='43)·10−1 min−1 as derived from the fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' Despite the Muon telescopeTime Angle 900 Coinoiclenza!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' Events Coincidenze totali->4 Total time coincidence events Last coincidenceMEASUREMENT OF THE COSMIC RAY FLUX BY AN ARDUSIPM-BASED MUON TELESCOPE IN THE FRAMEWORK OF THE LAB2GO PROJECT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content='3 large uncertainties due to poor statistics, the relation in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' 1 is qualitatively verified as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=': Flux interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' In addition, to further stress out the need for a quantitative validation and to allow the students to acquire the concept of agreement between measurements, we evaluated a test function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' In particular, we started assuming no-correlation between the estimated F0esti and measured F0meas = (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content='5 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content='1) · 10−1 min−1 flux parameters (zero-hypothesis) and we consider t = |F0esti−F0meas| � σ2 F0esti +σ2 F0meas , whose result (t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content='24) has been used to estimate a one-sided gaussian p-value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' We got p-value=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content='81 (> 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content='05, then verified at 95% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=') and an agreement within 1σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' – Students statisfaction At the end of the Lab2Go course, an anonymous satisfaction questionnaire has been proposed to the students, in order to evaluate their level of satisfaction concerning the lecture topics, their engagement, the quality of teaching and the overall project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' For each of the 10 questions, students could express their impression, from a minimum of 1 (un- satisfactory) to a maximum of 4 (very satisfactory).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' The grades have been summarised in an histogram (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' 3) resulting in a mean grade µ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content='435 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content='048.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=': Results of satisfaction questionnaires.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content='8 Angle (rad)Counts 90 80 70 60 50 40 30 20 10 0 2 3 4 Grade4V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' AGOSTINI, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' ARCESE, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' ASCANI, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' ASTONE, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' BOCCI, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' CAPERNA, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' CASABURO, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' CERICA, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' DAURIA, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' DE BONIS, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' DEDA, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' DI MAURO, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' DI VICO, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' FACCINI, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' FRASCA, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' GALUPPI, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' GIOVANNETTI, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' IACOANGELI, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' LUDOVICI, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' MARTONE, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' MARUCCI, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' MIZZONI, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' MORICONI, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' ORGANTINI, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' PIACENTINI, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' PIETROBONO, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' PONGELLI, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' SEVERA and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' VONA 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' – Conclusions During the school year 2021/22, in the framework of Lab2Go, the measurement of the cosmic muon flux by an ArduSiPM-based muon telescope has been proposed to high school students at Liceo ”L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' Pietrobono” in Alatri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' Thanks to this experiment we had the possibility to introduce many topics, as CRs, particle physics, muons, antimatter, special relativity, particle detectors and time coincidence, going far beyond the topics commonly taught in physics lectures at high schools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' Moreover, since students had to produce a laboratory report, they learned how to process, interpolate and show data, taking into account the uncertainties and estimating the agreement between measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' A satisfaction questionnaire proposed to students at the end of the course reported a positive evaluation of the proposed activity and, in general, of Lab2Go overall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' ∗ ∗ ∗ The authors acknowledge Mauro Mancini, Francesco Safai Therani, and Comitato di Coordinamento III missione (CC3M)-INFN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' REFERENCES [1] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' Kir´aly, “Two centenaries: The discovery of cosmic rays and the birth of Lajos J´anossy,” Journal of Physics: Conference Series, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' 409, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' 012001, feb 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' [2] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' Frankel, “Centennial of r¨ontgen’s discovery of x-rays,” The Western journal of medicine, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' 164, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' 497–501, Jun 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' 8764624[pmid].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' [3] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' Pacini, “Penetrating radiation at the surface of and in water.” https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content='org/abs/1002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content='1810, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' [4] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' Hess, “On the observations of the penetrating radiation during seven balloon flights.” https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content='org/abs/1808.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content='02927, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' [5] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' Bonolis, “Walther Bothe and Bruno Rossi: The birth and development of coincidence methods in cosmic-ray physics,” American Journal of Physics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' 79, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' 1133–1150, nov 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' [6] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' Haungs, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' Rebel, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' Roth, “Energy spectrum and mass composition of high-energy cosmic rays,” Reports on Progress in Physics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' 66, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' 1145, jun 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' [7] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' Bektasoglu and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' Arslan, “Investigation of the zenith angle dependence of cosmic-ray muons at sea level,” Pramana, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' 80, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' 837–846, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' [8] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' Andreotti and et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=', “Il progetto Lab2Go per la diffusione della pratica laboratoriale nelle scuole,” La Fisica nella Scuola, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' 3-4, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' [9] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' Bocci, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' Chiodi, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' Iacoangeli, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' Nuccetelli, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' Recchia, “The ArduSiPM a compact trasportable software/hardware data acquisition system for sipm detector,” in 2014 IEEE Nuclear Science Symposium and Medical Imaging Conference (NSS/MIC), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' 1–5, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' [10] Arduino collaboration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content='arduino.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content='cc/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' [11] M5Stack collaboration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content=' https://m5stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} +page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FOT4oBgHgl3EQf4DQT/content/2301.12948v1.pdf'} diff --git a/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf b/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..69a3fc308f16db1cbaa4bee72c74dcf390fb9179 --- /dev/null +++ b/ONAzT4oBgHgl3EQfk_3j/content/2301.01542v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:c85a29b0d364098d16a41abdadca0ee22286a6bc8464f1c7c57a19fcfb1695b6 +size 1264735 diff --git a/ONAzT4oBgHgl3EQfk_3j/vector_store/index.pkl b/ONAzT4oBgHgl3EQfk_3j/vector_store/index.pkl new file mode 100644 index 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b/PNE0T4oBgHgl3EQf1AIP/content/tmp_files/2301.02692v1.pdf.txt @@ -0,0 +1,1448 @@ +Isotonic Recalibration under a Low Signal-to-Noise Ratio +Mario V. W¨uthrich∗ +Johanna Ziegel† +Version of January 10, 2023 +Abstract +Insurance pricing systems should fulfill the auto-calibration property to ensure that there is +no systematic cross-financing between different price cohorts. Often, regression models are +not auto-calibrated. We propose to apply isotonic recalibration to a given regression model +to ensure auto-calibration. Our main result proves that under a low signal-to-noise ratio, +this isotonic recalibration step leads to explainable pricing systems because the resulting +isotonically recalibrated regression functions have a low complexity. +Keywords. Auto-calibration, isotonic regression, isotonic recalibration, low signal-to-noise +ratio, cross-financing, algorithmic solution, deep neural network, explainability. +1 +Introduction +There are two seemingly unrelated problems in insurance pricing that we are going to tackle in +this paper. First, an insurance pricing system should not have any systematic cross-financing +between different price cohorts. Systematic cross-financing implicitly means that some parts of +the portfolio are under-priced, and this is compensated by other parts of the portfolio that are +over-priced. We can prevent systematic cross-financing between price cohorts by ensuring that +the pricing system is auto-calibrated. We propose to apply isotonic recalibration which turns any +regression function into an auto-calibrated pricing system. +The second problem that we tackle is the explainability of complex algorithmic models for +insurance pricing. +In a first step, one may use any complex regression model to design an +insurance pricing system such as, e.g., a deep neural network. Such complex regression models +typically lack explainability and rather act as black boxes. For this reason, there are several +tools deployed to explain such complex solutions, we mention, for instance, SHAP by Lundberg– +Lee [22]. Since algorithmic solutions do not generally fulfill the aforementioned auto-calibration +property, we propose to apply isotonic recalibration to the algorithmic solution. If the signal-to- +noise ratio is low in the data, then the isotonic recalibration step leads to a coarse partition of the +covariate space and, as a consequence, it leads to an explainable version of the algorithmic model +used in the first place. Thus, explainability is a nice side result of applying isotonic recalibration +in low signal-to-noise ratio problems, which is typically the case in insurance pricing settings. +There are other methods for obtaining auto-calibration through a recalibration step; we men- +tion Lindholm et al. [21] and Denuit et al. [8]. These other methods often require tuning of +∗RiskLab, Department of Mathematics, ETH Zurich, mario.wuethrich@math.ethz.ch +†Institute of Mathematical Statistics and Actuarial Science, University of Bern, johanna.ziegel@stat.unibe.ch +1 +arXiv:2301.02692v1 [stat.ME] 6 Jan 2023 + +hyperparameters, e.g., using cross-validation. Isotonic recalibration does not involve any hy- +perparameters as it solves a constraint regression problem (ensuring monotonicity). As such, +isotonic recaliabration is universal because it also does not depend on the specific choice of the +loss function within the family of Bregman losses. +We formalize our proposal. Throughout, we assume that all considered random variables have +finite means. Consider a response variable Y that is equipped with covariate information X ∈ +X ⊆ Rq. The general goal is to determine the (true) regression function x �→ E[Y |X = x] +that describes the conditional mean of Y , given X. Typically, this true regression function +is unknown, and it needs to be determined from i.i.d. data (yi, xi)n +i=1, that is, a sample from +(Y, X). For this purpose, we try to select a regression function x �→ µ(x) from a (pre-chosen) +function class on X that approximates the conditional mean E[Y |X = ·] as well as possible. +Often, it is not possible to capture all features of the regression function from data. In financial +applications, a minimal important requirement for a well-selected regression function µ(·) is that +it fulfills the auto-calibration property. +Definition 1.1 The regression function µ is auto-calibrated for (Y, X) if +µ(X) = E [Y | µ(X)] , +P-a.s. +Auto-calibration is an important property in actuarial and financial applications because it +implies that, on average, the (price) cohorts µ(X) are self-financing for the corresponding claims +Y , i.e., there is no systematic cross-financing within the portfolio, if the structure of this portfolio +is described by the covariates X ∼ P and the price cohorts µ(X), respectively. In a Bernoulli +context, an early version of auto-calibration (called well-calibrated) has been introduced by +Schervish [28] to the community in statistics, and recently, it has been considered in detail by +Gneiting–Resin [12]. In an actuarial and financial context, the importance of auto-calibration +has been emphasized in Kr¨uger–Ziegel [17], Denuit et al. [8], W¨uthrich [30] and Lindholm et +al. [21]. +Many regression models do not satisfy the auto-calibration property. However, there is a sim- +ple and powerful method, which we call isotonic recalibration, to obtain an (in-sample) auto- +calibrated regression function starting from any candidate function π : X → R. +We apply +isotonic recalibration to the pseudo-sample (yi, π(xi))n +i=1 to obtain an isotonic regression func- +tion �µ. Then, +�µ(X′) = E +� +Y ′|�µ(X′) +� +, +Pn-a.s., +(1.1) +where (Y ′, X′) is distributed according to the empirical distribution Pn of (yi, xi)n +i=1; see Section +2.1 for details. Isotonic regression determines an adaptive partition of the covariate space X, +and �µ is determined by averaging y-values over the partition elements. Clearly, other binning +approaches can also be used on the pseudo-sample (yi, π(xi))n +i=1 to enforce (1.1), but we argue +that isotonic regression is preferable since it avoids subjective choices of tuning parameters and +leads to sensible regression functions under reasonable and verifiable assumptions. The only +assumption for isotonic recalibration to be informative is that the function π gets the rankings +of the conditional means right, that is, whenever E [Y |X = xi] ≤ E [Y |X = xj], we would like +to have π(xi) ≤ π(xj). +Using isotonic regression for recalibration is not new in the literature. In the case of binary +outcomes, it as already been proposed by Zadrozny–Elkan [32], Menon et al. [23] and recently +2 + +by Tasche [29, Section 5.3]. The monotone single index models of Balabdaoui et al. [2] follow +the same strategy as described above but the focus of their work is different from ours. They +specifically consider a linear regression model for the candidate function π, which is called the +index. In the case of distributional regression, that is, when interest is in determining the whole +conditional distribution of Y given covariate information X, Henzi et al. [13] have suggested to +first estimate an index function π that determines the ordering of the conditional distributions +w.r.t. first order stochastic dominance and then estimate conditional distributions using isotonic +distributional regression; see Henzi et al. [14]. +As a new contribution, we show that the size of the partition of the isotonic recalibration +may give insight concerning the information content of the recalibrated regression function �µ. +Furthermore, the partition of the isotonic recalibration allows to explain connections between +covariates and outcomes, in particular, when the signal-to-noise ratio is small which typically is +the case for insurance claims data. +In order to come up with a candidate function π : X → R, one may consider any regression +model such as, e.g., a generalized linear model, a regression tree, a tree boosting regression +model or a deep neural network regression model. The aim is that π(·) provides us with the +correct rankings of the conditional means E[Y |X = xi], i = 1, . . . , n. The details are discussed +in Section 3. +Organization. In Section 2, we formally introduce isotonic regression which is a constraint +optimization problem. This constraint optimization problem is usually solved with the pool +adjacent violators (PAV) algorithm, which is described in Appendix A.1. Our main result is +stated in Section 2.2. It relates the complexity of the isotonic recalibration solution to the signal- +to-noise ratio in the data. Section 3 gives practical guidance on the use of isotonic recalibration, +and in Section 4 we exemplify our results on a frequently used insurance data set. In this section +we also present graphic tools for interpreting the regression function. In Section 5, we conclude. +2 +Isotonic regression +2.1 +Definition and basic properties +For simplicity, we assume that the candidate function π : X → R does not lead to any ties in the +values π(x1), . . . , π(xn), and that the indices i = 1, . . . , n are chosen such that they are aligned +with the ranking, that is, π(x1) < . . . < π(xn). Remark 2.1 explains how to handle ties. The +isotonic regression of z = (yi, π(xi))n +i=1 with positive case weights (wi)n +i=1 is the solution �µ ∈ Rn +to the restricted minimization problem +�µ = +arg min +µ=(µ1,...,µn)⊤ +n +� +i=1 +wi (yi − µi)2 , +subject to µ1 ≤ . . . ≤ µn. +(2.1) +We can rewrite the side constraints as Aµ ≥ 0 (component-wise), where A = (ai,j)i,j ∈ Rn×(n−1) +is the matrix with the elements ai,j = 1i=j−1 − 1i=j. We define y = (y1, . . . , yn)⊤ ∈ Rn and +the (diagonal) case weight matrix W = diag(w1, . . . , wn). The above optimization problem then +reads as +�µ = �µ(z) = arg min +µ: Aµ≥0 +(y − µ)⊤W(y − µ). +(2.2) +3 + +This shows that the isotonic regression is solved by a convex minimization with linear side +constraints. It remains to verify that the auto-calibration property claimed in (1.1) holds. +Remark 2.1 If there are ties in the values π(x1), . . . , π(xn), for example, π(xi) = π(xj) for +some i ̸= j, we replace yi and yj with their weighted average (wiyi + wjyj)/(wi + wj) and assign +them weights (wi + wj)/2. The procedure is analogous for more than two tied values. This +corresponds to the second option of dealing with ties in Leeuw et al. [20, Section 2.1]. +Remark 2.2 Barlow et al. [3, Theorem 1.10] show that the square loss function in (2.1) can be +replaced by any Bregman loss function, Lφ(y, µ) = φ(y)−φ(µ)+φ′(µ)(y −µ), without changing +the optimal solution �µ. Here, φ is a strictly convex function with subgradient φ′. Bregman loss +functions are the only consistent loss functions for the mean; see Savage [27] and Gneiting [11, +Theorem 7]. If y and µ only take positive values, a Bregman loss function of relevance for this +paper is the gamma deviance loss, which is equivalent to the QLIKE loss that arises by choosing +φ(x) = − log(x); see Patton [25]. +The solution to the minimization problem (2.2) can be given explicitly as a min-max formula, +that is, +�µi = +min +ℓ=i,...,n max +k=1,...,ℓ +1 +�ℓ +j=k wj +ℓ +� +j=k +wjyj. +While the min-max formula is theoretically appealing and useful, the related minimum lower +sets (MLS) algorithm of Brunk et al. [6] is not efficient to compute the solution. The pool +adjacent violators (PAV) algorithm, which is due to Ayer et al. [1], Miles [24] and Kruskal [18], +allows for fast computation of the isotonic regression and provides us with the desired insights +about the solution. In Appendix A.1, we describe the PAV algorithm in detail. The solution is +obtained by suitably partitioning the index set I = {1, . . . , n} into (discrete) intervals +Ik = Ik(z) = {ik−1 + 1, . . . , ik} +for k = 1, . . . , K(z), +(2.3) +with z-dependent slicing points 0 = i0 < i1 < . . . < iK = n, and with K(z) ∈ {1, . . . , n} +denoting the number of discrete intervals Ik. The number K(z) of intervals and the slicing +points ik = ik(z), k = 1, . . . , K(z), for the partition of I depend on the observations z. On +each discrete interval Ik we then obtain the isotonic regression parameter estimate for instance +i ∈ Ik +�µi = �µik = +1 +� +j∈Ik wj +� +j∈Ik +wjyj, +(2.4) +see also (A.5). Thus, on each block Ik we have a constant estimate �µik, and the isotonic property +tells us that these estimates are strictly increasing over the block indices k = 1, . . . , K(z), +because these blocks have been chosen to be maximal. We call K(z) the complexity number of +the resulting isotonic regression. +Figure 1 gives an example for n = 20 and rankings π(xi) = i for i = 1, . . . , n. The resulting +(non-parametric) isotonic regression function �µ = �µ(z), which is only uniquely determined at +the observations (π(xi))n +i=1, can be interpolated by a step function. In Figure 1 this results in a +step function having K(z) − 1 = 9 steps, that is, we have K(z) = 10 blocks, and the estimated +regression function �µ takes only K(z) = 10 different values. This motivates to call K(z) the +complexity number of the resulting step function, see Figure 1. +4 + +5 +10 +15 +20 +5 +10 +15 +20 +isotonic regression +ranking pi +responses +observations Y +isotonic regression +Figure 1: Example of an isotonic regression with K(z) = 10 blocks. +The partition of the indices I into the isotonic blocks Ik is obtained naturally by requiring +monotonicity. This is different from the regression tree approach considered in Lindholm et +al. [21]. In fact, this latter reference does not require monotonicity but aims at minimizing the +“plain” square loss using, e.g., cross-validation for determining the optimal number of partitions. +In our context, the complexity number K(z) is fully determined through requiring monotonicity +and, in general, the results will differ. +In insurance applications, the blocks Ik ⊂ I provide us with the (empirical) price cohorts +�µi = �µik, for i ∈ Ik, and (2.4) leads to the (in-sample) auto-calibration property for Y +E +� +Y ′�� �µ(X′) = �µik +� += +1 +� +i∈Ik wi +� +i∈Ik +wiyi = �µik, +(2.5) +where (Y ′, X′) is distributed according to the weighted empirical distribution of (yi, xi)n +i=1 with +weights (wi)n +i=1. +Moreover, summing over the entire portfolio we have the (global) balance +property +n +� +i=1 +wi�µi = +K(z) +� +k=1 +� +i∈Ik +wi�µi = +K(z) +� +k=1 +�µik +� +i∈Ik +wi = +K(z) +� +k=1 +� +i∈Ik +wiyi = +n +� +i=1 +wiyi, +(2.6) +that is, in average the overall (price) level is correctly specified if we price the insurance policies +with covariates xi by wi�µi, where the weights wi > 0 now receive the interpretation of exposures. +2.2 +Monotonicity of the expected complexity number +In this section, we prove that the expected complexity number E[K(z)] is an increasing function +of the signal-to-noise ratio. For this, we assume a location-scale model for the responses Yi, that +is, we assume that +Yi = µi + σϵi, +i = 1, . . . , n, +(2.7) +with noise terms ϵi, location parameters µi ∈ R with µ1 ≤ . . . ≤ µn, and scale parameter +σ > 0. Here, µi takes the role of π(xi) in the previous section. The parameters µ1, . . . , µn are +unknown but it is known that they are labeled in increasing order. The signal-to-noise ratio is +5 + +then described by the scale parameter σ, i.e., we have a low signal-to-noise ratio for high σ and +vice-versa. The explicit location-scale structure (2.7) allows us to analyze +y = Y σ(ω) = µ + σϵ(ω) = (µ1, . . . , µn)⊤ + σ(ϵ1, . . . , ϵn)⊤(ω), +(2.8) +point-wise in the sample points ω ∈ Ω of the probability space (Ω, F, P) as a function of σ > 0; +this is similar to the re-parametrization trick of Kingma–Welling [16] that is frequently used to +explore variational auto-encoders. +In this section, we write K(y) = K(z), because the ranking of the outcomes y is clear from the +context (labeling), and we do not go via a ranking function π(·). +Theorem 2.3 Assume that the responses Yi, i = 1, . . . , n, follow the location-scale model (2.7) +with (unknown) ordered location parameters µ1 < . . . < µn, and scale parameter σ > 0. Then, +the expected complexity number E[K(Y )] of the isotonic regression of Y is a decreasing function +in σ > 0. If the distribution of the noise vector ϵ = (ϵ1, . . . , ϵn)⊤ has full support on Rn, then +E[K(Y )] is strictly decreasing in σ. +Theorem 2.3 proves that, under a specific but highly relevant model, the complexity number +K(Y ) of the isotonic regression is decreasing on average with a decreasing signal-to-noise ratio. +Implicitly, this means that more noisy data, which has a lower information ratio, leads to a less +granular regression function. Consequently, if the partition of the isotonic regression is used to +obtain a partition of the covariate space X via the candiate function π, this partition will be less +granular, the more noise of Y cannot be explained by π(X), see also Section 3.3 for a further +discussion. +To the best of our knowledge, our result is a new contribution to the literature on isotonic +regression. +While we focus on the finite sample case, a related result is the analysis of the +complexity number of the isotonic regression function as function of the sample size n, see +Dimitriadis et al. [9, Lemma 3.2]. +We are assuming strictly ordered location parameters in the formulation of Theorem 2.3. This +assumption simplifies the proof in the case where we show that the expected complexity num- +ber K(Y ) is strictly decreasing in σ. With some additional notation, the theorem could be +generalized to allow for ties between some (but not all) µi. +Figure 2 gives an example of a location-scale model (2.7) with i.i.d. standard Gaussian noise and +scale parameters σ = 2 (lhs) and σ = 20 (rhs), and both figures consider the same sample point +ω ∈ Ω in the noise term ϵ(ω), see (2.8). On the right-hand side of Figure 2, we have complexity +number K(y) = 13, and on the left-hand side K(y) = 46; the chosen sample size is n = 100. +3 +Isotonic recalibration for prediction and interpretation +3.1 +Prediction and estimation +In order to determine an auto-calibrated model for the true regression function x �→ E[Y |X = x] +from i.i.d. data (yi, xi)n +i=1, we are suggesting a two-step estimation procedure. First, we choose +a regression model and use the data (yi, xi)n +i=1 to obtain an estimate �π of a candidate function +π that should satisfy +π(x) ≤ π(x′) +⇐⇒ +E[Y |X = x] ≤ E[Y |X = x′], +(3.1) +6 + +0 +20 +40 +60 +80 +100 +0 +20 +40 +60 +80 +100 +isotonic regression sigma=2 +ranking mu_i +responses +observations Y +isotonic regression +0 +20 +40 +60 +80 +100 +20 +40 +60 +80 +isotonic regression sigma=20 +ranking mu_i +responses +observations Y +isotonic regression +Figure 2: Example of an isotonic regression of location-scale type with varying signal-to-noise +ratio for the identical sample point ω ∈ Ω: (lhs) σ = 2 with K(y) = 46 and (rhs) σ = 20 with +K(y) = 13. +for all x, x′ ∈ X. For example, in the case study in Section 4, a deep neural network model is +chosen for π. For sensible results, it is important that the estimation method for �π does not +overfit to the data. +In the second step, we apply isotonic regression to the pseudo-sample (yi, �π(xi))n +i=1 to obtain +an in-sample auto-calibrated regression function �µ defined on {�π(xi) : i = 1, . . . , n}. We call +this second step isotonic recalibration. +In order to obtain a prediction for a new covariate +value x ∈ X, we compute �π(x), find i such that �π(xi) < �π(x) ≤ �π(xi+1), and interpolate by +setting �µ(x) = (�µ(xi)+ �µ(xi+1))/2. This interpolation may be advantageous for prediction. For +interpretation and analysis, however, we prefer a step function interpolation as this leads to a +partition of the covariate space, see Section 3.3, below, and Figure 2. +This two-step estimation approach can be interpreted as a generalization of the monotone single +index models considered by Balabdaoui et al. [2]. They assume that the true regression function +is of the form E[Y |X = x] = ψ(α⊤x), with an increasing function ψ. +In contrast to our +proposal, the regression model π is fixed to be a linear model α⊤x in their approach. They +consider global least squares estimation jointly for (ψ, α), but find it computationally intensive. +As an alternative they suggest a two-step estimation procedure similar to our approach but +with a split of the data such that α and the isotonic regression are estimated on independent +samples. They find that if the rate of convergence of the estimator for α is sufficiently fast, then +the resulting estimator of the true regression function is consistent with a convergence rate of +order n1/3. +In a distributional regression framework, Henzi et al. [13] considered the described two-step +estimation procedure with an isotonic distributional regression [14], instead of a classical least +squares isotonic regression as described in Section 2.1. +They show that in both cases, with +and without sample splitting, the procedure leads to consistent estimation of the conditional +distribution of Y given X, as long as the index π can be estimated at a parametric rate. The +two options, with and without sample splitting, do not result in relevant differences in predictive +performance in the applications considered by Henzi et al. [13]. +Assumption (3.1) can be checked by diagnostic plots using binning similarly to the plots in Henzi +7 + +et al. [13, Figure 2] in the distributional regression case. Predictive performance should be as- +sessed on a test set of data disjoint from (yi, xi)n +i=1, that is, on data that has not been used in the +estimation procedure at all. Isotonic recalibration insures auto-calibration in-sample, and under +an i.i.d. assumption, auto-calibration will also hold approximately out-of-sample. Out-of-sample +auto-calibration can be diagnosed with CORP (consistent, optimally binned, reproducible and +PAV) mean reliability diagrams as suggested by Gneiting-Resin [12], and comparison of pre- +dictive performance can be done with the usual squared error loss function or deviance loss +functions. +3.2 +Over-fitting at the boundary +There is a small issue with the isotonic recalibration, namely, it tends to over-fit at the lower and +upper boundaries of the ranks �π(x1) < . . . < �π(xn). For instance, if yn is the largest observation +in the portfolio (which is not unlikely since the ranking �π is chosen response data-driven), then +we estimate �µiK = yn, where K = K((yi, �π(xi))n +i=1). +Often, this over-fits to the (smallest +and largest) observations, as such extreme values/estimates cannot be verified on out-of-sample +data. For this reason, we visually analyze the largest and smallest values in the estimates �µ, +and we may manually merge, say, the smallest block I1 with the second smallest one I2 (with +the resulting estimate (2.4) on the merged block). More rigorously, this pooling could be cross- +validated on out-of-sample data, but we refrain from doing so. We come back to this in Figure +5, below, where we merge the two blocks with the biggest estimates. +3.3 +Interpretation +In (2.3) we have introduced the complexity number K((yi, �π(xi))n +i=1) that counts the number of +different values in �µ, obtained by the isotonic regression (2.2) in the isotonic recalibration step. +This complexity number K((yi, �π(xi))n +i=1) allows one to assess the information content of the +model, or in other words, how much signal is explainable from the data. Theorem 2.3 shows that +the lower the signal-to-noise ratio, the lower the complexity number of the isotonic regression +that we can expect. Clearly, in Theorem 2.3 we assume that the ranking of the observations +is correct which will only be approximately satisfied since π has to be estimated. In general, +having large samples and flexible regression models for modeling π, it is reasonable to assume +that the statement remains qualitatively valid. However, in complex (algorithmic) regression +models, we need to ensure that we prevent from in-sample overfitting; this is typically controlled +by either using (independent) validation data or by performing a cross-validation analysis. +Typical claims data in non-life insurance have a low signal-to-noise ratio. Regarding claims +frequencies, this low signal-to-noise ratio is caused by the fact that claims are not very frequent +events, e.g., in car insurance annual claims frequencies range from 5% to 10%, that is, only one +out of 10 (or 20) drivers suffers a claim within a calendar year. A low signal-to-noise ratio also +applies to claim amounts, which are usually strongly driven by randomness and the explanatory +part from policyholder information is comparably limited. Therefore, we typically expect a low +complexity number K((yi, �π(xi))n +i=1) both for claims frequency and claim amounts modeling. +In case of a small to moderate complexity number K = K((yi, �π(xi))n +i=1), the regression function +�µ becomes interpretable through the isotonic recalibration step. For this, we extend the auto- +calibrated regression function �µ from the set {�π(x1), . . . , �π(xn)} to the entire covariate space X +8 + +by defining a step function +�µ(x) = �µik, +if +�π(xik) ≤ �π(x) < �π(xik+1), +for all x ∈ X, where 0 = i0 < i1 < · · · < iK = n are the slicing points of the isotonic regression +as defined in (2.3). Figure 1 illustrates this step function interpolation which is different from +an interpolation scheme that one would naturally use for prediction. +We define a partition +X1, . . . , XK of the original covariate space X by +Xk = {x ∈ X : �µ(x) = �µik}, +k = 1, . . . , K. +(3.2) +Figure 4 illustrates how this partition of X provides insights on the covariate-response relation- +ships in the model. This procedure has some analogy to regression trees and boosting trees that +rely on partitions of the covariate space X. In the case study in Section 4, we illustrate two +further possibilities to use the partition defined at (3.2) for understanding covariate-response +relationships. First, in Figure 7, the influence of individual covariates on the price cohorts is +analyzed, and second, Figure 9 gives a summary view of the whole covariate space for a chosen +price cohort. +4 +Swedish motorcycle data +We consider claim amounts modeling on the Swedish motorcycle data which was originally +presented in the text book of Ohlsson–Johansson [26] and which is also studied in W¨uthrich– +Merz [31].1 This data set comprises comprehensive insurance for motorcycles in Sweden. The +insurance product covers loss or damage of motorcycles other than collision, e.g., caused by theft, +fire or vandalism. The data contains claims aggregated per feature (covariate) combination for +the calendar years 1994–1998. There are 683 claims on 62,036 different covariates, thus, claims +are very sparse. We use exactly the same data pre-processing as described in [31, Listing 13.3], +and an excerpt of the pre-processed data is shown in Listing 1; for a description of the different +covariates we refer to [26, Section 2.4] and [31, Section 13.2]. The goal is to build a regression +model for these 683 positive claim amounts, and use isotonic recalibration for auto-calibration +and interpretation as described in Section 3.3. +Listing 1: Excerpt of the Swedish motorcycle data set. +1 +’data.frame ’: +62036 +obs. of +9 variables: +2 +$ OwnerAge +: num +18 18 18 18 18 18 18 18 18 18 ... +3 +$ Gender +: Factor w/ 2 levels "Female ","Male ": 1 1 1 1 1 1 1 1 1 1 ... +4 +$ Area +: Factor w/ 7 levels "Zone 1"," Zone +2" ,..: 1 1 1 1 2 2 2 3 3 3 ... +5 +$ RiskClass +: int +1 2 3 3 1 1 3 1 1 1 ... +6 +$ VehAge +: num +8 11 9 9 11 12 24 4 6 6 ... +7 +$ BonusClass : int +2 2 3 4 1 1 2 1 1 2 ... +8 +$ Exposure +: num +1 0.778 +0.499 +0.501 +0.929 +... +9 +$ ClaimNb +: int +0 0 0 0 0 0 0 0 0 0 ... +10 +$ ClaimAmount: int +0 0 0 0 0 0 0 0 0 0 ... +1The Swedish motorcycle data set is available through the R package CASdatasets [10]. +9 + +4.1 +Isotonic recalibration vs. binary regression trees +We start by considering the two covariate components RiskClass and VehAge only. +Since +the resulting covariate space X = {(RiskClass, VehAge)} ⊂ R2 is two-dimensional, we can +graphically illustrate the differences between the isotonic recalibration approach and a binary +regression tree (as a competing model) for interpretation. In Section 4.2, we consider all available +covariates. +We fit a deep feed-forward neural network (FFNN) regression model to these 683 claims. We +choose a network architecture of depth 3 with (20, 15, 10) neurons in the three hidden layers, the +hyperbolic tangent activation function in the hidden layers, and the log-link for the output layer. +The input has dimension 2, this results in a FFNN architecture with a network parameter of +dimension 546; for a more detailed discussion of FFNNs we refer to [31, Chapter 7], in particular, +to Listings 7.1-7.3 of that reference. We fit this model using the gamma deviance loss, see [31, +Section 5.3.7] and Remark 2.2, use the nadam version of stochastic gradient descent, and exercise +early stopping on a validation set being 20% of the entire data. Line (1a) of Table 1, called +gamma FFNN, shows the performance of the fitted FFNN regression model. This is compared to +the null model (empirical mean) on line (0) that does not consider any covariates.2 We observe a +decrease in gamma deviance loss and in root mean squared error (RMSE) which justifies the use +of a regression model; note that these are in-sample figures, but we use early stopping to prevent +the network from in-sample overfitting. The difficulty here is that, only having 683 claims, we +cannot provide a reasonable out-of-sample analysis. The last column of Table 1 called ’average’ +compares the average claims estimate of the FFNN to the empirical mean, and we observe a +slight positive bias in the FFNN prediction, i.e., 24, 932 > 24, 641. +gamma deviance +RMSE +average +(0) +null model +2.085 +35,311 +24,641 +(1a) +gamma FFNN +1.704 +32,562 +24,932 +(1b) +gamma FFNN recalibrated +1.640 +32,005 +24,641 +(2) +binary regression tree +1.761 +32,706 +24,641 +Table 1: Loss figures in the Swedish motorcycle example only considering RiskClass and VehAge +as covariates. +In the next step, we use the FFNN estimates as ranks �π(xi) for ordering the claims yi and +the covariates xi, respectively. Then we apply the non-parametric isotonic recalibration step +(2.2) to these ranks and claims. The Swedish motorcycle claims data is aggregated w.r.t. the +available covariate combinations, and the 683 positive claims come from 656 different covariate +combinations xi. This requires that we work with the weighted version of (2.2), where wi ∈ N +corresponds to the number of claims that have been observed for covariate xi, and yi corresponds +to the average observed claim amount on xi.3 We use the R package monotone [7] which provides +2In a gamma null model, i.e., assuming i.i.d. gamma distributed responses, we obtain that the MLE of the +mean is equal to the empirical mean of the observations; this generally holds true within the exponential dispersion +family. +3Since we only consider the two covariate components RiskClass and VehAge in this example, we further +aggregate the claims over these covariate combinations. This results in sufficient statistics for the gamma regression +model, and we only need to adjust the weights wi correspondingly. This is an elegant way of avoiding to deal with +ties for continuous regression functions (and supposed that the aggregation within different covariate combinations +10 + +a fast implementation of the PAV algorithm. The numerical results are presented on line (1b) +of Table 1. There is a slight decrease in average loss through the isotonic recalibration. This +is expected since the isotonic regression is optimizing the in-sample loss for any Bregman loss +function, see Remark 2.2. +The last column of Table 1 verifies that now the global balance +property (2.6) holds. +10000 +20000 +30000 +40000 +0 +20000 +40000 +60000 +gamma FFNN for RiskClass and VehAge +ranking pi +responses +observations Y +isotonic recalibrated +Figure 3: Isotonic recalibration in the Swedish motorcycle example only using RiskClass and +VehAge as covariates resulting in the complexity number K((yi, �π(xi))n +i=1) = 18. +Figure 3 provides the resulting step function from the isotonic recalibration (in red color) of +the ranking (�π(xi))n +i=1 given by the gamma FFNN; this is complemented with the observed +amounts yi (in blue color). The resulting complexity number is K = K((yi, �π(xi))n +i=1) = 18, +i.e., in this example the conditional expected claim amounts can be represented by 18 differ- +ent estimates �µik ∈ R, k = 1, . . . , K = 18; the FFNN regression function uses 6 · 21 = 126 +different values (ranks) which corresponds to the cardinality of the available covariate values +(RiskClass, VehAge) ∈ X. +The isotonic recalibration on the ranks �π(x) = �π(RiskClass, VehAge) of the FFNN leads to +a partition X1, . . . , X18 of the covariate space as defined at (3.2). We compare this partition +to the one that results from a binary split regression tree approach. +We use 10-fold cross- +validation to determine the optimal tree size. +In this example the optimal tree has only 3 +splits, and they all concern the variable VehAge. The resulting losses of this optimal tree are +shown on line (2) of Table 1, and we conclude that the regression tree approach is not fully +competitive, here. More interestingly, Figure 4 shows the resulting partitions of the covariate +space X = {(RiskClass, VehAge)} from the two approaches. The plot on the right-hand side +shows the three splits of the regression tree (all w.r.t. VehAge). From the isotonic recalibration +approach on the left-hand side, we learn that a good regression model should have diagonal +structures, emphasizing that the two covariates interact in a nontrivial way which cannot be +captured by the binary split regression tree in this case. +is computationally feasible). +11 + +isotonic recalibration +VehAge +RiskClass +1 +2 +3 +4 +5 +6 +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +11 +13 +15 +17 +19 +10000 +20000 +30000 +40000 +binary regression tree +VehAge +RiskClass +1 +2 +3 +4 +5 +6 +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +11 +13 +15 +17 +19 +10000 +20000 +30000 +40000 +50000 +Figure 4: (lhs) Isotonic recalibration and (rhs) binary regression tree, both only using RiskClass +and VehAge as covariates; the color scale is the same in both plots. +4.2 +Consideration of all covariates +We now consider all available covariate components, see lines 2-7 of Listing 1. We first fit a +FFNN to this data. This is done exacly as in the previous example with the only difference +that the input dimension changes from 2 to 6, when we consider all available information. We +transform the (ordered) Area code into real values, and also we also merge Area codes 5 to 7 +because of scarcity of claims for these Area codes, and we call this new variable Zone. The FFNN +has then a network parameter of dimension 626. The network is fitted with stochastic gradient +descent that is early stopped based on a validation loss analysis. The results are presented on +line (2a) of Table 2. +gamma deviance +RMSE +average +(0) +null model +2.085 +35,311 +24,641 +(1a) +gamma GLM +1.717 +32,562 +25,105 +(1b) +gamma GLM recalibrated with K = 24 +1.641 +31,578 +24,641 +(2a) +gamma FFNN +1.496 +29,673 +24,526 +(2b) +gamma FFNN recalibrated with K = 22 +1.452 +28,806 +24,641 +(2c) +gamma FFNN tree adjustment with 4 bins (seed 1) +1.508 +29,371 +24,641 +(2d) +gamma FFNN tree adjustment with 8 bins (seed 2) +1.466 +27,942 +24,641 +Table 2: Losses in the Swedish motorcycle example based on all available covariates. +We compare the fitted FFNN regression model to the null model (empirical mean) and a gamma +generalized linear model (GLM). The gamma GLM is identical to model Gamma GLM1 in [31, +Table 5.13]. We give some remarks on the results of Table 2. Firstly, the FFNN has the smallest +gamma deviance loss and the smallest RMSE of the three models on lines (0)-(2a). Thus, the +gamma FFNN adapts best to the data among the three model choices (we use early stopping in +the FFNN fitting). Interestingly, the gamma GLM and the FFNN both fail to have the global +balance property (2.6), see last column of Table 2. Stochastic gradient descent fitted models +with early stopping generally fail to satisfy the global balance property, whereas the gamma +12 + +GLM fails to have the global balance property because we work with the log-link and not with +the canonical link of the Gamma GLM, here. +0 +20000 +40000 +60000 +80000 +0 +50000 +100000 +150000 +200000 +isotonic recalibration: GLM +ranking pi +responses +observations Y +isotonic recalibrated +20000 +40000 +60000 +80000 +0 +50000 +100000 +150000 +200000 +isotonic recalibration: FFNN +ranking pi +responses +observations Y +isotonic recalibrated +20000 +40000 +60000 +80000 +0 +50000 +100000 +150000 +200000 +isotonic recalibration: FFNN corrected +ranking pi +responses +observations Y +isotonic recalibrated +Figure 5: Isotonically recalibrated regression models in the Swedish motorcycle example using +all covariates for the gamma GLM with complexity number K((yi, �π(xi))n +i=1) = 24 (lhs), for +the gamma FFNN with complexity number K((yi, �π(xi))n +i=1) = 23 (middle) and over-fitting +corrected (rhs). +In the next step, we use the FFNN predictions as ranks �π(xi) for ordering the responses and +covariates, and we label the claims yi such that �π(x1) < . . . < �π(xn). There are no ties in this +data, and we obtain n = 656 pairwise different values. The results of the isotonic recalibration +are presented in Figure 5 (middle). The complexity number is K = K((yi, �π(xi))n +i=1) = 23, +thus, the entire regression problem is encoded in 23 different values �µik, k = 1, . . . , K. In view +of this plot, it seems that the largest value �µiK over-fits to the corresponding observation, as +this estimate is determine by a single observation yn, being bigger than the weighted block mean +�µiK−1 on the previous block IK−1; compare Section 3.2. For this reason, we manually pool the +two last blocks IK−1 and IK. This provides us with a new estimate (2.4) on this merged block, +and reduces the complexity number by 1 to K = 22. The resulting isotonic recalibration is shown +in Figure 5 (rhs), and the empirical losses are provided on line (2b) of Table 2. Importantly, +this isotonic recalibrated regression is in-sample auto-calibrated (2.5) and, henceforth, it fulfills +the global balance property which can be verified in the last column of Table 2. +We perform the same isotonic recalibration to the ranks obtained from the gamma GLM in +Table 2. We observe that the isotonic recalibration step leads to a major decrease in average +loss in the gamma GLM, and it results in the complexity number K = 24, see also Figure 5 +(lhs). +We compare isotonic recalibration to a recent proposal of Lindholm et al. [21] that also achieves +auto-calibration in-sample. Isotonic regression provides a partition of the index set I = {1, . . . , n} +into disjoint blocks I1, . . . , IK on which the estimated regression function is constant. This can +also be achieved by considering a binary regression tree algorithm applied to the (rank) co- +variates {�π(xi); 1 ≤ i ≤ n} and corresponding responses yi; see Section 2.3.2 of Lindholm et +al. [21]. We call this latter approach the tree binning approach. There are two main differences +between the tree binning approach and the isotonic recalibration approach. First, generally, the +tree binning approach does not provide a regression function that has the same ranking as the +first regression step providing �π(xi). Second, in the isotonic regression approach, the complex- +ity number K((yi, �π(xi))n +i=1) is naturally given, i.e., the isotonic regression (2.2) automatically +13 + +extracts the degree of information contained in the responses y, and generally, this degree of +information is increasing for an increasing signal-to-noise ratio by Theorem 2.3. Conversely, +in the tree binning approach, we need to determine the optimal number of bins (leaves), e.g., +by k-fold cross-validation. The obtained number of bins depends on the hyperparameters of +the minimal leaf size and of the number of folds in cross-validation, as well as on the random +partition of the instances for cross-validation. We found that the number of bins is sensitive +to the tuning choices, and hence, contrary to isotonic recalibration, the resulting partition is +subject to potentially subjective choices and randomness. +For the results on the tree binning approach in Table 2 we have chosen k = 10 folds and a +minimal leaf size of 10, and only the random partitioning of the pseudo-sample is different +for the results in lines (2c)-(2d). A first random seed gives 4 bins and a second one 8 bins, +and we observe a considerable difference in the two models with respect to gamma deviance +loss and the RMSE. Figure 6 shows the isotonic recalibration and the tree binning approach +with 8 bins, corresponding to lines (2b) and (2d) of Table 2. From this plot, we conclude that +the tree binning approach does not necessarily preserve the rankings induces by �π(xi) as the +resulting step function (in blue color) is not monotonically increasing. We recommend isotonic +recalibration to achieve auto-calibration since it preserves monotonicity of the regression model +in the first estimation step, and there are no potentially influential tuning parameters. +20000 +40000 +60000 +80000 +0 +20000 +40000 +60000 +80000 +100000 +ranking pi +mean estimates +tree binning +isotonic recalib. +Figure 6: Tree binning vs. isotonic recalibration; the step functions correspond to lines (2d) and +(2b) of Table 2 with 8 bins for line (2d) and complexity number K = 22 for line (2b). +In Figure 7, we illustrate the resulting marginal plots if we project the estimated values �µ of +the isotonic recalibration to the corresponding covariate values, i.e., this is the marginal view of +the resulting covariate space partition (3.2). For a low complexity number K((yi, �π(xi))n +i=1) this +can be interpreted nicely. We see relevant differences in the distributions of the colors across the +different covariate levels of OwnerAge, Zone, RiskClass and VehAge. This indicates that these +variables are important for explaining claim sizes, with the reservation that this marginal view +ignores potential interactions. For the variable Gender we cannot make any conclusion as the +gender balance inequality is too large. The interpretation of BonusClass is less obvious. In fact, +from the gamma GLM we know that BonusClass is not significant, see [31, Table 5.13]. This is +because the BonusClass is related to collision claims, whereas our data studies comprehensive +insurance that excludes collision claims. Figure 8 shows the marginal view of the isotonically +14 + +18 +22 +26 +30 +34 +38 +42 +46 +50 +54 +58 +62 +66 +marginal view: OwnerAge +OwnerAge +frequency +0 +10 +20 +30 +40 +50 +60 +70 +20000 +40000 +60000 +80000 +Female +Male +marginal view: Gender +Gender +frequency +0 +100 +200 +300 +400 +500 +20000 +40000 +60000 +80000 +Zone 1 +Zone 2 +Zone 3 +Zone 4 +Zone 5 +marginal view: Zone +Zone +frequency +0 +50 +100 +150 +20000 +40000 +60000 +80000 +1 +2 +3 +4 +5 +6 +marginal view: RiskClass +RiskClass +frequency +0 +50 +100 +150 +20000 +40000 +60000 +80000 +0 +2 +4 +6 +8 +10 +12 +14 +16 +18 +20 +marginal view: VehAge +VehAge +frequency +0 +10 +20 +30 +40 +50 +60 +70 +20000 +40000 +60000 +80000 +1 +2 +3 +4 +5 +6 +7 +marginal view: BonusClass +BonusClass +frequency +0 +50 +100 +150 +200 +250 +20000 +40000 +60000 +80000 +Figure 7: Marginal view of the isotonically recalibrated gamma FFNN model of Table 2 of the +6 considered covariate components OwnerAge, Gender, Zone, RiskClass, VehAge, BonusClass. +recalibrated gamma FFNN (lhs) and the gamma GLM (rhs) for the covariate BonusClass. As +mentioned, BonusClass is not significant in the gamma GLM, and it seems from the figure that, +indeed, the color distribution across the different levels is rather similar for both models. +1 +2 +3 +4 +5 +6 +7 +marginal view: BonusClass +BonusClass +frequency +0 +50 +100 +150 +200 +250 +20000 +40000 +60000 +80000 +1 +2 +3 +4 +5 +6 +7 +marginal view: BonusClass +BonusClass +frequency +0 +50 +100 +150 +200 +250 +20000 +40000 +60000 +80000 +Figure 8: Marginal view of the isotonically recalibrated gamma FFNN model (lhs) and the +isotonically recalibrated gamma GLM (rhs) for the covariate components BonusClass. +Clearly, the VehAge is the most important variable showing the picture that claims on new +motorcycles are more expensive. There are substantial differences in claim size distributions +between the zones, Zone 1 being the three largest cities of Sweden having typically more big +claims. RiskClass corresponds to the size of the motorcycle which interacts with the OwnerAge, +the VehAge and the Zone, and it is therefore more difficult to interpret as we have relevant +interactions between these variables. +15 + +predicted claim amount mu=26,187 +feature levels +OwnerAge +Gender +Zone +RiskClass +VehAge +BonusCl +mu +predicted claim amount mu=59,851 +feature levels +OwnerAge +Gender +Zone +RiskClass +VehAge +BonusCl +mu +Figure 9: Partition (Xk)k=1,...,K of the covariate space X w.r.t. the isotonic recalibration for two +selected values of k = 12, 21. +Figure 9 gives an illustration of the partition (Xk)k=1,...,K of the 6-dimensional covariate space +X w.r.t. the isotonic recalibration (�µik)k=1,...,K for two selected values of k. The lines connect all +the covariate components in x that are observed within the data (xi)1≤i≤n for a given value �µik, +and the size of the black dots illustrates how often a certain covariate level is observed. E.g., +the figure on the right-hand side belongs to the second largest claim prediction �µiK−1 = 59, 851. +For this expected response level, the OwnerAge is comparably small (around 25 years), everyone +is Male mostly living in Zone 1 (three biggest cities of Sweden), having a motorcycle of a higher +RiskClass with a small VehAge. Similar conclusions can be drawn for the other parts Xk of the +covariate space X, thus, having a low complexity number K((yi, �π(xi))n +i=1) enables to explain +the regression model. +5 +Conclusions +We have tackled two problems. First, we have enforced that the regression model fulfills the +auto-calibration property by applying an isotonic recalibration to the ranks of a fitted (first) +regression model. This isotonic recalibration does not involve any hyperparameters, but it solely +assumes that the ranks from the first regression model are (approximately) correct. Isotonic +regression has the property that the complexity of the resulting (non-parametric) regression +function is small in low signal-to-noise ratio problems. Benefiting from this property, we have +shown that this leads to explainable regression functions because a low complexity is equivalent +to a coarse partition of the covariate space. In insurance pricing problems this is particularly +useful, as we typically face a low signal-to-noise ratio in insurance claims data. We can then fit +a complex (algorithmic) model to that data in a first step, and in a subsequent step we propose +to auto-calibrate the first regression function using isotonic recalibration, which also leads to a +substantial simplification of the regression function. +16 + +References +[1] Ayer, M., Brunk, H.D., Ewing, G.M., Reid, W.T., Silverman, E. (1955). An empirical distribution +function for sampling with incomplete information. Annals of Mathematical Statistics 26, 641-647. +[2] Balabdaoui, F., Durot, C., Jankowski, H. (2019). Least squares estimation in the monotone single +index model. Bernoulli 25, 3276-3310. +[3] Barlow, R.E., Bartholomew, D.J., Brenner, J.M., Brunk, H.D. (1972). Statistical Inference under +Order Restrictions. Wiley. +[4] Barlow, R.E., Brunk, H.D. (1972). The isotonic regression problem and its dual. Journal of the +American Statistical Association 67/337, 140-147. +[5] Best, M.J., Chakravarti, N. (1990). Active set algorithms for isotonic regression; A unifying frame- +work. Mathematical Programming 47 425-439. +[6] Brunk, H.D., Ewing, G.M., Utz, W.R. (1957). Minimizing integrals in certain classes of monotone +functions. Pacific Journal of Mathematics 7, 833-847. +[7] Busing, F.M.T.A. (2022). Monotone regression: a simple and fast O(n) PAVA implementation. +Journal of Statistical Software 102, Code Snipet 1. +[8] Denuit, M., Charpentier, A., Trufin, J. (2021). Autocalibration and Tweedie-dominance for insur- +ance pricing in machine learning. Insurance: Mathematics & Economics 101/B, 485-497. +[9] Dimitriadis, T., D¨umbgen, L., Henzi, A., Puke, M., Ziegel, J. (2022). Honest calibration assessment +for binary outcome predictions. arXiv:2203.04065. +[10] Dutang, C., Charpentier, A. (2018). CASdatasets R Package Vignette. Reference Manual. Version +1.0-8, packaged 2018-05-20. +[11] Gneiting, T. (2011). Making and evaluating point forecasts. Journal of the American Statistical +Association 106/494, 746-762. +[12] Gneiting, T. (2021). Regression diagnostics meets forecast evaluation: Conditional calibration, +reliability diagrams and coefficient of determination. arXiv:2108.03210. +[13] Henzi, A., Kleger, G.-R., Ziegel, J.F. (2021). Distributional (single) index models. Journal of the +American Statistical Association, to appear. +[14] Henzi, A., Ziegel, J.F., Gneiting, T. (2021). Isotonic distributional regression. Journal of the Royal +Statistical Society: Series B 85, 963-993. +[15] Karush, W. (1939). Minima of Functions of Several Variables with Inequalities as Side Constraints. +MSc Thesis. Department of Mathematics, University of Chicago. +[16] Kingma, D.P., Welling, M. (2019). An introduction to variational autoencoders. Foundations and +Trends in Machine Learning 12/4, 307-392. +[17] Kr¨uger, F., Ziegel, J.F. (2021). Generic conditions for forecast dominance. Journal of Business & +Economics Statistics 39/4, 972-983. +[18] Kruskal, J.B. (1964). Nonmetric multidimensional scaling. Psychometrica 29, 115-129. +[19] Kuhn, H.W., Tucker, A.W. (1951). Nonlinear programming. Proceedings of 2nd Berkeley Sympo- +sium. University of California Press, 481-492. +[20] Leeuw, de J., Hornik, K., Mair, P. (2009). Isotone optimization in R: pool-adjacent-violators algo- +rithm (PAVA) and active set methods. Journal of Statistical Software 32/5, 1-24. +17 + +[21] Lindholm, M., Lindskog, F., Palmquist, J. (2022). Local bias adjustment, duration-weighted prob- +abilities, and automatic construction of tariff cells. SSRN manuscript ID 4256876. +[22] Lundberg, S.M., Lee, S.-I. (2017). A unified approach to interpreting model predictions. In: Ad- +vances in Neural Information Processing Systems 30. Guyon, I., Luxburg, U.V., Bengio, S., Wal- +lach, H., Fergus, R., Vishwanathan, S., Garnett, R. (Eds.). Curran Associates, 4765-4774. +[23] Menon, A.K., Jiang, X., Vembu, S., Elkan, C., Ohno-Machado, L. (2012). Predicting accurate +probabilities with ranking loss. ICML’12: Proceedings of the 29th International Conference on +Machine Learning, 659-666. +[24] Miles, R.E. (1959). The complete amalgamation into blocks, by weighted means, of a finite set of +real numbers. Biometrika 46, 317-327. +[25] Patton, A.J. (2011). Volatility forecast comparison using imperfect volatility proxies. Journal of +Econometrics 160, 246-256. +[26] Ohlsson, E., Johansson, B. (2010). Non-Life Insurance Pricing with Generalized Linear Models. +Springer. +[27] Savage, L.J. (1971). Elicitable of personal probabilities and expectations. Journal of the American +Statistical Association 66/336, 783-810. +[28] Schervish, M.J. (1989). A general method of comparing probability assessors. The Annals of Statis- +tics 17/4, 1856-1879. +[29] Tasche, D. (2021). Calibrating sufficiently. Statistics: A Journal of Theoretical and Applied Statis- +tics 55/6, 1356-1386. +[30] W¨uthrich M.V. (2022). Model selection with Gini indices under auto-calibration. European Actu- +arial Journal, to appear. +[31] W¨uthrich, M.V., Merz, M. (2023). Statistical Foundations of Actuarial Learning and its Applica- +tions. Springer Actuarial. +[32] Zadrozny, B., Elkan, C. (2002). Transforming classifier scores into accurate multiclass probabil- +ity estimates. Proceedings of the Eighth ACM SIGKDD International Conference on Knowledge +Discovery and Data Mining, 694-699. +18 + +A +Appendix +A.1 +Pool adjacent violators algorithm +Minimization problem (2.2) is a quadratic optimization problem with linear side constraints, +and it can be solved using the method of Karush–Kuhn–Tucker (KKT) [15, 19]. We therefore +consider the Lagrangian +L(µ, η) = (y − µ)⊤W(y − µ) − η⊤Aµ, +with Lagrange multiplier η ∈ Rn−1. The KKT conditions are given by +0 += +∇µL(µ, η) = +− W(y − µ) − A⊤η, +(A.1) +0 +≥ +∇ηL(µ, η) = +− Aµ, +(A.2) +0 +≤ +η, +(A.3) +0 += +(η1(µ1 − µ2), . . . , ηn−1(µn−1 − µn))⊤ . +(A.4) +The solution to these KKT conditions (A.1)-(A.4) provides the isotonic estimate �µ. This solution +can be found by the PAV algorithm. The main idea is to compare raw estimates (�µi)i. If we +have an adjacent pair with �µi > �µi+1, it violates the monotonicity constraint. Such pairs are +recursively merged (pooled) to a block with an identical estimate, and iterating this pooling of +adjacent pairs and blocks, respectively, that violate the monotonicity constraint, yields the PAV +algorithm. +Pool Adjacent Violators (PAV) Algorithm +(0) Initialize the algorithm �µ(0) = y and define the blocks I(0) +k += {k} for k = 1, . . . , K(0) = n. +(1) Iterate for t ≥ 0: +(a) If �µ(t) fulfills KKT condition (A.2) go to item (2), otherwise go to the next step (1b). +(b) Select an index i = 1, . . . , n with �µ(t) +i +> �µ(t) +i+1, merge the two adjacent blocks with +i ∈ I(t) +k +and i + 1 ∈ I(t) +k+1, and leave all other blocks unchanged. This provides the +new blocks I(t+1) +k +with k = 1, . . . , K(t+1) = K(t) − 1. +(c) Set on each block k = 1, . . . , K(t+1) and for all indices i ∈ I(t+1) +k +the new estimates +�µ(t+1) +i += +1 +� +j∈I(t+1) +k +wj +� +j∈I(t+1) +k +wjyj. +(A.5) +(d) Increase t �→ t + 1. +(2) Set the isotonic regression estimate �µ = �µ(t) and merge adjacent blocks I(t) +k +and I(t) +k+1 if we +have the same estimates �µi on these blocks. Return the resulting partition of I denoted +by (Ik)k=1,...,K and �µ. +Remarks A.1 (PAV algorithm interpretation) +19 + +(0) We initialize with the unconstraint optimal solution, and setting η(0) = 0 ensures that the +KKT conditions (A.1), (A.3) and (A.4) are fulfilled, thus, only the monotonicity (A.2) is +not necessarily fulfilled. +(1a) We identify a pair �µ(t) +i +> �µ(t) +i+1 that violates the monotonicity constraint (A.2). +This +pair needs to belong to two adjacent blocks I(t) +k +and I(t) +k+1 because within blocks we have +constant estimates (2.4). We merge these two adjacent blocks to I(t+1) +k += I(t) +k +∪ I(t) +k+1, +which reduces the number of blocks K(t) by 1. +(1b) We set on each block the constant estimate (A.5) which satisfies the monotonicity con- +straint (A.2) within blocks, and also (A.4) is naturally fulfilled in this block. Conditions +(A.1) and (A.3) are achieved by changing the Lagrange parameter η(t) �→ η(t+1) ≥ 0 +correspondingly to account for the change in mean estimates (A.5) in (A.1). +(1c) On a sample of size n, this algorithm can be iterated at most n − 1 times, thus, the +algorithm will terminate. +(2) Since we have for i ∈ I(t) +k +and i + 1 ∈ I(t) +k+1 the inequality �µ(t) +i +≤ �µ(t) +i+1, the last step is to +ensure that the resulting blocks are maximal by merging blocks where we do not have a +strict inequality in the corresponding estimates. +A.2 +Proof of Theorem 2.3 +Proof of Theorem 2.3. +For given responses y = Y σ(ω), the solution to (2.2) gives the +partition (2.3) of the index set I with empirical weighted averages (2.4) on the blocks Ik. These +empirical weighted averages satisfy �µik < �µik+1 for all k = 1, . . . , K(Y ) − 1, because the blocks +Ik have been chosen maximal. We now consider how these blocks are constructed in the PAV +algorithm. Suppose that we are in iteration t ≥ 0, and in this iteration of the PAV algorithm, +we merge the two adjacent blocks I(t) +k +and I(t) +k+1 because �µ(t) +i +> �µ(t) +i+1 for i ∈ I(t) +k +and i+1 ∈ I(t) +k+1. +We analyze this inequality +1 +� +j∈I(t) +k wj +� +j∈I(t) +k +wjyj = �µ(t) +i +> �µ(t) +i+1 = +1 +� +l∈I(t) +k+1 wl +� +l∈I(t) +k+1 +wlyl. +We use the location-scale structure (2.4) which gives us the equivalent condition +1 +� +j∈I(t) +k wj +� +j∈I(t) +k +wj (µj + σϵj(ω)) > +1 +� +l∈I(t) +k+1 wl +� +l∈I(t) +k+1 +wl (µl + σϵl(ω)) . +Since for any indices j ∈ I(t) +k +and l ∈ I(t) +k+1 we have µj ≤ µl, it follows that the previous condition +for merging the two adjacent blocks I(t) +k +and I(t) +k+1 in iteration t of the PAV algorithm reads as +σ +� +� +� +j∈I(t) +k wjϵj(ω) +� +j∈I(t) +k wj +− +� +l∈I(t) +k+1 wlϵl(ω) +� +l∈I(t) +k+1 wl +� +� > +� +l∈I(t) +k+1 wlµl +� +l∈I(t) +k+1 wl +− +� +j∈I(t) +k wjµj +� +j∈I(t) +k wj +≥ 0. +(A.6) +The important observation is that if this condition is fulfilled for scale parameter σ > 0, then +it will also be fulfilled for any bigger scale parameter σ′ > σ > 0 (pointwise in ω ∈ Ω). Thus, +20 + +any pooling that happens for σ also happens for σ′ > σ > 0. Since this is pointwise on the +underlying probability space (Ω, F, P), it shows that E[K(Y )] is decreasing in σ > 0. +Suppose now that the distribution of ϵ has full support on Rn. Then, the event Aσ = {K(Y σ) = +n} occurs with positive probability, i.e., +0 +< +P[Aσ] = P [K(Y σ) = n] += +P [Y1 < Y2 < . . . < Yn] += +P [µ1 + σϵ1 < µ2 + σϵ2 < . . . < µn + σϵn] . +Consider +Aσ += +{µ1 + σϵ1 < µ2 + σϵ2 < . . . < µn + σϵn} += +n� +k=2 +{µk−1 + σϵk−1 < µk + σϵk} += +{σ (ϵ1 − ϵ2) < µ2 − µ1} ∩ +n� +k=3 +{µk−1 + σϵk−1 < µk + σϵk} . +We focus on the first event on the right-hand side. Note that µ2 − µ1 > 0, hence +� +ϵ ∈ Rn : ϵ1 − ϵ2 < µ2 − µ1 +σ +� +describes an open half space in Rn containing the origin and with bounding hyperplane that +moves further away from the origin when decreasing σ. Overall, the set ˜Aσ ⊂ Rn of values +of ϵ in Aσ is a non-empty open polyhedron containing the origin that scales with σ, that is, +˜Aσ = (1/σ) ˜A1. Therefore, since the distribution of ϵ has full support, the probability P[Aσ] is +strictly decreasing in σ. +2 +21 + diff --git a/PNE0T4oBgHgl3EQf1AIP/content/tmp_files/load_file.txt b/PNE0T4oBgHgl3EQf1AIP/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..70c11c31be7e24ef5c09cc02710627b31082891c --- /dev/null +++ b/PNE0T4oBgHgl3EQf1AIP/content/tmp_files/load_file.txt @@ -0,0 +1,1129 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf,len=1128 +page_content='Isotonic Recalibration under a Low Signal-to-Noise Ratio Mario V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' W¨uthrich∗ Johanna Ziegel† Version of January 10, 2023 Abstract Insurance pricing systems should fulfill the auto-calibration property to ensure that there is no systematic cross-financing between different price cohorts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Often, regression models are not auto-calibrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' We propose to apply isotonic recalibration to a given regression model to ensure auto-calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Our main result proves that under a low signal-to-noise ratio, this isotonic recalibration step leads to explainable pricing systems because the resulting isotonically recalibrated regression functions have a low complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Keywords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Auto-calibration, isotonic regression, isotonic recalibration, low signal-to-noise ratio, cross-financing, algorithmic solution, deep neural network, explainability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' 1 Introduction There are two seemingly unrelated problems in insurance pricing that we are going to tackle in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' First, an insurance pricing system should not have any systematic cross-financing between different price cohorts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Systematic cross-financing implicitly means that some parts of the portfolio are under-priced, and this is compensated by other parts of the portfolio that are over-priced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' We can prevent systematic cross-financing between price cohorts by ensuring that the pricing system is auto-calibrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' We propose to apply isotonic recalibration which turns any regression function into an auto-calibrated pricing system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' The second problem that we tackle is the explainability of complex algorithmic models for insurance pricing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' In a first step, one may use any complex regression model to design an insurance pricing system such as, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=', a deep neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Such complex regression models typically lack explainability and rather act as black boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' For this reason, there are several tools deployed to explain such complex solutions, we mention, for instance, SHAP by Lundberg– Lee [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Since algorithmic solutions do not generally fulfill the aforementioned auto-calibration property, we propose to apply isotonic recalibration to the algorithmic solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' If the signal-to- noise ratio is low in the data, then the isotonic recalibration step leads to a coarse partition of the covariate space and, as a consequence, it leads to an explainable version of the algorithmic model used in the first place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Thus, explainability is a nice side result of applying isotonic recalibration in low signal-to-noise ratio problems, which is typically the case in insurance pricing settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' There are other methods for obtaining auto-calibration through a recalibration step;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' we men- tion Lindholm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' [21] and Denuit et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' These other methods often require tuning of ∗RiskLab, Department of Mathematics, ETH Zurich, mario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='wuethrich@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='ethz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='ch †Institute of Mathematical Statistics and Actuarial Science, University of Bern, johanna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='ziegel@stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='unibe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='ch 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='02692v1 [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='ME] 6 Jan 2023 hyperparameters, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=', using cross-validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Isotonic recalibration does not involve any hy- perparameters as it solves a constraint regression problem (ensuring monotonicity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' As such, isotonic recaliabration is universal because it also does not depend on the specific choice of the loss function within the family of Bregman losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' We formalize our proposal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Throughout, we assume that all considered random variables have finite means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Consider a response variable Y that is equipped with covariate information X ∈ X ⊆ Rq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' The general goal is to determine the (true) regression function x �→ E[Y |X = x] that describes the conditional mean of Y , given X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Typically, this true regression function is unknown, and it needs to be determined from i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' data (yi, xi)n i=1, that is, a sample from (Y, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' For this purpose, we try to select a regression function x �→ µ(x) from a (pre-chosen) function class on X that approximates the conditional mean E[Y |X = ·] as well as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Often, it is not possible to capture all features of the regression function from data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' In financial applications, a minimal important requirement for a well-selected regression function µ(·) is that it fulfills the auto-calibration property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='1 The regression function µ is auto-calibrated for (Y, X) if µ(X) = E [Y | µ(X)] , P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Auto-calibration is an important property in actuarial and financial applications because it implies that, on average, the (price) cohorts µ(X) are self-financing for the corresponding claims Y , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=', there is no systematic cross-financing within the portfolio, if the structure of this portfolio is described by the covariates X ∼ P and the price cohorts µ(X), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' In a Bernoulli context, an early version of auto-calibration (called well-calibrated) has been introduced by Schervish [28] to the community in statistics, and recently, it has been considered in detail by Gneiting–Resin [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' In an actuarial and financial context, the importance of auto-calibration has been emphasized in Kr¨uger–Ziegel [17], Denuit et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' [8], W¨uthrich [30] and Lindholm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Many regression models do not satisfy the auto-calibration property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' However, there is a sim- ple and powerful method, which we call isotonic recalibration, to obtain an (in-sample) auto- calibrated regression function starting from any candidate function π : X → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' We apply isotonic recalibration to the pseudo-sample (yi, π(xi))n i=1 to obtain an isotonic regression func- tion �µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Then, �µ(X′) = E � Y ′|�µ(X′) � , Pn-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=', (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='1) where (Y ′, X′) is distributed according to the empirical distribution Pn of (yi, xi)n i=1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='1 for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Isotonic regression determines an adaptive partition of the covariate space X, and �µ is determined by averaging y-values over the partition elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Clearly, other binning approaches can also be used on the pseudo-sample (yi, π(xi))n i=1 to enforce (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='1), but we argue that isotonic regression is preferable since it avoids subjective choices of tuning parameters and leads to sensible regression functions under reasonable and verifiable assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' The only assumption for isotonic recalibration to be informative is that the function π gets the rankings of the conditional means right, that is, whenever E [Y |X = xi] ≤ E [Y |X = xj], we would like to have π(xi) ≤ π(xj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Using isotonic regression for recalibration is not new in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' In the case of binary outcomes, it as already been proposed by Zadrozny–Elkan [32], Menon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' [23] and recently 2 by Tasche [29, Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' The monotone single index models of Balabdaoui et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' [2] follow the same strategy as described above but the focus of their work is different from ours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' They specifically consider a linear regression model for the candidate function π, which is called the index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' In the case of distributional regression, that is, when interest is in determining the whole conditional distribution of Y given covariate information X, Henzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' [13] have suggested to first estimate an index function π that determines the ordering of the conditional distributions w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' first order stochastic dominance and then estimate conditional distributions using isotonic distributional regression;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' see Henzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' As a new contribution, we show that the size of the partition of the isotonic recalibration may give insight concerning the information content of the recalibrated regression function �µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Furthermore, the partition of the isotonic recalibration allows to explain connections between covariates and outcomes, in particular, when the signal-to-noise ratio is small which typically is the case for insurance claims data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' In order to come up with a candidate function π : X → R, one may consider any regression model such as, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=', a generalized linear model, a regression tree, a tree boosting regression model or a deep neural network regression model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' The aim is that π(·) provides us with the correct rankings of the conditional means E[Y |X = xi], i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' The details are discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' In Section 2, we formally introduce isotonic regression which is a constraint optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' This constraint optimization problem is usually solved with the pool adjacent violators (PAV) algorithm, which is described in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Our main result is stated in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' It relates the complexity of the isotonic recalibration solution to the signal- to-noise ratio in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Section 3 gives practical guidance on the use of isotonic recalibration, and in Section 4 we exemplify our results on a frequently used insurance data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' In this section we also present graphic tools for interpreting the regression function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' In Section 5, we conclude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' 2 Isotonic regression 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='1 Definition and basic properties For simplicity, we assume that the candidate function π : X → R does not lead to any ties in the values π(x1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' , π(xn), and that the indices i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' , n are chosen such that they are aligned with the ranking, that is, π(x1) < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' < π(xn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='1 explains how to handle ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' The isotonic regression of z = (yi, π(xi))n i=1 with positive case weights (wi)n i=1 is the solution �µ ∈ Rn to the restricted minimization problem �µ = arg min µ=(µ1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=',µn)⊤ n � i=1 wi (yi − µi)2 , subject to µ1 ≤ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' ≤ µn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='1) We can rewrite the side constraints as Aµ ≥ 0 (component-wise), where A = (ai,j)i,j ∈ Rn×(n−1) is the matrix with the elements ai,j = 1i=j−1 − 1i=j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' We define y = (y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' , yn)⊤ ∈ Rn and the (diagonal) case weight matrix W = diag(w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' , wn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' The above optimization problem then reads as �µ = �µ(z) = arg min µ: Aµ≥0 (y − µ)⊤W(y − µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='2) 3 This shows that the isotonic regression is solved by a convex minimization with linear side constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' It remains to verify that the auto-calibration property claimed in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='1) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='1 If there are ties in the values π(x1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' , π(xn), for example, π(xi) = π(xj) for some i ̸= j, we replace yi and yj with their weighted average (wiyi + wjyj)/(wi + wj) and assign them weights (wi + wj)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' The procedure is analogous for more than two tied values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' This corresponds to the second option of dealing with ties in Leeuw et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' [20, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='2 Barlow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' [3, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='10] show that the square loss function in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='1) can be replaced by any Bregman loss function, Lφ(y, µ) = φ(y)−φ(µ)+φ′(µ)(y −µ), without changing the optimal solution �µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Here, φ is a strictly convex function with subgradient φ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Bregman loss functions are the only consistent loss functions for the mean;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' see Savage [27] and Gneiting [11, Theorem 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' If y and µ only take positive values, a Bregman loss function of relevance for this paper is the gamma deviance loss, which is equivalent to the QLIKE loss that arises by choosing φ(x) = − log(x);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' see Patton [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' The solution to the minimization problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='2) can be given explicitly as a min-max formula, that is, �µi = min ℓ=i,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=',n max k=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=',ℓ 1 �ℓ j=k wj ℓ � j=k wjyj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' While the min-max formula is theoretically appealing and useful, the related minimum lower sets (MLS) algorithm of Brunk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' [6] is not efficient to compute the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' The pool adjacent violators (PAV) algorithm, which is due to Ayer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' [1], Miles [24] and Kruskal [18], allows for fast computation of the isotonic regression and provides us with the desired insights about the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' In Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='1, we describe the PAV algorithm in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' The solution is obtained by suitably partitioning the index set I = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' , n} into (discrete) intervals Ik = Ik(z) = {ik−1 + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' , ik} for k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' , K(z), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='3) with z-dependent slicing points 0 = i0 < i1 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' < iK = n, and with K(z) ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' , n} denoting the number of discrete intervals Ik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' The number K(z) of intervals and the slicing points ik = ik(z), k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' , K(z), for the partition of I depend on the observations z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' On each discrete interval Ik we then obtain the isotonic regression parameter estimate for instance i ∈ Ik �µi = �µik = 1 � j∈Ik wj � j∈Ik wjyj, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='4) see also (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Thus, on each block Ik we have a constant estimate �µik, and the isotonic property tells us that these estimates are strictly increasing over the block indices k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' , K(z), because these blocks have been chosen to be maximal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' We call K(z) the complexity number of the resulting isotonic regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Figure 1 gives an example for n = 20 and rankings π(xi) = i for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' The resulting (non-parametric) isotonic regression function �µ = �µ(z), which is only uniquely determined at the observations (π(xi))n i=1, can be interpolated by a step function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' In Figure 1 this results in a step function having K(z) − 1 = 9 steps, that is, we have K(z) = 10 blocks, and the estimated regression function �µ takes only K(z) = 10 different values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' This motivates to call K(z) the complexity number of the resulting step function, see Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' 4 5 10 15 20 5 10 15 20 isotonic regression ranking pi responses observations Y isotonic regression Figure 1: Example of an isotonic regression with K(z) = 10 blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' The partition of the indices I into the isotonic blocks Ik is obtained naturally by requiring monotonicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' This is different from the regression tree approach considered in Lindholm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' In fact, this latter reference does not require monotonicity but aims at minimizing the “plain” square loss using, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=', cross-validation for determining the optimal number of partitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' In our context, the complexity number K(z) is fully determined through requiring monotonicity and, in general, the results will differ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' In insurance applications, the blocks Ik ⊂ I provide us with the (empirical) price cohorts �µi = �µik, for i ∈ Ik, and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='4) leads to the (in-sample) auto-calibration property for Y E � Y ′�� �µ(X′) = �µik � = 1 � i∈Ik wi � i∈Ik wiyi = �µik, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='5) where (Y ′, X′) is distributed according to the weighted empirical distribution of (yi, xi)n i=1 with weights (wi)n i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Moreover, summing over the entire portfolio we have the (global) balance property n � i=1 wi�µi = K(z) � k=1 � i∈Ik wi�µi = K(z) � k=1 �µik � i∈Ik wi = K(z) � k=1 � i∈Ik wiyi = n � i=1 wiyi, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='6) that is, in average the overall (price) level is correctly specified if we price the insurance policies with covariates xi by wi�µi, where the weights wi > 0 now receive the interpretation of exposures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='2 Monotonicity of the expected complexity number In this section, we prove that the expected complexity number E[K(z)] is an increasing function of the signal-to-noise ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' For this, we assume a location-scale model for the responses Yi, that is, we assume that Yi = µi + σϵi, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' , n, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='7) with noise terms ϵi, location parameters µi ∈ R with µ1 ≤ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' ≤ µn, and scale parameter σ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Here, µi takes the role of π(xi) in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' The parameters µ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' , µn are unknown but it is known that they are labeled in increasing order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' The signal-to-noise ratio is 5 then described by the scale parameter σ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=', we have a low signal-to-noise ratio for high σ and vice-versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' The explicit location-scale structure (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='7) allows us to analyze y = Y σ(ω) = µ + σϵ(ω) = (µ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' , µn)⊤ + σ(ϵ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' , ϵn)⊤(ω), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='8) point-wise in the sample points ω ∈ Ω of the probability space (Ω, F, P) as a function of σ > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' this is similar to the re-parametrization trick of Kingma–Welling [16] that is frequently used to explore variational auto-encoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' In this section, we write K(y) = K(z), because the ranking of the outcomes y is clear from the context (labeling), and we do not go via a ranking function π(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='3 Assume that the responses Yi, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' , n, follow the location-scale model (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='7) with (unknown) ordered location parameters µ1 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' < µn, and scale parameter σ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Then, the expected complexity number E[K(Y )] of the isotonic regression of Y is a decreasing function in σ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' If the distribution of the noise vector ϵ = (ϵ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' , ϵn)⊤ has full support on Rn, then E[K(Y )] is strictly decreasing in σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='3 proves that, under a specific but highly relevant model, the complexity number K(Y ) of the isotonic regression is decreasing on average with a decreasing signal-to-noise ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Implicitly, this means that more noisy data, which has a lower information ratio, leads to a less granular regression function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Consequently, if the partition of the isotonic regression is used to obtain a partition of the covariate space X via the candiate function π, this partition will be less granular, the more noise of Y cannot be explained by π(X), see also Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='3 for a further discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' To the best of our knowledge, our result is a new contribution to the literature on isotonic regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' While we focus on the finite sample case, a related result is the analysis of the complexity number of the isotonic regression function as function of the sample size n, see Dimitriadis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' [9, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' We are assuming strictly ordered location parameters in the formulation of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' This assumption simplifies the proof in the case where we show that the expected complexity num- ber K(Y ) is strictly decreasing in σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' With some additional notation, the theorem could be generalized to allow for ties between some (but not all) µi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Figure 2 gives an example of a location-scale model (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='7) with i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' standard Gaussian noise and scale parameters σ = 2 (lhs) and σ = 20 (rhs), and both figures consider the same sample point ω ∈ Ω in the noise term ϵ(ω), see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' On the right-hand side of Figure 2, we have complexity number K(y) = 13, and on the left-hand side K(y) = 46;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' the chosen sample size is n = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' 3 Isotonic recalibration for prediction and interpretation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='1 Prediction and estimation In order to determine an auto-calibrated model for the true regression function x �→ E[Y |X = x] from i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' data (yi, xi)n i=1, we are suggesting a two-step estimation procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' First, we choose a regression model and use the data (yi, xi)n i=1 to obtain an estimate �π of a candidate function π that should satisfy π(x) ≤ π(x′) ⇐⇒ E[Y |X = x] ≤ E[Y |X = x′], (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='1) 6 0 20 40 60 80 100 0 20 40 60 80 100 isotonic regression sigma=2 ranking mu_i responses observations Y isotonic regression 0 20 40 60 80 100 20 40 60 80 isotonic regression sigma=20 ranking mu_i responses observations Y isotonic regression Figure 2: Example of an isotonic regression of location-scale type with varying signal-to-noise ratio for the identical sample point ω ∈ Ω: (lhs) σ = 2 with K(y) = 46 and (rhs) σ = 20 with K(y) = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' for all x, x′ ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' For example, in the case study in Section 4, a deep neural network model is chosen for π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' For sensible results, it is important that the estimation method for �π does not overfit to the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' In the second step, we apply isotonic regression to the pseudo-sample (yi, �π(xi))n i=1 to obtain an in-sample auto-calibrated regression function �µ defined on {�π(xi) : i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' We call this second step isotonic recalibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' In order to obtain a prediction for a new covariate value x ∈ X, we compute �π(x), find i such that �π(xi) < �π(x) ≤ �π(xi+1), and interpolate by setting �µ(x) = (�µ(xi)+ �µ(xi+1))/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' This interpolation may be advantageous for prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' For interpretation and analysis, however, we prefer a step function interpolation as this leads to a partition of the covariate space, see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='3, below, and Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' This two-step estimation approach can be interpreted as a generalization of the monotone single index models considered by Balabdaoui et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' They assume that the true regression function is of the form E[Y |X = x] = ψ(α⊤x), with an increasing function ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' In contrast to our proposal, the regression model π is fixed to be a linear model α⊤x in their approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' They consider global least squares estimation jointly for (ψ, α), but find it computationally intensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' As an alternative they suggest a two-step estimation procedure similar to our approach but with a split of the data such that α and the isotonic regression are estimated on independent samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' They find that if the rate of convergence of the estimator for α is sufficiently fast, then the resulting estimator of the true regression function is consistent with a convergence rate of order n1/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' In a distributional regression framework, Henzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' [13] considered the described two-step estimation procedure with an isotonic distributional regression [14], instead of a classical least squares isotonic regression as described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' They show that in both cases, with and without sample splitting, the procedure leads to consistent estimation of the conditional distribution of Y given X, as long as the index π can be estimated at a parametric rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' The two options, with and without sample splitting, do not result in relevant differences in predictive performance in the applications considered by Henzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Assumption (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='1) can be checked by diagnostic plots using binning similarly to the plots in Henzi 7 et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' [13, Figure 2] in the distributional regression case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Predictive performance should be as- sessed on a test set of data disjoint from (yi, xi)n i=1, that is, on data that has not been used in the estimation procedure at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Isotonic recalibration insures auto-calibration in-sample, and under an i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' assumption, auto-calibration will also hold approximately out-of-sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Out-of-sample auto-calibration can be diagnosed with CORP (consistent, optimally binned, reproducible and PAV) mean reliability diagrams as suggested by Gneiting-Resin [12], and comparison of pre- dictive performance can be done with the usual squared error loss function or deviance loss functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='2 Over-fitting at the boundary There is a small issue with the isotonic recalibration, namely, it tends to over-fit at the lower and upper boundaries of the ranks �π(x1) < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' < �π(xn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' For instance, if yn is the largest observation in the portfolio (which is not unlikely since the ranking �π is chosen response data-driven), then we estimate �µiK = yn, where K = K((yi, �π(xi))n i=1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Often, this over-fits to the (smallest and largest) observations, as such extreme values/estimates cannot be verified on out-of-sample data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' For this reason, we visually analyze the largest and smallest values in the estimates �µ, and we may manually merge, say, the smallest block I1 with the second smallest one I2 (with the resulting estimate (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='4) on the merged block).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' More rigorously, this pooling could be cross- validated on out-of-sample data, but we refrain from doing so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' We come back to this in Figure 5, below, where we merge the two blocks with the biggest estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='3 Interpretation In (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='3) we have introduced the complexity number K((yi, �π(xi))n i=1) that counts the number of different values in �µ, obtained by the isotonic regression (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='2) in the isotonic recalibration step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' This complexity number K((yi, �π(xi))n i=1) allows one to assess the information content of the model, or in other words, how much signal is explainable from the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='3 shows that the lower the signal-to-noise ratio, the lower the complexity number of the isotonic regression that we can expect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Clearly, in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='3 we assume that the ranking of the observations is correct which will only be approximately satisfied since π has to be estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' In general, having large samples and flexible regression models for modeling π, it is reasonable to assume that the statement remains qualitatively valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' However, in complex (algorithmic) regression models, we need to ensure that we prevent from in-sample overfitting;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' this is typically controlled by either using (independent) validation data or by performing a cross-validation analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Typical claims data in non-life insurance have a low signal-to-noise ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Regarding claims frequencies, this low signal-to-noise ratio is caused by the fact that claims are not very frequent events, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=', in car insurance annual claims frequencies range from 5% to 10%, that is, only one out of 10 (or 20) drivers suffers a claim within a calendar year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' A low signal-to-noise ratio also applies to claim amounts, which are usually strongly driven by randomness and the explanatory part from policyholder information is comparably limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Therefore, we typically expect a low complexity number K((yi, �π(xi))n i=1) both for claims frequency and claim amounts modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' In case of a small to moderate complexity number K = K((yi, �π(xi))n i=1), the regression function �µ becomes interpretable through the isotonic recalibration step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' For this, we extend the auto- calibrated regression function �µ from the set {�π(x1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' , �π(xn)} to the entire covariate space X 8 by defining a step function �µ(x) = �µik, if �π(xik) ≤ �π(x) < �π(xik+1), for all x ∈ X, where 0 = i0 < i1 < · · · < iK = n are the slicing points of the isotonic regression as defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Figure 1 illustrates this step function interpolation which is different from an interpolation scheme that one would naturally use for prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' We define a partition X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' , XK of the original covariate space X by Xk = {x ∈ X : �µ(x) = �µik}, k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' , K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='2) Figure 4 illustrates how this partition of X provides insights on the covariate-response relation- ships in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' This procedure has some analogy to regression trees and boosting trees that rely on partitions of the covariate space X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' In the case study in Section 4, we illustrate two further possibilities to use the partition defined at (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='2) for understanding covariate-response relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' First, in Figure 7, the influence of individual covariates on the price cohorts is analyzed, and second, Figure 9 gives a summary view of the whole covariate space for a chosen price cohort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' 4 Swedish motorcycle data We consider claim amounts modeling on the Swedish motorcycle data which was originally presented in the text book of Ohlsson–Johansson [26] and which is also studied in W¨uthrich– Merz [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='1 This data set comprises comprehensive insurance for motorcycles in Sweden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' The insurance product covers loss or damage of motorcycles other than collision, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=', caused by theft, fire or vandalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' The data contains claims aggregated per feature (covariate) combination for the calendar years 1994–1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' There are 683 claims on 62,036 different covariates, thus, claims are very sparse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' We use exactly the same data pre-processing as described in [31, Listing 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='3], and an excerpt of the pre-processed data is shown in Listing 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' for a description of the different covariates we refer to [26, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='4] and [31, Section 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' The goal is to build a regression model for these 683 positive claim amounts, and use isotonic recalibration for auto-calibration and interpretation as described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Listing 1: Excerpt of the Swedish motorcycle data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' 1 ’data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='frame ’: 62036 obs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' of 9 variables: 2 $ OwnerAge : num 18 18 18 18 18 18 18 18 18 18 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' 3 $ Gender : Factor w/ 2 levels "Female ","Male ": 1 1 1 1 1 1 1 1 1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' 4 $ Area : Factor w/ 7 levels "Zone 1"," Zone 2" ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='.: 1 1 1 1 2 2 2 3 3 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' 5 $ RiskClass : int 1 2 3 3 1 1 3 1 1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' 6 $ VehAge : num 8 11 9 9 11 12 24 4 6 6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' 7 $ BonusClass : int 2 2 3 4 1 1 2 1 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' 8 $ Exposure : num 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='778 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='499 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='501 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='929 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' 9 $ ClaimNb : int 0 0 0 0 0 0 0 0 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' 10 $ ClaimAmount: int 0 0 0 0 0 0 0 0 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' 1The Swedish motorcycle data set is available through the R package CASdatasets [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' 9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='1 Isotonic recalibration vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' binary regression trees We start by considering the two covariate components RiskClass and VehAge only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Since the resulting covariate space X = {(RiskClass, VehAge)} ⊂ R2 is two-dimensional, we can graphically illustrate the differences between the isotonic recalibration approach and a binary regression tree (as a competing model) for interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='2, we consider all available covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' We fit a deep feed-forward neural network (FFNN) regression model to these 683 claims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' We choose a network architecture of depth 3 with (20, 15, 10) neurons in the three hidden layers, the hyperbolic tangent activation function in the hidden layers, and the log-link for the output layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' The input has dimension 2, this results in a FFNN architecture with a network parameter of dimension 546;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' for a more detailed discussion of FFNNs we refer to [31, Chapter 7], in particular, to Listings 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='1-7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='3 of that reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' We fit this model using the gamma deviance loss, see [31, Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='7] and Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='2, use the nadam version of stochastic gradient descent, and exercise early stopping on a validation set being 20% of the entire data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Line (1a) of Table 1, called gamma FFNN, shows the performance of the fitted FFNN regression model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' This is compared to the null model (empirical mean) on line (0) that does not consider any covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='2 We observe a decrease in gamma deviance loss and in root mean squared error (RMSE) which justifies the use of a regression model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' note that these are in-sample figures, but we use early stopping to prevent the network from in-sample overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' The difficulty here is that, only having 683 claims, we cannot provide a reasonable out-of-sample analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' The last column of Table 1 called ’average’ compares the average claims estimate of the FFNN to the empirical mean, and we observe a slight positive bias in the FFNN prediction, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=', 24, 932 > 24, 641.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' gamma deviance RMSE average (0) null model 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='085 35,311 24,641 (1a) gamma FFNN 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='704 32,562 24,932 (1b) gamma FFNN recalibrated 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='640 32,005 24,641 (2) binary regression tree 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='761 32,706 24,641 Table 1: Loss figures in the Swedish motorcycle example only considering RiskClass and VehAge as covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' In the next step, we use the FFNN estimates as ranks �π(xi) for ordering the claims yi and the covariates xi, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Then we apply the non-parametric isotonic recalibration step (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='2) to these ranks and claims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' The Swedish motorcycle claims data is aggregated w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' the available covariate combinations, and the 683 positive claims come from 656 different covariate combinations xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' This requires that we work with the weighted version of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='2), where wi ∈ N corresponds to the number of claims that have been observed for covariate xi, and yi corresponds to the average observed claim amount on xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='3 We use the R package monotone [7] which provides 2In a gamma null model, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=', assuming i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' gamma distributed responses, we obtain that the MLE of the mean is equal to the empirical mean of the observations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' this generally holds true within the exponential dispersion family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' 3Since we only consider the two covariate components RiskClass and VehAge in this example, we further aggregate the claims over these covariate combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' This results in sufficient statistics for the gamma regression model, and we only need to adjust the weights wi correspondingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' This is an elegant way of avoiding to deal with ties for continuous regression functions (and supposed that the aggregation within different covariate combinations 10 a fast implementation of the PAV algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' The numerical results are presented on line (1b) of Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' There is a slight decrease in average loss through the isotonic recalibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' This is expected since the isotonic regression is optimizing the in-sample loss for any Bregman loss function, see Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' The last column of Table 1 verifies that now the global balance property (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='6) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' 10000 20000 30000 40000 0 20000 40000 60000 gamma FFNN for RiskClass and VehAge ranking pi responses observations Y isotonic recalibrated Figure 3: Isotonic recalibration in the Swedish motorcycle example only using RiskClass and VehAge as covariates resulting in the complexity number K((yi, �π(xi))n i=1) = 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Figure 3 provides the resulting step function from the isotonic recalibration (in red color) of the ranking (�π(xi))n i=1 given by the gamma FFNN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' this is complemented with the observed amounts yi (in blue color).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' The resulting complexity number is K = K((yi, �π(xi))n i=1) = 18, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=', in this example the conditional expected claim amounts can be represented by 18 differ- ent estimates �µik ∈ R, k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' , K = 18;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' the FFNN regression function uses 6 · 21 = 126 different values (ranks) which corresponds to the cardinality of the available covariate values (RiskClass, VehAge) ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' The isotonic recalibration on the ranks �π(x) = �π(RiskClass, VehAge) of the FFNN leads to a partition X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' , X18 of the covariate space as defined at (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' We compare this partition to the one that results from a binary split regression tree approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' We use 10-fold cross- validation to determine the optimal tree size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' In this example the optimal tree has only 3 splits, and they all concern the variable VehAge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' The resulting losses of this optimal tree are shown on line (2) of Table 1, and we conclude that the regression tree approach is not fully competitive, here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' More interestingly, Figure 4 shows the resulting partitions of the covariate space X = {(RiskClass, VehAge)} from the two approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' The plot on the right-hand side shows the three splits of the regression tree (all w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' VehAge).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' From the isotonic recalibration approach on the left-hand side, we learn that a good regression model should have diagonal structures, emphasizing that the two covariates interact in a nontrivial way which cannot be captured by the binary split regression tree in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' is computationally feasible).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' 11 isotonic recalibration VehAge RiskClass 1 2 3 4 5 6 0 1 2 3 4 5 6 7 8 9 11 13 15 17 19 10000 20000 30000 40000 binary regression tree VehAge RiskClass 1 2 3 4 5 6 0 1 2 3 4 5 6 7 8 9 11 13 15 17 19 10000 20000 30000 40000 50000 Figure 4: (lhs) Isotonic recalibration and (rhs) binary regression tree, both only using RiskClass and VehAge as covariates;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' the color scale is the same in both plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='2 Consideration of all covariates We now consider all available covariate components, see lines 2-7 of Listing 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' We first fit a FFNN to this data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' This is done exacly as in the previous example with the only difference that the input dimension changes from 2 to 6, when we consider all available information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' We transform the (ordered) Area code into real values, and also we also merge Area codes 5 to 7 because of scarcity of claims for these Area codes, and we call this new variable Zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' The FFNN has then a network parameter of dimension 626.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' The network is fitted with stochastic gradient descent that is early stopped based on a validation loss analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' The results are presented on line (2a) of Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' gamma deviance RMSE average (0) null model 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='085 35,311 24,641 (1a) gamma GLM 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='717 32,562 25,105 (1b) gamma GLM recalibrated with K = 24 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='641 31,578 24,641 (2a) gamma FFNN 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='496 29,673 24,526 (2b) gamma FFNN recalibrated with K = 22 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='452 28,806 24,641 (2c) gamma FFNN tree adjustment with 4 bins (seed 1) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='508 29,371 24,641 (2d) gamma FFNN tree adjustment with 8 bins (seed 2) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='466 27,942 24,641 Table 2: Losses in the Swedish motorcycle example based on all available covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' We compare the fitted FFNN regression model to the null model (empirical mean) and a gamma generalized linear model (GLM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' The gamma GLM is identical to model Gamma GLM1 in [31, Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' We give some remarks on the results of Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Firstly, the FFNN has the smallest gamma deviance loss and the smallest RMSE of the three models on lines (0)-(2a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Thus, the gamma FFNN adapts best to the data among the three model choices (we use early stopping in the FFNN fitting).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Interestingly, the gamma GLM and the FFNN both fail to have the global balance property (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='6), see last column of Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Stochastic gradient descent fitted models with early stopping generally fail to satisfy the global balance property, whereas the gamma 12 GLM fails to have the global balance property because we work with the log-link and not with the canonical link of the Gamma GLM, here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='20000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='40000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='60000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='80000 ' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='80000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='50000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='100000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='150000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='200000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='isotonic recalibration: FFNN corrected ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='ranking pi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='responses ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='observations Y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='isotonic recalibrated ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='Figure 5: Isotonically recalibrated regression models in the Swedish motorcycle example using ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='all covariates for the gamma GLM with complexity number K((yi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' �π(xi))n i=1) = 24 (lhs),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' for the gamma FFNN with complexity number K((yi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' �π(xi))n i=1) = 23 (middle) and over-fitting corrected (rhs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' In the next step, we use the FFNN predictions as ranks �π(xi) for ordering the responses and covariates, and we label the claims yi such that �π(x1) < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' < �π(xn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' There are no ties in this data, and we obtain n = 656 pairwise different values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' The results of the isotonic recalibration are presented in Figure 5 (middle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' The complexity number is K = K((yi, �π(xi))n i=1) = 23, thus, the entire regression problem is encoded in 23 different values �µik, k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' , K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' In view of this plot, it seems that the largest value �µiK over-fits to the corresponding observation, as this estimate is determine by a single observation yn, being bigger than the weighted block mean �µiK−1 on the previous block IK−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' compare Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' For this reason, we manually pool the two last blocks IK−1 and IK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' This provides us with a new estimate (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='4) on this merged block, and reduces the complexity number by 1 to K = 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' The resulting isotonic recalibration is shown in Figure 5 (rhs), and the empirical losses are provided on line (2b) of Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Importantly, this isotonic recalibrated regression is in-sample auto-calibrated (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='5) and, henceforth, it fulfills the global balance property which can be verified in the last column of Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' We perform the same isotonic recalibration to the ranks obtained from the gamma GLM in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' We observe that the isotonic recalibration step leads to a major decrease in average loss in the gamma GLM, and it results in the complexity number K = 24, see also Figure 5 (lhs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' We compare isotonic recalibration to a recent proposal of Lindholm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' [21] that also achieves auto-calibration in-sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Isotonic regression provides a partition of the index set I = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' , n} into disjoint blocks I1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' , IK on which the estimated regression function is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' This can also be achieved by considering a binary regression tree algorithm applied to the (rank) co- variates {�π(xi);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' 1 ≤ i ≤ n} and corresponding responses yi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='2 of Lindholm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' We call this latter approach the tree binning approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' There are two main differences between the tree binning approach and the isotonic recalibration approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' First, generally, the tree binning approach does not provide a regression function that has the same ranking as the first regression step providing �π(xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Second, in the isotonic regression approach, the complex- ity number K((yi, �π(xi))n i=1) is naturally given, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=', the isotonic regression (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='2) automatically 13 extracts the degree of information contained in the responses y, and generally, this degree of information is increasing for an increasing signal-to-noise ratio by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Conversely, in the tree binning approach, we need to determine the optimal number of bins (leaves), e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=', by k-fold cross-validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' The obtained number of bins depends on the hyperparameters of the minimal leaf size and of the number of folds in cross-validation, as well as on the random partition of the instances for cross-validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' We found that the number of bins is sensitive to the tuning choices, and hence, contrary to isotonic recalibration, the resulting partition is subject to potentially subjective choices and randomness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' For the results on the tree binning approach in Table 2 we have chosen k = 10 folds and a minimal leaf size of 10, and only the random partitioning of the pseudo-sample is different for the results in lines (2c)-(2d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' A first random seed gives 4 bins and a second one 8 bins, and we observe a considerable difference in the two models with respect to gamma deviance loss and the RMSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Figure 6 shows the isotonic recalibration and the tree binning approach with 8 bins, corresponding to lines (2b) and (2d) of Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' From this plot, we conclude that the tree binning approach does not necessarily preserve the rankings induces by �π(xi) as the resulting step function (in blue color) is not monotonically increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' We recommend isotonic recalibration to achieve auto-calibration since it preserves monotonicity of the regression model in the first estimation step, and there are no potentially influential tuning parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' 20000 40000 60000 80000 0 20000 40000 60000 80000 100000 ranking pi mean estimates tree binning isotonic recalib.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Figure 6: Tree binning vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' isotonic recalibration;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' the step functions correspond to lines (2d) and (2b) of Table 2 with 8 bins for line (2d) and complexity number K = 22 for line (2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' In Figure 7, we illustrate the resulting marginal plots if we project the estimated values �µ of the isotonic recalibration to the corresponding covariate values, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=', this is the marginal view of the resulting covariate space partition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' For a low complexity number K((yi, �π(xi))n i=1) this can be interpreted nicely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' We see relevant differences in the distributions of the colors across the different covariate levels of OwnerAge, Zone, RiskClass and VehAge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' This indicates that these variables are important for explaining claim sizes, with the reservation that this marginal view ignores potential interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' For the variable Gender we cannot make any conclusion as the gender balance inequality is too large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' The interpretation of BonusClass is less obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' In fact, from the gamma GLM we know that BonusClass is not significant, see [31, Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' This is because the BonusClass is related to collision claims, whereas our data studies comprehensive insurance that excludes collision claims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Figure 8 shows the marginal view of the isotonically ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='22 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='26 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='34 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='38 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='42 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='46 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='54 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='58 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='62 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='66 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='marginal view: OwnerAge ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='OwnerAge ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='frequency ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='40 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='40000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='60000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='80000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='Figure 7: Marginal view of the isotonically recalibrated gamma FFNN model of Table 2 of the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='6 considered covariate components OwnerAge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Gender,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Zone,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' RiskClass,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' VehAge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' BonusClass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' recalibrated gamma FFNN (lhs) and the gamma GLM (rhs) for the covariate BonusClass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' As mentioned, BonusClass is not significant in the gamma GLM, and it seems from the figure that, indeed, the color distribution across the different levels is rather similar for both models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' 1 2 3 4 5 6 7 marginal view: BonusClass BonusClass frequency 0 50 100 150 200 250 20000 40000 60000 80000 1 2 3 4 5 6 7 marginal view: BonusClass BonusClass frequency 0 50 100 150 200 250 20000 40000 60000 80000 Figure 8: Marginal view of the isotonically recalibrated gamma FFNN model (lhs) and the isotonically recalibrated gamma GLM (rhs) for the covariate components BonusClass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Clearly, the VehAge is the most important variable showing the picture that claims on new motorcycles are more expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' There are substantial differences in claim size distributions between the zones, Zone 1 being the three largest cities of Sweden having typically more big claims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' RiskClass corresponds to the size of the motorcycle which interacts with the OwnerAge, the VehAge and the Zone, and it is therefore more difficult to interpret as we have relevant interactions between these variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' 15 predicted claim amount mu=26,187 feature levels OwnerAge Gender Zone RiskClass VehAge BonusCl mu predicted claim amount mu=59,851 feature levels OwnerAge Gender Zone RiskClass VehAge BonusCl mu Figure 9: Partition (Xk)k=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=',K of the covariate space X w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' the isotonic recalibration for two selected values of k = 12, 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Figure 9 gives an illustration of the partition (Xk)k=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=',K of the 6-dimensional covariate space X w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' the isotonic recalibration (�µik)k=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=',K for two selected values of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' The lines connect all the covariate components in x that are observed within the data (xi)1≤i≤n for a given value �µik, and the size of the black dots illustrates how often a certain covariate level is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=', the figure on the right-hand side belongs to the second largest claim prediction �µiK−1 = 59, 851.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' For this expected response level, the OwnerAge is comparably small (around 25 years), everyone is Male mostly living in Zone 1 (three biggest cities of Sweden), having a motorcycle of a higher RiskClass with a small VehAge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Similar conclusions can be drawn for the other parts Xk of the covariate space X, thus, having a low complexity number K((yi, �π(xi))n i=1) enables to explain the regression model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' 5 Conclusions We have tackled two problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' First, we have enforced that the regression model fulfills the auto-calibration property by applying an isotonic recalibration to the ranks of a fitted (first) regression model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' This isotonic recalibration does not involve any hyperparameters, but it solely assumes that the ranks from the first regression model are (approximately) correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Isotonic regression has the property that the complexity of the resulting (non-parametric) regression function is small in low signal-to-noise ratio problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Benefiting from this property, we have shown that this leads to explainable regression functions because a low complexity is equivalent to a coarse partition of the covariate space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' In insurance pricing problems this is particularly useful, as we typically face a low signal-to-noise ratio in insurance claims data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' We can then fit a complex (algorithmic) model to that data in a first step, and in a subsequent step we propose to auto-calibrate the first regression function using isotonic recalibration, which also leads to a substantial simplification of the regression function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' 16 References [1] Ayer, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=', Brunk, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=', Ewing, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=', Reid, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=', Silverman, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' (1955).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' An empirical distribution function for sampling with incomplete information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Annals of Mathematical Statistics 26, 641-647.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' [2] Balabdaoui, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=', Durot, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=', Jankowski, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Least squares estimation in the monotone single index model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Bernoulli 25, 3276-3310.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' [3] Barlow, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=', Bartholomew, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=', Brenner, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=', Brunk, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' (1972).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Statistical Inference under Order Restrictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Wiley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' [4] Barlow, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=', Brunk, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' (1972).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' The isotonic regression problem and its dual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Journal of the American Statistical Association 67/337, 140-147.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' [5] Best, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=', Chakravarti, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' (1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Active set algorithms for isotonic regression;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' A unifying frame- work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Mathematical Programming 47 425-439.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' [6] Brunk, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=', Ewing, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=', Utz, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' (1957).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Minimizing integrals in certain classes of monotone functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Pacific Journal of Mathematics 7, 833-847.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' [7] Busing, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Monotone regression: a simple and fast O(n) PAVA implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Journal of Statistical Software 102, Code Snipet 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' [8] Denuit, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=', Charpentier, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=', Trufin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Autocalibration and Tweedie-dominance for insur- ance pricing in machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Insurance: Mathematics & Economics 101/B, 485-497.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' [9] Dimitriadis, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=', D¨umbgen, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=', Henzi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=', Puke, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=', Ziegel, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Honest calibration assessment for binary outcome predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' arXiv:2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='04065.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' [10] Dutang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=', Charpentier, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' CASdatasets R Package Vignette.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Reference Manual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Version 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='0-8, packaged 2018-05-20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' [11] Gneiting, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Making and evaluating point forecasts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Journal of the American Statistical Association 106/494, 746-762.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' [12] Gneiting, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Regression diagnostics meets forecast evaluation: Conditional calibration, reliability diagrams and coefficient of determination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' arXiv:2108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='03210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' [13] Henzi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=', Kleger, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=', Ziegel, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Distributional (single) index models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Journal of the American Statistical Association, to appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' [14] Henzi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=', Ziegel, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=', Gneiting, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Isotonic distributional regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Journal of the Royal Statistical Society: Series B 85, 963-993.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' [15] Karush, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' (1939).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Minima of Functions of Several Variables with Inequalities as Side Constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' MSc Thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Department of Mathematics, University of Chicago.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' [16] Kingma, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=', Welling, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' An introduction to variational autoencoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Foundations and Trends in Machine Learning 12/4, 307-392.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' [17] Kr¨uger, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=', Ziegel, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Generic conditions for forecast dominance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Journal of Business & Economics Statistics 39/4, 972-983.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' [18] Kruskal, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' (1964).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Nonmetric multidimensional scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Psychometrica 29, 115-129.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' [19] Kuhn, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=', Tucker, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' (1951).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Nonlinear programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Proceedings of 2nd Berkeley Sympo- sium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' University of California Press, 481-492.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' [20] Leeuw, de J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=', Hornik, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=', Mair, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Isotone optimization in R: pool-adjacent-violators algo- rithm (PAVA) and active set methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Journal of Statistical Software 32/5, 1-24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' 17 [21] Lindholm, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=', Lindskog, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=', Palmquist, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Local bias adjustment, duration-weighted prob- abilities, and automatic construction of tariff cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' SSRN manuscript ID 4256876.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' [22] Lundberg, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=', Lee, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='-I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' A unified approach to interpreting model predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' In: Ad- vances in Neural Information Processing Systems 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Guyon, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=', Luxburg, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=', Bengio, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=', Wal- lach, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=', Fergus, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=', Vishwanathan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=', Garnett, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' (Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Curran Associates, 4765-4774.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' [23] Menon, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=', Jiang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=', Vembu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=', Elkan, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=', Ohno-Machado, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Predicting accurate probabilities with ranking loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' ICML’12: Proceedings of the 29th International Conference on Machine Learning, 659-666.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' [24] Miles, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' (1959).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' The complete amalgamation into blocks, by weighted means, of a finite set of real numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Biometrika 46, 317-327.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' [25] Patton, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Volatility forecast comparison using imperfect volatility proxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Journal of Econometrics 160, 246-256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' [26] Ohlsson, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=', Johansson, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Non-Life Insurance Pricing with Generalized Linear Models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Springer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' [27] Savage, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' (1971).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Elicitable of personal probabilities and expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Journal of the American Statistical Association 66/336, 783-810.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' [28] Schervish, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' (1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' A general method of comparing probability assessors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' The Annals of Statis- tics 17/4, 1856-1879.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' [29] Tasche, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Calibrating sufficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Statistics: A Journal of Theoretical and Applied Statis- tics 55/6, 1356-1386.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' [30] W¨uthrich M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Model selection with Gini indices under auto-calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' European Actu- arial Journal, to appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' [31] W¨uthrich, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=', Merz, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' (2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Statistical Foundations of Actuarial Learning and its Applica- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Springer Actuarial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' [32] Zadrozny, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=', Elkan, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Transforming classifier scores into accurate multiclass probabil- ity estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Proceedings of the Eighth ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, 694-699.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' 18 A Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='1 Pool adjacent violators algorithm Minimization problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='2) is a quadratic optimization problem with linear side constraints, and it can be solved using the method of Karush–Kuhn–Tucker (KKT) [15, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' We therefore consider the Lagrangian L(µ, η) = (y − µ)⊤W(y − µ) − η⊤Aµ, with Lagrange multiplier η ∈ Rn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' The KKT conditions are given by 0 = ∇µL(µ, η) = − W(y − µ) − A⊤η, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='1) 0 ≥ ∇ηL(µ, η) = − Aµ, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='2) 0 ≤ η, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='3) 0 = (η1(µ1 − µ2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' , ηn−1(µn−1 − µn))⊤ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='4) The solution to these KKT conditions (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='1)-(A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='4) provides the isotonic estimate �µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' This solution can be found by the PAV algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' The main idea is to compare raw estimates (�µi)i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' If we have an adjacent pair with �µi > �µi+1, it violates the monotonicity constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Such pairs are recursively merged (pooled) to a block with an identical estimate, and iterating this pooling of adjacent pairs and blocks, respectively, that violate the monotonicity constraint, yields the PAV algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Pool Adjacent Violators (PAV) Algorithm (0) Initialize the algorithm �µ(0) = y and define the blocks I(0) k = {k} for k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' , K(0) = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' (1) Iterate for t ≥ 0: (a) If �µ(t) fulfills KKT condition (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='2) go to item (2), otherwise go to the next step (1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' (b) Select an index i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' , n with �µ(t) i > �µ(t) i+1, merge the two adjacent blocks with i ∈ I(t) k and i + 1 ∈ I(t) k+1, and leave all other blocks unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' This provides the new blocks I(t+1) k with k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' , K(t+1) = K(t) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' (c) Set on each block k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' , K(t+1) and for all indices i ∈ I(t+1) k the new estimates �µ(t+1) i = 1 � j∈I(t+1) k wj � j∈I(t+1) k wjyj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='5) (d) Increase t �→ t + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' (2) Set the isotonic regression estimate �µ = �µ(t) and merge adjacent blocks I(t) k and I(t) k+1 if we have the same estimates �µi on these blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Return the resulting partition of I denoted by (Ik)k=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=',K and �µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Remarks A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='1 (PAV algorithm interpretation) 19 (0) We initialize with the unconstraint optimal solution, and setting η(0) = 0 ensures that the KKT conditions (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='1), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='3) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='4) are fulfilled, thus, only the monotonicity (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='2) is not necessarily fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' (1a) We identify a pair �µ(t) i > �µ(t) i+1 that violates the monotonicity constraint (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' This pair needs to belong to two adjacent blocks I(t) k and I(t) k+1 because within blocks we have constant estimates (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' We merge these two adjacent blocks to I(t+1) k = I(t) k ∪ I(t) k+1, which reduces the number of blocks K(t) by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' (1b) We set on each block the constant estimate (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='5) which satisfies the monotonicity con- straint (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='2) within blocks, and also (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='4) is naturally fulfilled in this block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Conditions (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='1) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='3) are achieved by changing the Lagrange parameter η(t) �→ η(t+1) ≥ 0 correspondingly to account for the change in mean estimates (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='5) in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' (1c) On a sample of size n, this algorithm can be iterated at most n − 1 times, thus, the algorithm will terminate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' (2) Since we have for i ∈ I(t) k and i + 1 ∈ I(t) k+1 the inequality �µ(t) i ≤ �µ(t) i+1, the last step is to ensure that the resulting blocks are maximal by merging blocks where we do not have a strict inequality in the corresponding estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='2 Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='3 Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' For given responses y = Y σ(ω), the solution to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='2) gives the partition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='3) of the index set I with empirical weighted averages (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='4) on the blocks Ik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' These empirical weighted averages satisfy �µik < �µik+1 for all k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' , K(Y ) − 1, because the blocks Ik have been chosen maximal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' We now consider how these blocks are constructed in the PAV algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Suppose that we are in iteration t ≥ 0, and in this iteration of the PAV algorithm, we merge the two adjacent blocks I(t) k and I(t) k+1 because �µ(t) i > �µ(t) i+1 for i ∈ I(t) k and i+1 ∈ I(t) k+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' We analyze this inequality 1 � j∈I(t) k wj � j∈I(t) k wjyj = �µ(t) i > �µ(t) i+1 = 1 � l∈I(t) k+1 wl � l∈I(t) k+1 wlyl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' We use the location-scale structure (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='4) which gives us the equivalent condition 1 � j∈I(t) k wj � j∈I(t) k wj (µj + σϵj(ω)) > 1 � l∈I(t) k+1 wl � l∈I(t) k+1 wl (µl + σϵl(ω)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Since for any indices j ∈ I(t) k and l ∈ I(t) k+1 we have µj ≤ µl, it follows that the previous condition for merging the two adjacent blocks I(t) k and I(t) k+1 in iteration t of the PAV algorithm reads as σ � � � j∈I(t) k wjϵj(ω) � j∈I(t) k wj − � l∈I(t) k+1 wlϵl(ω) � l∈I(t) k+1 wl � � > � l∈I(t) k+1 wlµl � l∈I(t) k+1 wl − � j∈I(t) k wjµj � j∈I(t) k wj ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='6) The important observation is that if this condition is fulfilled for scale parameter σ > 0, then it will also be fulfilled for any bigger scale parameter σ′ > σ > 0 (pointwise in ω ∈ Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Thus, 20 any pooling that happens for σ also happens for σ′ > σ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Since this is pointwise on the underlying probability space (Ω, F, P), it shows that E[K(Y )] is decreasing in σ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Suppose now that the distribution of ϵ has full support on Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Then, the event Aσ = {K(Y σ) = n} occurs with positive probability, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=', 0 < P[Aσ] = P [K(Y σ) = n] = P [Y1 < Y2 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' < Yn] = P [µ1 + σϵ1 < µ2 + σϵ2 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' < µn + σϵn] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Consider Aσ = {µ1 + σϵ1 < µ2 + σϵ2 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' < µn + σϵn} = n� k=2 {µk−1 + σϵk−1 < µk + σϵk} = {σ (ϵ1 − ϵ2) < µ2 − µ1} ∩ n� k=3 {µk−1 + σϵk−1 < µk + σϵk} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' We focus on the first event on the right-hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Note that µ2 − µ1 > 0, hence � ϵ ∈ Rn : ϵ1 − ϵ2 < µ2 − µ1 σ � describes an open half space in Rn containing the origin and with bounding hyperplane that moves further away from the origin when decreasing σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Overall, the set ˜Aσ ⊂ Rn of values of ϵ in Aσ is a non-empty open polyhedron containing the origin that scales with σ, that is, ˜Aσ = (1/σ) ˜A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' Therefore, since the distribution of ϵ has full support, the probability P[Aσ] is strictly decreasing in σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} +page_content=' 2 21' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE0T4oBgHgl3EQf1AIP/content/2301.02692v1.pdf'} diff --git a/PtAzT4oBgHgl3EQfWvyr/content/tmp_files/2301.01307v1.pdf.txt b/PtAzT4oBgHgl3EQfWvyr/content/tmp_files/2301.01307v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..c59d90b3c56fbe0db10471ad27ff2b3f1aab5e4c --- /dev/null +++ b/PtAzT4oBgHgl3EQfWvyr/content/tmp_files/2301.01307v1.pdf.txt @@ -0,0 +1,1026 @@ +Draft version January 5, 2023 +Typeset using LATEX twocolumn style in AASTeX631 +The on-orbit performance of the Colorado Ultraviolet Transit Experiment (CUTE) Mission +Arika Egan +,1 Nicholas Nell +,1 Ambily Suresh,1 Kevin France +,1 Brian Fleming +,1 +Aickara Gopinathan Sreejith +,1, 2 Julian Lambert,1 and Nicholas DeCicco1 +1Laboratory for Atmospheric and Space Physics, University of Colorado Boulder, Boulder, CO 80303 +2Space Research Institute, Austrian Academy of Sciences, Schmiedlstrasse 6, 8042 Graz, Austria +(Received September 21, 2022; Revised November 4, 2022; Accepted November 6, 2022) +Submitted to AJ +ABSTRACT +We present the on-orbit performance of the Colorado Ultraviolet Transit Experiment (CUTE). +CUTE is a 6U CubeSat that launched on September 27th, 2021 and is obtaining near-ultraviolet +(NUV, 2480 ˚A – 3306 ˚A) transit spectroscopy of short-period exoplanets. The instrument comprises +a 20 cm × 8 cm rectangular Cassegrain telescope, an NUV spectrograph with a holographically ruled +aberration-correcting diffraction grating, and an NUV-optimized CCD detector. The telescope feeds +the spectrograph through an 18′ × 60′′ slit. The detector is a passively cooled, back-illuminated NUV- +enhanced CCD. The spacecraft bus is a Blue Canyon Technologies XB1, which has demonstrated ≤ 6′′ +jitter in 56% of CUTE science exposures. Following spacecraft commissioning, an on-orbit calibration +program was executed to characterize the CUTE instrument’s on-orbit performance. The results of +this calibration indicate that the effective area of CUTE is ≈ 19.0 – 27.5 cm2 and that the average +intrinsic resolution element is 2.9 ˚A across the bandpass. This paper describes the measurement of the +science instrument performance parameters as well as the thermal and pointing characteristics of the +observatory. +Keywords: +Exoplanet atmospheres (487) — Flux calibration (544) — Hot Jupiters (753) — Near +ultraviolet astronomy (1094) — Space telescopes (1547) — Ultraviolet telescopes (1743) +— Spectroscopy (1558) — Exoplanet atmospheric composition (2021) — Transmission +spectroscopy (2133) — Space observatories (1543) — Astronomical instrumentation (799) +1. INTRODUCTION +The +Colorado +Ultraviolet +Transit +Experiment +(CUTE) is a small satellite currently obtaining near- +ultraviolet transit spectroscopy of short-period exo- +planets around bright stars. CUTE’s spacecraft body, +measuring 11.2 cm × 23.7 cm × 36.2 cm, is a 6U Cube- +Sat, where a CubeSat is a class of small satellites with +external dimensions set by some multiple of a single +standardized unit (U) measuring 10 cm × 10 cm × 10 +cm. CUTE is NASA’s first ultraviolet astronomy Cube- +Sat and the first grant-funded small satellite dedicated +to the characterization of exoplanetary atmospheres. +The mission was developed at the Laboratory for At- +mospheric and Space Physics (LASP) at the University +of Colorado Boulder. This paper describes the in-flight +instrument performance while a companion paper in +this issue, France et al., details the CUTE science and +mission overview. +Two forthcoming papers will dis- +cuss the spacecraft and instrument commissioning (A. +Suresh et al. +2023, in preparation) and the data re- +duction pipeline Sreejith et al. (2022). +This paper is +organized as follows: Section 1.1 provides the science +motivation for the CUTE mission; Section 1.2 provides +a small overview of spacecraft testing and the first year +of operations; Section 2 describes the spacecraft and +payload; Section 3 details the instrument performance, +including the spectral bandpass and resolution, effective +area, background characteristics, and pointing stability; +and Section 4 describes how instrument performance +details drive the mission’s observing strategies. +1.1. Science Motivation +arXiv:2301.01307v1 [astro-ph.IM] 3 Jan 2023 + +ID2 +Near-ultraviolet (NUV) transit spectroscopy is a pow- +erful tool for characterizing the upper atmospheric layers +of highly-irradiated exoplanets. Planets with short or- +bital periods of just a few days are bathed in high-energy +photons and stellar winds from their host stars, swelling +their atmospheres to several planetary radii and poten- +tially past the planet’s gravitational boundary (Vidal- +Madjar et al. (2003); Lammer et al. (2003); Yelle (2004); +Munoz (2007); Murray-Clay et al. (2009); Ehrenreich, +D. & D´esert, J.-M. (2011)). Signatures of atmospheric +inflation and escape are evidenced in spectroscopic tran- +sit light curves created using several strong atomic and +ionic absorption features at NUV wavelengths, the depth +of which corresponds to the ion’s altitude and relative +abundance within the atmosphere. For example, Cos- +mic Origins Spectrograph (COS) NUV observations of +the hot Jupiter WASP-12b revealed a pseudo-continuum +of metal features throughout the bandpass, with deeper +transit depth in Mg II (2796/2803 ˚A) and at several Fe II +wavelengths (Fossati et al. (2010); Haswell et al. (2012); +Nichols et al. (2015)). WASP-121b displayed extended +absorption at Mg II, Fe I at 2484 ˚A and in several Fe II +lines, including 2381 ˚A and 2600 ˚A (Sing et al. (2019)). +HD 209458b has displayed Fe II at 2370 ˚A absorbing +beyond the planet’s Roche lobe (Cubillos et al. (2020)). +The shape of a planet’s NUV transmission spectrum +have the ability to provide constraints on the presence of +high-altitude clouds or hazes (Lothringer et al. (2022); +Wakeford et al. (2020); Cubillos et al. (2020)). Models +using these light curves seek to identify the main drivers +of atmospheric outflows and provide estimates for the +atmosphere’s mass-loss rates. +The CUTE mission was designed to observe these +NUV absorption features to explore exoplanetary com- +position and the drivers of atmospheric mass-loss for +several known exoplanets: it is observing 6 – 10 transits +of each target to constrain the atmospheric composition +properties, identify variability among transits, and pro- +vide mass-loss rates for approximately 10 targets over +the mission’s lifetime. +1.2. CUTE Mission Overview +The CUTE CubeSat, shown in Figure 1, was devel- +oped, assembled, and tested at LASP from Summer 2017 +to Summer 2021. Long lead time items were ordered in +the first year, component level calibration and testing +occurred in years two and three (including a 10-month +delay due to the COVID-19 pandemic), and the major- +ity of assembly and spacecraft testing took place in the +year before launch. +The spacecraft was delivered to the Vandenberg Space +Force Base on July 21st, 2021 and launched on Septem- +Figure 1. +The CUTE spacecraft with solar panels fully +deployed and communication cables attached. +Solar cells +are opposite the telescope boresight. The telescope is visible +in the left 2/3 of the spacecraft and the star tracker and +avionics are contained in the right 1/3. The startracker has +a red “remove before flight” panel covering its aperture. The +UHF antenna is deployed on the bottom left of the spacecraft +chassis and is noted with the yellow arrow. + +90.6 cm +36.2 cm +11.2 cm +UHF Antenna3 +ber 27th, 2021 into an orbit with an average altitude +of 560 km, a 97.6◦ inclination, a 10 am local ascending +node, and an approximately 95-minute orbital period. +Data is downlinked with an S-band radio to the LASP +ground station. +Spacecraft and payload commissioning took place +shortly after launch through February 2022. We used +two main stars to conduct initial payload characteriza- +tion: ζ Puppis (HD 66811; O4 star, V = 2.25 mag) and +Castor (α Gem, HD 60178; A1 star, V = 1.58 mag). +These two stars have International Ultraviolet Explorer +(IUE) data against which we can calculate CUTE’s ef- +fective area; they were chosen for their brightness and +expected high signal-to-noise ratio that were calculated +using CUTE pre-flight effective area estimates and CCD +background rates obtained from commissioning expo- +sures. CUTE’s on-orbit effective area a representative +flux calibrated spectrum are presented in Section 3.1. +CUTE science and dark exposures are typically 300s +while the readout time takes an additional 32s; we fur- +ther command two CCD erasures, or two full transfers +using vertical clocking only, before each science, dark, +or bias exposure. +There are opportunities for up to +five 300s exposures per CUTE orbit, though constraints +due to solar and lunar keep-out angles, telescope eleva- +tion lower limits, and avoidance windows around the +north/south poles and South Atlantic Anomaly occa- +sionally reduce that number (France et al. - this issue). +Considering the full range of keep-out angles, between +20% and 30% of an orbit is used to obtain science and +calibration frames. Additional mission operation details +will be presented in a forthcoming paper (A. Suresh et +al. 2023 - in prep). Full frame images with no process- +ing, called NOPROCs, have 2200 x 515 pixels and are +occasionally downlinked to assess the full CCD health. +However, due to constrained downlink capacity, the typ- +ical science data product is a 2200 x 100 pixel sub-image +that is centered on the spectral trace, called a TRIM2D. +CUTE science operations were adjusted post-launch +to accommodate damages to the thermoelectric cooler +(TEC) and the electronics board that controls both the +TEC and the shutter. +During thermal vacuum test- +ing, the TEC was damaged. +As a secondary payload +on a rideshare, it was not possible to replace the TEC +and re-test before the delivery date. The CCD now re- +lies on passive cooling (see Figure 10). The damaged +TEC likely deposited a small contamination layer on +the CCD and nearby optics, potentially degrading the +instrument’s effective area (Section 3.1). +The TEC and shutter share the same electronics +board. +While the TEC was damaged pre-launch, the +shutter operated nominally and showed no indication of +Table 1. CUTE Optical Summary +Instrument Metric +Value +Primary Dimensions +206 x 84 mm +Primary Radius +300 mm +Secondary Dimensions +68 × 26 mm +Secondary Radius +-129.6 mm +Telescope Focal Ratio +f/2.6 +Telescope PSF FWHMa +6′′ +Instrument Focal Ratio +f/5.5 in cross-dispersion +Slit Dimensions +18 ′× 120′′, 60′′, or 30′′ +Grating Dimensions +31 × 31 mm +Grating Radius +86.1 mm +Grating groove density +1713.4 gr mm−1 +Mirror Coating +Al + MgF2 +Grating Coating +Bare Al +CCD Pixel Size +13.5 µm +CCD Format +515 × 2048 active area +CCD Readout Time +32 s +aPoint spread function, full width at half maximum. +Measured pre-flight only +damage. However, during on-orbit payload commission- +ing, the 12V rail on TEC/shutter electronics board ex- +hibited spikes in its current within a few hours of being +powered; these current spikes would trigger the space- +craft’s fault protection and reset the spacecraft and pay- +load. The cause of the damage is unclear. +The TEC/shutter electronics board contains a capac- +itor that will close the shutter whenever the 12V rail +loses power, meaning that each spacecraft/payload re- +set closed the shutter. To prevent the electronics board +from continuing to interrupt spacecraft operations, we +used a 10 minute pass over the LASP ground station to +power the TEC/shutter board, open the shutter, and re- +move power from the board’s 12V line slowly to drain the +shutter capacitor to a low enough charge that it would +not be able to close the shutter. +The shutter is now +permanently open. Details about how an open shutter +affects CUTE science operations and data reduction are +outlined in Sections 3.3 and 4. +2. INSTRUMENT DESCRIPTION +The CUTE instrument, shown in Figure 2, is a rect- +angular Cassegrain telescope provided by Nu-Tek Pre- +cision Optical Corporation with an NUV spectrograph +and passively cooled CCD. The rectangular primary +mirror provides 3× the surface area of a standard cir- + +4 +cular mirror fitting into the same volume and was de- +signed to maximize the instrument’s light-collecting area +in an otherwise small payload volume. The four non- +diffractive mirrors are coated in Al + MgF2 while the +grating is coated in bare Al. The primary mirror serves +as the mounting structure for both the secondary mirror +and the spectrograph. +Light from the secondary mirror passes through a +ridge-baffled central spire and reflects off of a 45◦ fold +mirror before reaching a slit at the Cassegrain focus. +The slit, manufactured by OSH Stencils, is 18′ long in +the spatial dimension with three separate sky-projected +widths, 30′′, 60′′, and 120′′ that were chosen to ac- +commodate differently crowded target fields. We have +placed the star in the middle of the 60′′section for all ob- +servations. A spectral projection of the slit on the CCD +is shown in Figure 3. After passing through the slit, light +is diffracted off of the bare Al coated, holographically +ruled, aberration-correcting grating from Horiba-JY and +is additionally focused with a cylindrical fold mirror be- +fore the spectrum is recorded on a NUV-enhanced back- +illuminated e2v CCD42-10 with a 2048 × 515 active area +(CCD details are in Section 3.2 and Nell et al. (2021)). +The ridge baffling inside the central spire and additional +baffles inside of the spectrograph (Figure 2) mitigate +strong scattered light paths, but there are additional +signatures of scattered light we identified once in orbit +(see Section 3.3). Optical design details are provided in +Table 1. +The CCD is passively cooled; a copper heatsink and +thermal strap made of several silver-coated copper wires +connect the CCD to a radiator on the side of the space- +craft chassis (Egan et al. (2022)). The payload is housed +in a Blue Canyon Technologies (BCT) XB1: a 6U space- +craft with 4U housing the payload, 0.5U for instrument +electronics, and 1.5U for avionics including the attitude +determination and control system (ADCS), batteries, +and radios. Four 2U × 3U solar panels provide power +to the spacecraft bus. +3. INSTRUMENT PERFORMANCE +The CUTE instrument was assembled and tested in +LASP vacuum chamber facilities (France et al. (2016), +Egan et al. (2020)). Table 2 details performance param- +eters between laboratory and in-flight measurements. +Bandpass, spectral and spatial resolution, and effective +area were all measured with two calibration stars, Cas- +tor and ζ Puppis, and compared against IUE data early +in CUTE’s commissioning phase. These stars were cho- +sen for their brightness and visibility. Background rates, +limiting flux, thermal cycles, and pointing stability were +Figure 2. +CAD renderings of the CUTE telescope and +spectrograph. Top: front view of the Cassegrain telescope. +Middle: view of the spectrograph internals. Bottom: view +of the fully closed-out spectrograph, including the detector +and the thermal strap attached to the spacecraft radiator. +measured in-flight using dark, bias, and science frames +from CUTE’s first science target, WASP-189b. +As CUTE operates without a shutter, CCD pixels re- +main exposed to light during the 32s readout, and the +CCD region above the stellar spectrum contains both +background counts and residual spectral counts. An ex- +ample of this is shown in Figure 4. +The spectrum is +seen in the center of the image, and the residual readout +exposure is evident above the spectrum. The normal- +ized one-dimensional cross-section plotted on the right +of Figure 4 illustrates the residual readout exposure. In + +Spectrograph +Primary +enclosure +Copper +mirror +heatsink +Heatstrap +Central +Secondary +Primary +spire +mirror +mount flexure +Shutter +Fold focusing +Shutter state +mirror +sensor +Grating +Baffle +Folding +Slit +Slitjaw +flat +aperture +Thermal +stackup5 +Figure 3. A 2048 × 515 CUTE CCD image with the slit fully illuminated with a diffuse, uncollimated Hg light source, obtained +during thermal vacuum testing. The projected image of the slit is shown at several mercury lines, including 2536 ˚A, 2967 ˚A, +and a doublet at 3125 and 3131 ˚A. The exposure time for this image was set to sample the fainter spectral features of the light +source; as a result, many pixels in the 2536 ˚A line reached saturation levels and some side effects of this saturation can be seen +in the image. +Figure 4. Left: 2048 × 515 CCD image of Castor’s spectra with the the background subtracted. The spectrum is the bright +green line in the image’s center. Below the spectrum is residual background subtraction noise, and above the spectrum is residual +spectral light as the CCD is read out while the shutter remains open. CCD register clocking moves charges to the right and +vertical clocking moves the charges down. Right: One-dimensional cross-section of the frame to illustrate the readout streak. +The vertical dashed line marks the median value of the readout residual exposure at 2.2% of the normalized counts cross-section. +this example, the increase in counts above the spectrum +due to the residual readout is on the order of 2.2% of +the total CCD counts from an observation. For a typ- +ical CUTE observation with a 300s exposure time and +an average of 150 counts per pixel, the readout time in- +troduces an additional 0.031 counts to each pixel. This +is about 0.90% of the read noise as measured from the +blank CCD pixels, and thus the error from the exposed +readout is well below the error from other background +and noise sources. +3.1. Spectrograph +In this section, we characterize CUTE’s in-flight spec- +trograph and present the measured effective area, the +bandpass, and the spectral and spatial resolutions. We +used IUE observations of Castor to measure the spectral +and spatial resolution; CUTE’s two-dimensional spec- +trum of Castor and one-dimensional dispersion profiles +are shown in Figure 5. The tilt of the spectral trace, +the asymmetric out-of-focus two-dimensional spectrum, +the bandpass, and the dispersion have different values +in-flight than measured in the laboratory, indicating +shifts in the optical system which likely occurred during +launch. +The tilt of the spectral trace is largely a result of the +fold focusing mirror’s position that was set during the +focusing process, though the CCD placement also affects +the footprint of the spectrum on the detector. The fold +focusing mirror (Figure 2, middle panel) has three ad- +justment screws on three of the four corners that were +used to focus the spectrograph (see Egan et al. (2020) +for more details). +The spectrograph’s best focus was +found with a spectral trace tilt of 1.08◦; in-flight, the +trace’s tilt measures 0.85◦. +The spectrograph defocused during launch and now +has a double-lobe feature at the blue end that merges +into a single lobe at the red end. Despite the change in +profile across the detector, the total spectral extraction + +500 +400 +200 +100 +2595 +2705 +2811 +2913 +3012 +3109 +3206 +3303 +Wavelength, A500- +400 +300 +200 +100 +0 +2511 +2648 +2777 +2901 +3024 +3150 +3283 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Normalized Counts +Wavelength, A6 +Table 2. CUTE Laboratory and measured in-flight performance +Metric +Laboratory Value +In-flight Value +Bandpass +2480 – 3322 ˚ +A +2480 – 3306 ˚ +A +Spectral Tilt +1.05◦ +0.85◦ +Intrinsic Spectral Resolutiona +2.1 ˚ +A +2.9 ˚ +A +Cross-Dispersion Resolutiona +12.5′′ +30′′ +Aeff at 2500 ˚ +A +28.8 cm2 +27.5 cm2 +Background Limiting Fluxb +N/A +5 × 10−14 erg s−1 cm−2 ˚ +A−1 +a Average resolution over the bandpass. In-flight value has ADCS jitter effects removed. +b In 300s, measured on orbit only, evaluated at 3000˚ +A. +region to fully capture the extent of the cross-dispersion +spectrum remains at a constant value of about 25 pix- +els across the bandpass for observations with jitter less +than 6′′ RMS, compared to a 13-pixel extraction region +measured in the laboratory. +For each additional row +added to the extraction region, the total noise increases +by about 4%. +Figure 5. Top: A 2048 x 515 CUTE spectral image of Cas- +tor with background effects removed to show the spectrum. +Middle left: Zoom of a blue portion of the CUTE spectrum +to highlight the out-of-focus lobes. Middle Right: Zoom of +a red portion of the CUTE spectrum to highlight the less- +out-of-focus lobes. Bottom: One-dimensional cross-sections +of the middle panels. +We used IUE observations of ζ Puppis to measure +the in-flight bandpass; the vertical tick marks in Figure 7 +show the stellar features used to that end. The measured +in-flight bandpass is 2480 – 3306 ˚A, a change from the +laboratory-measured bandpass of 2479 – 3322 ˚A. The +spectrograph’s dispersion was also affected by launch; +in the laboratory, the bandpass-averaged dispersion was +measured between 0.47 ˚A/pixel to 0.39 ˚A/pixel, and in- +flight it is measured to be 0.45 ˚A/pixel to 0.35 ˚A/pixel. +Altogether, the change in trace tilt, the shifted band- +pass, the change in dispersion, and the spectrograph’s +defocus indicate shifts occurring in the optical chain af- +ter the Cassegrain focus. +The end-to-end effective area of the optical system was +not directly measured prior to launch, but rather cal- +culated using a combination of measured and provided +quantities: the telescope’s geometric collecting area, op- +tical reflectivities from coating witness samples provided +by Nu-Tek for the four non-diffractive reflecting optics, +the measured detector quantum efficiency (QE) from +Nell et al. (2021), and laboratory-measured efficiency +of the bare Al-coated diffraction grating (Egan et al. +(2020)). +The in-flight effective area was measured using a com- +bination of IUE flux-calibrated spectra of ζ Puppis for +λ < 3100 ˚A and a linear model for λ > 3100 ˚A. IUE +sensitivity falls off significantly after 3100 ˚A and there- +fore our initial calibration program could not directly +measure CUTE’s effective area beyond 3100 ˚A. We in- +stead made the assumption that the slope of the stellar +flux from λ ≈ 2500 – 3100 ˚A remained the same for +λ ≈ 3100 – 3306 ˚A and used that slope to approximate +the stellar flux red-ward of 3100 ˚A. We combined the +IUE flux and the model to calculate CUTE’s effective +area. The IUE ζ Puppis spectra were obtained from the +Mikulski Archive for Space Telescopes (MAST) at the +Space Telescope Science Institute and can be accessed +via 10.17909/a8wa-vc91. +To arrive at the CUTE effective area in units of cm2, +the CUTE two-dimensional ζ Puppis spectrum image +was background subtracted, the spectral trace was ex- +tracted and summed into a one-dimensional counts spec- +trum, converted into erg s−1 ˚A−1, and finally divided + +2511 +2648 +2777 +2901 +3024 +3150 +3283 +Wavelength,A +523.2 +Arcseconds +457.8 +Arcseconds +479.6 +414.2 +436.0 +370.6 +392.4 +327.0 +2508 +2532 +2555 +3303 +3326 +3350 +3374 +Wavelength, A +Wavelength,A +1.00 +1.00 +Normalized +0.75 +Normalized +0.75 +Counts +Counts +0.50 +0.50 +0.25 +0.25 +0.00 +0.00 +180 +200 +220 +240 +150 +170 +190 +210 +Pixels +Pixels7 +by the IUE spectra in units of erg s−1 cm−2 ˚A−1. The +resulting flux-calibrated CUTE spectrum of ζ Puppis +is shown in Figure 7. +The effective area ranges from +19.0 – 27.5 cm2 across the bandpass and a smoothed ef- +fective area curve is shown in Figure 6. The in-flight +measured effective area is lower than the laboratory- +calculated values by a median of 12% across the band- +pass. We attribute this change to two possible causes: a +contamination layer deposited onto the CCD and optics +during the TEC failure (Section 1.2, Egan et al. (2022)) +and/or atmospheric contamination of the optics during +the two month period when CUTE sat in the launch +dispenser. However, CUTE science measurements are +relative in nature and an individual light curve is made +from observations taken within ≈24 hours of each other; +changes in the instrument’s sensitivity between delivery +and science operations do not affect CUTE light curve +creation, other than the raised noise floor from the TEC +failure and launch-induced defocus. +The spectral and spatial resolution were measured us- +ing IUE observations of Castor. We define the spatial +resolution as the full width at half-maximum (FWHM) +of the spectrum in the cross-dispersion direction. This +varies by a few pixels across the bandpass (as shown in +Figure 5) with an average of 30′′ at 3000 ˚A for observa- +tions with jitter less than 6′′ RMS. +CUTE’s spectral resolution as measured from its +one-dimensional spectrum comes from the combination +of the spectrograph’s intrinsic resolution and spectral +smearing from spacecraft jitter. Assuming a Gaussian +profile for both the spectral line shape and the jitter +distribution, the FWHM of each add in quadrature to +arrive at the spectral resolution measured directly from +CUTE one-dimensional spectra. +We convolved IUE +spectra of Castor until it matched CUTE’s spectra. Us- +ing the observation’s jitter of 6′′ RMS, we arrive at a 2.9 +˚A intrinsic spectral resolution element. +3.2. CCD Characterization +The detector has a 2200 × 515 pixel format with the +following layout for each row: 51 true overscan pixels, +50 blank pixels, 2048 active pixels, 50 blank pixels, and +1 null element (a readout artifact used in testing to sim- +plify waveform memory implementation in the FPGA), +as can be seen in the full frame images in Figure 8. The +100 blank pixels are not able to sample dark current. We +use the horizontal blank and overscan pixels to measure +the read noise and bias levels and implement a 5-pixel +buffer to avoid charge transfer effects at the active/blank +pixel boundary. For example, a full 2200 × 515 CCD im- +age will have on one side a 50 × 515 blank region, and +a 40 × 505 sub-region is used to calculate bias and read +Figure 6. Effective area of the CUTE instrument as pro- +posed (blue), estimated with laboratory measurements (or- +ange), and calculated in-flight and smoothed (black). The +in-flight effective area beyond λ ≈ 3100 ˚A is estimated based +on a model fit to the IUE ζ Puppis one-dimensional spec- +trum. +Figure 7. Flux-calibrated spectrum of ζ Pup. A CUTE +spectrum is in black, plotted over the blue IUE spectrum. +Black vertical tick marks note the spectral features used to +calculate CUTE’s wavelength solution. IUE sensitivity falls +off beyond 3100 ˚A, so a model was fit to the IUE data in +order to calculate CUTE’s effective area and flux spectrum. +noise levels. The same 5-pixel buffer enacted on the 102 +× 515 blank + overscan side produces a 92 × 505 region. +The height of these regions is reduced to 90 pixels for +a TRIM2D frame. The read noise is considered to be +the RMS of these sub-regions and the bias level is cal- +culated from the median. The CCD has a single output +channel; register clocking moves charges from the blue +end to the red end of the detector, and vertical clocking + +30 +28 +Estimatedpre-flight +26 +冏忆 +Proposed +20 +Observedin-flight +18 +2600 +2800 +3000 +3200 +Wavelength, A1e-9 +IUE Pup +CUTE +A +4 +2 +2600 +2800 +3000 +3200 +Wavelength,A8 +moves charges from the top to the bottom of the images +shown in Figures 3, 4, and 5. +Much of the CCD characterization took place in the +laboratory and is detailed in Nell et al. (2021). Quan- +tities including gain, photon-response non-uniformity +(PRNU), and non-linearity were not able to be mea- +sured in flight due to the time and budget constraints +of a suborbital-class mission. In-flight flat fields cannot +be obtained as there is no onboard calibration lamp and +true flats cannot be obtained with a dispersing element. +CUTE science and calibration targets fill the pixel wells +to about 5% of full capacity, levels where non-linearity is +not a concern. PRNU, gain, and any dust effects are ac- +counted for by (1) the flux calibration and (2) the nature +of light curves being fundamentally a relative measure- +ment. +3.3. Background +A single dark frame has contributions from dark noise, +read noise, detector bias, and the sky due to the unshut- +tered spectrograph. The background exhibits additional +variation dependent on (1) the CCD’s temperature- +dependent dark rate and (2) a scattered light feature +with a brightness that is correlated with the telescope’s +elevation angle; an extreme example of this scattered +light feature is shown in Figure 8. Both dark and bias +frames have some level of contribution to all of the above +phenomena. +Figure 10 shows the detector’s periodic temperature +cycle over several orbits, cycling between ≈ −11◦C and +−6◦C. These temperature swings are evident in the av- +erage background frame count rates, shown in Figure +9. The background count rate increases as temperature +increases, which is the expected dark rate behavior of +the detector. However, the background count rate addi- +tionally varies at similar temperatures due to scattered +light entering the spectrograph; the angles between the +telescope and the Earth’s limb, the Sun, and the Moon +all have an effect on the level of scatted light present +in background frames. An example is shown in Figure +9 where the Sun’s elevation angle is correlated with in- +creased background counts. +In CCD regions that are +more prone to scattered light (e.g. the left side of the +CCD as shown in the top panel of Figure 8), background +count rates differ on the order of about 0.15 photons s−1 +pixel−1 when telescope elevation is maintained above +10◦. +Background subtraction for a single science frame +must consider the orbital and thermal environment +of the exposure. +The temperature- and pointing- +dependent nature of the background (e.g. +Figure 9) +means that each science frame must have tailored cal- +Figure 8. Two 2200 × 515 CUTE frames of different tele- +scope elevation angles, 3◦ and 12◦, to demonstrate the scat- +tered light feature present at low telescope elevation angles. +The elevation of each exposure is marked in the left-side +overscan region. +The top and middle frames compare the +3◦ and 12◦ elevation angles with the same colorbar. +The +bottom frame shows the 12◦ frame with a colorbar scaled +to the frame. Typical frame features are evident on the 12◦ +frame: an increase in counts from the bottom to the top of +the frame and hot pixel streaks due to the 32s CCD readout +time. The horizontal dark bars in the top image are missing +data packets that were unsuccessfully downlinked. +Figure 9. CUTE CCD dark frame background rates for the +detector, plotted against detector temperature and colored +by the elevation of the Sun with respect to the telescope bore- +sight. In general, higher CCD temperatures produce higher +background count rates. Background rates are additionally +influenced by the position of astronomical bodies like the Sun +as exemplified here, as well as the Moon and Earth. At the +time of writing, we are still exploring sources of contribution +to background rates. + +1.2 +-20S +1.0 +Rate, +-300 + Elevation Angle, +a +0.8 +40 +0.6 +50 +ledian +un +0.4 +M +-10 +-8 +-6 +-4 +-2 +Temperature,C400 +30000 +ixels +3° +20000 +P +200 +100008 +0 +0 +400 +S +xel +12° +20000 +P +200 +0 +0 +800 +400 +600号 +S +xel +12 +Counts, +P +200 +400 +200 +0 +2479 +2592 +2702 +2808 +2910 +3010 +3106 +3199 +Wavelength,A9 +Figure 10. A typical CCD temperature profile over several +CUTE orbits. +ibration frames. +Efforts are underway to model each +pixel’s value as a function of both its intrinsic behavior +and its environment in order to create individual back- +ground frames for a given science exposure, based on +the frame’s exposure conditions; these efforts will be de- +scribed in detail in Egan et al. 2023 −in prep and we +currently use a combination of median-combined dark +and bias frames with similar telescope pointings and +CCD temperatures to approximate the background in +each science frame. +Finally, we present CUTE’s background flux limit. +After a science frame is background-subtracted, we de- +fine a background region below the spectrum that has +the same size as the frame’s spectral extraction region +and calculate the background flux limit, or minimum +flux below which a source cannot be detected above +the background. The counts in the background region +are converted to flux and then averaged for 67 × 300s +WASP-189 science frames. The background flux limit is +≈ 5 × 10−14 erg s−1 cm−2 ˚A−1. Background subtrac- +tion and handling of hot pixels are discussed in more +detail in Sreejith et al. (2022). +3.4. Pointing Stability +The quality of CUTE spectra is strongly influenced by +the spacecraft pointing jitter. Stability about the com- +manded coordinates can vary between 3′′ and more than +20′′ and thereby smear the spectrum across a greater +CCD area, increase the spectral extraction region, and +reduce the signal-to-noise of the one-dimensional spec- +trum. Significant jitter can cause vignetting of the tele- +scope point-spread-function by the slit mask and ren- +der those data unusable for transit spectroscopy. +We +currently use a 6′′ jitter cutoff for science frames to un- +dergo data reduction and observations with higher jitter +Figure 11. Jitter histogram for 254 × 300s science expo- +sures from the WASP-189b campaign. The pointing sample +rate is 5s for each 300s observation. Histogram bin widths +are 0.5′′ for jitter < 10′′ and 1′′ for jitter > 10′′. +A ver- +tical dashed line indicates the cutoff used to eliminate low +signal-to-noise observations from light curve analysis. +are not used Sreejith et al. (2022). Figure 11 shows a +histogram of ADCS jitter in all of the WASP-189 science +frames which include complete jitter telemetry; jitter is +less than 6′′ for about 56% of CUTE observations. In +our preliminary data reduction and light curve creation, +we remove frames with higher jitter to eliminate low +signal-to-noise observations; this cutoff will be honed as +we continue to refine our data reduction pipeline (Sree- +jith et al. (2022)). +4. MISSION OPERATIONS +CUTE’s standard mission operation sequence in- +volves observing a given science target for 6−10 transits. +Each visit’s duration is centered on the planet’s mid- +transit time and lasts for a span of time equal to five +times the transit duration in order to establish an out- +of-transit stellar baseline pre- and post-transit. CUTE +target transits last between 3 and 6 hours, or about 2 +to 4 CUTE orbits, while the duration of each visit is +typically less than 24 hours. +As discussed in Section 3, CUTE data quality has +pointing and temperature dependencies: (1) CCD ex- +posures are captured without a shutter, meaning that +calibration frames are not truly “dark” and include sky +background; (2) CCD dark rates scale with the ther- +mal changes present due to the passive cooling method; +(3) there is a pointing-dependent scattered light feature +present in all frames. We create observation plans that +take these into account which are detailed below. +The roll angle for a given science target observing cam- +paign is set with two considerations. First, we prioritize + +Temperature (°C) +8 +9 +-10 +0 +Time(DOY:HHUTC)Observations +10- +Number of +100 +4 +6810 +20 +40 +100 +200 +PointingRMS,arcseconds10 +keeping the CCD as cool as possible by orienting the +radiator panel into space and away from the Earth and +Sun. Once that angle is set, we additionally check the +target field to make sure no bright stars other than the +target are within the slit; this has been the case for +all science observations obtained since launch. The roll +angle is maintained for all calibration frames taken ad- +jacent to a given set of science exposures. To capture +scattered light features from the Sun, Moon, or Earth +limb that may appear in science frames (Figure 9), dark +and bias calibration frames are planned to occur at ap- +proximately the same orbital position as science frames +with a pointing offset from the target star of 0.75◦; this +has an additional benefit of obtaining a calibration frame +in a similar thermal state. +Fully characterizing the background is a continued +effort. +Background trends related to scattered light +and telescope pointing did not become apparent until +a sufficient number of science and dark frames from the +WASP-189 observing campaign were analyzed. Model- +ing efforts are underway to model each pixel’s behavior +in its environment to better produce calibration frames +for background subtraction (Egan et al. 2023 – in prep.). +5. CONCLUSION +CUTE is currently obtaining NUV transit spec- +troscopy of short-period exoplanets around bright stars. +We have detailed the NUV payload’s spectroscopic per- +formance as calculated within the first six months of +calibration and science observations, including the spec- +tral and spatial resolution, the effective area, and the +limiting flux level. We have demonstrated stable point- +ing for about 56% of our observations. Science opera- +tions will continue through June 2023, during which we +will conduct additional calibration campaigns to mea- +sure time-dependent sensitivity, as well as continue to +hone our understanding of the background. CUTE is +expected to reenter the atmosphere within 3 years from +launch. +Acknowledgments: CUTE was developed and op- +erated with the support to two NASA/APRA awards +to the Laboratory for Atmospheric and Space Physics +at the University of Colorado Boulder, NNX17AI84G +and 80NSSC21K1667s. +A. G. S. was supported +by a Schr¨odinger Fellowship through the Austrian +Science Fund (FWF) [J 4596-N] and additionally +acknowledges +financial +support +from +the +Austrian +Forschungsf¨orderungsgesellschaft FFG project 859718 +and 865968. The CUTE team acknowledges the numer- +ous invaluable discussions with colleagues excited about +ultraviolet transit science and the potential to do science +with small satellites. The CUTE team wishes to specif- +ically recognize the amateur radio operator community +for hosting numerous telemetry tracking tools that have +improved the mission’s ability to recover from faults and +understand long-term spacecraft trends much more effi- +ciently than would have been otherwise possible. +REFERENCES +Cubillos, P. E., Fossati, L., Koskinen, T., et al. 2020, AJ, +159, 111, doi: 10.3847/1538-3881/ab6a0b +Egan, A., France, K., Nell, N., et al. 2022, in Small Satellite +Conference, Vol. SS22, Small Satellite, WKII–02. https: +//digitalcommons.usu.edu/smallsat/2022/all2022/52/ +Egan, A., France, K., Fleming, B. T., et al. 2020, in Space +Telescopes and Instrumentation 2020: Ultraviolet to +Gamma Ray, Vol. 11444, International Society for Optics +and Photonics, 1144406, doi: 10.1117/12.2559983 +Ehrenreich, D., & D´esert, J.-M. 2011, A&A, 529, A136, +doi: 10.1051/0004-6361/201016356 +Fossati, L., Haswell, C. A., Froning, C. S., et al. 2010, The +Astrophysical Journal Letters, 714, L222, +doi: 10.1088/2041-8205/714/2/L222 +France, K., Hoadley, K., Fleming, B. T., et al. 2016, +Journal of Astronomical Instrumentation, 05, 1640001, +doi: 10.1142/S2251171716400018 +Haswell, C. A., Fossati, L., Ayres, T., et al. 2012, The +Astrophysical Journal, 760, 79, +doi: 10.1088/0004-637X/760/1/79 +Lammer, H., Selsis, F., Ribas, I., et al. 2003, The +Astrophysical Journal Letters, 598, L121, +doi: 10.1086/380815 +Lothringer, J. D., Sing, D., Rustamkulov, Z., et al. 2022, +Nature, 604 +Munoz, A. G. 2007, Planetary and Space Science, 55, 1426 , +doi: https://doi.org/10.1016/j.pss.2007.03.007 +Murray-Clay, R. A., Chiang, E. I., & Murray, N. 2009, The +Astrophysical Journal, 693, 23, +doi: 10.1088/0004-637X/693/1/23 +Nell, N., DeCicco, N., Ulrich, S., France, K., & Fleming, B. +2021, in UV, X-Ray, and Gamma-Ray Space +Instrumentation for Astronomy XXII, ed. O. H. +Siegmund, Vol. 11821, International Society for Optics +and Photonics (SPIE), 391 – 400, +doi: 10.1117/12.2594743 + +11 +Nichols, J. D., Wynn, G. A., Goad, M., et al. 2015, The +Astrophysical Journal, 803, 9, +doi: 10.1088/0004-637X/803/1/9 +Sing, D. K., Lavvas, P., Ballester, G. E., et al. 2019, AJ, +158, 91, doi: 10.3847/1538-3881/ab2986 +Sreejith, A. G., Fossati, L., Ambily, S., et al. 2022, +Publications of the Astronomical Society of the Pacific, +134, 114506, doi: 10.1088/1538-3873/aca17d +Vidal-Madjar, A., des Etangs, A. L., D´esert, J.-M., et al. +2003, Nature, 422, 143 EP . +http://dx.doi.org/10.1038/nature01448 +Wakeford, H. R., Sing, D. K., Stevenson, K. B., et al. 2020, +AJ, 159, 204, doi: 10.3847/1538-3881/ab7b78 +Yelle, R. V. 2004, Icarus, 170, 167 , +doi: https://doi.org/10.1016/j.icarus.2004.02.008 + diff --git a/PtAzT4oBgHgl3EQfWvyr/content/tmp_files/load_file.txt b/PtAzT4oBgHgl3EQfWvyr/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..258a387f107d405431cb6793da35937f5e31ab45 --- /dev/null +++ b/PtAzT4oBgHgl3EQfWvyr/content/tmp_files/load_file.txt @@ -0,0 +1,553 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf,len=552 +page_content='Draft version January 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' 2023 Typeset using LATEX twocolumn style in AASTeX631 The on-orbit performance of the Colorado Ultraviolet Transit Experiment (CUTE) Mission Arika Egan ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='1 Nicholas Nell ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='1 Ambily Suresh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='1 Kevin France ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='1 Brian Fleming ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='1 Aickara Gopinathan Sreejith ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' 2 Julian Lambert,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='1 and Nicholas DeCicco1 1Laboratory for Atmospheric and Space Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' University of Colorado Boulder,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' Boulder,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' CO 80303 2Space Research Institute,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' Austrian Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' Schmiedlstrasse 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' 8042 Graz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' Austria (Received September 21,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' Revised November 4, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' Accepted November 6, 2022) Submitted to AJ ABSTRACT We present the on-orbit performance of the Colorado Ultraviolet Transit Experiment (CUTE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' CUTE is a 6U CubeSat that launched on September 27th, 2021 and is obtaining near-ultraviolet (NUV, 2480 ˚A – 3306 ˚A) transit spectroscopy of short-period exoplanets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' The instrument comprises a 20 cm × 8 cm rectangular Cassegrain telescope, an NUV spectrograph with a holographically ruled aberration-correcting diffraction grating, and an NUV-optimized CCD detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' The telescope feeds the spectrograph through an 18′ × 60′′ slit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' The detector is a passively cooled, back-illuminated NUV- enhanced CCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' The spacecraft bus is a Blue Canyon Technologies XB1, which has demonstrated ≤ 6′′ jitter in 56% of CUTE science exposures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' Following spacecraft commissioning, an on-orbit calibration program was executed to characterize the CUTE instrument’s on-orbit performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' The results of this calibration indicate that the effective area of CUTE is ≈ 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='0 – 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='5 cm2 and that the average intrinsic resolution element is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='9 ˚A across the bandpass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' This paper describes the measurement of the science instrument performance parameters as well as the thermal and pointing characteristics of the observatory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' Keywords: Exoplanet atmospheres (487) — Flux calibration (544) — Hot Jupiters (753) — Near ultraviolet astronomy (1094) — Space telescopes (1547) — Ultraviolet telescopes (1743) — Spectroscopy (1558) — Exoplanet atmospheric composition (2021) — Transmission spectroscopy (2133) — Space observatories (1543) — Astronomical instrumentation (799) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' INTRODUCTION The Colorado Ultraviolet Transit Experiment (CUTE) is a small satellite currently obtaining near- ultraviolet transit spectroscopy of short-period exo- planets around bright stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' CUTE’s spacecraft body, measuring 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='2 cm × 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='7 cm × 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='2 cm, is a 6U Cube- Sat, where a CubeSat is a class of small satellites with external dimensions set by some multiple of a single standardized unit (U) measuring 10 cm × 10 cm × 10 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' CUTE is NASA’s first ultraviolet astronomy Cube- Sat and the first grant-funded small satellite dedicated to the characterization of exoplanetary atmospheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' The mission was developed at the Laboratory for At- mospheric and Space Physics (LASP) at the University of Colorado Boulder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' This paper describes the in-flight instrument performance while a companion paper in this issue, France et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=', details the CUTE science and mission overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' Two forthcoming papers will dis- cuss the spacecraft and instrument commissioning (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' Suresh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' 2023, in preparation) and the data re- duction pipeline Sreejith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' This paper is organized as follows: Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='1 provides the science motivation for the CUTE mission;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='2 provides a small overview of spacecraft testing and the first year of operations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' Section 2 describes the spacecraft and payload;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' Section 3 details the instrument performance, including the spectral bandpass and resolution, effective area, background characteristics, and pointing stability;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' and Section 4 describes how instrument performance details drive the mission’s observing strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' Science Motivation arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='01307v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='IM] 3 Jan 2023 ID2 Near-ultraviolet (NUV) transit spectroscopy is a pow- erful tool for characterizing the upper atmospheric layers of highly-irradiated exoplanets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' Planets with short or- bital periods of just a few days are bathed in high-energy photons and stellar winds from their host stars, swelling their atmospheres to several planetary radii and poten- tially past the planet’s gravitational boundary (Vidal- Madjar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' (2003);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' Lammer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' (2003);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' Yelle (2004);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' Munoz (2007);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' Murray-Clay et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' (2009);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' Ehrenreich, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' & D´esert, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' (2011)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' Signatures of atmospheric inflation and escape are evidenced in spectroscopic tran- sit light curves created using several strong atomic and ionic absorption features at NUV wavelengths, the depth of which corresponds to the ion’s altitude and relative abundance within the atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' For example, Cos- mic Origins Spectrograph (COS) NUV observations of the hot Jupiter WASP-12b revealed a pseudo-continuum of metal features throughout the bandpass, with deeper transit depth in Mg II (2796/2803 ˚A) and at several Fe II wavelengths (Fossati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' (2010);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' Haswell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' (2012);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' Nichols et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' (2015)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' WASP-121b displayed extended absorption at Mg II, Fe I at 2484 ˚A and in several Fe II lines, including 2381 ˚A and 2600 ˚A (Sing et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' (2019)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' HD 209458b has displayed Fe II at 2370 ˚A absorbing beyond the planet’s Roche lobe (Cubillos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' (2020)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' The shape of a planet’s NUV transmission spectrum have the ability to provide constraints on the presence of high-altitude clouds or hazes (Lothringer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' Wakeford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' Cubillos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' (2020)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' Models using these light curves seek to identify the main drivers of atmospheric outflows and provide estimates for the atmosphere’s mass-loss rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' The CUTE mission was designed to observe these NUV absorption features to explore exoplanetary com- position and the drivers of atmospheric mass-loss for several known exoplanets: it is observing 6 – 10 transits of each target to constrain the atmospheric composition properties, identify variability among transits, and pro- vide mass-loss rates for approximately 10 targets over the mission’s lifetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' CUTE Mission Overview The CUTE CubeSat, shown in Figure 1, was devel- oped, assembled, and tested at LASP from Summer 2017 to Summer 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' Long lead time items were ordered in the first year, component level calibration and testing occurred in years two and three (including a 10-month delay due to the COVID-19 pandemic), and the major- ity of assembly and spacecraft testing took place in the year before launch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' The spacecraft was delivered to the Vandenberg Space Force Base on July 21st, 2021 and launched on Septem- Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' The CUTE spacecraft with solar panels fully deployed and communication cables attached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' Solar cells are opposite the telescope boresight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' The telescope is visible in the left 2/3 of the spacecraft and the star tracker and avionics are contained in the right 1/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' The startracker has a red “remove before flight” panel covering its aperture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' The UHF antenna is deployed on the bottom left of the spacecraft chassis and is noted with the yellow arrow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='6 cm 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='2 cm 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='2 cm UHF Antenna3 ber 27th, 2021 into an orbit with an average altitude of 560 km, a 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='6◦ inclination, a 10 am local ascending node, and an approximately 95-minute orbital period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' Data is downlinked with an S-band radio to the LASP ground station.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' Spacecraft and payload commissioning took place shortly after launch through February 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' We used two main stars to conduct initial payload characteriza- tion: ζ Puppis (HD 66811;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' O4 star, V = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='25 mag) and Castor (α Gem, HD 60178;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' A1 star, V = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='58 mag).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' These two stars have International Ultraviolet Explorer (IUE) data against which we can calculate CUTE’s ef- fective area;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' they were chosen for their brightness and expected high signal-to-noise ratio that were calculated using CUTE pre-flight effective area estimates and CCD background rates obtained from commissioning expo- sures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' CUTE’s on-orbit effective area a representative flux calibrated spectrum are presented in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' CUTE science and dark exposures are typically 300s while the readout time takes an additional 32s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' we fur- ther command two CCD erasures, or two full transfers using vertical clocking only, before each science, dark, or bias exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' There are opportunities for up to five 300s exposures per CUTE orbit, though constraints due to solar and lunar keep-out angles, telescope eleva- tion lower limits, and avoidance windows around the north/south poles and South Atlantic Anomaly occa- sionally reduce that number (France et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' - this issue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' Considering the full range of keep-out angles, between 20% and 30% of an orbit is used to obtain science and calibration frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' Additional mission operation details will be presented in a forthcoming paper (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' Suresh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' 2023 - in prep).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' Full frame images with no process- ing, called NOPROCs, have 2200 x 515 pixels and are occasionally downlinked to assess the full CCD health.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' However, due to constrained downlink capacity, the typ- ical science data product is a 2200 x 100 pixel sub-image that is centered on the spectral trace, called a TRIM2D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' CUTE science operations were adjusted post-launch to accommodate damages to the thermoelectric cooler (TEC) and the electronics board that controls both the TEC and the shutter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' During thermal vacuum test- ing, the TEC was damaged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' As a secondary payload on a rideshare, it was not possible to replace the TEC and re-test before the delivery date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' The CCD now re- lies on passive cooling (see Figure 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' The damaged TEC likely deposited a small contamination layer on the CCD and nearby optics, potentially degrading the instrument’s effective area (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' The TEC and shutter share the same electronics board.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' While the TEC was damaged pre-launch, the shutter operated nominally and showed no indication of Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' CUTE Optical Summary Instrument Metric Value Primary Dimensions 206 x 84 mm Primary Radius 300 mm Secondary Dimensions 68 × 26 mm Secondary Radius 129.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='6 mm Telescope Focal Ratio f/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='6 Telescope PSF FWHMa 6′′ Instrument Focal Ratio f/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='5 in cross-dispersion Slit Dimensions 18 ′× 120′′, 60′′, or 30′′ Grating Dimensions 31 × 31 mm Grating Radius 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='1 mm Grating groove density 1713.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='4 gr mm−1 Mirror Coating Al + MgF2 Grating Coating Bare Al CCD Pixel Size 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='5 µm CCD Format 515 × 2048 active area CCD Readout Time 32 s aPoint spread function, full width at half maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' Measured pre-flight only damage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' However, during on-orbit payload commission- ing, the 12V rail on TEC/shutter electronics board ex- hibited spikes in its current within a few hours of being powered;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' these current spikes would trigger the space- craft’s fault protection and reset the spacecraft and pay- load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' The cause of the damage is unclear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' The TEC/shutter electronics board contains a capac- itor that will close the shutter whenever the 12V rail loses power, meaning that each spacecraft/payload re- set closed the shutter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' To prevent the electronics board from continuing to interrupt spacecraft operations, we used a 10 minute pass over the LASP ground station to power the TEC/shutter board, open the shutter, and re- move power from the board’s 12V line slowly to drain the shutter capacitor to a low enough charge that it would not be able to close the shutter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' The shutter is now permanently open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' Details about how an open shutter affects CUTE science operations and data reduction are outlined in Sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' INSTRUMENT DESCRIPTION The CUTE instrument, shown in Figure 2, is a rect- angular Cassegrain telescope provided by Nu-Tek Pre- cision Optical Corporation with an NUV spectrograph and passively cooled CCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' The rectangular primary mirror provides 3× the surface area of a standard cir- 4 cular mirror fitting into the same volume and was de- signed to maximize the instrument’s light-collecting area in an otherwise small payload volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' The four non- diffractive mirrors are coated in Al + MgF2 while the grating is coated in bare Al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' The primary mirror serves as the mounting structure for both the secondary mirror and the spectrograph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' Light from the secondary mirror passes through a ridge-baffled central spire and reflects off of a 45◦ fold mirror before reaching a slit at the Cassegrain focus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' The slit, manufactured by OSH Stencils, is 18′ long in the spatial dimension with three separate sky-projected widths, 30′′, 60′′, and 120′′ that were chosen to ac- commodate differently crowded target fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' We have placed the star in the middle of the 60′′section for all ob- servations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' A spectral projection of the slit on the CCD is shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' After passing through the slit, light is diffracted off of the bare Al coated, holographically ruled, aberration-correcting grating from Horiba-JY and is additionally focused with a cylindrical fold mirror be- fore the spectrum is recorded on a NUV-enhanced back- illuminated e2v CCD42-10 with a 2048 × 515 active area (CCD details are in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='2 and Nell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' (2021)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' The ridge baffling inside the central spire and additional baffles inside of the spectrograph (Figure 2) mitigate strong scattered light paths, but there are additional signatures of scattered light we identified once in orbit (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' Optical design details are provided in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' The CCD is passively cooled;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' a copper heatsink and thermal strap made of several silver-coated copper wires connect the CCD to a radiator on the side of the space- craft chassis (Egan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' (2022)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' The payload is housed in a Blue Canyon Technologies (BCT) XB1: a 6U space- craft with 4U housing the payload, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='5U for instrument electronics, and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='5U for avionics including the attitude determination and control system (ADCS), batteries, and radios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' Four 2U × 3U solar panels provide power to the spacecraft bus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' INSTRUMENT PERFORMANCE The CUTE instrument was assembled and tested in LASP vacuum chamber facilities (France et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' (2016), Egan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' (2020)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' Table 2 details performance param- eters between laboratory and in-flight measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' Bandpass, spectral and spatial resolution, and effective area were all measured with two calibration stars, Cas- tor and ζ Puppis, and compared against IUE data early in CUTE’s commissioning phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' These stars were cho- sen for their brightness and visibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' Background rates, limiting flux, thermal cycles, and pointing stability were Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' CAD renderings of the CUTE telescope and spectrograph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' Top: front view of the Cassegrain telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' Middle: view of the spectrograph internals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' Bottom: view of the fully closed-out spectrograph, including the detector and the thermal strap attached to the spacecraft radiator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' measured in-flight using dark, bias, and science frames from CUTE’s first science target, WASP-189b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' As CUTE operates without a shutter, CCD pixels re- main exposed to light during the 32s readout, and the CCD region above the stellar spectrum contains both background counts and residual spectral counts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' An ex- ample of this is shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' The spectrum is seen in the center of the image, and the residual readout exposure is evident above the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' The normal- ized one-dimensional cross-section plotted on the right of Figure 4 illustrates the residual readout exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' In Spectrograph Primary enclosure Copper mirror heatsink Heatstrap Central Secondary Primary spire mirror mount flexure Shutter Fold focusing Shutter state mirror sensor Grating Baffle Folding Slit Slitjaw flat aperture Thermal stackup5 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' A 2048 × 515 CUTE CCD image with the slit fully illuminated with a diffuse, uncollimated Hg light source, obtained during thermal vacuum testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' The projected image of the slit is shown at several mercury lines, including 2536 ˚A, 2967 ˚A, and a doublet at 3125 and 3131 ˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' The exposure time for this image was set to sample the fainter spectral features of the light source;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' as a result, many pixels in the 2536 ˚A line reached saturation levels and some side effects of this saturation can be seen in the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' Left: 2048 × 515 CCD image of Castor’s spectra with the the background subtracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' The spectrum is the bright green line in the image’s center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' Below the spectrum is residual background subtraction noise, and above the spectrum is residual spectral light as the CCD is read out while the shutter remains open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' CCD register clocking moves charges to the right and vertical clocking moves the charges down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' Right: One-dimensional cross-section of the frame to illustrate the readout streak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' The vertical dashed line marks the median value of the readout residual exposure at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='2% of the normalized counts cross-section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' this example, the increase in counts above the spectrum due to the residual readout is on the order of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='2% of the total CCD counts from an observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' For a typ- ical CUTE observation with a 300s exposure time and an average of 150 counts per pixel, the readout time in- troduces an additional 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='031 counts to each pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' This is about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='90% of the read noise as measured from the blank CCD pixels, and thus the error from the exposed readout is well below the error from other background and noise sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' Spectrograph In this section, we characterize CUTE’s in-flight spec- trograph and present the measured effective area, the bandpass, and the spectral and spatial resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' We used IUE observations of Castor to measure the spectral and spatial resolution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' CUTE’s two-dimensional spec- trum of Castor and one-dimensional dispersion profiles are shown in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' The tilt of the spectral trace, the asymmetric out-of-focus two-dimensional spectrum, the bandpass, and the dispersion have different values in-flight than measured in the laboratory, indicating shifts in the optical system which likely occurred during launch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' The tilt of the spectral trace is largely a result of the fold focusing mirror’s position that was set during the focusing process, though the CCD placement also affects the footprint of the spectrum on the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' The fold focusing mirror (Figure 2, middle panel) has three ad- justment screws on three of the four corners that were used to focus the spectrograph (see Egan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' (2020) for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' The spectrograph’s best focus was found with a spectral trace tilt of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='08◦;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' in-flight, the trace’s tilt measures 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='85◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' The spectrograph defocused during launch and now has a double-lobe feature at the blue end that merges into a single lobe at the red end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' Despite the change in profile across the detector, the total spectral extraction 500 400 200 100 2595 2705 2811 2913 3012 3109 3206 3303 Wavelength, A500- 400 300 200 100 0 2511 2648 2777 2901 3024 3150 3283 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='0 Normalized Counts Wavelength, A6 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' CUTE Laboratory and measured in-flight performance Metric Laboratory Value In-flight Value Bandpass 2480 – 3322 ˚ A 2480 – 3306 ˚ A Spectral Tilt 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='05◦ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='85◦ Intrinsic Spectral Resolutiona 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='1 ˚ A 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='9 ˚ A Cross-Dispersion Resolutiona 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='5′′ 30′′ Aeff at 2500 ˚ A 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='8 cm2 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='5 cm2 Background Limiting Fluxb N/A 5 × 10−14 erg s−1 cm−2 ˚ A−1 a Average resolution over the bandpass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' In-flight value has ADCS jitter effects removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' b In 300s, measured on orbit only, evaluated at 3000˚ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' region to fully capture the extent of the cross-dispersion spectrum remains at a constant value of about 25 pix- els across the bandpass for observations with jitter less than 6′′ RMS, compared to a 13-pixel extraction region measured in the laboratory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' For each additional row added to the extraction region, the total noise increases by about 4%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' Top: A 2048 x 515 CUTE spectral image of Cas- tor with background effects removed to show the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' Middle left: Zoom of a blue portion of the CUTE spectrum to highlight the out-of-focus lobes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' Middle Right: Zoom of a red portion of the CUTE spectrum to highlight the less- out-of-focus lobes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' Bottom: One-dimensional cross-sections of the middle panels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' We used IUE observations of ζ Puppis to measure the in-flight bandpass;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' the vertical tick marks in Figure 7 show the stellar features used to that end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' The measured in-flight bandpass is 2480 – 3306 ˚A, a change from the laboratory-measured bandpass of 2479 – 3322 ˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' The spectrograph’s dispersion was also affected by launch;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' in the laboratory, the bandpass-averaged dispersion was measured between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='47 ˚A/pixel to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='39 ˚A/pixel, and in- flight it is measured to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='45 ˚A/pixel to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='35 ˚A/pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' Altogether, the change in trace tilt, the shifted band- pass, the change in dispersion, and the spectrograph’s defocus indicate shifts occurring in the optical chain af- ter the Cassegrain focus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' The end-to-end effective area of the optical system was not directly measured prior to launch, but rather cal- culated using a combination of measured and provided quantities: the telescope’s geometric collecting area, op- tical reflectivities from coating witness samples provided by Nu-Tek for the four non-diffractive reflecting optics, the measured detector quantum efficiency (QE) from Nell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' (2021), and laboratory-measured efficiency of the bare Al-coated diffraction grating (Egan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' (2020)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' The in-flight effective area was measured using a com- bination of IUE flux-calibrated spectra of ζ Puppis for λ < 3100 ˚A and a linear model for λ > 3100 ˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' IUE sensitivity falls off significantly after 3100 ˚A and there- fore our initial calibration program could not directly measure CUTE’s effective area beyond 3100 ˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' We in- stead made the assumption that the slope of the stellar flux from λ ≈ 2500 – 3100 ˚A remained the same for λ ≈ 3100 – 3306 ˚A and used that slope to approximate the stellar flux red-ward of 3100 ˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' We combined the IUE flux and the model to calculate CUTE’s effective area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' The IUE ζ Puppis spectra were obtained from the Mikulski Archive for Space Telescopes (MAST) at the Space Telescope Science Institute and can be accessed via 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='17909/a8wa-vc91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' To arrive at the CUTE effective area in units of cm2, the CUTE two-dimensional ζ Puppis spectrum image was background subtracted, the spectral trace was ex- tracted and summed into a one-dimensional counts spec- trum, converted into erg s−1 ˚A−1, and finally divided 2511 2648 2777 2901 3024 3150 3283 Wavelength,A 523.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='2 Arcseconds 457.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='8 Arcseconds 479.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='6 414.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='2 436.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='0 370.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='6 392.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='4 327.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='0 2508 2532 2555 3303 3326 3350 3374 Wavelength, A Wavelength,A 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='00 Normalized 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='75 Normalized 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='75 Counts Counts 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='00 180 200 220 240 150 170 190 210 Pixels Pixels7 by the IUE spectra in units of erg s−1 cm−2 ˚A−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' The resulting flux-calibrated CUTE spectrum of ζ Puppis is shown in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' The effective area ranges from 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='0 – 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='5 cm2 across the bandpass and a smoothed ef- fective area curve is shown in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' The in-flight measured effective area is lower than the laboratory- calculated values by a median of 12% across the band- pass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' We attribute this change to two possible causes: a contamination layer deposited onto the CCD and optics during the TEC failure (Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='2, Egan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' (2022)) and/or atmospheric contamination of the optics during the two month period when CUTE sat in the launch dispenser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' However, CUTE science measurements are relative in nature and an individual light curve is made from observations taken within ≈24 hours of each other;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' changes in the instrument’s sensitivity between delivery and science operations do not affect CUTE light curve creation, other than the raised noise floor from the TEC failure and launch-induced defocus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' The spectral and spatial resolution were measured us- ing IUE observations of Castor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' We define the spatial resolution as the full width at half-maximum (FWHM) of the spectrum in the cross-dispersion direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' This varies by a few pixels across the bandpass (as shown in Figure 5) with an average of 30′′ at 3000 ˚A for observa- tions with jitter less than 6′′ RMS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' CUTE’s spectral resolution as measured from its one-dimensional spectrum comes from the combination of the spectrograph’s intrinsic resolution and spectral smearing from spacecraft jitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' Assuming a Gaussian profile for both the spectral line shape and the jitter distribution, the FWHM of each add in quadrature to arrive at the spectral resolution measured directly from CUTE one-dimensional spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' We convolved IUE spectra of Castor until it matched CUTE’s spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' Us- ing the observation’s jitter of 6′′ RMS, we arrive at a 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='9 ˚A intrinsic spectral resolution element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' CCD Characterization The detector has a 2200 × 515 pixel format with the following layout for each row: 51 true overscan pixels, 50 blank pixels, 2048 active pixels, 50 blank pixels, and 1 null element (a readout artifact used in testing to sim- plify waveform memory implementation in the FPGA), as can be seen in the full frame images in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' The 100 blank pixels are not able to sample dark current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' We use the horizontal blank and overscan pixels to measure the read noise and bias levels and implement a 5-pixel buffer to avoid charge transfer effects at the active/blank pixel boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' For example, a full 2200 × 515 CCD im- age will have on one side a 50 × 515 blank region, and a 40 × 505 sub-region is used to calculate bias and read Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' Effective area of the CUTE instrument as pro- posed (blue), estimated with laboratory measurements (or- ange), and calculated in-flight and smoothed (black).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' The in-flight effective area beyond λ ≈ 3100 ˚A is estimated based on a model fit to the IUE ζ Puppis one-dimensional spec- trum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' Flux-calibrated spectrum of ζ Pup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' A CUTE spectrum is in black, plotted over the blue IUE spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' Black vertical tick marks note the spectral features used to calculate CUTE’s wavelength solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' IUE sensitivity falls off beyond 3100 ˚A, so a model was fit to the IUE data in order to calculate CUTE’s effective area and flux spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' noise levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' The same 5-pixel buffer enacted on the 102 × 515 blank + overscan side produces a 92 × 505 region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' The height of these regions is reduced to 90 pixels for a TRIM2D frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' The read noise is considered to be the RMS of these sub-regions and the bias level is cal- culated from the median.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' The CCD has a single output channel;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' register clocking moves charges from the blue end to the red end of the detector, and vertical clocking 30 28 Estimatedpre-flight 26 冏忆 Proposed 20 Observedin-flight 18 2600 2800 3000 3200 Wavelength, A1e-9 IUE Pup CUTE A 4 2 2600 2800 3000 3200 Wavelength,A8 moves charges from the top to the bottom of the images shown in Figures 3, 4, and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' Much of the CCD characterization took place in the laboratory and is detailed in Nell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' Quan- tities including gain, photon-response non-uniformity (PRNU), and non-linearity were not able to be mea- sured in flight due to the time and budget constraints of a suborbital-class mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' In-flight flat fields cannot be obtained as there is no onboard calibration lamp and true flats cannot be obtained with a dispersing element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' CUTE science and calibration targets fill the pixel wells to about 5% of full capacity, levels where non-linearity is not a concern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' PRNU, gain, and any dust effects are ac- counted for by (1) the flux calibration and (2) the nature of light curves being fundamentally a relative measure- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' Background A single dark frame has contributions from dark noise, read noise, detector bias, and the sky due to the unshut- tered spectrograph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' The background exhibits additional variation dependent on (1) the CCD’s temperature- dependent dark rate and (2) a scattered light feature with a brightness that is correlated with the telescope’s elevation angle;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' an extreme example of this scattered light feature is shown in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' Both dark and bias frames have some level of contribution to all of the above phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' Figure 10 shows the detector’s periodic temperature cycle over several orbits, cycling between ≈ −11◦C and −6◦C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' These temperature swings are evident in the av- erage background frame count rates, shown in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' The background count rate increases as temperature increases, which is the expected dark rate behavior of the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' However, the background count rate addi- tionally varies at similar temperatures due to scattered light entering the spectrograph;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' the angles between the telescope and the Earth’s limb, the Sun, and the Moon all have an effect on the level of scatted light present in background frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' An example is shown in Figure 9 where the Sun’s elevation angle is correlated with in- creased background counts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' In CCD regions that are more prone to scattered light (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' the left side of the CCD as shown in the top panel of Figure 8), background count rates differ on the order of about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='15 photons s−1 pixel−1 when telescope elevation is maintained above 10◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' Background subtraction for a single science frame must consider the orbital and thermal environment of the exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' The temperature- and pointing- dependent nature of the background (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' Figure 9) means that each science frame must have tailored cal- Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' Two 2200 × 515 CUTE frames of different tele- scope elevation angles, 3◦ and 12◦, to demonstrate the scat- tered light feature present at low telescope elevation angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' The elevation of each exposure is marked in the left-side overscan region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' The top and middle frames compare the 3◦ and 12◦ elevation angles with the same colorbar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' The bottom frame shows the 12◦ frame with a colorbar scaled to the frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' Typical frame features are evident on the 12◦ frame: an increase in counts from the bottom to the top of the frame and hot pixel streaks due to the 32s CCD readout time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' The horizontal dark bars in the top image are missing data packets that were unsuccessfully downlinked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' CUTE CCD dark frame background rates for the detector, plotted against detector temperature and colored by the elevation of the Sun with respect to the telescope bore- sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' In general, higher CCD temperatures produce higher background count rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' Background rates are additionally influenced by the position of astronomical bodies like the Sun as exemplified here, as well as the Moon and Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' At the time of writing, we are still exploring sources of contribution to background rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='2 20S 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='0 Rate, 300 Elevation Angle, a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='8 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='6 50 ledian un 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='4 M 10 8 6 4 2 Temperature,C400 30000 ixels 3° 20000 P 200 100008 0 0 400 S xel 12° 20000 P 200 0 0 800 400 600号 S xel 12 Counts, P 200 400 200 0 2479 2592 2702 2808 2910 3010 3106 3199 Wavelength,A9 Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' A typical CCD temperature profile over several CUTE orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' ibration frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' Efforts are underway to model each pixel’s value as a function of both its intrinsic behavior and its environment in order to create individual back- ground frames for a given science exposure, based on the frame’s exposure conditions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' these efforts will be de- scribed in detail in Egan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' 2023 −in prep and we currently use a combination of median-combined dark and bias frames with similar telescope pointings and CCD temperatures to approximate the background in each science frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' Finally, we present CUTE’s background flux limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' After a science frame is background-subtracted, we de- fine a background region below the spectrum that has the same size as the frame’s spectral extraction region and calculate the background flux limit, or minimum flux below which a source cannot be detected above the background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' The counts in the background region are converted to flux and then averaged for 67 × 300s WASP-189 science frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' The background flux limit is ≈ 5 × 10−14 erg s−1 cm−2 ˚A−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' Background subtrac- tion and handling of hot pixels are discussed in more detail in Sreejith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' Pointing Stability The quality of CUTE spectra is strongly influenced by the spacecraft pointing jitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' Stability about the com- manded coordinates can vary between 3′′ and more than 20′′ and thereby smear the spectrum across a greater CCD area, increase the spectral extraction region, and reduce the signal-to-noise of the one-dimensional spec- trum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' Significant jitter can cause vignetting of the tele- scope point-spread-function by the slit mask and ren- der those data unusable for transit spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' We currently use a 6′′ jitter cutoff for science frames to un- dergo data reduction and observations with higher jitter Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' Jitter histogram for 254 × 300s science expo- sures from the WASP-189b campaign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' The pointing sample rate is 5s for each 300s observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' Histogram bin widths are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='5′′ for jitter < 10′′ and 1′′ for jitter > 10′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' A ver- tical dashed line indicates the cutoff used to eliminate low signal-to-noise observations from light curve analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' are not used Sreejith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' Figure 11 shows a histogram of ADCS jitter in all of the WASP-189 science frames which include complete jitter telemetry;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' jitter is less than 6′′ for about 56% of CUTE observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' In our preliminary data reduction and light curve creation, we remove frames with higher jitter to eliminate low signal-to-noise observations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' this cutoff will be honed as we continue to refine our data reduction pipeline (Sree- jith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' (2022)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' MISSION OPERATIONS CUTE’s standard mission operation sequence in- volves observing a given science target for 6−10 transits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' Each visit’s duration is centered on the planet’s mid- transit time and lasts for a span of time equal to five times the transit duration in order to establish an out- of-transit stellar baseline pre- and post-transit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' CUTE target transits last between 3 and 6 hours, or about 2 to 4 CUTE orbits, while the duration of each visit is typically less than 24 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' As discussed in Section 3, CUTE data quality has pointing and temperature dependencies: (1) CCD ex- posures are captured without a shutter, meaning that calibration frames are not truly “dark” and include sky background;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' (2) CCD dark rates scale with the ther- mal changes present due to the passive cooling method;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' (3) there is a pointing-dependent scattered light feature present in all frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' We create observation plans that take these into account which are detailed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' The roll angle for a given science target observing cam- paign is set with two considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' First, we prioritize Temperature (°C) 8 9 10 0 Time(DOY:HHUTC)Observations 10- Number of 100 4 6810 20 40 100 200 PointingRMS,arcseconds10 keeping the CCD as cool as possible by orienting the radiator panel into space and away from the Earth and Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' Once that angle is set, we additionally check the target field to make sure no bright stars other than the target are within the slit;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' this has been the case for all science observations obtained since launch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' The roll angle is maintained for all calibration frames taken ad- jacent to a given set of science exposures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' To capture scattered light features from the Sun, Moon, or Earth limb that may appear in science frames (Figure 9), dark and bias calibration frames are planned to occur at ap- proximately the same orbital position as science frames with a pointing offset from the target star of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='75◦;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' this has an additional benefit of obtaining a calibration frame in a similar thermal state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' Fully characterizing the background is a continued effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' Background trends related to scattered light and telescope pointing did not become apparent until a sufficient number of science and dark frames from the WASP-189 observing campaign were analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' Model- ing efforts are underway to model each pixel’s behavior in its environment to better produce calibration frames for background subtraction (Egan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' 2023 – in prep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' CONCLUSION CUTE is currently obtaining NUV transit spec- troscopy of short-period exoplanets around bright stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' We have detailed the NUV payload’s spectroscopic per- formance as calculated within the first six months of calibration and science observations, including the spec- tral and spatial resolution, the effective area, and the limiting flux level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' We have demonstrated stable point- ing for about 56% of our observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' Science opera- tions will continue through June 2023, during which we will conduct additional calibration campaigns to mea- sure time-dependent sensitivity, as well as continue to hone our understanding of the background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' CUTE is expected to reenter the atmosphere within 3 years from launch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' Acknowledgments: CUTE was developed and op- erated with the support to two NASA/APRA awards to the Laboratory for Atmospheric and Space Physics at the University of Colorado Boulder, NNX17AI84G and 80NSSC21K1667s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' was supported by a Schr¨odinger Fellowship through the Austrian Science Fund (FWF) [J 4596-N] and additionally acknowledges financial support from the Austrian Forschungsf¨orderungsgesellschaft FFG project 859718 and 865968.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' The CUTE team acknowledges the numer- ous invaluable discussions with colleagues excited about ultraviolet transit science and the potential to do science with small satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' The CUTE team wishes to specif- ically recognize the amateur radio operator community for hosting numerous telemetry tracking tools that have improved the mission’s ability to recover from faults and understand long-term spacecraft trends much more effi- ciently than would have been otherwise possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' REFERENCES Cubillos, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=', Fossati, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=', Koskinen, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' 2020, AJ, 159, 111, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='3847/1538-3881/ab6a0b Egan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=', France, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=', Nell, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' 2022, in Small Satellite Conference, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' SS22, Small Satellite, WKII–02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' https: //digitalcommons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='usu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='edu/smallsat/2022/all2022/52/ Egan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=', France, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=', Fleming, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' 2020, in Space Telescopes and Instrumentation 2020: Ultraviolet to Gamma Ray, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' 11444, International Society for Optics and Photonics, 1144406, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='1117/12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='2559983 Ehrenreich, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=', & D´esert, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' 2011, A&A, 529, A136, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='1051/0004-6361/201016356 Fossati, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=', Haswell, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=', Froning, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' 2010, The Astrophysical Journal Letters, 714, L222, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='1088/2041-8205/714/2/L222 France, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=', Hoadley, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=', Fleming, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' 2016, Journal of Astronomical Instrumentation, 05, 1640001, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='1142/S2251171716400018 Haswell, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=', Fossati, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=', Ayres, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' 2012, The Astrophysical Journal, 760, 79, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='1088/0004-637X/760/1/79 Lammer, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=', Selsis, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=', Ribas, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' 2003, The Astrophysical Journal Letters, 598, L121, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='1086/380815 Lothringer, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=', Sing, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=', Rustamkulov, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' 2022, Nature, 604 Munoz, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' 2007, Planetary and Space Science, 55, 1426 , doi: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='pss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='007 Murray-Clay, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=', Chiang, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=', & Murray, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' 2009, The Astrophysical Journal, 693, 23, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='1088/0004-637X/693/1/23 Nell, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=', DeCicco, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=', Ulrich, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=', France, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=', & Fleming, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' 2021, in UV, X-Ray, and Gamma-Ray Space Instrumentation for Astronomy XXII, ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' Siegmund, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' 11821, International Society for Optics and Photonics (SPIE), 391 – 400, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='1117/12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='2594743 11 Nichols, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=', Wynn, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=', Goad, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' 2015, The Astrophysical Journal, 803, 9, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='1088/0004-637X/803/1/9 Sing, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=', Lavvas, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=', Ballester, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' 2019, AJ, 158, 91, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='3847/1538-3881/ab2986 Sreejith, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=', Fossati, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=', Ambily, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' 2022, Publications of the Astronomical Society of the Pacific, 134, 114506, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='1088/1538-3873/aca17d Vidal-Madjar, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=', des Etangs, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=', D´esert, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' 2003, Nature, 422, 143 EP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' http://dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='1038/nature01448 Wakeford, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=', Sing, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=', Stevenson, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' 2020, AJ, 159, 204, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='3847/1538-3881/ab7b78 Yelle, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content=' 2004, Icarus, 170, 167 , doi: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='icarus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} +page_content='008' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAzT4oBgHgl3EQfWvyr/content/2301.01307v1.pdf'} diff --git a/R9E0T4oBgHgl3EQfUgAs/content/tmp_files/2301.02250v1.pdf.txt b/R9E0T4oBgHgl3EQfUgAs/content/tmp_files/2301.02250v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..64fc226886926e2105909ec7fcf720a0e51f8d47 --- /dev/null +++ b/R9E0T4oBgHgl3EQfUgAs/content/tmp_files/2301.02250v1.pdf.txt @@ -0,0 +1,1113 @@ +Draft version January 9, 2023 +Typeset using LATEX twocolumn style in AASTeX631 +The Colorado Ultraviolet Transit Experiment (CUTE) Mission Overview +Kevin France,1 Brian Fleming,1 Arika Egan,1 Jean-Michel Desert,2 Luca Fossati,3 Tommi T. Koskinen,4 +Nicholas Nell,1 Pascal Petit,5 Aline A. Vidotto,6 Matthew Beasley,7 Nicholas DeCicco,1 +Aickara Gopinathan Sreejith,1, 3 Ambily Suresh,1 Jared Baumert,1 P. Wilson Cauley,1 +Carolina Villarreal D’Angelo,8 Keri Hoadley,9 Robert Kane,1, 10 Richard Kohnert,1 Julian Lambert,1 and +Stefan Ulrich1 +1Laboratory for Atmospheric and Space Physics, University of Colorado Boulder, Boulder, CO 80303 +2Anton Pannekoek Institute for Astronomy, University of Amsterdam, P.O. Box 94249, Noord Holland, NL-1090GE Amsterdam, the +Netherlands +3Space Research Institute, Austrian Academy of Sciences, Schmiedlstrasse 6, 8042 Graz, Austria +4Lunar and Planetary Laboratory, University of Arizona, Tucson, AZ 85721–0092 +5Institut de Recherche en Astrophysique et Plan´etologie, Universit´e de Toulouse, CNRS, CNES, 14 avenue Edouard Belin, 31400 +Toulouse, France +6Leiden Observatory, Leiden University, PO Box 9513, 2300 RA, Leiden, the Netherlands +7Southwest Research Institute, Boulder, CO 80302 +8Instituto de Astronom´ıa Te´orica y Experimental, CONICET-UNC. Laprida 854, C`ordoba, Argentina +9The University of Iowa, Dept. of Physics & Astronomy, Van Allen Hall, Iowa City, IA 52242 +10Blue Canyon Technologies 2550 Crescent Dr, Lafayette, CO 80026 +ABSTRACT +Atmospheric escape is a fundamental process that affects the structure, composition, and evolution +of many planets. +The signatures of escape are detectable on close-in, gaseous exoplanets orbiting +bright stars, owing to the high levels of extreme-ultraviolet irradiation from their parent stars. The +Colorado Ultraviolet Transit Experiment (CUTE) is a CubeSat mission designed to take advantage of +the near-ultraviolet stellar brightness distribution to conduct a survey of the extended atmospheres +of nearby close-in planets. The CUTE payload is a magnifying NUV (2479 – 3306 ˚A) spectrograph +fed by a rectangular Cassegrain telescope (206mm × 84mm); the spectrogram is recorded on a back- +illuminated, UV-enhanced CCD. The science payload is integrated into a 6U Blue Canyon Technology +XB1 bus. CUTE was launched into a polar, low-Earth orbit on 27 September 2021 and has been +conducting this transit spectroscopy survey following an on-orbit commissioning period. This paper +presents the mission motivation, development path, and demonstrates the potential for small satellites +to conduct this type of science by presenting initial on-orbit science observations. The primary science +mission is being conducted in 2022 – 2023, with a publicly available data archive coming on line in +2023. +1. INTRODUCTION +The history of observational astronomy has been +marked by the push to ever larger and more capable +telescopes and instruments. +The 2010s witnessed the +development of a new generation of large astronomical +observatories. Both on the ground and in space, facilities +2 – 3 times the primary mirror diameter of the previous +state-of-the-art were brought closer to fruition for im- +plementation in the 2020s, 2030s, and 2040s, including +Corresponding author: Kevin France +kevin.france@colorado.edu +the James Webb Space Telescope (Gardner et al. 2006; +Rigby et al. 2022), thirty-meter class ground-based tele- +scopes (e.g., Simard et al. 2016), and advanced ultravio- +let/optical (UV/O) facilities such as the Large Ultravio- +let/ Optical/ Infrared Surveyor (LUVOIR; LUVOIR Fi- +nal Report 2019). The large mission studies conducted +ahead of the 2020 Decadal Survey on Astronomy and As- +trophysics drove the recommendation for NASA’s suite +of Future Great Observatories, a series of probe- and +flagship-class missions offering many order-of-magnitude +gains in the scientific grasp across numerous areas of as- +trophysics. In parallel with this large observatory de- +velopment, numerous small telescope arrays have come +on-line or have been expanded, and NASA’s science divi- +arXiv:2301.02250v1 [astro-ph.IM] 5 Jan 2023 + +2 +sions made significant new investments in small satellites +covering a range of scientific topics. +Small telescopes at ground-based sites have excelled +at detecting and characterizing new objects in the time- +variable sky, including supernovae eruptions (Dong et al. +2016) and tidal disruption events (Hammerstein et al. +2022) from the Zwicky Transient Facility (Masci et al. +2019) and All Sky Automated Survey for SuperNovae +(ASAS-SN) (Holoien et al. 2017). The impact of small +telescopes has also been powerful for the detection of +extrasolar planets, including many Jovian-sized plan- +ets from Wide-Angle Search for Planets (WASP) (Pol- +lacco et al. 2006) and the Kilodegree Extremely Lit- +tle Telescope (KELT) (Pepper et al. 2007), and some +of the most promising rocky planets for study with +JWST from the MEarth (Charbonneau et al. 2009), +Transiting Planets and Planetesimals Small Telescope +(TRAPPIST) (Gillon et al. 2011), and Search for hab- +itable Planets EClipsing ULtra-cOOl Stars (SPECU- +LOOS) (Burdanov et al. 2018) facilities. +The recent decadal survey highlighted the power of +small space-based telescopes, astronomical CubeSats +and smallsats, for “monitoring of sources for weeks +or months at time, and at wavelengths not acces- +sible from the ground”, complementing the Hubble +Space Telescope’s surveys in areas of transmission spec- +troscopy (Sing et al. 2019; Cubillos et al. 2020) and ex- +oplanet host star radiation fields (France et al. 2016b; +Loyd et al. 2018; Ramiaramanantsoa et al. 2021). NASA +has embraced this opportunity with a dedicated fund- +ing line for astrophysics CubeSats (full mission life cycle +cost < $10M) and the Pioneers program (mission cost +$10M – $20M). In this paper, we present an overview of +NASA’s first UV astronomy CubeSat and the first grant- +funded small satellite dedicated to the characterization +of extrasolar planetary atmospheres, the Colorado Ul- +traviolet Transit Experiment (CUTE). +CUTE conducts transit spectroscopy of short-period, +giant planets in the near-UV (2479 – 3306 ˚A) bandpass +to access strong atomic transitions tracing atmospheric +escape and the near-UV spectral slope of giant planet +atmospheres that provide constraints on their compo- +sition. This paper presents the science background for +and the technical implementation of the mission. The +manuscript is laid out as follows: the scientific motiva- +tion for CUTE and its science objectives are presented +in Section 2. Because CUTE is one of the first astron- +omy missions to be developed in a CubeSat framework, +we present a description of the mission development and +implementation path in Section 3. Section 4 presents the +instrument design and high-level performance specifica- +tions (see also Fleming et al. 2018 for a description of +CUTE’s science payload). Section 5 describes CUTE’s +mission operations and we present early-release exam- +ples of the mission’s on-orbit science data in Section +6. We conclude with a brief summary in Section 7. A +detailed description of CUTE’s science instrument and +on-orbit performance is presented in a companion paper +by A. Egan. Mission operations and on-orbit commis- +sioning (Suresh et al. – in prep), CUTE’s on-orbit data +pipeline (Sreejith et al. – in press), and Early Release +Science results (Egan et al. – in prep; Sreejith et al. – +in prep) will be described in forthcoming papers. +2. CUTE SCIENCE OBJECTIVES +Planetary escape processes play a key role in de- +termining the chemical and physical state of plan- +ets both within and beyond our solar system. +At- +mospheric escape is thought to create the radius gap +observed in the distribution of short-period exoplan- +ets (Fulton & Petigura 2018), likely driven by a com- +bination of photoevaporative (Owen & Wu 2017) and +core-powered (Ginzburg et al. 2018) mass loss. Escape +is also a fundamental process in the evolution of terres- +trial worlds. For a planet to be habitable, our current +view is that it must lose its primordial hydrogen atmo- +sphere and acquire/generate (and retain) a secondary +atmosphere (Lammer et al. 2018). Atmospheric escape +is known to have shaped the early atmospheres of Venus, +Earth, and Mars, which subsequently followed differ- +ent evolutionary paths. The rapid hydrodynamic escape +that is believed to have affected Venus, Earth and Mars +in the past no longer takes place on any planet in the +solar system. Therefore, we turn to short-period extra- +solar planets as laboratories on which to study vigorous +atmospheric loss. +The first detection of exoplanet atmospheric escape +was achieved by Vidal-Madjar et al. (2003) who used HI +Lyα transit observations in the far-ultraviolet (FUV) +to observe the extended atmosphere of the Hot Jupiter +HD209458b. This was followed by the detection of O I, +C II, Si III and Mg I on the same planet (Vidal-Madjar +et al. 2004, 2013; Linsky et al. 2010). These initial obser- +vations inspired several independent groups to develop +1D and 3D models to study both the physical character- +istics of the upper atmospheres of close-in planets and +the escaping gas and plasma surrounding them (e.g., +Koskinen et al. 2007; Murray-Clay et al. 2009; Koskinen +et al. 2013a,b; Bourrier & Lecavelier des Etangs 2013; +Bourrier et al. 2016; Villarreal D’Angelo et al. 2018; Car- +olan et al. 2021). +The interpretation of FUV transit measurements has +often been controversial (see Fossati et al. 2015 for +a discussion). +Recently, several atmospheric escape + +3 +Figure 1. The CUTE instrument development from con- +cept (instrument schematic, top) to telescope characteriza- +tion (CUTE flight telescope in the test facilities at the Uni- +versity of Colorado, middle), to pre-delivery in-band spectral +resolution test data (bottom). +studies have shifted to the near-ultraviolet (NUV), +where the stellar flux is much higher than in the FUV +and the light curves are measured against a better- +understood intensity distribution from the stellar photo- +sphere (e.g.,Haswell et al. 2012; Llama & Shkolnik 2015). +The NUV includes the Fe II complexes near 2400 and +2600 ˚A, the Mg II doublet at 2796/2803 ˚A, the Mg I +line at 2852 ˚A, some of which have been detected on +the Hot Jupiters WASP-12b, HD209458b, and WASP- +121b (Fossati et al. 2010; Sing et al. 2019; Cubillos et al. +2020). We note that the Fe II and Mg II resonance lines +in the near-UV trace the highly extended (and poten- +tially escaping) exoplanet atmosphere, whereas optical +band metal line detections made with ground-based tele- +scopes trace the lower, bound atmospheric layers (Hoei- +jmakers et al. 2019; Casasayas-Barris et al. 2019; Cauley +et al. 2019; Turner et al. 2020; Hoeijmakers et al. 2020; +Casasayas-Barris et al. 2021; Deibert et al. 2021). The +NUV also contains a pseudo-continuum that can probe +scattering by high altitude clouds and gas phase silicon +and magnesium (Lothringer et al. 2022), as well as the +A – X bands of OH (3100 ˚A). Furthermore, NUV trans- +mission spectra give the unique opportunity to constrain +the composition of the aerosols lying in the lower atmo- +spheres (Cubillos et al. 2020). +Depending on the temperature profile in the atmo- +sphere, species like Si, Mg, and Fe are expected to +condense to form clouds in the lower atmosphere, how- +ever, the calculations indicate that strong mixing, either +by turbulence or global circulation, can inhibit cloud +formation or allow for these species to be present in +the upper atmosphere where they can escape (Koski- +nen et al. 2013b; Cubillos et al. 2020; Koskinen et al. +2022). The comparison of continuum and atomic line +absorption therefore acts as a diagnostic of cloud for- +mation, elemental abundances and mass loss on close-in +exoplanets (Lothringer et al. 2020; Cubillos et al. 2020). +Model outputs can be used to translate observed plane- +tary transit light curves into global mass-loss rates: the +depth and shape of the light curves directly relate to the +atmospheric parameters. +Finally, UV transits with HST have provided ev- +idence for time-variability, +potentially arising from +changing stellar high-energy input, orbital timescale +changes in the planet’s atmosphere, or variation in +the star-planet magnetic environment. +Lecavelier des +Etangs et al. (2012) observed time-variable neutral hy- +drogen absorption in FUV transit observations of HD +189733b, possible due to the influence of high-energy +stellar flares. NUV transit observations of the close-in +giant planet WASP-12b by Fossati et al. (2010) found +that the transit light curve of WASP-12b presents both +an early ingress when compared to its optical transit and +excess absorption during the transit (see also Haswell +et al. 2012; Nichols et al. 2015). Possible explanations +include atmospheric hydrodynamic mass-loss support- +ing a shock upstream of the planet’s orbit or generating + +2536.6 +2705 +2967.3 +310g +3125.6 +3131.4 +an accretion stream that produces an early ingress (Lai +et al. 2010; Bisikalo et al. 2013; Turner et al. 2016) and a +magnetically supported bow-shock 4 – 5 planetary radii +upstream of the planet’s orbital motion (analogous to +the Earth-Sun system; Vidotto et al. 2010; Llama et al. +2011). +CUTE’s primary science goal is to provide new con- +straints on the physics and chemistry of hot, Jovian-size +exoplanets. +The CUTE +mission addresses this goal +with the following observing program: +1. Measure NUV transmission spectra for a small sur- +vey of approximately 10 short period planets +2. Infer atmospheric escape rates and constrain the +composition of the upper atmospheres of hot giant +planets +3. Measure temporal variability in UV transit light +curves by observing 6 – 10 transit observations per +planet +4. Measure out-of-transit baseline fluxes to better +characterize the stellar inputs to the planet’s at- +mosphere and to capture light curve asymmetries +CUTE’s instrument design and mission implementa- +tion was developed to enable the four key goals of the ob- +serving program. The spectral coverage and resolution +of the CUTE (∆ v < 300 km s−1) spectrograph provides +ample separation of the relevant atomic, molecular, and +continuum bands in this range (see, e.g., Figure 8 of Sing +et al. 2019). CUTE’s mission design complements the +instrument to meet the science goals of the mission. (1) +We couple observations of the NUV continuum opacity, +individual ionic tracers (Fe II, Mg II) with atmospheric +chemistry, and hydrodynamic escape models to deter- +mine mass loss rates for CUTE’s targets. The sample +size is driven by a combination of mission lifetime and +instrumental sensitivity considerations. (2) CUTE mea- +sures the amplitude and slope of the NUV transmission +curve to provide constraints on the chemistry and struc- +ture of the escaping atmosphere. The instrumental ef- +fective area was specified to enable multiple, wavelength +resolved, NUV bands with sufficient photometric pre- +cision to distinguish the NUV transit radius from the +white-light radius of the planet on all 10 targets and +isolate transit spectra of the strongest absorption lines +(e.g., Fe II and Mg II) on the brightest targets (address- +ing goals 1 and 2). The target sample was defined by +estimating the detectability of excess NUV absorption; +a combination of stellar brightness (V-magnitude), spec- +tral type (A- and F-type stars have spectral energy dis- +tributions peaked in the NUV), planetary radius, effec- +tive planetary surface temperature, and gravity (hotter, +lower-mass planets being more likely to exhibit extended +atmospheres). (3) CUTE’s point-stare-repeat concept +of operations is designed to make numerous visits to the +same planet over the course of 4 to 8 weeks, building +signal-to-noise for fainter targets and enabling measure- +ments of light curve variability for brighter targets. (4) +The same point-stare-repeat observing mode provides a +wide stellar baseline to measure changes in the Mg II ac- +tivity and the increased dispersion of the photospheric +and chromospheric continuum flux that indicate vari- +ability in the star’s escape-driving XUV output. +3. MISSION IMPLEMENTATION PATH +CUTE is NASA’s first grant-funded UV/ Optical/ In- +frared small satellite and first dedicated exoplanet spec- +troscopy mission. Given the novelty of this mission for- +mat for astrophysics science missions, we present a brief +overview of the process, schedule, and cost of the mission +here. The initial motivation for CUTE was discussed at +a Keck/KISS workshop on exoplanet magnetic fields in +August 2013, with the final science and measurement +concept in place by the summer of 2015 following nu- +merous informal discussions at science conferences. Fall +2015 was spent on science measurement definition and +the development of the CUTE instrument design. +CUTE was proposed as a four-year program through +NASA’s ROSES2015 call (submitted in March 2016), +at an initial cost-to-launch of $3.3M, comparable to an +astrophysics sounding rocket proposal but considerably +lower cost than a stratospheric balloon program. CUTE +was proposed and selected prior to the initiation of ded- +icated funding for astrophysics CubeSats, leading to a +long delay between proposal submission and the start of +funding (approximately 16 months; there was no Phase +A or Concept Study period for CUTE). Long lead-time +items such as the CUTE spacecraft bus, the rectan- +gular telescope, holographically-ruled diffraction grat- +ing, and NUV-optimized CCD detectors were ordered +a few months after selection in fall 2017. The CUTE +spacecraft (Blue Canyon Technology, BCT) costs in- +creased relative to the quote provided for the pro- +posal. +To accommodate the cost increases in a mis- +sion class without reserves, several descopes were im- +plemented, including scaling back the spacecraft’s atti- +tude control system to a single star-tracker and elim- +inating engineering model radios. +In 2018 and 2019, +we developed the hardware test facilities that comple- +mented the University of Colorado’s existing UV vac- +uum calibration facilities (France et al. 2016a) and con- +ducted component-level characterization (e.g., groove ef- +ficiency of the diffraction gratings, trade study of Al vs. +Al+MgF2 grating coatings). Instrument assembly and + +5 +characterization, integration into the spacecraft, and +pre-delivery environmental testing (e.g., vibration test- +ing, comprehensive performance testing, thermal vac- +uum testing, etc.) were completed in 2020 and 2021. +The duration from the start of CUTE funding to de- +livery of the completed observatory was almost exactly +four years, although approximately 10 months of sched- +ule were lost to the COVID-19 pandemic. +CUTE proposed to NASA’s CubeSat Launch Initia- +tive (CSLI) for launch support in fall 2017 and was +selected for flight. The proposed spacecraft orbit, in- +cluding the initial CUTE mission requirement docu- +mentation submitted to CSLI, requested a dawn-dusk +(terminator), sun-synchronous orbit to enable uninter- +rupted orbital phase coverage of transiting planets and +to minimize day/night thermal variations. Orbital alti- +tude (450 - 600km) was a secondary consideration driven +by desired mission lifetime. CSLI was unable to accom- +modate the requested sun-synchronous orbit within the +time-period covered by the mission funding and CUTE +was instead manifested in November 2019 as a secondary +payload on NASA’s Landsat 9 mission. +The Landsat +9 launch was scheduled 8 months after CUTE’s tar- +geted launch window. As a result, NASA provided a +6 month, $0.5M extension to the CUTE program in +Fall 2019. Starting in Spring 2020, COVID delayed the +Landsat 9 launch by another 9 months. COVID also im- +peded CUTE’s development timescale owing to supply- +chain delays and the challenges of getting students, sci- +entists, and engineers into CUTE’s labs for continued +testing and development. The CUTE mission submit- +ted a follow-on, competed APRA proposal in December +2020 to conduct mission operations and carry out the +science program (bringing CUTE’s cost to complete the +full science mission to approximately $5M). Of the 18 +CubeSat missions originally manifested with Landsat 9, +only four (including CUTE) would ultimately deliver +and be flown on the mission. +The integrated and tested observatory was delivered +to NASA at Vandenberg Space Force Base (VSFB) in +July 2021, and the CUTE team supported installation +into the CubeSat dispenser on the ESPA ring. The mis- +sion was launched on September 27 2021 into a sun- +synchronous orbit (≈98◦, 10am Local Time of Ascend- +ing Node; LTAN) with a 560km apogee. +CUTE de- +ployed from the dispenser approximately two hours af- +ter launch; solar arrays deployed and the communi- +cation beacon started approximately 30 minutes later. +CUTE’s beacons were identified by the amateur RF +community on the first orbit and communications were +established with the ground station at the University of +Colorado on September 28 2021. We refer the reader +to Section 4 and Suresh et al. +(in preparation) for a +description of the CUTE ground segment and commis- +sioning program. We refer the reader to the SmallSat +Conference proceeding by Egan et al. 2022 for a discus- +sion of lessons learned during CUTE development and +early on-orbit operations. +CUTE is part of NASA’s suborbital program, where +student training and early-career mentorship are key in- +gredients to the definition of mission success. CUTE’s +approach was built off of the framework of the NASA +Sounding Rocket Program, which has a long history in +the professional development of NASA’s space scientist +workforce. The core science and instrument team (de- +fined as those working with CUTE for more than 2 years +of the implementation phase) included two Ph.D. candi- +dates (in astrophysics and aerospace engineering), four +undergraduates, two postdoctoral researchers, two early +career engineers (CUTE was the first job post-bachelors +degree for the mission’s lead mechanical and electrical +engineers), and the early-career project scientist (Dr. +Brian Fleming, who became the PI of a NASA sound- +ing rocket program and the SPRITE CubeSat (Fleming +2022) mission during the course of the CUTE develop- +ment phase). Over the course of CUTE’s component- +level and instrument test phase, the project employed +another six undergraduate students in various labora- +tory and science program development tasks (e.g., tar- +get field checking for crowded fields). In addition to this, +the operations team for CUTE (see Section 5) included +an additional two undergraduate students, two graduate +students, and one flight software undergraduate student. +Taken as a whole, CUTE supported the mentoring and +training for over 20 early-career scientists and engineers +through the completion of the on-orbit commissioning +phase. +4. IMPLEMENTATION: THE CUTE SCIENCE +PAYLOAD +The CUTE payload is a magnifying NUV spectro- +graph fed by a rectangular Cassegrain telescope. The +spectrogram is recorded on a back-illuminated, UV- +enhanced e2v CCD42-10 that is maintained at a nom- +inal operating temperature (−15 – −5◦ C) by passive +cooling through a radiator panel. CUTE employs the +BCT XB1 bus to provide critical subsystems including +power, command and data handling, communications, +and attitude control (ADCS). Figure 1 shows an instru- +ment schematic, optical testing of the telescope, and an +in-band calibration spectrum from the flight instrument +prior to instrument integration into the BCT chassis. +The CUTE instrument is housed in 4U of the 6U +spacecraft. +The CUTE aperture is a 206 × 84 mm, + +6 +Figure 2. CUTE calibration observations of the O4 supergiant +ζ Puppis. The top plot shows the full-frame (2048 × 515 pixel) +calibration image and the bottom plot shows the extracted one- +dimensional spectrum (flux calibrated against archival HST and +IUE spectra). ζ Puppis was selected as a calibration target be- +cause of the wealth of archival NUV spectra and the high photo- +spheric temperature ensures that iron and magnesium lines in the +spectrum are narrow, interstellar features. +f/0.75 (in the cross-dispersion axis) primary mirror that +is part of the f/2.6 Cassegrain telescope. The rectan- +gular shape of the primary is matched to the long axis +of the 6U CubeSat chassis and allows for 3 times more +throughput than a 1U circular aperture (Fleming et al. +2018). The hyperbolic secondary mirror is cantilevered +off of the primary mirror, which serves as the bench for +the optical instrument, by means of an Invar tower (see +Figure 1). A 15 × 6 mm fold mirror redirects the beam +90◦ through a 141 µm × 3.5 mm (60′′ × 1400′′ projected) +slit at the Cassegrain focus. The slit, manufactured by +OSH Stencils, was polished on the incident side and an- +gled 45◦ about the slit axis to redirect the field to an +aspect camera for use in telescope performance testing +and alignment with the BCT spacecraft during integra- +tion. +The rectangular telescope design optimizes col- +lecting area within the mass-volume constraints of the +cubesat form factor, while the large sky field-of-view, +increased cost, and mechanical stress at the primary +mirror-secondary tower interface add design complica- +tions. +Once through the slit, the starlight is diffracted, redi- +rected, and magnified by a spherical, R = 86.1mm ra- +dius, 1714 gr mm−1 aberration correcting, ion-etched +holographic grating fabricated by Horiba Jobin-Yvon +(Horiba J-Y). The holographic grating design was +adopted to minimize scattered light in the system. A +second fold mirror with an Rx = 300 mm radius of curva- +ture about the cross-dispersion dimension provides ad- +ditional aberration corrections before the beam reaches +the CCD. The final beam focal ratio is f/5.5 in the +Table 1. CUTE Instrument Specifications +Instrument Metric +On-orbit Value +Bandpass +2479 – 3306 ˚A +Spectral Resolutiona +3.9 ˚A +Cross-Dispersion Resolutionb +≈ 30′′ +Peak Aeff +27.5 cm2 at 2500 ˚A +Background Flux Limit +5 × 10−14 erg cm−2 s−1 ˚A−1 +in 300sb +aAverage resolution over the bandpass, including spacecraft jit- +ter. +bEvaluated at 3000 ˚A. +cross-dispersion axis, with a detector plate scale of 186′′ +mm−1. The detector and custom avionics were tested +and flight ruggedized by the CUTE team in their on- +campus laboratories at the University of Colorado (Nell +et al. 2021). +The telescope was delivered fully assembled to CU by +Nu-Tek Precision Optical. All mirrors are coated with +MgF2 + Al to prevent the formation of an oxide layer +(AlO3). We elected to receive flight and flight-backup +gratings coated in bare Al and MgF2 + Al, respectively +(coated by Horiba J-Y), to control for a potential effi- +ciency anomaly similar to that seen on the COS NUV +gratings (Wilkinson 2002). Detailed pre-flight efficiency +and environmental testing showed better performance +with the bare Al grating, without a measurable loss in +efficiency over time. As a result, the instrument team +elected to fly the bare Al-coated mirror on the flight in- +strument. The design-prediction flight instrument per- +formance curves are presented in Fleming et al. (2018) +and the on-orbit instrument performance of the CUTE +payload is presented in Egan et al. (2022 – this volume); +we provide a brief summary of the key performance met- +rics in the following subsection. +4.1. Instrument Specifications +The final bandpass recorded by the CCD detector is +2479 – 3306 ˚A (see Table 1), which is a slight change +from the pre-flight projection owing to shifts in the op- +tical system during ascent. +The exact bandpass also +varies by several ˚A depending on the alignment of the +stellar point spread function (PSF) in the spectrograph +slit. The spectral resolving power of the instrument is +≈ 750 (∆λ ≈ 3.3 – 4.5 ˚A across the bandpass, including +the effects of spacecraft pointing jitter). Figure 2 shows +a representative calibration spectrum from the CUTE +on-orbit commissioning program. + +400 +200 +0 +-1 +A +1e-9 +z- +2 +2500 +2600 +2700 +2800 +2900 +3000 +3100 +3200 +3300 +Wavelength,A7 +Figure 3. Schematic description of CUTE science and calibration observations. +The system effective area is a function of the reflec- +tion efficiency of the optics (R), efficiency of the grating +(ϵg), and quantum efficiency of the detector (DQE), mul- +tiplied by the geometric collecting area of the telescope +(Ageo): +Aeff(λ) = AgeoR5(λ)ϵg(λ)DQE(λ). +(1) +The on-orbit effective area was measured by comparing +CUTE’s observations (in units of electrons s−1 ˚A−1) +with flux-calibrated observations from the IUE and +HST archives. We measured Aeff = 27.5 – 19.0 cm2 +across the CUTE spectral range with a peak at ap- +proximately 2500 ˚A. Component-level efficiencies were +measured prior to instrument assembly in the UV cali- +bration facilities at the University of Colorado (France +et al. 2016a; Egan et al. 2020). The component-based, +pre-flight Aeff estimate was about 12% higher than the +median effective area subsequently measured on-orbit +(Egan et al. – this volume). We attribute the loss of +sensitivity to two possible causes: particulate contam- +ination during the failure of CUTE’s thermal-electric +cooling system (which occurred during thermal vacuum +testing) and contamination during the ∼2 months that +CUTE sat in the CubeSat dispenser at VSFB prior to +launch. A dry nitrogen purge was requested in order +to minimize optical degradation following dispenser in- +tegration, but was not made available. The difference +in effective area does not have a significant impact on +target selection and detectability, however, the larger +and variable thermal environment resulting from the loss +of the active cooling system removes most of the stars +fainter than the nominal target list. +Combining the CUTE effective area with the on-orbit +instrumental background level and the nominal 300 sec- +ond exposure time for CUTE’s exoplanet surveys, we +calculate the typical dispersion in the residual flux fol- +lowing background subtraction. This sets the minimum +flux level that can be detected above the noise in a +300 second spectrum, which we refer to as the back- +ground flux limit. We measure a background flux limit +of ≈ 5 × 10−14 erg cm−2 s−1 ˚A−1 at 3000 ˚A. +5. CUTE MISSION OPERATIONS +The CUTE spacecraft includes a UHF (437.25 MHz) +antenna with both transmission and receiving capabili- +ties. The UHF link is used for uploading commands to +the spacecraft and monitoring real-time telemetry dur- +ing ground passes. +CUTE also has an S-band (2402 +MHz) downlink-only mode for primary science data +transmission. The mission operations and ground sta- +tion for CUTE are located at the Laboratory for Atmo- +spheric and Space Physics in Boulder, Colorado. CUTE +typically has 1 – 2 high-elevation (> 50◦) passes and 1 – +2 low-elevation passes per day over the Boulder ground +station, resulting in approximately 10 minutes per day of +optimal downlink time. Figure 3 presents an illustration +of the CUTE science operations observing mode, includ- +ing science data acquisition, approximately monthly cal- +ibration activities, and data downlinks over the Boulder +ground station. + +CUTE Observing Summary +Sun-Synch Orbit permits high-efficiency observing & +positive power. Targets are seasonally available +Targets: roughly anti- +Primary survey will return to +sunward +same target over the course +3 - 8 weeks to complete 6 - +Primary +10 orbit campaign +Target +Instrumental dark and bias frames are +acquired at similar temperature and +illumination conditions as science +Science observing mode is +observations. Data downlink over +staring. Exposure times are +Boulder +typically 5 min, lightcurves +created over multiple orbits. +Calibration targets acquired +at approximately one-month +cadence to monitor +Calibration +sensitivity and detector +Target +health.8 +The CubeSat Operations Center at LASP utilizes the +LASP ground station initially built for CSSWE and +MinXSS, and the recently completed CSIM CubeSat +mission for NASA’s heliophysics division (Mason et al. +2016), using a combination of HYDRA and OASIS- +CC (Flynn et al. 2021) for command and control. The +mission operations are conducted by a team of profes- +sionals with experience from larger NASA flight missions +(e.g., Kepler, IXPE, and several heliophysics missions) +and a dedicated student operations group. The under- +graduate and graduate student operators perform mis- +sion planning, operations, and health and status moni- +toring for the spacecraft; the science team has developed +a graphical user interface tool to determine optimal tar- +get visibility and viewing conditions. The output of the +science planning tool is processed into CUTE’s weekly +operations plan to define the observing, charging, and +communication activities for the week. +To maximize operational simplicity, CUTE conducts +single-target campaigns: we schedule multiple transit +observations in a single command block (typically last- +ing 3 - 7 days) that is uploaded to the spacecraft. Each +command block includes calibration exposures, science +exposures, and data downlink periods. We repeat this +exercise until 6 – 10 transits of a given planet have been +executed. Interleaved with the transit observing blocks +are dedicated downlink periods, typically between 3 and +5 days, to re-transmit science and calibration data that +experienced low data-completion fractions in the initial +downlink or were lost to spacecraft resets. +The need +to execute 1 – 3 additional data downlinks per transit +campaign, driven by the frequent loss of fine-pointing +control and resets on the spacecraft bus (1 – 2 events +per week), is the limiting factor to CUTE’s operational +efficiency. +Pre-flight observation planning predicted 3 +transits per week could be successfully executed and +downlinked, which would complete 10 transits per tar- +get of the 10 target sample in approximately 8 months. +The realized mission efficiency is projected to complete +an average of 6 transits per target over a science mission +lifetime of ∼ 15 months, or approximately a factor of 3 +reduction in efficiency compared to pre-flight estimates. +A summary of the CUTE on-orbit operations and pay- +load commissioning will be presented in a forthcoming +paper (A. Suresh et al. - in prep.). +6. CUTE SCIENCE DATA EXAMPLE +In this section we present representative samples of +CUTE’s individual science data products and a prelim- +inary reduced transit light curve from the Early Release +Science program. Detailed analyses of the wavelength +dependent transit depths, interpretation and quantifica- +tion of atmospheric composition and escape rates, and +inter-comparison between different transit visits for the +Early Release Data program will be presented in up- +coming works (Egan et al. and Sreejith et al. 2022 – in +prep). The goal here is to present flight data of exoplan- +ets and their host stars to illustrate the features (and +limitations) of CUTE observations as revealed by the +first two targets of the Early Release Science program. +Figure 4 (top) presents a standard spectral data prod- +uct that CUTE transmits over the S-band downlink. +These are “TRIM2D” data products, 2048 × 100 pixel +two-dimensional spectra with a 5 minute exposure time. +The images are trimmed to reduce downlink volume. +The 5 minute exposure time is typical of all CUTE +spectra and is a balance of signal-to-noise for our tar- +get brightness, number of exposures possible per orbital +night, and simplicity of operational planning. +Each +transit visit is buffered by a number of bias and dark +exposures taken at similar celestial pointing, orbital po- +sition (latitude and longitude, and therefore similar tem- +perature and illumination conditions), and elevation an- +gle of the telescope with respect to the Earth limb. +These calibration files are used to remove thermal and +readout noise effects, as described in Egan et al. (2022). +Data processing beyond the downlink of the TRIM2D +data products occurs on the ground. +The two- +dimensional data are collapsed along a diagonal extrac- +tion region; the spectra are wavelength and flux cal- +ibrated using observations from the on-orbit commis- +sioning phase. Figure 4 (bottom) shows calibrated one- +dimensional spectra of WASP-189 and KELT-20, taken +outside of transit. The spectra are typical of NUV obser- +vations from A-type stars in the IUE archive, with the +most prominent feature being Mg II absorption in the +photosphere of these intermediate temperature stars. +The reader will also notice the defocus seen in the cross- +dispersion direction, manifest as the double-lobe struc- +ture that increases to shorter wavelengths across the +band. This defocus was introduced during an additional +payload vibration test that was not part of the original +test specifications but later required by NASA just prior +to the delivery of the spacecraft. This defocus was then +exacerbated during the powered ascent; there is no de- +tectable “breathing” of the focus with orbital location +although the background levels are strongly driven by +the spacecraft thermal and illumination conditions. +CUTE’s two-dimensional spectra are calibrated with +master bias frames before cosmic ray correction using +LAcosmic algorithm (van Dokkum 2001). We extract +the one-dimensional spectra from this corrected image +as described in Sreejith et al. (2022 – in prep). The +one-dimensional spectra are obtained by subtracting the + +9 +Figure 4. CUTE spectral observations of WASP-189 and KELT-20, A-type host stars of CUTE’s first Early Release Science targets. +The top plots shows a ”TRIM2D” 2048 × 100 pixel two-dimensional data product (exposure time = 5 minutes). +The bottom plots +show the one-dimensional spectral collapse including wavelength and flux calibration. A 10 pixel boxcar smooth has been applied to the +one-dimensional spectra for display purposes. +background, which are then wavelength calibrated. The +spectra are integrated over the wavelength region of +≈2540˚A to ≈3300˚A to create a light curve point. These +light curves can be created down to a wavelength resolu- +tion limit of ∼ 4 ˚A per bin, but for initial demonstration +purposes we display a broad NUV bandpass. +Figure 5 presents CUTE’s approximately 2540 – +3300 ˚A light curves for 3 different visits of the ultra- +hot Jupiter WASP-189b. +The best fit transit model, +taking into account wavelength-dependent stellar limb +darkening (shown in gray), will be presented in detail +in Sreejith et al. (2022 – in prep). The optical transit +light curve from CHEOPS (Lendl et al. 2020) is shown +for comparison and suggests excess transit absorption at +UV wavelength compared with the broadband geomet- +ric size of the planet. We demonstrate self-consistent +transit depth recoveries of ≈ 1.0 – 1.1 % over three sep- +arate transit observations of WASP-189b separated by +several weeks. +Excess planetary absorption at NUV wavelengths +is consistent with previous observations of ultra-hot +Jupiters observed with HST (Sing et al. 2019; Cubil- +los et al. 2020; Lothringer et al. 2022). A transit depth +of 1% in WASP-189b would indicate that the NUV tran- +sit observations are probing the extended upper atmo- +sphere of the planet that is subject to stellar high energy +radiation and escape. This is because the 1 microbar +level, used here as a rough proxy for the base of the +thermosphere, has a radius of about 1.1 Rp and a tran- +sit depth of 0.6%, based on an effective temperature of +2410 K and an atmosphere with solar abundances. In +contrast, a transit depth of 1% corresponds to a larger +radius of 1.4 Rp. If we assume a temperature of 8000 +K in the thermosphere, the pressure at 1.4 Rp would be +between 0.1 and 1 nbar. Given that this pressure is too +low for significant clouds and hazes, a pseudo-continuum +by these absorbers is unlikely and the broadband NUV +transit depth likely arises from a forest of metal ion lines +(e.g., Fossati et al. 2010; Sing et al. 2019). The individ- +ual absorption lines responsible would have to extend to +much higher radii than 1.4 Rp in transit to be detectable +in the broad NUV band. +We note that the preliminary lightcurves show sig- +nificant scatter beyond the photon noise limit at +this stage of the reduction. +Work is ongoing to +model the temperature- and orbital position-dependent +background to reduce the observed dispersion in the +lightcurves. +7. CONCLUSIONS +The CUTE CubeSat mission was launched in Septem- +ber 2021 and is currently carrying out its primary sci- +ence mission to collect NUV spectroscopy of transit- +ing planets. CUTE has successfully completed space- +craft and instrument commissioning and has completed +initial science observations on a number of exoplane- +tary systems. Optimal science targets are short-period, +Jovian-sized planets orbiting bright (V < 8) F and A +stars. +The science instrument has demonstrated sen- +sitivity to NUV white light transit depth of < 1% and +wavelength-dependent exoplanet atmosphere opacity in- +crease at λ ≲ 3300 ˚A. +We have presented the motivation for, mission design +of, and on-orbit characteristics of the CUTE mission. +The companion paper by Egan et al. present the details +of the on-orbit instrument performance of CUTE and +future papers will present mission science results as well +as information about the CUTE ground-segment data +pipeline, commissioning, and operations of the mission. +CUTE has supported mentoring and training for over 20 +early-career scientists and engineers. Mission operations +are planned to continue until June 2023 (currently lim- +ited by project funding) and the initial release of CUTE +data will be delivered to the NexSci archive in 2023. + +WASP-189 +KELT-20 +le-12 +1e-12 +4 +A +ergs s-1 +0.5 +Flux, +2500 +2600 +2700 +280029003000 +3100 +3200 +3300 +2500 +2600 +2700280029003000 +31003200 +3300 +Wavelength,A +Wavelength,A10 +Figure 5. Initial CUTE light curves of WASP-189b, showing three independent NUV (approximately 2540 – 3300 ˚A) light curves (black +points) and the best-fit transit models in gray. The plots compare the NUV band light curves with the optical light curve (in red) from +CHEOPS (Lendl et al. 2020). The NUV transits are significantly deeper than their broadband optical counterparts, indicating an effective +planetary radius increase of RP,NUV ≈ 1.5 RP,opt. +Acknowledgments: CUTE was developed and oper- +ated with the support to two NASA/APRA awards +to the Laboratory for Atmospheric and Space Physics +at the University of Colorado Boulder, NNX17AI84G +and +80NSSC21K1667. +A. +G. +S. +was +supported +by a Schr¨odinger Fellowship through the Austrian +Science +Fund +(FWF) +[J +4596-N]. +A. +G. +S. +and +L.F. acknowledge financial support from the Austrian +Forschungsf¨orderungsgesellschaft FFG project 859718 +and 865968. +AAV acknowledges funding from the +European Research Council (ERC) under the Euro- +pean Union’s Horizon 2020 research and innovation pro- +gramme (grant agreement No 817540, ASTROFLOW). +K.F. acknowledges the numerous and invaluable discus- +sions with colleagues excited about ultraviolet transit +science and the potential to do science with small satel- +lites. +The CUTE team wishes to specifically recog- +nize the amateur radio operator community and, and +SatNOGS network specifically, for hosting numerous +telemetry tracking tools that have improved the mis- +sion’s ability to recover from faults and understand long- +term spacecraft trends much more efficiently than would +have been otherwise possible. + +CUTE Transit 1 +CUTE Transit 2 +CUTE Transit 3 +Best Fit Model +Best Fit Model +Best Fit Model +1.01 +Peak Depth = 1.03% +1.01 +Peak Depth = 1.13% +1.01 +Peak Depth = 1.10% +NUV +1.00 +1.00 +1.00 +Normalized +0.99 +0.99 +0.99 +0.98 +0.98 +0.98 +-0.1 +0.0 +0.1 +-0.1 +0.0 +0.1 +-0.1 +0.0 +0.1 +Orbital Phase +Orbital Phase +Orbital Phase11 +REFERENCES +Bisikalo, D., Kaygorodov, P., Ionov, D., et al. 2013, ApJ, +764, 19, doi: 10.1088/0004-637X/764/1/19 +Bourrier, V., & Lecavelier des Etangs, A. 2013, A&A, 557, +A124, doi: 10.1051/0004-6361/201321551 +Bourrier, V., Lecavelier des Etangs, A., Ehrenreich, D., +Tanaka, Y. A., & Vidotto, A. A. 2016, A&A, 591, A121, +doi: 10.1051/0004-6361/201628362 +Burdanov, A., Delrez, L., Gillon, M., & Jehin, E. 2018, in +Handbook of Exoplanets, ed. H. J. Deeg & J. A. +Belmonte, 130, doi: 10.1007/978-3-319-55333-7 130 +Carolan, S., Vidotto, A. A., Villarreal D’Angelo, C., & +Hazra, G. 2021, MNRAS, 500, 3382, +doi: 10.1093/mnras/staa3431 +Casasayas-Barris, N., Pall´e, E., Yan, F., et al. 2019, A&A, +628, A9, doi: 10.1051/0004-6361/201935623 +Casasayas-Barris, N., Orell-Miquel, J., Stangret, M., et al. +2021, A&A, 654, A163, +doi: 10.1051/0004-6361/202141669 +Cauley, P. W., Shkolnik, E. L., Llama, J., & Lanza, A. F. +2019, Nature Astronomy, 3, 1128, +doi: 10.1038/s41550-019-0840-x +Charbonneau, D., Berta, Z. K., Irwin, J., et al. 2009, +Nature, 462, 891, doi: 10.1038/nature08679 +Cubillos, P. E., Fossati, L., Koskinen, T., et al. 2020, AJ, +159, 111, doi: 10.3847/1538-3881/ab6a0b +Deibert, E. K., de Mooij, E. J. W., Jayawardhana, R., et al. +2021, ApJL, 919, L15, doi: 10.3847/2041-8213/ac2513 +Dong, S., Shappee, B. J., Prieto, J. L., et al. 2016, Science, +351, 257, doi: 10.1126/science.aac9613 +Egan, A., France, K., Nell, N., et al. 2022, in Small Satellite +Conference, Vol. SS22, Small Satellite, WKII–02 +Egan, A., France, K., Fleming, B. T., et al. 2020, in Society +of Photo-Optical Instrumentation Engineers (SPIE) +Conference Series, Vol. 11444, Society of Photo-Optical +Instrumentation Engineers (SPIE) Conference Series, +1144406, doi: 10.1117/12.2559983 +Fleming, B. 2022, Nature Astronomy, 6, 1213, +doi: 10.1038/s41550-022-01807-8 +Fleming, B. T., France, K., Nell, N., et al. 2018, Journal of +Astronomical Telescopes, Instruments, and Systems, 4, +014004, doi: 10.1117/1.JATIS.4.1.014004 +Flynn, S., Byrant, K., Pilinski, E. B., et al. 2021, in Small +Satellite Conference, Vol. SS21, Small Satellite, +SSC21–S1–25 +Fossati, L., France, K., & Koskinen, T. 2015, +ApJ-submitted, 000, 0 +Fossati, L., Haswell, C. A., Froning, C. S., et al. 2010, +ApJL, 714, L222, doi: 10.1088/2041-8205/714/2/L222 +France, K., Hoadley, K., Fleming, B. T., et al. 2016a, +Journal of Astronomical Instrumentation, 5, 1640001, +doi: 10.1142/S2251171716400018 +France, K., Loyd, R. O. P., Youngblood, A., et al. 2016b, +ApJ, 820, 89, doi: 10.3847/0004-637X/820/2/89 +Fulton, B. J., & Petigura, E. A. 2018, AJ, 156, 264, +doi: 10.3847/1538-3881/aae828 +Gardner, J. P., Mather, J. C., Clampin, M., et al. 2006, +SSRv, 123, 485, doi: 10.1007/s11214-006-8315-7 +Gillon, M., Jehin, E., Magain, P., et al. 2011, in European +Physical Journal Web of Conferences, Vol. 11, European +Physical Journal Web of Conferences, 06002, +doi: 10.1051/epjconf/20101106002 +Ginzburg, S., Schlichting, H. E., & Sari, R. 2018, MNRAS, +476, 759, doi: 10.1093/mnras/sty290 +Hammerstein, E., van Velzen, S., Gezari, S., et al. 2022, +arXiv e-prints, arXiv:2203.01461. +https://arxiv.org/abs/2203.01461 +Haswell, C. A., Fosatti, L., & et al., A. 2012, ApJ, 0, 1 +Hoeijmakers, H. J., Ehrenreich, D., Kitzmann, D., et al. +2019, A&A, 627, A165, +doi: 10.1051/0004-6361/201935089 +Hoeijmakers, H. J., Seidel, J. V., Pino, L., et al. 2020, +A&A, 641, A123, doi: 10.1051/0004-6361/202038365 +Holoien, T. W. S., Stanek, K. Z., Kochanek, C. S., et al. +2017, MNRAS, 464, 2672, doi: 10.1093/mnras/stw2273 +Koskinen, T. T., Aylward, A. D., Smith, C. G. A., & +Miller, S. 2007, ApJ, 661, 515, doi: 10.1086/513594 +Koskinen, T. T., Harris, M. J., Yelle, R. V., & Lavvas, P. +2013a, Icarus, 226, 1678, doi: 10.1016/j.icarus.2012.09.027 +Koskinen, T. T., Lavvas, P., Huang, C., et al. 2022, ApJ, +929, 52, doi: 10.3847/1538-4357/ac4f45 +Koskinen, T. T., Yelle, R. V., Harris, M. J., & Lavvas, P. +2013b, Icarus, 226, 1695, doi: 10.1016/j.icarus.2012.09.026 +Lai, D., Helling, C., & van den Heuvel, E. P. J. 2010, ApJ, +721, 923, doi: 10.1088/0004-637X/721/2/923 +Lammer, H., Zerkle, A. L., Gebauer, S., et al. 2018, +A&A Rv, 26, 2, doi: 10.1007/s00159-018-0108-y +Lecavelier des Etangs, A., Bourrier, V., Wheatley, P. J., +et al. 2012, A&A, 543, L4, +doi: 10.1051/0004-6361/201219363 +Lendl, M., Csizmadia, S., Deline, A., et al. 2020, A&A, 643, +A94, doi: 10.1051/0004-6361/202038677 +Linsky, J. L., Yang, H., France, K., et al. 2010, ApJ, 717, +1291, doi: 10.1088/0004-637X/717/2/1291 +Llama, J., & Shkolnik, E. L. 2015, ApJ, 802, 41, +doi: 10.1088/0004-637X/802/1/41 +Llama, J., Wood, K., Jardine, M., et al. 2011, MNRAS, +416, L41, doi: 10.1111/j.1745-3933.2011.01093.x + +12 +Lothringer, J. D., Fu, G., Sing, D. K., & Barman, T. S. +2020, ApJL, 898, L14, doi: 10.3847/2041-8213/aba265 +Lothringer, J. D., Sing, D. K., Rustamkulov, Z., et al. 2022, +Nature, 604, 49, doi: 10.1038/s41586-022-04453-2 +Loyd, R. O. P., France, K., Youngblood, A., et al. 2018, +ApJ, 867, 71, doi: 10.3847/1538-4357/aae2bd +Masci, F. J., Laher, R. R., Rusholme, B., et al. 2019, +PASP, 131, 018003, doi: 10.1088/1538-3873/aae8ac +Mason, J. P., Woods, T. N., Webb, D. F., et al. 2016, ApJ, +830, 20, doi: 10.3847/0004-637X/830/1/20 +Murray-Clay, R. A., Chiang, E. I., & Murray, N. 2009, +ApJ, 693, 23, doi: 10.1088/0004-637X/693/1/23 +Nell, N., DeCicco, N., Ulrich, S., France, K., & Fleming, B. +2021, in Society of Photo-Optical Instrumentation +Engineers (SPIE) Conference Series, Vol. 11821, Society +of Photo-Optical Instrumentation Engineers (SPIE) +Conference Series, 1182117, doi: 10.1117/12.2594743 +Nichols, J. D., Wynn, G. A., Goad, M., et al. 2015, ApJ, +803, 9, doi: 10.1088/0004-637X/803/1/9 +Owen, J. E., & Wu, Y. 2017, ApJ, 847, 29, +doi: 10.3847/1538-4357/aa890a +Pepper, J., Pogge, R. W., DePoy, D. L., et al. 2007, PASP, +119, 923, doi: 10.1086/521836 +Pollacco, D. L., Skillen, I., Collier Cameron, A., et al. 2006, +PASP, 118, 1407, doi: 10.1086/508556 +Ramiaramanantsoa, T., Bowman, J. D., Shkolnik, E. L., +et al. 2021, arXiv e-prints, arXiv:2111.02443. +https://arxiv.org/abs/2111.02443 +Rigby, J., Perrin, M., McElwain, M., et al. 2022, arXiv +e-prints, arXiv:2207.05632. +https://arxiv.org/abs/2207.05632 +Simard, L., Ellerbroek, B., Bhatia, R., Radovan, M., & +Chisholm, E. 2016, in Society of Photo-Optical +Instrumentation Engineers (SPIE) Conference Series, +Vol. 9908, Ground-based and Airborne Instrumentation +for Astronomy VI, ed. C. J. Evans, L. Simard, & +H. Takami, 99081V, doi: 10.1117/12.2234058 +Sing, D. K., Lavvas, P., Ballester, G. E., et al. 2019, AJ, +158, 91, doi: 10.3847/1538-3881/ab2986 +Turner, J. D., Christie, D., Arras, P., Johnson, R. E., & +Schmidt, C. 2016, MNRAS, 458, 3880, +doi: 10.1093/mnras/stw556 +Turner, J. D., de Mooij, E. J. W., Jayawardhana, R., et al. +2020, ApJL, 888, L13, doi: 10.3847/2041-8213/ab60a9 +van Dokkum, P. G. 2001, PASP, 113, 1420, +doi: 10.1086/323894 +Vidal-Madjar, A., Lecavelier des Etangs, A., D´esert, J.-M., +et al. 2003, Nature, 422, 143, doi: 10.1038/nature01448 +Vidal-Madjar, A., D´esert, J.-M., Lecavelier des Etangs, A., +et al. 2004, ApJL, 604, L69, doi: 10.1086/383347 +Vidal-Madjar, A., Huitson, C. M., Bourrier, V., et al. 2013, +A&A, 560, A54, doi: 10.1051/0004-6361/201322234 +Vidotto, A. A., Jardine, M., & Helling, C. 2010, ApJL, 722, +L168, doi: 10.1088/2041-8205/722/2/L168 +Villarreal D’Angelo, C., Esquivel, A., Schneiter, M., & +Sgr´o, M. A. 2018, MNRAS, 479, 3115, +doi: 10.1093/mnras/sty1544 +Wilkinson, E. 2002, HST-COS Instrument Performance +Reports, 25, COS + diff --git a/R9E0T4oBgHgl3EQfUgAs/content/tmp_files/load_file.txt b/R9E0T4oBgHgl3EQfUgAs/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7696b612884a5fbe94a83c9f67e8eafa742c6316 --- /dev/null +++ b/R9E0T4oBgHgl3EQfUgAs/content/tmp_files/load_file.txt @@ -0,0 +1,927 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf,len=926 +page_content='Draft version January 9, 2023 Typeset using LATEX twocolumn style in AASTeX631 The Colorado Ultraviolet Transit Experiment (CUTE) Mission Overview Kevin France,1 Brian Fleming,1 Arika Egan,1 Jean-Michel Desert,2 Luca Fossati,3 Tommi T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Koskinen,4 Nicholas Nell,1 Pascal Petit,5 Aline A.' 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+page_content=' The Colorado Ultraviolet Transit Experiment (CUTE) is a CubeSat mission designed to take advantage of the near-ultraviolet stellar brightness distribution to conduct a survey of the extended atmospheres of nearby close-in planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' The CUTE payload is a magnifying NUV (2479 – 3306 ˚A) spectrograph fed by a rectangular Cassegrain telescope (206mm × 84mm);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' the spectrogram is recorded on a back- illuminated, UV-enhanced CCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' The science payload is integrated into a 6U Blue Canyon Technology XB1 bus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' CUTE was launched into a polar, low-Earth orbit on 27 September 2021 and has been conducting this transit spectroscopy survey following an on-orbit commissioning period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' This paper presents the mission motivation, development path, and demonstrates the potential for small satellites to conduct this type of science by presenting initial on-orbit science observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' The primary science mission is being conducted in 2022 – 2023, with a publicly available data archive coming on line in 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' INTRODUCTION The history of observational astronomy has been marked by the push to ever larger and more capable telescopes and instruments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' The 2010s witnessed the development of a new generation of large astronomical observatories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Both on the ground and in space, facilities 2 – 3 times the primary mirror diameter of the previous state-of-the-art were brought closer to fruition for im- plementation in the 2020s, 2030s, and 2040s, including Corresponding author: Kevin France kevin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='france@colorado.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='edu the James Webb Space Telescope (Gardner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Rigby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2022), thirty-meter class ground-based tele- scopes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Simard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2016), and advanced ultravio- let/optical (UV/O) facilities such as the Large Ultravio- let/ Optical/ Infrared Surveyor (LUVOIR;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' LUVOIR Fi- nal Report 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' The large mission studies conducted ahead of the 2020 Decadal Survey on Astronomy and As- trophysics drove the recommendation for NASA’s suite of Future Great Observatories, a series of probe- and flagship-class missions offering many order-of-magnitude gains in the scientific grasp across numerous areas of as- trophysics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' In parallel with this large observatory de- velopment, numerous small telescope arrays have come on-line or have been expanded, and NASA’s science divi- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='02250v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='IM] 5 Jan 2023 2 sions made significant new investments in small satellites covering a range of scientific topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Small telescopes at ground-based sites have excelled at detecting and characterizing new objects in the time- variable sky, including supernovae eruptions (Dong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2016) and tidal disruption events (Hammerstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2022) from the Zwicky Transient Facility (Masci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2019) and All Sky Automated Survey for SuperNovae (ASAS-SN) (Holoien et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' The impact of small telescopes has also been powerful for the detection of extrasolar planets, including many Jovian-sized plan- ets from Wide-Angle Search for Planets (WASP) (Pol- lacco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2006) and the Kilodegree Extremely Lit- tle Telescope (KELT) (Pepper et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2007), and some of the most promising rocky planets for study with JWST from the MEarth (Charbonneau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2009), Transiting Planets and Planetesimals Small Telescope (TRAPPIST) (Gillon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2011), and Search for hab- itable Planets EClipsing ULtra-cOOl Stars (SPECU- LOOS) (Burdanov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2018) facilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' The recent decadal survey highlighted the power of small space-based telescopes, astronomical CubeSats and smallsats, for “monitoring of sources for weeks or months at time, and at wavelengths not acces- sible from the ground”, complementing the Hubble Space Telescope’s surveys in areas of transmission spec- troscopy (Sing et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Cubillos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2020) and ex- oplanet host star radiation fields (France et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2016b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Loyd et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Ramiaramanantsoa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' NASA has embraced this opportunity with a dedicated fund- ing line for astrophysics CubeSats (full mission life cycle cost < $10M) and the Pioneers program (mission cost $10M – $20M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' In this paper, we present an overview of NASA’s first UV astronomy CubeSat and the first grant- funded small satellite dedicated to the characterization of extrasolar planetary atmospheres, the Colorado Ul- traviolet Transit Experiment (CUTE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' CUTE conducts transit spectroscopy of short-period, giant planets in the near-UV (2479 – 3306 ˚A) bandpass to access strong atomic transitions tracing atmospheric escape and the near-UV spectral slope of giant planet atmospheres that provide constraints on their compo- sition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' This paper presents the science background for and the technical implementation of the mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' The manuscript is laid out as follows: the scientific motiva- tion for CUTE and its science objectives are presented in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Because CUTE is one of the first astron- omy missions to be developed in a CubeSat framework, we present a description of the mission development and implementation path in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Section 4 presents the instrument design and high-level performance specifica- tions (see also Fleming et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2018 for a description of CUTE’s science payload).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Section 5 describes CUTE’s mission operations and we present early-release exam- ples of the mission’s on-orbit science data in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' We conclude with a brief summary in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' A detailed description of CUTE’s science instrument and on-orbit performance is presented in a companion paper by A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Egan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Mission operations and on-orbit commis- sioning (Suresh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' – in prep), CUTE’s on-orbit data pipeline (Sreejith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' – in press), and Early Release Science results (Egan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' – in prep;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Sreejith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' – in prep) will be described in forthcoming papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' CUTE SCIENCE OBJECTIVES Planetary escape processes play a key role in de- termining the chemical and physical state of plan- ets both within and beyond our solar system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' At- mospheric escape is thought to create the radius gap observed in the distribution of short-period exoplan- ets (Fulton & Petigura 2018), likely driven by a com- bination of photoevaporative (Owen & Wu 2017) and core-powered (Ginzburg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2018) mass loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Escape is also a fundamental process in the evolution of terres- trial worlds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' For a planet to be habitable, our current view is that it must lose its primordial hydrogen atmo- sphere and acquire/generate (and retain) a secondary atmosphere (Lammer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Atmospheric escape is known to have shaped the early atmospheres of Venus, Earth, and Mars, which subsequently followed differ- ent evolutionary paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' The rapid hydrodynamic escape that is believed to have affected Venus, Earth and Mars in the past no longer takes place on any planet in the solar system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Therefore, we turn to short-period extra- solar planets as laboratories on which to study vigorous atmospheric loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' The first detection of exoplanet atmospheric escape was achieved by Vidal-Madjar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' (2003) who used HI Lyα transit observations in the far-ultraviolet (FUV) to observe the extended atmosphere of the Hot Jupiter HD209458b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' This was followed by the detection of O I, C II, Si III and Mg I on the same planet (Vidal-Madjar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2004, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Linsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' These initial obser- vations inspired several independent groups to develop 1D and 3D models to study both the physical character- istics of the upper atmospheres of close-in planets and the escaping gas and plasma surrounding them (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Koskinen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Murray-Clay et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Koskinen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2013a,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Bourrier & Lecavelier des Etangs 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Bourrier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Villarreal D’Angelo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Car- olan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' The interpretation of FUV transit measurements has often been controversial (see Fossati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2015 for a discussion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Recently, several atmospheric escape 3 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' The CUTE instrument development from con- cept (instrument schematic, top) to telescope characteriza- tion (CUTE flight telescope in the test facilities at the Uni- versity of Colorado, middle), to pre-delivery in-band spectral resolution test data (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' studies have shifted to the near-ultraviolet (NUV), where the stellar flux is much higher than in the FUV and the light curves are measured against a better- understood intensity distribution from the stellar photo- sphere (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=',Haswell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Llama & Shkolnik 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' The NUV includes the Fe II complexes near 2400 and 2600 ˚A, the Mg II doublet at 2796/2803 ˚A, the Mg I line at 2852 ˚A, some of which have been detected on the Hot Jupiters WASP-12b, HD209458b, and WASP- 121b (Fossati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Sing et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Cubillos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' We note that the Fe II and Mg II resonance lines in the near-UV trace the highly extended (and poten- tially escaping) exoplanet atmosphere, whereas optical band metal line detections made with ground-based tele- scopes trace the lower, bound atmospheric layers (Hoei- jmakers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Casasayas-Barris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Cauley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Turner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Hoeijmakers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Casasayas-Barris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Deibert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' The NUV also contains a pseudo-continuum that can probe scattering by high altitude clouds and gas phase silicon and magnesium (Lothringer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2022), as well as the A – X bands of OH (3100 ˚A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Furthermore, NUV trans- mission spectra give the unique opportunity to constrain the composition of the aerosols lying in the lower atmo- spheres (Cubillos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Depending on the temperature profile in the atmo- sphere, species like Si, Mg, and Fe are expected to condense to form clouds in the lower atmosphere, how- ever, the calculations indicate that strong mixing, either by turbulence or global circulation, can inhibit cloud formation or allow for these species to be present in the upper atmosphere where they can escape (Koski- nen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2013b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Cubillos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Koskinen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' The comparison of continuum and atomic line absorption therefore acts as a diagnostic of cloud for- mation, elemental abundances and mass loss on close-in exoplanets (Lothringer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Cubillos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Model outputs can be used to translate observed plane- tary transit light curves into global mass-loss rates: the depth and shape of the light curves directly relate to the atmospheric parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Finally, UV transits with HST have provided ev- idence for time-variability, potentially arising from changing stellar high-energy input, orbital timescale changes in the planet’s atmosphere, or variation in the star-planet magnetic environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Lecavelier des Etangs et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' (2012) observed time-variable neutral hy- drogen absorption in FUV transit observations of HD 189733b, possible due to the influence of high-energy stellar flares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' NUV transit observations of the close-in giant planet WASP-12b by Fossati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' (2010) found that the transit light curve of WASP-12b presents both an early ingress when compared to its optical transit and excess absorption during the transit (see also Haswell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Nichols et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Possible explanations include atmospheric hydrodynamic mass-loss support- ing a shock upstream of the planet’s orbit or generating 2536.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='6 2705 2967.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='3 310g 3125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='6 3131.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='4 an accretion stream that produces an early ingress (Lai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Bisikalo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Turner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2016) and a magnetically supported bow-shock 4 – 5 planetary radii upstream of the planet’s orbital motion (analogous to the Earth-Sun system;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Vidotto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Llama et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' CUTE’s primary science goal is to provide new con- straints on the physics and chemistry of hot, Jovian-size exoplanets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' The CUTE mission addresses this goal with the following observing program: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Measure NUV transmission spectra for a small sur- vey of approximately 10 short period planets 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Infer atmospheric escape rates and constrain the composition of the upper atmospheres of hot giant planets 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Measure temporal variability in UV transit light curves by observing 6 – 10 transit observations per planet 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Measure out-of-transit baseline fluxes to better characterize the stellar inputs to the planet’s at- mosphere and to capture light curve asymmetries CUTE’s instrument design and mission implementa- tion was developed to enable the four key goals of the ob- serving program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' The spectral coverage and resolution of the CUTE (∆ v < 300 km s−1) spectrograph provides ample separation of the relevant atomic, molecular, and continuum bands in this range (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Figure 8 of Sing et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' CUTE’s mission design complements the instrument to meet the science goals of the mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' (1) We couple observations of the NUV continuum opacity, individual ionic tracers (Fe II, Mg II) with atmospheric chemistry, and hydrodynamic escape models to deter- mine mass loss rates for CUTE’s targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' The sample size is driven by a combination of mission lifetime and instrumental sensitivity considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' (2) CUTE mea- sures the amplitude and slope of the NUV transmission curve to provide constraints on the chemistry and struc- ture of the escaping atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' The instrumental ef- fective area was specified to enable multiple, wavelength resolved, NUV bands with sufficient photometric pre- cision to distinguish the NUV transit radius from the white-light radius of the planet on all 10 targets and isolate transit spectra of the strongest absorption lines (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Fe II and Mg II) on the brightest targets (address- ing goals 1 and 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' The target sample was defined by estimating the detectability of excess NUV absorption;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' a combination of stellar brightness (V-magnitude), spec- tral type (A- and F-type stars have spectral energy dis- tributions peaked in the NUV), planetary radius, effec- tive planetary surface temperature, and gravity (hotter, lower-mass planets being more likely to exhibit extended atmospheres).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' (3) CUTE’s point-stare-repeat concept of operations is designed to make numerous visits to the same planet over the course of 4 to 8 weeks, building signal-to-noise for fainter targets and enabling measure- ments of light curve variability for brighter targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' (4) The same point-stare-repeat observing mode provides a wide stellar baseline to measure changes in the Mg II ac- tivity and the increased dispersion of the photospheric and chromospheric continuum flux that indicate vari- ability in the star’s escape-driving XUV output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' MISSION IMPLEMENTATION PATH CUTE is NASA’s first grant-funded UV/ Optical/ In- frared small satellite and first dedicated exoplanet spec- troscopy mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Given the novelty of this mission for- mat for astrophysics science missions, we present a brief overview of the process, schedule, and cost of the mission here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' The initial motivation for CUTE was discussed at a Keck/KISS workshop on exoplanet magnetic fields in August 2013, with the final science and measurement concept in place by the summer of 2015 following nu- merous informal discussions at science conferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Fall 2015 was spent on science measurement definition and the development of the CUTE instrument design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' CUTE was proposed as a four-year program through NASA’s ROSES2015 call (submitted in March 2016), at an initial cost-to-launch of $3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='3M, comparable to an astrophysics sounding rocket proposal but considerably lower cost than a stratospheric balloon program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' CUTE was proposed and selected prior to the initiation of ded- icated funding for astrophysics CubeSats, leading to a long delay between proposal submission and the start of funding (approximately 16 months;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' there was no Phase A or Concept Study period for CUTE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Long lead-time items such as the CUTE spacecraft bus, the rectan- gular telescope, holographically-ruled diffraction grat- ing, and NUV-optimized CCD detectors were ordered a few months after selection in fall 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' The CUTE spacecraft (Blue Canyon Technology, BCT) costs in- creased relative to the quote provided for the pro- posal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' To accommodate the cost increases in a mis- sion class without reserves, several descopes were im- plemented, including scaling back the spacecraft’s atti- tude control system to a single star-tracker and elim- inating engineering model radios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' In 2018 and 2019, we developed the hardware test facilities that comple- mented the University of Colorado’s existing UV vac- uum calibration facilities (France et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2016a) and con- ducted component-level characterization (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', groove ef- ficiency of the diffraction gratings, trade study of Al vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Al+MgF2 grating coatings).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Instrument assembly and 5 characterization, integration into the spacecraft, and pre-delivery environmental testing (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', vibration test- ing, comprehensive performance testing, thermal vac- uum testing, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=') were completed in 2020 and 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' The duration from the start of CUTE funding to de- livery of the completed observatory was almost exactly four years, although approximately 10 months of sched- ule were lost to the COVID-19 pandemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' CUTE proposed to NASA’s CubeSat Launch Initia- tive (CSLI) for launch support in fall 2017 and was selected for flight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' The proposed spacecraft orbit, in- cluding the initial CUTE mission requirement docu- mentation submitted to CSLI, requested a dawn-dusk (terminator), sun-synchronous orbit to enable uninter- rupted orbital phase coverage of transiting planets and to minimize day/night thermal variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Orbital alti- tude (450 - 600km) was a secondary consideration driven by desired mission lifetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' CSLI was unable to accom- modate the requested sun-synchronous orbit within the time-period covered by the mission funding and CUTE was instead manifested in November 2019 as a secondary payload on NASA’s Landsat 9 mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' The Landsat 9 launch was scheduled 8 months after CUTE’s tar- geted launch window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' As a result, NASA provided a 6 month, $0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='5M extension to the CUTE program in Fall 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Starting in Spring 2020, COVID delayed the Landsat 9 launch by another 9 months.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' COVID also im- peded CUTE’s development timescale owing to supply- chain delays and the challenges of getting students, sci- entists, and engineers into CUTE’s labs for continued testing and development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' The CUTE mission submit- ted a follow-on, competed APRA proposal in December 2020 to conduct mission operations and carry out the science program (bringing CUTE’s cost to complete the full science mission to approximately $5M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Of the 18 CubeSat missions originally manifested with Landsat 9, only four (including CUTE) would ultimately deliver and be flown on the mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' The integrated and tested observatory was delivered to NASA at Vandenberg Space Force Base (VSFB) in July 2021, and the CUTE team supported installation into the CubeSat dispenser on the ESPA ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' The mis- sion was launched on September 27 2021 into a sun- synchronous orbit (≈98◦, 10am Local Time of Ascend- ing Node;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' LTAN) with a 560km apogee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' CUTE de- ployed from the dispenser approximately two hours af- ter launch;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' solar arrays deployed and the communi- cation beacon started approximately 30 minutes later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' CUTE’s beacons were identified by the amateur RF community on the first orbit and communications were established with the ground station at the University of Colorado on September 28 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' We refer the reader to Section 4 and Suresh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' (in preparation) for a description of the CUTE ground segment and commis- sioning program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' We refer the reader to the SmallSat Conference proceeding by Egan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2022 for a discus- sion of lessons learned during CUTE development and early on-orbit operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' CUTE is part of NASA’s suborbital program, where student training and early-career mentorship are key in- gredients to the definition of mission success.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' CUTE’s approach was built off of the framework of the NASA Sounding Rocket Program, which has a long history in the professional development of NASA’s space scientist workforce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' The core science and instrument team (de- fined as those working with CUTE for more than 2 years of the implementation phase) included two Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' candi- dates (in astrophysics and aerospace engineering), four undergraduates, two postdoctoral researchers, two early career engineers (CUTE was the first job post-bachelors degree for the mission’s lead mechanical and electrical engineers), and the early-career project scientist (Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Brian Fleming, who became the PI of a NASA sound- ing rocket program and the SPRITE CubeSat (Fleming 2022) mission during the course of the CUTE develop- ment phase).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Over the course of CUTE’s component- level and instrument test phase, the project employed another six undergraduate students in various labora- tory and science program development tasks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', tar- get field checking for crowded fields).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' In addition to this, the operations team for CUTE (see Section 5) included an additional two undergraduate students, two graduate students, and one flight software undergraduate student.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Taken as a whole, CUTE supported the mentoring and training for over 20 early-career scientists and engineers through the completion of the on-orbit commissioning phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' IMPLEMENTATION: THE CUTE SCIENCE PAYLOAD The CUTE payload is a magnifying NUV spectro- graph fed by a rectangular Cassegrain telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' The spectrogram is recorded on a back-illuminated, UV- enhanced e2v CCD42-10 that is maintained at a nom- inal operating temperature (−15 – −5◦ C) by passive cooling through a radiator panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' CUTE employs the BCT XB1 bus to provide critical subsystems including power, command and data handling, communications, and attitude control (ADCS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Figure 1 shows an instru- ment schematic, optical testing of the telescope, and an in-band calibration spectrum from the flight instrument prior to instrument integration into the BCT chassis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' The CUTE instrument is housed in 4U of the 6U spacecraft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' The CUTE aperture is a 206 × 84 mm, 6 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' CUTE calibration observations of the O4 supergiant ζ Puppis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' The top plot shows the full-frame (2048 × 515 pixel) calibration image and the bottom plot shows the extracted one- dimensional spectrum (flux calibrated against archival HST and IUE spectra).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' ζ Puppis was selected as a calibration target be- cause of the wealth of archival NUV spectra and the high photo- spheric temperature ensures that iron and magnesium lines in the spectrum are narrow, interstellar features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' f/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='75 (in the cross-dispersion axis) primary mirror that is part of the f/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='6 Cassegrain telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' The rectan- gular shape of the primary is matched to the long axis of the 6U CubeSat chassis and allows for 3 times more throughput than a 1U circular aperture (Fleming et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' The hyperbolic secondary mirror is cantilevered off of the primary mirror, which serves as the bench for the optical instrument, by means of an Invar tower (see Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' A 15 × 6 mm fold mirror redirects the beam 90◦ through a 141 µm × 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='5 mm (60′′ × 1400′′ projected) slit at the Cassegrain focus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' The slit, manufactured by OSH Stencils, was polished on the incident side and an- gled 45◦ about the slit axis to redirect the field to an aspect camera for use in telescope performance testing and alignment with the BCT spacecraft during integra- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' The rectangular telescope design optimizes col- lecting area within the mass-volume constraints of the cubesat form factor, while the large sky field-of-view, increased cost, and mechanical stress at the primary mirror-secondary tower interface add design complica- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Once through the slit, the starlight is diffracted, redi- rected, and magnified by a spherical, R = 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='1mm ra- dius, 1714 gr mm−1 aberration correcting, ion-etched holographic grating fabricated by Horiba Jobin-Yvon (Horiba J-Y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' The holographic grating design was adopted to minimize scattered light in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' A second fold mirror with an Rx = 300 mm radius of curva- ture about the cross-dispersion dimension provides ad- ditional aberration corrections before the beam reaches the CCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' The final beam focal ratio is f/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='5 in the Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' CUTE Instrument Specifications Instrument Metric On-orbit Value Bandpass 2479 – 3306 ˚A Spectral Resolutiona 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='9 ˚A Cross-Dispersion Resolutionb ≈ 30′′ Peak Aeff 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='5 cm2 at 2500 ˚A Background Flux Limit 5 × 10−14 erg cm−2 s−1 ˚A−1 in 300sb aAverage resolution over the bandpass, including spacecraft jit- ter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' bEvaluated at 3000 ˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' cross-dispersion axis, with a detector plate scale of 186′′ mm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' The detector and custom avionics were tested and flight ruggedized by the CUTE team in their on- campus laboratories at the University of Colorado (Nell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' The telescope was delivered fully assembled to CU by Nu-Tek Precision Optical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' All mirrors are coated with MgF2 + Al to prevent the formation of an oxide layer (AlO3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' We elected to receive flight and flight-backup gratings coated in bare Al and MgF2 + Al, respectively (coated by Horiba J-Y), to control for a potential effi- ciency anomaly similar to that seen on the COS NUV gratings (Wilkinson 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Detailed pre-flight efficiency and environmental testing showed better performance with the bare Al grating, without a measurable loss in efficiency over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' As a result, the instrument team elected to fly the bare Al-coated mirror on the flight in- strument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' The design-prediction flight instrument per- formance curves are presented in Fleming et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' (2018) and the on-orbit instrument performance of the CUTE payload is presented in Egan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' (2022 – this volume);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' we provide a brief summary of the key performance met- rics in the following subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Instrument Specifications The final bandpass recorded by the CCD detector is 2479 – 3306 ˚A (see Table 1), which is a slight change from the pre-flight projection owing to shifts in the op- tical system during ascent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' The exact bandpass also varies by several ˚A depending on the alignment of the stellar point spread function (PSF) in the spectrograph slit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' The spectral resolving power of the instrument is ≈ 750 (∆λ ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='3 – 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='5 ˚A across the bandpass, including the effects of spacecraft pointing jitter).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Figure 2 shows a representative calibration spectrum from the CUTE on-orbit commissioning program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 400 200 0 1 A 1e-9 z- 2 2500 2600 2700 2800 2900 3000 3100 3200 3300 Wavelength,A7 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Schematic description of CUTE science and calibration observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' The system effective area is a function of the reflec- tion efficiency of the optics (R), efficiency of the grating (ϵg), and quantum efficiency of the detector (DQE), mul- tiplied by the geometric collecting area of the telescope (Ageo): Aeff(λ) = AgeoR5(λ)ϵg(λ)DQE(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' (1) The on-orbit effective area was measured by comparing CUTE’s observations (in units of electrons s−1 ˚A−1) with flux-calibrated observations from the IUE and HST archives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' We measured Aeff = 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='5 – 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='0 cm2 across the CUTE spectral range with a peak at ap- proximately 2500 ˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Component-level efficiencies were measured prior to instrument assembly in the UV cali- bration facilities at the University of Colorado (France et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2016a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Egan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' The component-based, pre-flight Aeff estimate was about 12% higher than the median effective area subsequently measured on-orbit (Egan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' – this volume).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' We attribute the loss of sensitivity to two possible causes: particulate contam- ination during the failure of CUTE’s thermal-electric cooling system (which occurred during thermal vacuum testing) and contamination during the ∼2 months that CUTE sat in the CubeSat dispenser at VSFB prior to launch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' A dry nitrogen purge was requested in order to minimize optical degradation following dispenser in- tegration, but was not made available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' The difference in effective area does not have a significant impact on target selection and detectability, however, the larger and variable thermal environment resulting from the loss of the active cooling system removes most of the stars fainter than the nominal target list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Combining the CUTE effective area with the on-orbit instrumental background level and the nominal 300 sec- ond exposure time for CUTE’s exoplanet surveys, we calculate the typical dispersion in the residual flux fol- lowing background subtraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' This sets the minimum flux level that can be detected above the noise in a 300 second spectrum, which we refer to as the back- ground flux limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' We measure a background flux limit of ≈ 5 × 10−14 erg cm−2 s−1 ˚A−1 at 3000 ˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' CUTE MISSION OPERATIONS The CUTE spacecraft includes a UHF (437.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='25 MHz) antenna with both transmission and receiving capabili- ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' The UHF link is used for uploading commands to the spacecraft and monitoring real-time telemetry dur- ing ground passes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' CUTE also has an S-band (2402 MHz) downlink-only mode for primary science data transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' The mission operations and ground sta- tion for CUTE are located at the Laboratory for Atmo- spheric and Space Physics in Boulder, Colorado.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' CUTE typically has 1 – 2 high-elevation (> 50◦) passes and 1 – 2 low-elevation passes per day over the Boulder ground station, resulting in approximately 10 minutes per day of optimal downlink time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Figure 3 presents an illustration of the CUTE science operations observing mode, includ- ing science data acquisition, approximately monthly cal- ibration activities, and data downlinks over the Boulder ground station.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' CUTE Observing Summary Sun-Synch Orbit permits high-efficiency observing & positive power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Targets are seasonally available Targets: roughly anti- Primary survey will return to sunward same target over the course 3 - 8 weeks to complete 6 - Primary 10 orbit campaign Target Instrumental dark and bias frames are acquired at similar temperature and illumination conditions as science Science observing mode is observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Data downlink over staring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Exposure times are Boulder typically 5 min, lightcurves created over multiple orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Calibration targets acquired at approximately one-month cadence to monitor Calibration sensitivity and detector Target health.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='8 The CubeSat Operations Center at LASP utilizes the LASP ground station initially built for CSSWE and MinXSS, and the recently completed CSIM CubeSat mission for NASA’s heliophysics division (Mason et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2016), using a combination of HYDRA and OASIS- CC (Flynn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2021) for command and control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' The mission operations are conducted by a team of profes- sionals with experience from larger NASA flight missions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Kepler, IXPE, and several heliophysics missions) and a dedicated student operations group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' The under- graduate and graduate student operators perform mis- sion planning, operations, and health and status moni- toring for the spacecraft;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' the science team has developed a graphical user interface tool to determine optimal tar- get visibility and viewing conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' The output of the science planning tool is processed into CUTE’s weekly operations plan to define the observing, charging, and communication activities for the week.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' To maximize operational simplicity, CUTE conducts single-target campaigns: we schedule multiple transit observations in a single command block (typically last- ing 3 - 7 days) that is uploaded to the spacecraft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Each command block includes calibration exposures, science exposures, and data downlink periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' We repeat this exercise until 6 – 10 transits of a given planet have been executed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Interleaved with the transit observing blocks are dedicated downlink periods, typically between 3 and 5 days, to re-transmit science and calibration data that experienced low data-completion fractions in the initial downlink or were lost to spacecraft resets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' The need to execute 1 – 3 additional data downlinks per transit campaign, driven by the frequent loss of fine-pointing control and resets on the spacecraft bus (1 – 2 events per week), is the limiting factor to CUTE’s operational efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Pre-flight observation planning predicted 3 transits per week could be successfully executed and downlinked, which would complete 10 transits per tar- get of the 10 target sample in approximately 8 months.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' The realized mission efficiency is projected to complete an average of 6 transits per target over a science mission lifetime of ∼ 15 months, or approximately a factor of 3 reduction in efficiency compared to pre-flight estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' A summary of the CUTE on-orbit operations and pay- load commissioning will be presented in a forthcoming paper (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Suresh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' - in prep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' CUTE SCIENCE DATA EXAMPLE In this section we present representative samples of CUTE’s individual science data products and a prelim- inary reduced transit light curve from the Early Release Science program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Detailed analyses of the wavelength dependent transit depths, interpretation and quantifica- tion of atmospheric composition and escape rates, and inter-comparison between different transit visits for the Early Release Data program will be presented in up- coming works (Egan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' and Sreejith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2022 – in prep).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' The goal here is to present flight data of exoplan- ets and their host stars to illustrate the features (and limitations) of CUTE observations as revealed by the first two targets of the Early Release Science program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Figure 4 (top) presents a standard spectral data prod- uct that CUTE transmits over the S-band downlink.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' These are “TRIM2D” data products, 2048 × 100 pixel two-dimensional spectra with a 5 minute exposure time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' The images are trimmed to reduce downlink volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' The 5 minute exposure time is typical of all CUTE spectra and is a balance of signal-to-noise for our tar- get brightness, number of exposures possible per orbital night, and simplicity of operational planning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Each transit visit is buffered by a number of bias and dark exposures taken at similar celestial pointing, orbital po- sition (latitude and longitude, and therefore similar tem- perature and illumination conditions), and elevation an- gle of the telescope with respect to the Earth limb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' These calibration files are used to remove thermal and readout noise effects, as described in Egan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Data processing beyond the downlink of the TRIM2D data products occurs on the ground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' The two- dimensional data are collapsed along a diagonal extrac- tion region;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' the spectra are wavelength and flux cal- ibrated using observations from the on-orbit commis- sioning phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Figure 4 (bottom) shows calibrated one- dimensional spectra of WASP-189 and KELT-20, taken outside of transit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' The spectra are typical of NUV obser- vations from A-type stars in the IUE archive, with the most prominent feature being Mg II absorption in the photosphere of these intermediate temperature stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' The reader will also notice the defocus seen in the cross- dispersion direction, manifest as the double-lobe struc- ture that increases to shorter wavelengths across the band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' This defocus was introduced during an additional payload vibration test that was not part of the original test specifications but later required by NASA just prior to the delivery of the spacecraft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' This defocus was then exacerbated during the powered ascent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' there is no de- tectable “breathing” of the focus with orbital location although the background levels are strongly driven by the spacecraft thermal and illumination conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' CUTE’s two-dimensional spectra are calibrated with master bias frames before cosmic ray correction using LAcosmic algorithm (van Dokkum 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' We extract the one-dimensional spectra from this corrected image as described in Sreejith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' (2022 – in prep).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' The one-dimensional spectra are obtained by subtracting the 9 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' CUTE spectral observations of WASP-189 and KELT-20, A-type host stars of CUTE’s first Early Release Science targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' The top plots shows a ”TRIM2D” 2048 × 100 pixel two-dimensional data product (exposure time = 5 minutes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' The bottom plots show the one-dimensional spectral collapse including wavelength and flux calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' A 10 pixel boxcar smooth has been applied to the one-dimensional spectra for display purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' background, which are then wavelength calibrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' The spectra are integrated over the wavelength region of ≈2540˚A to ≈3300˚A to create a light curve point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' These light curves can be created down to a wavelength resolu- tion limit of ∼ 4 ˚A per bin, but for initial demonstration purposes we display a broad NUV bandpass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Figure 5 presents CUTE’s approximately 2540 – 3300 ˚A light curves for 3 different visits of the ultra- hot Jupiter WASP-189b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' The best fit transit model, taking into account wavelength-dependent stellar limb darkening (shown in gray), will be presented in detail in Sreejith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' (2022 – in prep).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' The optical transit light curve from CHEOPS (Lendl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2020) is shown for comparison and suggests excess transit absorption at UV wavelength compared with the broadband geomet- ric size of the planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' We demonstrate self-consistent transit depth recoveries of ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='0 – 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='1 % over three sep- arate transit observations of WASP-189b separated by several weeks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Excess planetary absorption at NUV wavelengths is consistent with previous observations of ultra-hot Jupiters observed with HST (Sing et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Cubil- los et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Lothringer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' A transit depth of 1% in WASP-189b would indicate that the NUV tran- sit observations are probing the extended upper atmo- sphere of the planet that is subject to stellar high energy radiation and escape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' This is because the 1 microbar level, used here as a rough proxy for the base of the thermosphere, has a radius of about 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='1 Rp and a tran- sit depth of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='6%, based on an effective temperature of 2410 K and an atmosphere with solar abundances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' In contrast, a transit depth of 1% corresponds to a larger radius of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='4 Rp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' If we assume a temperature of 8000 K in the thermosphere, the pressure at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='4 Rp would be between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='1 and 1 nbar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Given that this pressure is too low for significant clouds and hazes, a pseudo-continuum by these absorbers is unlikely and the broadband NUV transit depth likely arises from a forest of metal ion lines (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Fossati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Sing et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' The individ- ual absorption lines responsible would have to extend to much higher radii than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='4 Rp in transit to be detectable in the broad NUV band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' We note that the preliminary lightcurves show sig- nificant scatter beyond the photon noise limit at this stage of the reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Work is ongoing to model the temperature- and orbital position-dependent background to reduce the observed dispersion in the lightcurves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' CONCLUSIONS The CUTE CubeSat mission was launched in Septem- ber 2021 and is currently carrying out its primary sci- ence mission to collect NUV spectroscopy of transit- ing planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' CUTE has successfully completed space- craft and instrument commissioning and has completed initial science observations on a number of exoplane- tary systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Optimal science targets are short-period, Jovian-sized planets orbiting bright (V < 8) F and A stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' The science instrument has demonstrated sen- sitivity to NUV white light transit depth of < 1% and wavelength-dependent exoplanet atmosphere opacity in- crease at λ ≲ 3300 ˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' We have presented the motivation for, mission design of, and on-orbit characteristics of the CUTE mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' The companion paper by Egan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' present the details of the on-orbit instrument performance of CUTE and future papers will present mission science results as well as information about the CUTE ground-segment data pipeline, commissioning, and operations of the mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' CUTE has supported mentoring and training for over 20 early-career scientists and engineers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Mission operations are planned to continue until June 2023 (currently lim- ited by project funding) and the initial release of CUTE data will be delivered to the NexSci archive in 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' WASP-189 KELT-20 le-12 1e-12 4 A ergs s-1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='5 Flux, 2500 2600 2700 280029003000 3100 3200 3300 2500 2600 2700280029003000 31003200 3300 Wavelength,A Wavelength,A10 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Initial CUTE light curves of WASP-189b, showing three independent NUV (approximately 2540 – 3300 ˚A) light curves (black points) and the best-fit transit models in gray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' The plots compare the NUV band light curves with the optical light curve (in red) from CHEOPS (Lendl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' The NUV transits are significantly deeper than their broadband optical counterparts, indicating an effective planetary radius increase of RP,NUV ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='5 RP,opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Acknowledgments: CUTE was developed and oper- ated with the support to two NASA/APRA awards to the Laboratory for Atmospheric and Space Physics at the University of Colorado Boulder, NNX17AI84G and 80NSSC21K1667.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' was supported by a Schr¨odinger Fellowship through the Austrian Science Fund (FWF) [J 4596-N].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' acknowledge financial support from the Austrian Forschungsf¨orderungsgesellschaft FFG project 859718 and 865968.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' AAV acknowledges funding from the European Research Council (ERC) under the Euro- pean Union’s Horizon 2020 research and innovation pro- gramme (grant agreement No 817540, ASTROFLOW).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' acknowledges the numerous and invaluable discus- sions with colleagues excited about ultraviolet transit science and the potential to do science with small satel- lites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' The CUTE team wishes to specifically recog- nize the amateur radio operator community and, and SatNOGS network specifically, for hosting numerous telemetry tracking tools that have improved the mis- sion’s ability to recover from faults and understand long- term spacecraft trends much more efficiently than would have been otherwise possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' CUTE Transit 1 CUTE Transit 2 CUTE Transit 3 Best Fit Model Best Fit Model Best Fit Model 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='01 Peak Depth = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='03% 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='01 Peak Depth = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='13% 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='01 Peak Depth = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='10% NUV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='00 Normalized 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='1 Orbital Phase Orbital Phase Orbital Phase11 REFERENCES Bisikalo, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Kaygorodov, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Ionov, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2013, ApJ, 764, 19, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='1088/0004-637X/764/1/19 Bourrier, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', & Lecavelier des Etangs, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2013, A&A, 557, A124, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='1051/0004-6361/201321551 Bourrier, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Lecavelier des Etangs, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Ehrenreich, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Tanaka, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', & Vidotto, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2016, A&A, 591, A121, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='1051/0004-6361/201628362 Burdanov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Delrez, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Gillon, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', & Jehin, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2018, in Handbook of Exoplanets, ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Deeg & J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Belmonte, 130, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='1007/978-3-319-55333-7 130 Carolan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Vidotto, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Villarreal D’Angelo, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', & Hazra, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2021, MNRAS, 500, 3382, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='1093/mnras/staa3431 Casasayas-Barris, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Pall´e, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Yan, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2019, A&A, 628, A9, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='1051/0004-6361/201935623 Casasayas-Barris, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Orell-Miquel, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Stangret, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2021, A&A, 654, A163, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='1051/0004-6361/202141669 Cauley, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Shkolnik, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Llama, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', & Lanza, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2019, Nature Astronomy, 3, 1128, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='1038/s41550-019-0840-x Charbonneau, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Berta, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Irwin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2009, Nature, 462, 891, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='1038/nature08679 Cubillos, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Fossati, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Koskinen, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2020, AJ, 159, 111, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='3847/1538-3881/ab6a0b Deibert, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', de Mooij, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Jayawardhana, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2021, ApJL, 919, L15, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='3847/2041-8213/ac2513 Dong, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Shappee, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Prieto, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2016, Science, 351, 257, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='1126/science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='aac9613 Egan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', France, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Nell, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2022, in Small Satellite Conference, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' SS22, Small Satellite, WKII–02 Egan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', France, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Fleming, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2020, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 11444, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, 1144406, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='1117/12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='2559983 Fleming, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2022, Nature Astronomy, 6, 1213, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='1038/s41550-022-01807-8 Fleming, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', France, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Nell, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2018, Journal of Astronomical Telescopes, Instruments, and Systems, 4, 014004, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='1117/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='JATIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='014004 Flynn, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Byrant, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Pilinski, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2021, in Small Satellite Conference, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' SS21, Small Satellite, SSC21–S1–25 Fossati, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', France, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', & Koskinen, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2015, ApJ-submitted, 000, 0 Fossati, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Haswell, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Froning, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2010, ApJL, 714, L222, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='1088/2041-8205/714/2/L222 France, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Hoadley, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Fleming, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2016a, Journal of Astronomical Instrumentation, 5, 1640001, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='1142/S2251171716400018 France, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Loyd, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Youngblood, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2016b, ApJ, 820, 89, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='3847/0004-637X/820/2/89 Fulton, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', & Petigura, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2018, AJ, 156, 264, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='3847/1538-3881/aae828 Gardner, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Mather, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Clampin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2006, SSRv, 123, 485, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='1007/s11214-006-8315-7 Gillon, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Jehin, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Magain, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2011, in European Physical Journal Web of Conferences, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 11, European Physical Journal Web of Conferences, 06002, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='1051/epjconf/20101106002 Ginzburg, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Schlichting, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', & Sari, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2018, MNRAS, 476, 759, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='1093/mnras/sty290 Hammerstein, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', van Velzen, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Gezari, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2022, arXiv e-prints, arXiv:2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='01461.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='org/abs/2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='01461 Haswell, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Fosatti, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', & et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2012, ApJ, 0, 1 Hoeijmakers, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Ehrenreich, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Kitzmann, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2019, A&A, 627, A165, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='1051/0004-6361/201935089 Hoeijmakers, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Seidel, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Pino, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2020, A&A, 641, A123, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='1051/0004-6361/202038365 Holoien, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Stanek, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Kochanek, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2017, MNRAS, 464, 2672, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='1093/mnras/stw2273 Koskinen, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Aylward, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Smith, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', & Miller, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2007, ApJ, 661, 515, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='1086/513594 Koskinen, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Harris, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Yelle, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', & Lavvas, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2013a, Icarus, 226, 1678, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='icarus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='027 Koskinen, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Lavvas, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Huang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2022, ApJ, 929, 52, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='3847/1538-4357/ac4f45 Koskinen, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Yelle, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Harris, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', & Lavvas, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2013b, Icarus, 226, 1695, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='icarus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='026 Lai, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Helling, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', & van den Heuvel, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2010, ApJ, 721, 923, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='1088/0004-637X/721/2/923 Lammer, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Zerkle, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Gebauer, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2018, A&A Rv, 26, 2, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='1007/s00159-018-0108-y Lecavelier des Etangs, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Bourrier, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Wheatley, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2012, A&A, 543, L4, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='1051/0004-6361/201219363 Lendl, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Csizmadia, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Deline, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2020, A&A, 643, A94, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='1051/0004-6361/202038677 Linsky, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Yang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', France, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2010, ApJ, 717, 1291, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='1088/0004-637X/717/2/1291 Llama, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', & Shkolnik, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2015, ApJ, 802, 41, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='1088/0004-637X/802/1/41 Llama, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Wood, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Jardine, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2011, MNRAS, 416, L41, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='1111/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='1745-3933.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='01093.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='x 12 Lothringer, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Fu, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Sing, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', & Barman, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2020, ApJL, 898, L14, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='3847/2041-8213/aba265 Lothringer, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Sing, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Rustamkulov, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2022, Nature, 604, 49, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='1038/s41586-022-04453-2 Loyd, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', France, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Youngblood, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2018, ApJ, 867, 71, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='3847/1538-4357/aae2bd Masci, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Laher, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Rusholme, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2019, PASP, 131, 018003, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='1088/1538-3873/aae8ac Mason, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Woods, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Webb, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2016, ApJ, 830, 20, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='3847/0004-637X/830/1/20 Murray-Clay, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Chiang, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', & Murray, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2009, ApJ, 693, 23, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='1088/0004-637X/693/1/23 Nell, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', DeCicco, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Ulrich, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', France, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', & Fleming, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2021, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 11821, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, 1182117, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='1117/12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='2594743 Nichols, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Wynn, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Goad, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2015, ApJ, 803, 9, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='1088/0004-637X/803/1/9 Owen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', & Wu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2017, ApJ, 847, 29, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='3847/1538-4357/aa890a Pepper, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Pogge, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', DePoy, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2007, PASP, 119, 923, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='1086/521836 Pollacco, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Skillen, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Collier Cameron, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2006, PASP, 118, 1407, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='1086/508556 Ramiaramanantsoa, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Bowman, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Shkolnik, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2021, arXiv e-prints, arXiv:2111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='02443.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='org/abs/2111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='02443 Rigby, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Perrin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', McElwain, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2022, arXiv e-prints, arXiv:2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='05632.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='org/abs/2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='05632 Simard, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Ellerbroek, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Bhatia, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Radovan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', & Chisholm, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2016, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 9908, Ground-based and Airborne Instrumentation for Astronomy VI, ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Evans, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Simard, & H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' Takami, 99081V, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='1117/12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='2234058 Sing, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Lavvas, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Ballester, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2019, AJ, 158, 91, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='3847/1538-3881/ab2986 Turner, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Christie, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Arras, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Johnson, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', & Schmidt, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2016, MNRAS, 458, 3880, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='1093/mnras/stw556 Turner, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', de Mooij, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Jayawardhana, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2020, ApJL, 888, L13, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='3847/2041-8213/ab60a9 van Dokkum, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2001, PASP, 113, 1420, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='1086/323894 Vidal-Madjar, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Lecavelier des Etangs, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', D´esert, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2003, Nature, 422, 143, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='1038/nature01448 Vidal-Madjar, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', D´esert, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Lecavelier des Etangs, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2004, ApJL, 604, L69, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='1086/383347 Vidal-Madjar, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Huitson, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Bourrier, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2013, A&A, 560, A54, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='1051/0004-6361/201322234 Vidotto, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Jardine, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', & Helling, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2010, ApJL, 722, L168, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='1088/2041-8205/722/2/L168 Villarreal D’Angelo, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Esquivel, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', Schneiter, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=', & Sgr´o, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2018, MNRAS, 479, 3115, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content='1093/mnras/sty1544 Wilkinson, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} +page_content=' 2002, HST-COS Instrument Performance Reports, 25, COS' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfUgAs/content/2301.02250v1.pdf'} diff --git a/R9E0T4oBgHgl3EQfkwEq/vector_store/index.pkl b/R9E0T4oBgHgl3EQfkwEq/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..9cf0a00acd695ae41b711c22cbd399b7d7061cec --- /dev/null +++ b/R9E0T4oBgHgl3EQfkwEq/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:dfeb623f1c7c5923e1e9c1b1f4be444736c44164ff137531aa26f5f6940fbc7c +size 381764 diff --git a/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf b/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf new file mode 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file mode 100644 index 0000000000000000000000000000000000000000..9a6aa0ec9721299516d743307ed80ed6a7a5ea63 --- /dev/null +++ b/XdFJT4oBgHgl3EQf5i00/content/tmp_files/2301.11670v1.pdf.txt @@ -0,0 +1,5841 @@ +LHCHWG-2022-003 +December 15, 2022 +LHC Higgs Working Groupa +Public Note +Study of t¯tb¯b and t¯tW background modelling for t¯tH analyses +Lars Ferencz 1,b, Kirill Grevtsov 1,c, Judith Katzy 1,d, Andrea Knue 2,e, +Jan van der Linden 3,f, Josh McFayden 4,g, Gianna Moenig 4,h, Emanuel Pfeffer 3,i, +Andrej Saibel 5,j, Matthias Schr¨oder 6,k, Joshuha Thomas-Wilsker 7,l +1 DESY +2 Universit¨at Freiburg +3 KIT +4 University of Sussex +5 Instituto de F´ısica Corpuscular, Consejo Superior de Investigaciones Cient´ıficas +6 Universit¨at Hamburg +7 Institute of High Energy Physics, Chinese Academy of Sciences +work done on behalf of the LHCHWG +Reproduction of this article or parts of it is allowed as specified in the CC-BY-4.0 license. +ahttps://twiki.cern.ch/twiki/bin/view/LHCPhysics/LHCHWG +blars.ferencz@desy.de +ckirill.grevtsov@desy.de +djudith.katzy@desy.de +eandrea.knue@physik.uni-freiburg.de +fjan.linden@kit.edu +gjoshua.angus.mcfayden@cern.ch +hgianna.moenig@cern.ch +iemanuel.pfeffer@kit.edu +jandrej.saibel@cern.ch +kmatthias.schroeder@uni-hamburg.de +ljoshuha.thomas-wilsker@cern.ch +arXiv:2301.11670v1 [hep-ex] 27 Jan 2023 + +Abstract +This note presents Monte Carlo generator comparisons of the t¯tb¯b and t¯tW processes at particle +level. The aim is to compare the modelling of important backgrounds to t¯tH measurements in multi- +lepton final states and in the t¯tH(H → b¯b) decay channel and the treatment of the associated theory +uncertainties for a combination of the full Run-2 t¯tH results from ATLAS and CMS. As a first +step, modelling and theory uncertainties as used in ATLAS an CMS are compared in the relevant +analysis regions. Significant differences in the treatment of systematic uncertainties between the +experiments have been observed in t¯tb¯b and t¯tW. As a first step, ATLAS and CMS agreed on a +common reference value of the inclusive t¯tW cross section to allow direct comparisons between +experiments. + +Contents +1 +Introduction +1 +2 +Comparisons of Monte Carlo predictions for the t¯tb¯b process +2 +2.1 +MC generator set-ups +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +3 +2.2 +Object reconstruction, fiducial volume and observables . . . . . . . . . . . . . . . . . +7 +2.3 +Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +7 +2.4 +Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +8 +3 +Comparisons of Monte Carlo predictions for the t¯tW process +14 +3.1 +MC generator set-ups +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +15 +3.2 +Object reconstruction, fiducial volume and observables . . . . . . . . . . . . . . . . . +17 +3.3 +Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +20 +3.4 +Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +32 + +1 +Introduction +The search for Higgs boson production in association with a top quark pair (t¯tH) has been per- +formed in the H → b¯b [1, 2, 3, 4] decay channel and in multi-lepton final states [5, 6] which are +primarily sensitive to the decays of H→ WW ∗, H→ ττ and H→ ZZ∗. These searches are limited +by the modelling uncertainties of the main backgrounds, t¯tb¯b and t¯tW, respectively. Examples of +tree-level diagrams of the background processes are shown in Fig. 1. +A comparison of Monte Carlo (MC) generators used by ATLAS and CMS is thus performed to +compare the background modelling and the estimates of modelling uncertainties in view of future +combinations of the experimental results. The goals is to provide input to a discussion between the +experiments and between experiments and theorists to define modelling uncertainties. Furthermore, +the experiments aim to develop a common strategy for combination of the t¯tH(H → b¯b) and +t¯tH(multi-lepton) analyses of the full Run-2 data set. Comparisons of observables relevant for the +analyses are made at stable particle level, in a phase space similar to the reference measurements +using the Rivet analysis toolkit [7]. +The note is structured as follows: comparisons of t¯tb¯b distributions will be presented in Section 2 +and comparisons of t¯tW distributions in Section 3. +Figure 1: Examples of tree-level Feynman diagrams for t¯tb¯b (left) and t¯tW (right). +1 + +d +g +000000 +七 +d +n2 +Comparisons of Monte Carlo predictions for the t¯tb¯b process +In the following section t¯tb¯b background predictions and variations considered to estimate their +uncertainties used by ATLAS and CMS in published and future analyses of t¯tH(H → b¯b) are +compared. The first Run-2 t¯tH(H → b¯b) analyses of both experiments [2, 4] based on partial +data sets predicted the t¯t + jets background with a t¯t matrix element (ME) calculated at next- +to-leading-order (NLO) accuracy in QCD in the five-flavour scheme (5FS) and matched to the +Pythia8 parton shower (PS) [8] in the Powheg framework +[9, 10, 11, 12, 13]. In this set-up, +b-quarks not originating in the top quark decay chain are produced by Pythia8. +The first predictions using a t¯tb¯b ME at NLO have been performed with stable top quarks in +5FS some time ago [14, 15, 16]. They have been matched subsequently to parton shower programs +[17]. Very recently complete calculations for the t¯tb¯b process in di-lepton top quark decay channel +have been carried out in 5FS without matching to PS by two independent groups [18, 19, 20]. +Such computations are based on e+νeµ−νµbbbb matrix elements and include all resonant and non- +resonant Feynman diagrams, interferences and off-shell effects of the top quark and the W gauge +boson. +The first t¯tH(H → b¯b) analysis based on the full Run-2 data set from ATLAS [1] (”first full +Run-2 analysis”) used as nominal generator a calculation where the t¯tb¯b ME is calculated at NLO +with massive b-quarks1 in the four-flavour scheme (4FS) [21] and matched to Pythia8 in the +Powheg-Box-Res framework [21], referred to as t¯tb¯b-Powheg in the following. +For future analyses both experiments consider to use the calculations of t¯tb¯b-Powheg matched +to Pythia8 as nominal generator however with different settings of the renormalisation and fac- +torisation scale compared to the original paper [21] and slightly different settings of the internal +parameters based on more recent studies [22] as will be discussed below. +The estimation of systematic uncertainties differs significantly between the two experiments +for the published analyses. ATLAS considered uncertainties due the particular choice of matching +algorithm and of the parton shower generator. For the analyses based on partial and first full Run-2 +data set, these differences were derived from 5FS t¯t sample predicted by MG5 aMC@NLO [23, 24] +matched to Pythia8 for the first and a sample where Powheg is matched to Herwig7 [25] for +the latter. Since the nominal generator in the first full Run-2 analysis was based on a t¯tb¯b-Powheg +calculation, the relative uncertainties derived from the 5FS t¯t samples were used. Uncertainties +due to higher order effects were estimated by varying the renormalisation and factorisation scales +in the ME, µR and µF, simultaneously up and down by a factor of two. Correlations between the +scale settings in the ME and αs in the PS ISR were considered by simultaneous variation with µR +and µF to cover the effects of PS variations in the presence of matching [26]. +In the first Run-2 analysis, CMS considered the uncertainty due to the choice of generator +settings by varying the hdamp parameter in Powheg which controls the transverse momentum (pT) +of the first additional emission beyond the leading-order Feynman diagram in the PS and therefore +regulates the high-pT emission against which the t¯t system recoils. +Comparisons with Sherpa +were done internally but not added to the list of systematic uncertainties. The renormalisation and +factorisation scales µR and µF as well as αs in both the PS ISR and FSR were varied independently, +i.e. one parameter was changed at a time while keeping the other parameters at their nominal values. +For future analyses, both experiments consider predictions with varied µR and µF scales and +varied PS αs as well as different settings of t¯tb¯b-Powheg internal parameters, however ATLAS +studies additional uncertainties due to parton shower and matching. To estimate the dependence on +t¯tb¯b-Powheg internal parameters, ATLAS varies the parameter hbzd which regulates the splitting +1“quarks” refers to both quarks and anti-quarks +2 + +into the finite and the singular part of the real emission in the Powheg framework. Variations +of the parameter hdamp were studied in Ref. [22] but no significant differences were found and +therefore this variation is not further considered for uncertainty estimates. Uncertainties due to +the particular setting of PS are estimated with set-ups of t¯tb¯b-Powheg matched to Herwig7 and +Pythia8 with a dipole recoil. The dependence on the particular choice of generator and the NLO +matching algorithm is studied by comparing to NLO 4FS predictions of t¯tb¯b generated with Sherpa +2.2.10 [27, 28, 29]. Details of the studies are given in Ref. [22]. +In case of CMS, the dependence on t¯tb¯b-Powheg internal parameters is estimated by varying +the matching parameter hdamp. +Both experiments consider PDF uncertainties in the published and future analyses, however +they are neglected in the studies presented here due to the smallness of the effect. Finally, in order +to get comparable results, the scale uncertainties are treated the same way for both experiments +in all studies presented here, i.e. µR and µF, PS ISR and PS FSR are changed individually by a +factor 0.5 (2) while keeping the other parameters at their nominal values. +All comparisons are performed using stable final-state particles in a fiducial phase space simi- +lar to the experimental measurements implemented in a dedicated routine in the Rivet analysis +toolkit [7, 30]. +The chapter is organised as follows. Section 2.1 describes the samples used for the comparison +and the technical set-up of their generation. Section 2.2 describes the observables and the fiducial +phase space used for the comparison and finally, Sec. 2.3 displays the resulting comparisons. +2.1 +MC generator set-ups +The set-ups used to generate t¯tb¯b predictions with t¯tb¯b-Powheg, Powheg, MG5 aMC@NLO +and Sherpa are described in the following. The generator configurations and version numbers are +summarised in Table 1 and their scale settings are given in Table 2. The systematic uncertainty +estimates due to scale and αs variations are summarised in Table 3. +The b-quark mass is set to 4.75 GeV for CMS samples and for Sherpa, and to 4.95 GeV for +all other ATLAS samples. The top quark mass is set to 172.5 GeV. The decay of the top quark +is calculated by the corresponding generators (Powheg, Sherpa) respecting the spin correlation. +The PDF sets used in the ME calculation are selected from the NNPDF family for all samples, +where ATLAS uses version 3.0 while CMS uses version 3.1. The ATLAS t¯tb¯b-Powheg, Powheg +and MG5 aMC@NLO samples use EvtGen [31] for simulation of the B-hadron decays, while the +Sherpa sample and all CMS samples calculate the decays within the corresponding PS codes. All +samples were produced for final states with one or two leptons. +t¯tb¯b-Powheg samples: +Nominal t¯tb¯b predictions are calculated using the Powheg-Box-Res framework at NLO with +massive b-quarks [21] with the “4FS NLO as 0118” PDF sets. The renormalisation scale is +set to half of the geometric average of the transverse mass of top- and b-quarks defined as +mT,i = +� +m2 +i + p2 +T,i, where mi refers to the mass, pT,i to the transverse momentum and i +to the top or b-quark. +The factorisation scale is related to the average of the transverse +mass of the outgoing partons in the ME calculation, see Table 2. For ATLAS, it follows +Ref. [21], while it is set to a factor two smaller in CMS following Ref. [32]. The t¯tb¯b-Powheg +internal parameters differ between the experiments: hbzd is set to 5 for ATLAS and to 2 for +CMS, hdamp is set to HT/2 for ATLAS and to 1.379 times the top quark mass for CMS. The +Pythia8 parameters for PS and hadronisation modelling are set to the A14 [33] and CP5 [34] +3 + +tunes for ATLAS and CMS and the samples are referred to as ATLAS and CMS PP8 t¯tb¯b +samples, respectively. +To vary t¯tb¯b-Powheg internal parameters, ATLAS sets the parameter hbzd to 2. CMS varies +in its set-up the hdamp parameter to 2.305 times the top quark mass for the “hdamp up” +variation and to 0.8738 times the top quark mass for the “hdamp down” variation. +The ATLAS t¯tb¯b-Powheg calculation was performed using a special option where virtual +corrections are switched off and then reweighted with virtual corrections switched on2, while +the CMS samples used default calculation. +For the PS variations, ATLAS uses the set of LHE files which store the results of the ME +calculation by t¯tb¯b-Powheg for the PP8 t¯tb¯b sample and matches them to a different PS +prediction. For the prediction with the Pythia8 dipole shower only the treatment of the +recoil of the radiated parton in the shower is changed and all other parameters are kept as +the A14 tuned values. Another sample is produced where Herwig7 is used with the default +tune provided with this generator version. +Sherpa t¯tb¯b samples: +A t¯tb¯b sample was generated using Sherpa version 2.2.10 [27, 28, 29]. The t¯tb¯b MEs were +calculated with massive b-quarks at NLO, using the COMIX [35] and Openloops [29] ME +generators, and merged with the Sherpa PS, tuned by the authors [36]. The same renormal- +isation and factorisation scales and PDFs are used as for the ATLAS PP8 t¯tb¯b prediction. +Inclusive t¯t samples: +The inclusive t¯t samples are generated with the Powheg v2 NLO event generator [9, 10, 12, +13, 37] and MG5 aMC@NLO using a 5FS PDF set. The renormalisation and factorisation +scales were set to the average transverse mass of the top quark and antiquark. +For the Powheg samples of both experiments, the PS and hadronisation is modeled by +Pythia8 with the same versions and settings as for the PP8 t¯tb¯b samples above. The hdamp +parameter was set to the 1.5 times the top quark mass for ATLAS and to 1.379 times the top +quark mass for CMS. Another ATLAS sample is generated using Herwig7 for the PS and +hadronization. These samples are referred to as ATLAS (CMS) PP8 t¯t and ATLAS PH7 t¯t +samples. +The inclusive MG5 aMC@NLO t¯t sample uses the same scale settings and the same Pythia8 +version as the ATLAS PP8 t¯t sample and is referred to as ATLAS aMC+P8 t¯t sample. +2steered via ”for reweight 1” +4 + +Table 1: +Configurations used for the event generation of the t¯tb¯b process and the predicted total cross section for events with at least +one lepton. +name +ME +Generator +ME order +Shower +Tunea +NNPDF PDF set (ME) +hdamp +hbzd +σ≥1lep [pb] +ATLAS +PP8 t¯tb¯b +t¯tb¯b +t¯tb¯b-Powheg +NLO +Pythia 8.224 +A14 +4FS 3.0 NLO as 0118 +HT /2 +5 +18.72 +CMS +PP8 t¯tb¯b +t¯tb¯b +t¯tb¯b-Powheg +NLO +Pythia 8.230 +CP5 +4FS 3.1 NLO as 0118 +1.379 · mt +2 +23.86 +ATLAS +PP8 t¯tb¯b hbzd 2 +t¯tb¯b +t¯tb¯b-Powheg +NLO +Pythia 8.224 +A14 +4FS 3.0 NLO as 0118 +HT/2 +2 +18.46 +ATLAS +PP8 t¯tb¯b dipole +t¯tb¯b +t¯tb¯b-Powheg +NLO +Pythia 8.224 +A14, dipoleRecoilb +4FS 3.0 NLO as 0118 +HT/2 +2 +18.72 +ATLAS +PH7 t¯tb¯b +t¯tb¯b +t¯tb¯b-Powheg +NLO +Herwig 7.1.6 +default +4FS 3.0 NLO as 0118 +HT/2 +5 +18.47 +ATLAS +Sherpa t¯tb¯b +t¯tb¯b +Sherpa 2.2.10 +NLO +Sherpa +default +4FS 3.0 NNLO as 0118 +— +— +20.24 +CMS +PP8 t¯tb¯b hdamp up +t¯tb¯b +t¯tb¯b-Powheg +NLO +Pythia 8.230 +CP5 +4FS 3.1 NLO as 0118 +2.305 · mt +5 +23.86 +CMS +PP8 t¯tb¯b hdamp down +t¯tb¯b +t¯tb¯b-Powheg +NLO +Pythia 8.230 +CP5 +4FS 3.1 NLO as 0118 +0.8738 · mt +5 +23.86 +ATLAS +PP8 t¯t +t¯t +Powheg v2 +NLO +Pythia 8.210 +A14 +5FS 3.0 NLO +1.5 · mt +5 +451.78c +CMS +PP8 t¯t +t¯t +Powheg v2 +NLO +Pythia 8.230 +CP5 +5FS 3.1 NLO +1.5 · mt +5 +451.78c +ATLAS +PH7 t¯t +t¯t +Powheg v2 +NLO +Herwig 7.13 +default +5FS 3.0 NLO +1.5 · mt +5 +451.78c +ATLAS +aMC+P8 t¯t +t¯t +MG5 aMC@NLO +NLO +Pythia 8.210 +A14 +5FS 3.0 NLO +— +— +451.78c +CMS +PP8 t¯t hdamp up +t¯t +Powheg v2 +NLO +Pythia 8.230 +CP5 +5FS 3.1 NLO +2.305 · mt +5 +451.78c +CMS +PP8 t¯t hdamp down +t¯t +Powheg v2 +NLO +Pythia 8.230 +CP5 +5FS 3.1 NLO +0.8738 · mt +5 +451.78c +a“default” refers to the generator’s default tune +bcalled by SpaceShower::dipoleRecoil “on” +ccross section predicted by NNLO calculation +5 + +Table 2: +Scale choices used in the event generation of t¯tb¯b and t¯t processes for the different generators. +ME Generator +µR +µF +ATLAS t¯tb¯b-Powheg t¯tb¯b +1 +2 +4√mT,t · mT,¯t · mT,b · mT,¯b +1 +2(mT,t + mT,¯t + mT,b + mT,¯b + mT,g) +CMS t¯tb¯b-Powheg t¯tb¯b +1 +2 +4√mT,t · mT,¯t · mT,b · mT,¯b +1 +4(mT,t + mT,¯t + mT,b + mT,¯b + mT,g) +Sherpa 2.2.10 +1 +2 +4√mT,t · mT,¯t · mT,b · mT,¯b +1 +2(mT,t + mT,¯t + mT,b + mT,¯b + mT,g) +ATLAS Powheg t¯t +� +0.5 · (m2 +T,t + m2 +T,¯t) +� +0.5 · (m2 +T,t + m2 +T,¯t) +CMS Powheg t¯t +� +0.5 · (m2 +T,t + m2 +T,¯t) +� +0.5 · (m2 +T,t + m2 +T,¯t) +ATLAS aMC t¯t +� +0.5 · (m2 +T,t + m2 +T,¯t) +� +0.5 · (m2 +T,t + m2 +T,¯t) +6 + +Table 3: Systematic variations of scales in the ME and PS codes used for all comparisons presented +here. +Variation +Scale variation ME +µR × 0.5, µF × 0.5; µR × 2, µF × 2 +ISR variation (PS) +αISR +s +× 0.5; αsISR × 2.0 +FSR variation (PS) +αFSR +s +× 0.5; αsFSR × 2.0 +2.2 +Object reconstruction, fiducial volume and observables +The object definition and event selection applied in this comparison study is defined at particle level +and is the same for ATLAS and CMS. All objects are defined using stable final-state particles with +a mean lifetime of τ > 3 × 10−11 s. Jets are reconstructed from all stable final-state particles (but +excluding leptons and neutrinos from the top quark decay chain) using the anti-kt jet algorithm [38, +39] with a radius parameter of R = 0.4. Jets which contain at least one ghost-associated [40] B- +hadron with pT > 5 GeV are defined as b-jets, all other jets are considered “light” jets. +The +four-momentum of the bare leptons from top quark decay are modified (“dressed”) by adding the +four-momenta of all radiated photons within a cone of size ∆R = 0.1. All objects are considered +within pseudo-rapidity |η| < 2.5 and with pT > 27 GeV for leptons and pT > 25 GeV for jets and +b-jets. +Leptons are removed if they are separated from a jet by less than ∆R = 0.4, where ∆R = +� +(∆η)2 + (∆φ)2. +Events are selected with at least four b-jets, and further separated into two +analysis regions: events with exactly one lepton and at least six jets (single lepton channel) and +events with exactly two leptons and at least four jets (dilepton channel). +A set of observables relevant for the t¯tH(H → b¯b) analysis is studied within this fiducial phase +space. All observables are studied for both the single lepton and the dilepton channel, however +only the variables listed in Table 4 are shown in the following figures, as no significant qualitative +difference is observed between the different top quark decay channels. +Table 4: +The list of observables used for the comparison of the generators for the t¯tb¯b process. +Variable +Description +Channel +∆Rmin∆R +bb +∆R of the two b-jets in the event which are closest in ∆R +dilepton +mmin∆R +bb +Invariant mass of the two b-jets closest in ∆R +dilepton +Njets +Number of jets in the event (all jet flavours) +dilepton +Light jet pT +Transverse momentum of the light jets in the event +dilepton +Nb-jets +Number of b-jets in the event +single lepton +Hjets +T +Scalar sum of pT of jets in the event (all jet flavours) +single lepton +Leading b-jet pT +pT of b-jet with largest pT in the event +single lepton +Fourth b-jet pT +pT of b-jet with fourth largest pT in the event +single lepton +2.3 +Results +Three sets of generator predictions are compared for the observables given in Table 4 as follows. +All comparisons are performed with respect to the t¯tb¯b PP8 sample. The PP8 t¯tb¯b sample and the +7 + +alternative predictions are normalised to an integral of one, after all selections and in each histogram +individually for a shape-only comparison. The scale uncertainty variations on PP8 t¯tb¯b are derived +as listed in Table 3 and the differences are added in quadrature to the statistical uncertainties to +form the shaded area displayed in the figures. +Figure 2 shows the nominal t¯tb¯b predictions from ATLAS and CMS to be used in future analyses +compared to the nominal predictions used in the early Run-2 analyses. The differences between +ATLAS and CMS set-ups cause only minor differences between the predictions. However, significant +differences between the PP8 t¯tb¯b predictions and the PP8 t¯t predictions are observed in ∆Rmin∆R +b¯b +, +the jet multiplicity and in the number of events with more than four b-jets. Furthermore, the +uncertainty band is slightly larger in the CMS t¯tb¯b predictions, potentially caused by the lower +factorisation scale. +In Fig. 3, the ATLAS nominal PP8 t¯tb¯b prediction is compared to all generator variations +potentially considered as modelling uncertainties for future ATLAS t¯tH(H → b¯b) analyses, i.e. +variations in t¯tb¯b-Powheg and Pythia8 parameter settings as well as Sherpa as alternative +generator. As already discussed in Ref. [22], the parameter hbzd has only a minor influence on +the observables. Interestingly, predictions of t¯tb¯b-Powheg matched to Pythia8 using the dipole +shower and matched to the Herwig7 PS both show a significant decrease with respect to the +nominal PP8 t¯tb¯b in the jet multiplicity and HT. Sherpa differs up to 10–20 % in all distributions +with significant differences in shape. +In Fig. 4, the CMS nominal PP8 t¯tb¯b prediction is compared to generator variations potentially +considered for the CMS t¯tH(H → b¯b) analysis. The scale uncertainties, which include the scale +variations in the ME and the PS, are significantly larger than the differences observed for the +different hdamp variations, except at very low HT and low leading b-jet pT where the hdamp down +variations shows up to 20 % differences. Significant statistical fluctuations are observed at regions +of low event yields, which are, however, not expected to be relevant for the analysis. +Finally, Fig. 5 shows the distributions used to estimate the systematic modelling uncertainties +of the first Run-2 analysis by CMS [4] and of the first full Run-2 analysis by ATLAS [1]. In addition +to the scale and PS αs variations, the uncertainty on the t¯tb¯b PP8 prediction is estimated in case +of ATLAS by assigning the relative difference between PP8 t¯t and alternative t¯t predictions listed +in Table 1 to the t¯tb¯b prediction, and in case of CMS by the hdamp variations, where also a cross +check with Sherpa t¯t has been made but was not included in the fit. Due to displaying purposes, +the ATLAS PP8 t¯t prediction, which is very similar to the CMS t¯t prediction as demonstrated in +Fig. 2, is not shown. +2.4 +Conclusions +Comparisons of generator predictions used by ATLAS and CMS in a typical phase space of the +t¯tH(H → b¯b) measurement were presented. Two sets are used for comparison: the generators +used in the most recent published analyses involving t¯t inclusive predictions based on 5FS PDFs to +estimate uncertainties and the set of generators in the future effort using t¯tb¯b calculations at NLO +based on 4FS PDFs. +The difference between the predictions exceeds the uncertainties from the scale variations both +for the uncertainties considered in the published t¯tH(H → b¯b) analysis and for the future analyses. +The uncertainties due to the choice of PS and NLO generator are reduced when estimating them +based on t¯tb¯b ME predictions compared to the previously used t¯t ME matched predictions. +Despite differences in the set-ups between the experiments for the nominal PP8 t¯tb¯b generator, +only small differences are observed in the predictions. However, the considerations of the mod- +elling uncertainties differs significantly: CMS considers inherent variations of the chosen model +8 + +as uncertainty, while ATLAS studied inherent variations and differences obtained with alternative +generator choices and the latter dominates the uncertainties. Scale variations are applied by both +experiments, however the details of the estimates differ between ATLAS and CMS in the published +analysis but the effect of the different treatment are not yet studied for the future analyses. +The presented studies shall be used as input to discussions between the experiments and theo- +rists to define theory uncertainties for future combinations of ATLAS and CMS. +9 + +0.05 +0.1 +0.15 +0.2 +0.25 +Arbitrary Units +b +b +t +ATLAS PP8 t +b +b +t +CMS PP8 t +t +ATLAS PP8 t +t +CMS PP8 t +ATLAS + CMS +Generator Level + 4j +≥ + 4b, +≥ +=13 TeV, +s +Dilepton channel +0.5 +1 +1.5 +2 +2.5 +3 + R +∆ +min +bb + R +∆ +0.8 +1 +1.2 +b ++b +t +ATLAS t +Ratio to PP8 +0.05 +0.1 +0.15 +0.2 +0.25 +Arbitrary Units +b +b +t +ATLAS PP8 t +b +b +t +CMS PP8 t +t +ATLAS PP8 t +t +CMS PP8 t +ATLAS + CMS +Generator Level + 4j +≥ + 4b, +≥ +=13 TeV, +s +Dilepton channel +50 +100 +150 +200 +250 +300 +350 + [GeV] + R +∆ +min +bb +m +0.8 +1 +1.2 +b ++b +t +ATLAS t +Ratio to PP8 +0.05 +0.1 +0.15 +0.2 +0.25 +0.3 +0.35 +0.4 +Arbitrary Units +b +b +t +ATLAS PP8 t +b +b +t +CMS PP8 t +t +ATLAS PP8 t +t +CMS PP8 t +ATLAS + CMS +Generator Level + 4j +≥ + 4b, +≥ +=13 TeV, +s +Dilepton channel +4 +5 +6 +7 +8 +9 + 10 +≥ +jets +N +0.8 +1 +1.2 +b ++b +t +ATLAS t +Ratio to PP8 +0.02 +0.04 +0.06 +0.08 +0.1 +0.12 +0.14 +0.16 +0.18 +0.2 +0.22 +Arbitrary Units +b +b +t +ATLAS PP8 t +b +b +t +CMS PP8 t +t +ATLAS PP8 t +t +CMS PP8 t +ATLAS + CMS +Generator Level + 4j +≥ + 4b, +≥ +=13 TeV, +s +Dilepton channel +50 +100 +150 +200 +250 +300 +350 + [GeV] +T +Light jet p +0.8 +1 +1.2 +b ++b +t +ATLAS t +Ratio to PP8 +4 +− +10 +3 +− +10 +2 +− +10 +1 +− +10 +1 +Arbitrary Units +b +b +t +ATLAS PP8 t +b +b +t +CMS PP8 t +t +ATLAS PP8 t +t +CMS PP8 t +ATLAS + CMS +Generator Level + 6j +≥ + 4b, +≥ +=13 TeV, +s +Single lepton channel +4 +5 +6 + 7 +≥ +b-jets +N +0.6 +0.8 +1 +1.2 +1.4 +b ++b +t +ATLAS t +Ratio to PP8 +0.01 +0.02 +0.03 +0.04 +0.05 +0.06 +Arbitrary Units +b +b +t +ATLAS PP8 t +b +b +t +CMS PP8 t +t +ATLAS PP8 t +t +CMS PP8 t +ATLAS + CMS +Generator Level + 6j +≥ + 4b, +≥ +=13 TeV, +s +Single lepton channel +400 +600 +800 +1000 +1200 +1400 + [GeV] +jets +T +H +0.8 +1 +1.2 +b ++b +t +ATLAS t +Ratio to PP8 +0.01 +0.02 +0.03 +0.04 +0.05 +0.06 +0.07 +0.08 +Arbitrary Units +b +b +t +ATLAS PP8 t +b +b +t +CMS PP8 t +t +ATLAS PP8 t +t +CMS PP8 t +ATLAS + CMS +Generator Level + 6j +≥ + 4b, +≥ +=13 TeV, +s +Single lepton channel +50 +100 +150 +200 +250 +300 +350 +400 +450 +500 + [GeV] +T +Leading b-jet p +0.8 +1 +1.2 +b ++b +t +ATLAS t +Ratio to PP8 +0.05 +0.1 +0.15 +0.2 +0.25 +0.3 +0.35 +Arbitrary Units +b +b +t +ATLAS PP8 t +b +b +t +CMS PP8 t +t +ATLAS PP8 t +t +CMS PP8 t +ATLAS + CMS +Generator Level + 6j +≥ + 4b, +≥ +=13 TeV, +s +Single lepton channel +20 +40 +60 +80 +100 +120 +140 +160 +180 +200 + [GeV] +T +4th b-jet p +0.8 +1 +1.2 +b ++b +t +ATLAS t +Ratio to PP8 +Figure 2: +Comparison of PP8 predictions for t¯tb¯b and t¯t with the described settings using the +observables defined in Table 4 in the fiducial analysis phase space. All predictions are normalised +to one. The error bands are constructed from the statistical uncertainties and the scale variations +(ME and PS) for the ATLAS PP8 t¯tb¯b (blue) and the CMS PP8 t¯tb¯b (red) samples. Statistical +uncertainties are indicated by vertical lines. The ratio shows the different curves divided by the +ATLAS PP8 t¯tb¯b prediction. +10 + +0.05 +0.1 +0.15 +0.2 +0.25 +Arbitrary Units +b +b +t +ATLAS PP8 t +b +b +t +ATLAS PH7 t + dipole +b +b +t +ATLAS PP8 t + 2 +bzd + h +b +b +t +ATLAS PP8 t +b +b +t +ATLAS Sherpa t +ATLAS +Generator Level + 4j +≥ + 4b, +≥ +=13 TeV, +s +Dilepton channel +0.5 +1 +1.5 +2 +2.5 +3 + R +∆ +min +bb + R +∆ +0.8 +1 +1.2 +b ++b +t +ATLAS t +Ratio to PP8 +0.05 +0.1 +0.15 +0.2 +0.25 +Arbitrary Units +b +b +t +ATLAS PP8 t +b +b +t +ATLAS PH7 t + dipole +b +b +t +ATLAS PP8 t + 2 +bzd + h +b +b +t +ATLAS PP8 t +b +b +t +ATLAS Sherpa t +ATLAS +Generator Level + 4j +≥ + 4b, +≥ +=13 TeV, +s +Dilepton channel +50 +100 +150 +200 +250 +300 +350 + [GeV] + R +∆ +min +bb +m +0.8 +1 +1.2 +b ++b +t +ATLAS t +Ratio to PP8 +0.05 +0.1 +0.15 +0.2 +0.25 +0.3 +0.35 +0.4 +Arbitrary Units +b +b +t +ATLAS PP8 t +b +b +t +ATLAS PH7 t + dipole +b +b +t +ATLAS PP8 t + 2 +bzd + h +b +b +t +ATLAS PP8 t +b +b +t +ATLAS Sherpa t +ATLAS +Generator Level + 4j +≥ + 4b, +≥ +=13 TeV, +s +Dilepton channel +4 +5 +6 +7 +8 +9 + 10 +≥ +jets +N +0.8 +1 +1.2 +b ++b +t +ATLAS t +Ratio to PP8 +0.02 +0.04 +0.06 +0.08 +0.1 +0.12 +0.14 +0.16 +0.18 +0.2 +0.22 +0.24 +Arbitrary Units +b +b +t +ATLAS PP8 t +b +b +t +ATLAS PH7 t + dipole +b +b +t +ATLAS PP8 t + 2 +bzd + h +b +b +t +ATLAS PP8 t +b +b +t +ATLAS Sherpa t +ATLAS +Generator Level + 4j +≥ + 4b, +≥ +=13 TeV, +s +Dilepton channel +50 +100 +150 +200 +250 +300 +350 + [GeV] +T +Light jet p +0.8 +1 +1.2 +b ++b +t +ATLAS t +Ratio to PP8 +4 +− +10 +3 +− +10 +2 +− +10 +1 +− +10 +1 +Arbitrary Units +b +b +t +ATLAS PP8 t +b +b +t +ATLAS PH7 t + dipole +b +b +t +ATLAS PP8 t + 2 +bzd + h +b +b +t +ATLAS PP8 t +b +b +t +ATLAS Sherpa t +ATLAS +Generator Level + 6j +≥ + 4b, +≥ +=13 TeV, +s +Single lepton channel +4 +5 +6 + 7 +≥ +b-jets +N +0.6 +0.8 +1 +1.2 +1.4 +b ++b +t +ATLAS t +Ratio to PP8 +0.01 +0.02 +0.03 +0.04 +0.05 +0.06 +0.07 +Arbitrary Units +b +b +t +ATLAS PP8 t +b +b +t +ATLAS PH7 t + dipole +b +b +t +ATLAS PP8 t + 2 +bzd + h +b +b +t +ATLAS PP8 t +b +b +t +ATLAS Sherpa t +ATLAS +Generator Level + 6j +≥ + 4b, +≥ +=13 TeV, +s +Single lepton channel +400 +600 +800 +1000 +1200 +1400 + [GeV] +jets +T +H +0.8 +1 +1.2 +b ++b +t +ATLAS t +Ratio to PP8 +0.01 +0.02 +0.03 +0.04 +0.05 +0.06 +0.07 +0.08 +Arbitrary Units +b +b +t +ATLAS PP8 t +b +b +t +ATLAS PH7 t + dipole +b +b +t +ATLAS PP8 t + 2 +bzd + h +b +b +t +ATLAS PP8 t +b +b +t +ATLAS Sherpa t +ATLAS +Generator Level + 6j +≥ + 4b, +≥ +=13 TeV, +s +Single lepton channel +50 +100 +150 +200 +250 +300 +350 +400 +450 +500 + [GeV] +T +Leading b-jet p +0.8 +1 +1.2 +b ++b +t +ATLAS t +Ratio to PP8 +0.05 +0.1 +0.15 +0.2 +0.25 +0.3 +0.35 +Arbitrary Units +b +b +t +ATLAS PP8 t +b +b +t +ATLAS PH7 t + dipole +b +b +t +ATLAS PP8 t + 2 +bzd + h +b +b +t +ATLAS PP8 t +b +b +t +ATLAS Sherpa t +ATLAS +Generator Level + 6j +≥ + 4b, +≥ +=13 TeV, +s +Single lepton channel +20 +40 +60 +80 +100 +120 +140 +160 +180 +200 + [GeV] +T +4th b-jet p +0.8 +1 +1.2 +b ++b +t +ATLAS t +Ratio to PP8 +Figure 3: +Comparison of ATLAS PP8 predictions for t¯tb¯b with different matching and PS settings +and Sherpa. All distributions are normalised to one. The ratio shows the different curves divided +by PP8 t¯tb¯b. The error band contains the statistical uncertainty and the scale variations (ME and +PS) for the ATLAS PP8 t¯tb¯b sample. Statistical uncertainties are indicated by vertical lines. +11 + +0.05 +0.1 +0.15 +0.2 +0.25 +Arbitrary Units +b +b +t +CMS PP8 t + up +damp + h +b +b +t +CMS PP8 t + down +damp + h +b +b +t +CMS PP8 t +CMS +Generator Level + 4j +≥ + 4b, +≥ +=13 TeV, +s +Dilepton channel +0.5 +1 +1.5 +2 +2.5 +3 + R +∆ +min +bb + R +∆ +0.8 +1 +1.2 +b ++b +t + CMS t +Ratio to PP8 +0.05 +0.1 +0.15 +0.2 +0.25 +Arbitrary Units +b +b +t +CMS PP8 t + up +damp + h +b +b +t +CMS PP8 t + down +damp + h +b +b +t +CMS PP8 t +CMS +Generator Level + 4j +≥ + 4b, +≥ +=13 TeV, +s +Dilepton channel +50 +100 +150 +200 +250 +300 +350 + [GeV] + R +∆ +min +bb +m +0.8 +1 +1.2 +b ++b +t + CMS t +Ratio to PP8 +0.05 +0.1 +0.15 +0.2 +0.25 +0.3 +0.35 +0.4 +Arbitrary Units +b +b +t +CMS PP8 t + up +damp + h +b +b +t +CMS PP8 t + down +damp + h +b +b +t +CMS PP8 t +CMS +Generator Level + 4j +≥ + 4b, +≥ +=13 TeV, +s +Dilepton channel +4 +5 +6 +7 +8 +9 + 10 +≥ +jets +N +0.8 +1 +1.2 +b ++b +t + CMS t +Ratio to PP8 +0.02 +0.04 +0.06 +0.08 +0.1 +0.12 +0.14 +0.16 +0.18 +0.2 +0.22 +Arbitrary Units +b +b +t +CMS PP8 t + up +damp + h +b +b +t +CMS PP8 t + down +damp + h +b +b +t +CMS PP8 t +CMS +Generator Level + 4j +≥ + 4b, +≥ +=13 TeV, +s +Dilepton channel +50 +100 +150 +200 +250 +300 +350 + [GeV] +T +Light jet p +0.8 +1 +1.2 +b ++b +t + CMS t +Ratio to PP8 +4 +− +10 +3 +− +10 +2 +− +10 +1 +− +10 +1 +Arbitrary Units +b +b +t +CMS PP8 t + up +damp + h +b +b +t +CMS PP8 t + down +damp + h +b +b +t +CMS PP8 t +CMS +Generator Level + 6j +≥ + 4b, +≥ +=13 TeV, +s +Single lepton channel +4 +5 +6 + 7 +≥ +b-jets +N +0.6 +0.8 +1 +1.2 +1.4 +b ++b +t + CMS t +Ratio to PP8 +0.01 +0.02 +0.03 +0.04 +0.05 +0.06 +Arbitrary Units +b +b +t +CMS PP8 t + up +damp + h +b +b +t +CMS PP8 t + down +damp + h +b +b +t +CMS PP8 t +CMS +Generator Level + 6j +≥ + 4b, +≥ +=13 TeV, +s +Single lepton channel +400 +600 +800 +1000 +1200 +1400 + [GeV] +jets +T +H +0.8 +1 +1.2 +b ++b +t + CMS t +Ratio to PP8 +0.01 +0.02 +0.03 +0.04 +0.05 +0.06 +0.07 +0.08 +Arbitrary Units +b +b +t +CMS PP8 t + up +damp + h +b +b +t +CMS PP8 t + down +damp + h +b +b +t +CMS PP8 t +CMS +Generator Level + 6j +≥ + 4b, +≥ +=13 TeV, +s +Single lepton channel +50 +100 +150 +200 +250 +300 +350 +400 +450 +500 + [GeV] +T +Leading b-jet p +0.8 +1 +1.2 +b ++b +t + CMS t +Ratio to PP8 +0.05 +0.1 +0.15 +0.2 +0.25 +0.3 +0.35 +Arbitrary Units +b +b +t +CMS PP8 t + up +damp + h +b +b +t +CMS PP8 t + down +damp + h +b +b +t +CMS PP8 t +CMS +Generator Level + 6j +≥ + 4b, +≥ +=13 TeV, +s +Single lepton channel +20 +40 +60 +80 +100 +120 +140 +160 +180 +200 + [GeV] +T +4th b-jet p +0.8 +1 +1.2 +b ++b +t + CMS t +Ratio to PP8 +Figure 4: +Comparison of CMS PP8 predictions for t¯tb¯b with different settings of the parameter +hdamp. All predictions are normalised to one. The ratio shows the different curves divided by PP8 +t¯tb¯b. The error band contains the statistical uncertainty and the scale variations (ME and PS) for +the CMS PP8 t¯tb¯b sample. Statistical uncertainties are indicated by vertical lines. +12 + +0.05 +0.1 +0.15 +0.2 +0.25 +Arbitrary Units +b +b +t +ATLAS PP8 t +t +CMS PP8 t + up +damp + h +t +CMS PP8 t + down +damp + h +t +CMS PP8 t +t +ATLAS PH7 t +t +ATLAS aMC+P8 t +ATLAS + CMS +Generator Level + 4j +≥ + 4b, +≥ +=13 TeV, +s +Dilepton channel +0.5 +1 +1.5 +2 +2.5 +3 + R +∆ +min +bb + R +∆ +0.8 +1 +1.2 +b ++b +t +ATLAS t +Ratio to PP8 +0.05 +0.1 +0.15 +0.2 +0.25 +Arbitrary Units +b +b +t +ATLAS PP8 t +t +CMS PP8 t + up +damp + h +t +CMS PP8 t + down +damp + h +t +CMS PP8 t +t +ATLAS PH7 t +t +ATLAS aMC+P8 t +ATLAS + CMS +Generator Level + 4j +≥ + 4b, +≥ +=13 TeV, +s +Dilepton channel +50 +100 +150 +200 +250 +300 +350 + [GeV] + R +∆ +min +bb +m +0.8 +1 +1.2 +b ++b +t +ATLAS t +Ratio to PP8 +0.05 +0.1 +0.15 +0.2 +0.25 +0.3 +0.35 +0.4 +Arbitrary Units +b +b +t +ATLAS PP8 t +t +CMS PP8 t + up +damp + h +t +CMS PP8 t + down +damp + h +t +CMS PP8 t +t +ATLAS PH7 t +t +ATLAS aMC+P8 t +ATLAS + CMS +Generator Level + 4j +≥ + 4b, +≥ +=13 TeV, +s +Dilepton channel +4 +5 +6 +7 +8 +9 + 10 +≥ +jets +N +0.8 +1 +1.2 +b ++b +t +ATLAS t +Ratio to PP8 +0.02 +0.04 +0.06 +0.08 +0.1 +0.12 +0.14 +0.16 +0.18 +0.2 +0.22 +Arbitrary Units +b +b +t +ATLAS PP8 t +t +CMS PP8 t + up +damp + h +t +CMS PP8 t + down +damp + h +t +CMS PP8 t +t +ATLAS PH7 t +t +ATLAS aMC+P8 t +ATLAS + CMS +Generator Level + 4j +≥ + 4b, +≥ +=13 TeV, +s +Dilepton channel +50 +100 +150 +200 +250 +300 +350 + [GeV] +T +Light jet p +0.8 +1 +1.2 +b ++b +t +ATLAS t +Ratio to PP8 +4 +− +10 +3 +− +10 +2 +− +10 +1 +− +10 +1 +Arbitrary Units +b +b +t +ATLAS PP8 t +t +CMS PP8 t + up +damp + h +t +CMS PP8 t + down +damp + h +t +CMS PP8 t +t +ATLAS PH7 t +t +ATLAS aMC+P8 t +ATLAS + CMS +Generator Level + 6j +≥ + 4b, +≥ +=13 TeV, +s +Single lepton channel +4 +5 +6 + 7 +≥ +b-jets +N +0.6 +0.8 +1 +1.2 +1.4 +b ++b +t +ATLAS t +Ratio to PP8 +0.01 +0.02 +0.03 +0.04 +0.05 +0.06 +Arbitrary Units +b +b +t +ATLAS PP8 t +t +CMS PP8 t + up +damp + h +t +CMS PP8 t + down +damp + h +t +CMS PP8 t +t +ATLAS PH7 t +t +ATLAS aMC+P8 t +ATLAS + CMS +Generator Level + 6j +≥ + 4b, +≥ +=13 TeV, +s +Single lepton channel +400 +600 +800 +1000 +1200 +1400 + [GeV] +jets +T +H +0.8 +1 +1.2 +b ++b +t +ATLAS t +Ratio to PP8 +0.01 +0.02 +0.03 +0.04 +0.05 +0.06 +0.07 +0.08 +Arbitrary Units +b +b +t +ATLAS PP8 t +t +CMS PP8 t + up +damp + h +t +CMS PP8 t + down +damp + h +t +CMS PP8 t +t +ATLAS PH7 t +t +ATLAS aMC+P8 t +ATLAS + CMS +Generator Level + 6j +≥ + 4b, +≥ +=13 TeV, +s +Single lepton channel +50 +100 +150 +200 +250 +300 +350 +400 +450 +500 + [GeV] +T +Leading b-jet p +0.8 +1 +1.2 +b ++b +t +ATLAS t +Ratio to PP8 +0.05 +0.1 +0.15 +0.2 +0.25 +0.3 +0.35 +Arbitrary Units +b +b +t +ATLAS PP8 t +t +CMS PP8 t + up +damp + h +t +CMS PP8 t + down +damp + h +t +CMS PP8 t +t +ATLAS PH7 t +t +ATLAS aMC+P8 t +ATLAS + CMS +Generator Level + 6j +≥ + 4b, +≥ +=13 TeV, +s +Single lepton channel +20 +40 +60 +80 +100 +120 +140 +160 +180 +200 + [GeV] +T +4th b-jet p +0.8 +1 +1.2 +b ++b +t +ATLAS t +Ratio to PP8 +Figure 5: Comparison of predictions used for the systematic uncertainties of the first Run-2 analysis +by CMS [4] and of the first full Run-2 analysis by ATLAS [1]. All distributions are normalised +to one. The ratio shows the different curves divided by ATLAS PP8 t¯tb¯b. The error bands are +constructed from the statistical uncertainties and the scale variations (ME and PS) for the ATLAS +PP8 t¯tb¯b (blue) and the CMS PP8 t¯tb¯b (red) samples. Statistical uncertainties are indicated by +vertical lines. +13 + +3 +Comparisons of Monte Carlo predictions for the t¯tW process +The ATLAS [5] and CMS [6] experiments measured the t¯tH production cross section in multi-lepton +final states, which are primarily sensitive to the decays of H → WW ∗, H → ττ and H → ZZ∗. +The dominant background in these measurements stems from t¯tW production. These measurements +along with the recent CMS measurement of t¯tW production [41] show some tension with the SM +t¯tW predictions which were used to calculate the inclusive cross section and the acceptance in the +analysis phase space. +Different nominal MC predictions were used by the experiments for these measurements, AT- +LAS used Sherpa 2.2.1 [27] and CMS used MG5 aMC@NLO 2.4.2 matched to Pythia8 using +the FxFx merging scheme [24] and including sub-leading electroweak (EW) corrections of the order +αsα3 where α (αs) refers to the EW (QCD) coupling constant. The experiments applied different +corrections to predict the theoretical inclusive t¯tW cross section that entered the calculation of +the scale factor to data, resulting in a value of 727 fb for ATLAS [5] and 650 fb for CMS [6]. Both +experiments estimate the uncertainty of the MC prediction related to missing higher order correc- +tions by varying the renormalisation and factorisation scales in the ME. However, ATLAS considers +additionally uncertainties associated with the modelling of additional QCD radiation by comparing +the nominal t¯tW prediction with that of MG5 aMC@NLO+Pythia8 as alternative MC generator +differing in particular in the number of additional partons in the ME calculation, the parton shower +and merging algorithm. +In recent times there have been significant theoretical developments in t¯tW modelling despite +the challenges associated with calculations of t¯tW with higher order corrections in the QCD, αs, +and EWK, α, couplings. +Even at LO in αs, complications arise because t¯tW is a qq-initiated +process in which the radiation of the W-boson from one of the initial state quarks polarises the +incoming quark, making spin correlations all the more important [42]. Initial calculations of t¯tW +production at next-to-leading order (NLO) in QCD at fixed order [43] and later matched to a +parton shower [44, 45] were later augmented with NLO EWK corrections (of order α2α2 +s) [46] to +provide the higher order cross sections used across the LHC programme for a number of years [47]. +Furthermore, full NLO calculations including fixed-order corrections matched to parton shower +in the POWHEG-BOX framework and accounting for LO spin-correlation of decay product have +recently been provided in [48]. +Since then there has been significant theoretical progress in calculating more complex and precise +predictions. Higher order QCD corrections including t¯tW production with additional partons open +gluon-initiated production modes with significant contributions to the total cross section. Recent +studies show that these contributions also have large next-to-leading order (NLO) corrections [23] +and that t¯tW jj can be large [49], both of which require NLO-merged calculations [50] for such +effects to be properly included. Furthermore, beyond the traditionally “leading” NLO EWK cor- +rections (of order α2α2 +s) there are even larger contributions from traditionally “sub-leading” NLO +corrections (of order α3αs) [51, 52, 48] due to the existence of tW scattering contributions embedded +in to the t¯tWj process. Calculations at NLO in QCD accounting for next-to-next-to-leading loga- +rithmic effects (NNLL) are also available [53] as well as recent predictions at NLO+NNLL in QCD +also with NLO EWK corrections [54, 55]. Full off-shell calculations at NLO in QCD [56, 57, 58] are +also now available and more recently the NLO EWK corrections have also been incorporated [59] +into these calculations, along with the development of procedures to apply the off-shell corrections +to NLO+PS setups [60]. +A first attempt to formulate an uncertainty estimate in view of these theoretical predictions +has been made in [48] where different generator codes at NLO QCD are compared with fixed order +calculations to demonstrate that a robust theoretical prediction of hadronic t¯tW production cannot +14 + +be expressed as a simple recipe covering the specifics of all experimental observables. Therefore the +value of comparing several well tested tools is emphasised. +For future analyses, updated MC models will be used and the estimate of systematic uncertainty +is under development. In particular, ATLAS is considering Sherpa predictions including several +higher order EW corrections in addition to the predictions at NLO in the strong coupling, namely of +the order α3, α2α2 +s and α3αs. Furthermore, calculations of MG5 aMC@NLO+Pythia8 employing +the FxFx merging scheme will be considered. For inclusive predictions, Powheg predictions [48] +are also considered. CMS will continue to use MG5 aMC@NLO+Pythia8 with the FxFx merging +scheme including subleading EW corrections however the EW corrections are not included in the +present document in order to facilitate the comparison between the setups used by each experiment. +The samples will be described in the following and an overview with detailed information on the +samples is given in Table 5. The use of other theoretical developments, already outlined, will also +be considered in future but are beyond the scope of this document. +Comparisons are performed using stable final-state particles in a fiducial phase space similar +to the experimental measurements in the two same-sign leptons (2lSS) channel as implemented +in a dedicated routine in the Rivet analysis toolkit [7]. Two sets of distributions are presented, +one where the histograms are normalised to unit area to asses shape differences in the differential +distributions and another set where the generator cross sections are set to 600.8 fb the value reported +in Ref. [47]. This allows to study differences in acceptance for the different generator predictions. +The chapter is organised as follows: Section 3.1 gives the detailed set-up for the generator +samples, Section 3.2 describes the object reconstruction and event selection, Section 3.3 gives the +two sets of results and finally conclusions are drawn in Section 3.4. +3.1 +MC generator set-ups +This chapter describes in detail the set-up of the MC generator set-ups used for the ATLAS and +CMS samples. +ATLAS setup +The nominal sample for the comparison of this note was generated using the Sherpa 2.2.10 [27, 61] +generator with the NNPDF3.0 NLO PDF set. The t¯tW matrix element was calculated for up to one +additional parton at NLO and up to two partons at leading order (LO) accuracy using Comix [35] +and OpenLoops [29], and merged with the Sherpa parton shower [36] using the MEPs@NLO +prescription [62] with a merging scale of 30 GeV. The choice of renormalisation and factorisation +scales of the core process is µR = µF = HT/2, where HT is defined as the scalar sum of the +transverse masses +� +p2 +T + m2 of all final state particles. Systematic uncertainties due to missing +higher-order QCD corrections are estimated in the nominal sample by varying the factorisation and +renormalisation scales together with αs in the parton shower by a factor of 0.5 (2.0) with respect +to the central value. +In addition to this nominal prediction at NLO in the strong coupling, a separate sample is +produced which contains also higher order corrections relating to EW contributions. These are +added in two ways. First, event-by-event correction factors are applied that provide virtual NLO +EW corrections of the order α2α2 +s derived using the formalism described in Ref. [63] along with +LO corrections of order α3, both are implemented using the prescription outlined in Refs. [27, 64]. +Second, sub-leading EW corrections at order α3αs [52] are partially accounted for (only the real +emission contribution) via the addition of an independent Sherpa 2.2.10 sample produced at LO +in QCD for this final state. This sample is marked as “QCD+EW” in the following. +15 + +Alternative t¯tW predictions are produced using the MG5 aMC@NLO 2.3.3 program to gen- +erate t¯tW production with up to one additional parton in the final state at NLO accuracy in the +strong coupling. The renormalisation and factorisation scales are the same as in the nominal sam- +ple. Another sample is generated using MG5 aMC@NLO 2.9.3 for up to one additional parton +at NLO accuracy and up to two additional partons at LO accuracy in the ME and merging the +different jet multiplicities using the FxFx NLO matrix-element and parton-shower merging pre- +scription [24], see detailed description in [65]. As part of the FxFx merging algorithm, scales are +dynamically chosen and set to the characteristic scale of the hard process. In both samples, spin +correlation effects between the ME decay products are accounted for by Madspin [66] and the +showering and subsequent hadronization is performed using Pythia 8.210 and Pythia 8.245 [8], +respectively, with the A14 tune [33]. These samples are referred to as “ATLAS MG5 aMC+Py8” +and “ATLAS MG5 aMC+Py8 FxFx” in the following. +CMS setup +CMS simulates proton-proton to t¯tℓν processes at NLO accuracy in the matrix element calcula- +tion using MG5 aMC@NLO 2.4.2. Spin correlation effects between the ME decay products are +accounted for by Madspin [66]. The ME calculation includes diagrams with up to one additional +parton at NLO and any further partons are generated by the parton shower. The renormalisation +and factorisation scales are set to the characteristic scale of the hard process. They are chosen +dynamically and are dependent kinematics of the event after the FxFx merging prescription3. +Theoretical uncertainties associated with missing higher-order QCD corrections from the ME +calculation are estimated by varying the renormalisation and factorisation scale by a factor of +0.5 and 2.0. All possible combinations of these variations, implemented using a dedicated set of +per-event weights, are then used to construct the uncertainty envelope. +The parton shower, hadronization processes and decays of τ leptons (including polarisation +effects) are modelled using Pythia 8.226 with the CP5 tune. +The samples is called “CMS +MG5 aMC+Py8 FxFx” in the following. +3see in particular section 2.2.3 of Ref. [24] where elements of Refs. [67, 68] are taken into account +16 + +3.2 +Object reconstruction, fiducial volume and observables +Object and event selection is defined at stable particle-level that closely matches the detector-level +described in reference [5] (ATLAS) and [6] (CMS). Jets are reconstructed from all stable final state +particles with a mean lifetime of τ > 3 × 10−11 s (but excluding leptons and neutrinos from the +top quark decay chain), using the anti-kt algorithm with a radius parameter of R = 0.4. Jets are +required to satisfy pT > 25 GeV and |η| < 2.5. Jets that are matched to a b-hadron4 by ghost +matching [40] are referred to as b-jets. Electrons and muons, referred to as light leptons ℓ, are +required to be separated from selected jets by ∆R > 0.4 and are otherwise removed. Hadronically +decaying τ leptons are required to satisfy pT > 25 GeV and |η| < 2.5. Events are selected with +exactly two light leptons. +The four-momentum of the bare leptons from top quark decay are +modified (”dressed”) by adding the four-momenta of all radiated photons within a cone of size +∆R = 0.1. Leptons are required to have |η| < 2.5 and pT > 25(20) GeV for leading ℓ0 (subleading +ℓ1) lepton (pT ordered). Leptons are required to have same charge, targeting the semi-leptonic t¯t +decay and leptonic W decay. +Events with at least 3 jets and at least one of them being a b-jet are considered in the fiducial +volume. The object definition and event selection is summarised in Tables 6 and 7. These are then +split into five regions, categorized by the number of jets of any flavour (three or ≥4), Nb−jets (one +or ≥2) as well as the presence of hadronically decaying τ lepton, as summarised in Table 8. +The definitions of the regions are motivated by the t¯tH multi-lepton analysis strategy. Regions +1 and 2 corresponds to the signal regions5 and Regions 3 and 4 are used as control regions in the +2ℓ same-sign 0-τhad t¯tH channel. Definition of Region 5 is closely followed6 by the selections in the +2ℓ same-sign 1-τhad t¯tH channel. +The list of variables for the comparison of the t¯tW generators presented in this note are sum- +marised in Table 9. +4no pT cut is applied +5slightly different then in Ref. [5], in order to define a common selection with the CMS Collaboration. +6requirement on jet multiplicity is relaxed. +17 + +Table 5: +The configurations used for the event generation of the t¯tW processes. Scale settings given in terms of HT = �N +i=0 +� +p2 +T,i + m2 +i , +where N corresponds to the number of final state particles. +Label +ATLAS Sherpa 2.2.10 +ATLAS Sherpa 2.2.10 +ATLAS MG5 aMC+Py8 FxFx +ATLAS MG5 aMC+Py8 +CMS MG5 aMC+Py8 FxFx +QCD+EW +Process +t¯tW inclusive +t¯tW inclusive +t¯tW inclusive +t¯tW inclusive +t¯tℓν (t¯tW inclusive) +Generator +Sherpa 2.2.10 [27] +Sherpa 2.2.10 [27] +MG5 aMC@NLO 2.9.3 [69] +MG5 aMC@NLO 2.3.3 [70] +MG5 aMC@NLO 2.4.2 +order of QCD ME +0,1 j@NLOa +0,1 j@NLOa +0,1 j@NLO +NLO +0,1 j@NLO +ME or core scale +µR = µF = HT/2 +µR = µF = HT/2 +dynamic scale choice [24, 67, 68] +µR = µF = HT/2 +dynamic scale choice [24, 67, 68] +order of EW corr. +- +α3, α2α2 +s, α3αs +- +- +- +Parton Shower +Sherpa 2.2.10 +Sherpa 2.2.10 +Pythia 8.245 [8] +Pythia 8.210 [8] +Pythia 8.226 +Merging Scheme +MEPs@NLO [62] +MEPs@NLO [62] +FxFx [24] +- +FxFx +Merging Scale +30 GeV +30 GeV +30 GeV +- +42 GeV +PDF +NNPDF3.0 NNLO [71] +NNPDF3.0 NNLO +NNPDF3.0 NLO +NNPDF3.0 NLO +NNPDF3.1 NLO [72] +Tune +Sherpa default +Sherpa default +A14 [33] +A14 +CP5 [34] +Cross sectionb +597 fb +615 fb +613 fb +548 fb +220 fb (666 fbc) +aIn addition to the implicit 2j@LO contribution from the real emission part of the 1j@NLO calculation, Sherpa adds the 2j@LO as an explicit separate process +within the merging such that the ME is supplemented with higher-order improvements such as the CKKW scale choice and Sudakov factors.” +bσtot=600.8 fb from YR4 is used for all samples in the generator comparisons in section 3.3.2 except for Sherpa QCD+EW +ccalculated from t¯tℓν as 0.2198 x (1/ (3 x 0.11) ) +18 + +Table 6: +The object reconstruction used in the Rivet analysis of the t¯tW processes. Leptons are +ordered in pT. +Object +reconstruction and selection +jets +stable final state particles with anti-kt algorithm, radius R = 0.4 +prompt ”dressed” leptons and neutrinos are vetoed from jet +pT > 25 GeV and |η| < 2.5 +b-jets +jets ghost matched to B-hadrons +pT > 25 GeV and |η| < 2.5 +light leptons (electrons and muons) +dressed with photons within ∆R < 0.1 +|η| < 2.5 and pT > 25(20) GeV for leading (subleading) lepton +overlap removal +remove light lepton if ∆R(jet, lepton) < 0.4 +hadronicaly decaying τ leptons (before decay) pT > 25 GeV and |η| < 2.5 +Table 7: +The event selection used in the Rivet analysis for the t¯tW processes. Njets refers to all +jets independent of jet flavour, i.e. b-jets are included. +Event selection for 2ℓSS +exactly 2 leptons with same charge +Njets ≥3 +Nb−jets ≥1 +Table 8: +The region definitions used in the Rivet analysis for the t¯tW processes. +Region +Selection +1 +Nb−jets =1, Njets ≥4, 0-τhad +2 +Nb−jets ≥2, Njets ≥4, 0-τhad +3 +Nb−jets =1, Njets=3, 0-τhad +4 +Nb−jets ≥2, Njets=3, 0-τhad +5 +Nb−jets ≥1, Njets ≥3, 1-τhad +Table 9: +List of the observables for the comparison of t¯tW predictions. Leptons and b-jets are +ordered in pT. +Variable Description +Regions +Njets +Jet multiplicity +1,2,5 +Nb−jets +Number of b-jets +1,2,5 +Hjets +T +Scalar sum of transverse momentum of all jets in the event +1,2,3,4 +pb0 +T +Leading b-jet transverse momentum +1,2 +pℓ0 +T +Leading lepton transverse momentum +1,2,5 +∆Rℓ0jets Minimum angular separation between the leading lepton ℓ0 and the nearest jet +1,2 +∆Rℓ0ℓ1 +Angular distance between the two leptons +1,2,5 +max|ηℓ| Value of the highest lepton’s pseudorapidity in the event +1,2 +19 + +3.3 +Results +The samples described in Table 5 are compared in the following. The ratio plots show the ratios +of the all MC samples with respect to ATLAS Sherpa 2.2.10, the shaded band represents scale +variations. The same set of distributions are presented twice with different focus: in Sect. 3.3.1 +shapes are compared and in Sect. 3.3.2 acceptance effects are studied. +3.3.1 +Shape comparison +In the following, shape comparisons between nominal and alternative generators will be presented, +i.e. the distributions are normalised to unit area. The modelling of jet based distributions are +presented in Fig. 6 for the regions without hadronic τ leptons. Sizeable discrepancies in the mod- +elling of high jet multiplicities can be observed between the ATLAS and CMS MG5 aMC@NLO +FxFx predictions which are in opposite direction compared to Sherpa t¯tW. All predictions except +ATLAS MG5 aMC@NLO+Pythia8 agree well on HT in regions with at least four jets, but larger +discrepancies are observed for the three jet regions. The distributions of b-jet pT differ more in the +regions with one b-jet, as shown in Fig. 7. +Only ATLAS MG5 aMC@NLO+Pythia8 shows significant differences for the angular distance +between the two leptons and the value of lepton’s pseudo-rapidity as demonstrated in Fig. 8. The +lepton pT distributions are similar, but their distance to the closest jet vary at as seen in Fig. 9. +Distributions of the jet multiplicity, number of b-jets, the leading lepton transverse momentum +and the angular distance between the two leptons ∆Rℓ0ℓ1 for the Region 5 with Nτhad = 1 selection +are presented in Fig. 10. The jet multiplicity predictions of MG5 aMC+Py8 FxFx with the ATLAS +and CMS set-ups differ most from the other predictions in this region. +20 + +Number of jets +3 +4 +5 +6 +7 +8 +9 +Normalised to unit area +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +ATLAS + CMS +Generator Level + = 13 TeV, ttW +s + j4 +≥ + +b + 1 +had +τ +SS 0 +l2 +ATLAS Sherpa 2.2.10 +ATLAS Sherpa 2.2.10 QCD+EW +ATLAS MG5_aMC+Py8 FxFx +ATLAS MG5_aMC+Py8 +CMS MG5_aMC+Py8 FxFx +ATLAS Sherpa scale var ME+PS +CMS scale variation ME-only +Number of jets +3 +4 +5 +6 +7 +8 +9 +Ratio to Sherpa +0.5 +1 +1.5 +Number of jets +3 +4 +5 +6 +7 +8 +9 +Normalised to unit area +0 +0.2 +0.4 +0.6 +0.8 +1 +ATLAS + CMS +Generator Level + = 13 TeV, ttW +s + j4 +≥ + +b +2 +≥ + +had +τ +SS 0 +l2 +ATLAS Sherpa 2.2.10 +ATLAS Sherpa 2.2.10 QCD+EW +ATLAS MG5_aMC+Py8 FxFx +ATLAS MG5_aMC+Py8 +CMS MG5_aMC+Py8 FxFx +ATLAS Sherpa scale var ME+PS +CMS scale variation ME-only +Number of jets +3 +4 +5 +6 +7 +8 +9 +Ratio to Sherpa +0.5 +1 +1.5 + [GeV] +jets +T +H +0 +200 +400 +600 +800 +1000 +1200 +1400 +Normalised to unit area +0 +0.05 +0.1 +0.15 +0.2 +0.25 +0.3 +0.35 +0.4 +0.45 +ATLAS + CMS +Generator Level + = 13 TeV, ttW +s + j4 +≥ + +b + 1 +had +τ +SS 0 +l2 +ATLAS Sherpa 2.2.10 +ATLAS Sherpa 2.2.10 QCD+EW +ATLAS MG5_aMC+Py8 FxFx +ATLAS MG5_aMC+Py8 +CMS MG5_aMC+Py8 FxFx +ATLAS Sherpa scale var ME+PS +CMS scale variation ME-only + [GeV] +jets +T +H +0 +200 +400 +600 +800 +1000 +1200 +1400 +Ratio to Sherpa +0.5 +1 +1.5 + [GeV] +jets +T +H +0 +200 +400 +600 +800 +1000 +1200 +1400 +Normalised to unit area +0 +0.05 +0.1 +0.15 +0.2 +0.25 +0.3 +0.35 +0.4 +0.45 +ATLAS + CMS +Generator Level + = 13 TeV, ttW +s + j4 +≥ + +b +2 +≥ + +had +τ +SS 0 +l2 +ATLAS Sherpa 2.2.10 +ATLAS Sherpa 2.2.10 QCD+EW +ATLAS MG5_aMC+Py8 FxFx +ATLAS MG5_aMC+Py8 +CMS MG5_aMC+Py8 FxFx +ATLAS Sherpa scale var ME+PS +CMS scale variation ME-only + [GeV] +jets +T +H +0 +200 +400 +600 +800 +1000 +1200 +1400 +Ratio to Sherpa +0.5 +1 +1.5 + [GeV] +jets +T +H +0 +200 +400 +600 +800 +1000 +1200 +1400 +Normalised to unit area +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +ATLAS + CMS +Generator Level + = 13 TeV, ttW +s + j + 3 +b + 1 +had +τ +SS 0 +l2 +ATLAS Sherpa 2.2.10 +ATLAS Sherpa 2.2.10 QCD+EW +ATLAS MG5_aMC+Py8 FxFx +ATLAS MG5_aMC+Py8 +CMS MG5_aMC+Py8 FxFx +ATLAS Sherpa scale var ME+PS +CMS scale variation ME-only + [GeV] +jets +T +H +0 +200 +400 +600 +800 +1000 +1200 +1400 +Ratio to Sherpa +0.5 +1 +1.5 + [GeV] +jets +T +H +0 +200 +400 +600 +800 +1000 +1200 +1400 +Normalised to unit area +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +ATLAS + CMS +Generator Level + = 13 TeV, ttW +s + j + 3 +b +2 +≥ + +had +τ +SS 0 +l2 +ATLAS Sherpa 2.2.10 +ATLAS Sherpa 2.2.10 QCD+EW +ATLAS MG5_aMC+Py8 FxFx +ATLAS MG5_aMC+Py8 +CMS MG5_aMC+Py8 FxFx +ATLAS Sherpa scale var ME+PS +CMS scale variation ME-only + [GeV] +jets +T +H +0 +200 +400 +600 +800 +1000 +1200 +1400 +Ratio to Sherpa +0.5 +1 +1.5 +Figure 6: Distribution of the jet multiplicities (top) and the scalar sum of jets transverse momentum, +Hjets +T +(middle), for the Region 1 with Nb−jets =1 (left) and Region 2 with Nb−jets ≥2 (right) selection +requiring four and more jets, and for the Region 3 Nb−jets = 1 (bottom, left) and Region 4 with +Nb−jets ≥2 (bottom, right) selection requiring exactly three jets. +21 + +-jets +b +Number of +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +Normalised to unit area +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +1.4 +1.6 +1.8 +2 +2.2 +ATLAS + CMS +Generator Level + = 13 TeV, ttW +s + j4 +≥ + +b +2 +≥ + +had +τ +SS 0 +l2 +ATLAS Sherpa 2.2.10 +ATLAS Sherpa 2.2.10 QCD+EW +ATLAS MG5_aMC+Py8 FxFx +ATLAS MG5_aMC+Py8 +CMS MG5_aMC+Py8 FxFx +ATLAS Sherpa scale var ME+PS +CMS scale variation ME-only +-jets +b +Number of +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +Ratio to Sherpa +0.5 +1 +1.5 + [GeV] +T +p +-jet +b +Leading +0 +50 +100 +150 +200 +250 +300 +350 +400 +450 +500 +Normalised to unit area +0 +0.05 +0.1 +0.15 +0.2 +0.25 +0.3 +0.35 +0.4 +ATLAS + CMS +Generator Level + = 13 TeV, ttW +s + j4 +≥ + +b + 1 +had +τ +SS 0 +l2 +ATLAS Sherpa 2.2.10 +ATLAS Sherpa 2.2.10 QCD+EW +ATLAS MG5_aMC+Py8 FxFx +ATLAS MG5_aMC+Py8 +CMS MG5_aMC+Py8 FxFx +ATLAS Sherpa scale var ME+PS +CMS scale variation ME-only + [GeV] +T +p +-jet +b +Leading +0 +50 +100 +150 +200 +250 +300 +350 +400 +450 +500 +Ratio to Sherpa +0.5 +1 +1.5 + [GeV] +T +p +-jet +b +Leading +0 +50 +100 +150 +200 +250 +300 +350 +400 +450 +500 +Normalised to unit area +0 +0.1 +0.2 +0.3 +0.4 +0.5 +ATLAS + CMS +Generator Level + = 13 TeV, ttW +s + j4 +≥ + +b +2 +≥ + +had +τ +SS 0 +l2 +ATLAS Sherpa 2.2.10 +ATLAS Sherpa 2.2.10 QCD+EW +ATLAS MG5_aMC+Py8 FxFx +ATLAS MG5_aMC+Py8 +CMS MG5_aMC+Py8 FxFx +ATLAS Sherpa scale var ME+PS +CMS scale variation ME-only + [GeV] +T +p +-jet +b +Leading +0 +50 +100 +150 +200 +250 +300 +350 +400 +450 +500 +Ratio to Sherpa +0.5 +1 +1.5 +Figure 7: Distribution of the b-jet multiplicities (top) and the leading b-jet transverse momentum +(bottom), for the Region 1 with Nb−jets=1 (left) and Region 2 with Nb−jets ≥2 (right) selection +requiring four and more jets. +22 + + [GeV] +T +p +Leading lepton +0 +50 +100 +150 +200 +250 +300 +350 +400 +450 +500 +Normalised to unit area +0 +0.05 +0.1 +0.15 +0.2 +0.25 +0.3 +0.35 +0.4 +0.45 +ATLAS + CMS +Generator Level + = 13 TeV, ttW +s + j4 +≥ + +b + 1 +had +τ +SS 0 +l2 +ATLAS Sherpa 2.2.10 +ATLAS Sherpa 2.2.10 QCD+EW +ATLAS MG5_aMC+Py8 FxFx +ATLAS MG5_aMC+Py8 +CMS MG5_aMC+Py8 FxFx +ATLAS Sherpa scale var ME+PS +CMS scale variation ME-only + [GeV] +T +p +Leading lepton +0 +50 +100 +150 +200 +250 +300 +350 +400 +450 +500 +Ratio to Sherpa +0.5 +1 +1.5 + [GeV] +T +p +Leading lepton +0 +50 +100 +150 +200 +250 +300 +350 +400 +450 +500 +Normalised to unit area +0 +0.05 +0.1 +0.15 +0.2 +0.25 +0.3 +0.35 +0.4 +0.45 +ATLAS + CMS +Generator Level + = 13 TeV, ttW +s + j4 +≥ + +b +2 +≥ + +had +τ +SS 0 +l2 +ATLAS Sherpa 2.2.10 +ATLAS Sherpa 2.2.10 QCD+EW +ATLAS MG5_aMC+Py8 FxFx +ATLAS MG5_aMC+Py8 +CMS MG5_aMC+Py8 FxFx +ATLAS Sherpa scale var ME+PS +CMS scale variation ME-only + [GeV] +T +p +Leading lepton +0 +50 +100 +150 +200 +250 +300 +350 +400 +450 +500 +Ratio to Sherpa +0.5 +1 +1.5 +,jet +0l + R +∆ + +min +0 +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +4 +4.5 +Normalised to unit area +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +ATLAS + CMS +Generator Level + = 13 TeV, ttW +s + j4 +≥ + +b + 1 +had +τ +SS 0 +l2 +ATLAS Sherpa 2.2.10 +ATLAS Sherpa 2.2.10 QCD+EW +ATLAS MG5_aMC+Py8 FxFx +ATLAS MG5_aMC+Py8 +CMS MG5_aMC+Py8 FxFx +ATLAS Sherpa scale var ME+PS +CMS scale variation ME-only +,jet +0l + R +∆ + +min +0 +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +4 +4.5 +Ratio to Sherpa +0.5 +1 +1.5 +,jet +0l + R +∆ + +min +0 +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +4 +4.5 +Normalised to unit area +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +ATLAS + CMS +Generator Level + = 13 TeV, ttW +s + j4 +≥ + +b +2 +≥ + +had +τ +SS 0 +l2 +ATLAS Sherpa 2.2.10 +ATLAS Sherpa 2.2.10 QCD+EW +ATLAS MG5_aMC+Py8 FxFx +ATLAS MG5_aMC+Py8 +CMS MG5_aMC+Py8 FxFx +ATLAS Sherpa scale var ME+PS +CMS scale variation ME-only +,jet +0l + R +∆ + +min +0 +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +4 +4.5 +Ratio to Sherpa +0.5 +1 +1.5 +Figure 8: Distribution of the leading lepton transverse momentum (top) and the minimum angular +separation between the leading lepton and the nearest jet (bottom), for the Region 1 with Nb−jets=1 +(left) and Region 2 with Nb−jets ≥2 (right) selection requiring four and more jets. +23 + +1 +,l +0l + R +∆ +0 +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +4 +4.5 +Normalised to unit area +0 +0.05 +0.1 +0.15 +0.2 +0.25 +0.3 +0.35 +0.4 +ATLAS + CMS +Generator Level + = 13 TeV, ttW +s + j4 +≥ + +b + 1 +had +τ +SS 0 +l2 +ATLAS Sherpa 2.2.10 +ATLAS Sherpa 2.2.10 QCD+EW +ATLAS MG5_aMC+Py8 FxFx +ATLAS MG5_aMC+Py8 +CMS MG5_aMC+Py8 FxFx +ATLAS Sherpa scale var ME+PS +CMS scale variation ME-only +1 +,l +0l + R +∆ +0 +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +4 +4.5 +Ratio to Sherpa +0.5 +1 +1.5 +1 +,l +0l + R +∆ +0 +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +4 +4.5 +Normalised to unit area +0 +0.05 +0.1 +0.15 +0.2 +0.25 +0.3 +0.35 +0.4 +ATLAS + CMS +Generator Level + = 13 TeV, ttW +s + j4 +≥ + +b +2 +≥ + +had +τ +SS 0 +l2 +ATLAS Sherpa 2.2.10 +ATLAS Sherpa 2.2.10 QCD+EW +ATLAS MG5_aMC+Py8 FxFx +ATLAS MG5_aMC+Py8 +CMS MG5_aMC+Py8 FxFx +ATLAS Sherpa scale var ME+PS +CMS scale variation ME-only +1 +,l +0l + R +∆ +0 +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +4 +4.5 +Ratio to Sherpa +0.5 +1 +1.5 +|l +η + | +max +0 +0.5 +1 +1.5 +2 +2.5 +Normalised to unit area +0 +0.05 +0.1 +0.15 +0.2 +0.25 +ATLAS + CMS +Generator Level + = 13 TeV, ttW +s + j4 +≥ + +b + 1 +had +τ +SS 0 +l2 +ATLAS Sherpa 2.2.10 +ATLAS Sherpa 2.2.10 QCD+EW +ATLAS MG5_aMC+Py8 FxFx +ATLAS MG5_aMC+Py8 +CMS MG5_aMC+Py8 FxFx +ATLAS Sherpa scale var ME+PS +CMS scale variation ME-only +|l +η + | +max +0 +0.5 +1 +1.5 +2 +2.5 +Ratio to Sherpa +0.5 +1 +1.5 +|l +η + | +max +0 +0.5 +1 +1.5 +2 +2.5 +Normalised to unit area +0.05 +0.1 +0.15 +0.2 +0.25 +ATLAS + CMS +Generator Level + = 13 TeV, ttW +s + j4 +≥ + +b +2 +≥ + +had +τ +SS 0 +l2 +ATLAS Sherpa 2.2.10 +ATLAS Sherpa 2.2.10 QCD+EW +ATLAS MG5_aMC+Py8 FxFx +ATLAS MG5_aMC+Py8 +CMS MG5_aMC+Py8 FxFx +ATLAS Sherpa scale var ME+PS +CMS scale variation ME-only +|l +η + | +max +0 +0.5 +1 +1.5 +2 +2.5 +Ratio to Sherpa +0.5 +1 +1.5 +Figure 9: Distribution of the angular distance between the two leptons (top), maximum of lepton +|ηℓ0| and |ηℓ1| (bottom) , for the Region 1 with Nb−jets=1 (left) and Region 2 with Nb−jets ≥2 (right) +selection requiring four and more jets. +24 + +Number of jets +3 +4 +5 +6 +7 +8 +9 +Normalised to unit area +0 +0.2 +0.4 +0.6 +0.8 +1 +ATLAS + CMS +Generator Level + = 13 TeV, ttW +s + j3 +≥ + +b +1 +≥ + +had +τ +SS 1 +l2 +ATLAS Sherpa 2.2.10 +ATLAS Sherpa 2.2.10 QCD+EW +ATLAS MG5_aMC+Py8 FxFx +ATLAS MG5_aMC+Py8 +CMS MG5_aMC+Py8 FxFx +ATLAS Sherpa scale var ME+PS +CMS scale variation ME-only +Number of jets +3 +4 +5 +6 +7 +8 +9 +Ratio to Sherpa +0.5 +1 +1.5 +-jets +b +Number of +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +Normalised to unit area +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +1.4 +1.6 +1.8 +ATLAS + CMS +Generator Level + = 13 TeV, ttW +s + j3 +≥ + +b +1 +≥ + +had +τ +SS 1 +l2 +ATLAS Sherpa 2.2.10 +ATLAS Sherpa 2.2.10 QCD+EW +ATLAS MG5_aMC+Py8 FxFx +ATLAS MG5_aMC+Py8 +CMS MG5_aMC+Py8 FxFx +ATLAS Sherpa scale var ME+PS +CMS scale variation ME-only +-jets +b +Number of +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +Ratio to Sherpa +0.5 +1 +1.5 + [GeV] +T +p +Leading lepton +0 +50 +100 +150 +200 +250 +300 +350 +400 +450 +500 +Normalised to unit area +0 +0.05 +0.1 +0.15 +0.2 +0.25 +0.3 +0.35 +0.4 +0.45 +ATLAS + CMS +Generator Level + = 13 TeV, ttW +s + j3 +≥ + +b +1 +≥ + +had +τ +SS 1 +l2 +ATLAS Sherpa 2.2.10 +ATLAS Sherpa 2.2.10 QCD+EW +ATLAS MG5_aMC+Py8 FxFx +ATLAS MG5_aMC+Py8 +CMS MG5_aMC+Py8 FxFx +ATLAS Sherpa scale var ME+PS +CMS scale variation ME-only + [GeV] +T +p +Leading lepton +0 +50 +100 +150 +200 +250 +300 +350 +400 +450 +500 +Ratio to Sherpa +0.5 +1 +1.5 +1 +,l +0l + R +∆ +0 +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +4 +4.5 +Normalised to unit area +0 +0.05 +0.1 +0.15 +0.2 +0.25 +0.3 +0.35 +ATLAS + CMS +Generator Level + = 13 TeV, ttW +s + j3 +≥ + +b +1 +≥ + +had +τ +SS 1 +l2 +ATLAS Sherpa 2.2.10 +ATLAS Sherpa 2.2.10 QCD+EW +ATLAS MG5_aMC+Py8 FxFx +ATLAS MG5_aMC+Py8 +CMS MG5_aMC+Py8 FxFx +ATLAS Sherpa scale var ME+PS +CMS scale variation ME-only +1 +,l +0l + R +∆ +0 +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +4 +4.5 +Ratio to Sherpa +0.5 +1 +1.5 +Figure 10: Distribution of the the jet multiplicity, number of b-jets, the leading lepton transverse +momentum and the angular distance between the two leptons ∆Rℓℓ for the Region 5 with 1τhad +selection. +25 + +3.3.2 +Comparisons of predictions including acceptance effects +In the following section, a comparison of the generators will be given in the fiducial phase space, +i.e. the predicted distributions include acceptance effects. For this comparison, all distributions are +normalised to a common total cross section value of σYR4 +tot += 600.8 fb as given in the Yellow Report +4 [47], except the distributions of Sherpa 2.2.10 QCD+EW which is normalised to its generator +cross section of 614.7 fb. The same set of distributions as discussed in Section 3.3.1 are presented. +In all distributions, a significant increase of scale uncertainties is observed, reaching up to 50 % +at high jet multiplicity. The observables related to jet multiplicity and HT show similar trends as +in the shape comparisons, see Fig. 11. Only the discrepancy of the jet multiplicity prediction in +MG5 aMC+Py8 FxFx is significantly enhanced. +26 + +Number of jets +3 +4 +5 +6 +7 +8 +9 + [fb] +jet +N +d +/ +fid +σ +d +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +1.4 +1.6 +1.8 +ATLAS + CMS +Generator Level + = 13 TeV, ttW +s + j4 +≥ + +b + 1 +had +τ +SS 0 +l2 +ATLAS Sherpa 2.2.10 +ATLAS Sherpa 2.2.10 QCD+EW +ATLAS MG5_aMC+Py8 FxFx +ATLAS MG5_aMC+Py8 +CMS MG5_aMC+Py8 FxFx +ATLAS Sherpa scale var ME+PS +CMS scale variation ME-only +Number of jets +3 +4 +5 +6 +7 +8 +9 +Ratio to Sherpa +0.5 +1 +1.5 +Number of jets +3 +4 +5 +6 +7 +8 +9 + [fb] +jet +N +d +/ +fid +σ +d +0 +1 +2 +3 +4 +5 +6 +ATLAS + CMS +Generator Level + = 13 TeV, ttW +s + j4 +≥ + +b +2 +≥ + +had +τ +SS 0 +l2 +ATLAS Sherpa 2.2.10 +ATLAS Sherpa 2.2.10 QCD+EW +ATLAS MG5_aMC+Py8 FxFx +ATLAS MG5_aMC+Py8 +CMS MG5_aMC+Py8 FxFx +ATLAS Sherpa scale var ME+PS +CMS scale variation ME-only +Number of jets +3 +4 +5 +6 +7 +8 +9 +Ratio to Sherpa +0.5 +1 +1.5 + [GeV] +jets +T +H +0 +200 +400 +600 +800 +1000 +1200 +1400 + [fb/GeV] +jets +T +H +d +/ +fid +σ +d +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +ATLAS + CMS +Generator Level + = 13 TeV, ttW +s + j4 +≥ + +b + 1 +had +τ +SS 0 +l2 +ATLAS Sherpa 2.2.10 +ATLAS Sherpa 2.2.10 QCD+EW +ATLAS MG5_aMC+Py8 FxFx +ATLAS MG5_aMC+Py8 +CMS MG5_aMC+Py8 FxFx +ATLAS Sherpa scale var ME+PS +CMS scale variation ME-only + [GeV] +jets +T +H +0 +200 +400 +600 +800 +1000 +1200 +1400 +Ratio to Sherpa +0.5 +1 +1.5 + [GeV] +jets +T +H +0 +200 +400 +600 +800 +1000 +1200 +1400 + [fb/GeV] +jets +T +H +d +/ +fid +σ +d +0 +0.5 +1 +1.5 +2 +2.5 +ATLAS + CMS +Generator Level + = 13 TeV, ttW +s + j4 +≥ + +b +2 +≥ + +had +τ +SS 0 +l2 +ATLAS Sherpa 2.2.10 +ATLAS Sherpa 2.2.10 QCD+EW +ATLAS MG5_aMC+Py8 FxFx +ATLAS MG5_aMC+Py8 +CMS MG5_aMC+Py8 FxFx +ATLAS Sherpa scale var ME+PS +CMS scale variation ME-only + [GeV] +jets +T +H +0 +200 +400 +600 +800 +1000 +1200 +1400 +Ratio to Sherpa +0.5 +1 +1.5 + [GeV] +jets +T +H +0 +200 +400 +600 +800 +1000 +1200 +1400 + [fb/GeV] +jets +T +H +d +/ +fid +σ +d +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +ATLAS + CMS +Generator Level + = 13 TeV, ttW +s + j + 3 +b + 1 +had +τ +SS 0 +l2 +ATLAS Sherpa 2.2.10 +ATLAS Sherpa 2.2.10 QCD+EW +ATLAS MG5_aMC+Py8 FxFx +ATLAS MG5_aMC+Py8 +CMS MG5_aMC+Py8 FxFx +ATLAS Sherpa scale var ME+PS +CMS scale variation ME-only + [GeV] +jets +T +H +0 +200 +400 +600 +800 +1000 +1200 +1400 +Ratio to Sherpa +0.5 +1 +1.5 + [GeV] +jets +T +H +0 +200 +400 +600 +800 +1000 +1200 +1400 + [fb/GeV] +jets +T +H +d +/ +fid +σ +d +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +1.4 +ATLAS + CMS +Generator Level + = 13 TeV, ttW +s + j + 3 +b +2 +≥ + +had +τ +SS 0 +l2 +ATLAS Sherpa 2.2.10 +ATLAS Sherpa 2.2.10 QCD+EW +ATLAS MG5_aMC+Py8 FxFx +ATLAS MG5_aMC+Py8 +CMS MG5_aMC+Py8 FxFx +ATLAS Sherpa scale var ME+PS +CMS scale variation ME-only + [GeV] +jets +T +H +0 +200 +400 +600 +800 +1000 +1200 +1400 +Ratio to Sherpa +0.5 +1 +1.5 +Figure 11: Distribution of jet multiplicities (top) and scalar sum of jets transverse momentum, Hjets +T +(middle), for the Region 1 with Nb−jets=1 (middle, left) and Region 2 with Nb−jets ≥2 (middle, right) +selection requiring four and more jets. Hjets +T , for the Region 3 with Nb−jets=1 (bottom, left) and +Region 4 with Nb−jets ≥2 (bottom, right) selection requiring exactly three jets. All distributions +are normalised to the YR4 cross section of 600.8 fb, except Sherpa 2.2.10 QCD+EW which is +normalised to 614.7 fb. +27 + +-jets +b +Number of +0.5 +1 +1.5 +2 +2.5 +3 +3.5 + [fb] +-jet +b +N +d +/ +fid +σ +d +0 +0.5 +1 +1.5 +2 +2.5 +3 +ATLAS + CMS +Generator Level + = 13 TeV, ttW +s + j4 +≥ + +b + 1 +had +τ +SS 0 +l2 +ATLAS Sherpa 2.2.10 +ATLAS Sherpa 2.2.10 QCD+EW +ATLAS MG5_aMC+Py8 FxFx +ATLAS MG5_aMC+Py8 +CMS MG5_aMC+Py8 FxFx +ATLAS Sherpa scale var ME+PS +CMS scale variation ME-only +-jets +b +Number of +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +Ratio to Sherpa +0.5 +1 +1.5 +-jets +b +Number of +0.5 +1 +1.5 +2 +2.5 +3 +3.5 + [fb] +-jet +b +N +d +/ +fid +σ +d +0 +2 +4 +6 +8 +10 +12 +ATLAS + CMS +Generator Level + = 13 TeV, ttW +s + j4 +≥ + +b +2 +≥ + +had +τ +SS 0 +l2 +ATLAS Sherpa 2.2.10 +ATLAS Sherpa 2.2.10 QCD+EW +ATLAS MG5_aMC+Py8 FxFx +ATLAS MG5_aMC+Py8 +CMS MG5_aMC+Py8 FxFx +ATLAS Sherpa scale var ME+PS +CMS scale variation ME-only +-jets +b +Number of +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +Ratio to Sherpa +0.5 +1 +1.5 + [GeV] +T +p +-jet +b +Leading +0 +50 +100 +150 +200 +250 +300 +350 +400 +450 +500 + [fb/GeV] +0 +b +T +p +d +/ +fid +σ +d +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +ATLAS + CMS +Generator Level + = 13 TeV, ttW +s + j4 +≥ + +b + 1 +had +τ +SS 0 +l2 +ATLAS Sherpa 2.2.10 +ATLAS Sherpa 2.2.10 QCD+EW +ATLAS MG5_aMC+Py8 FxFx +ATLAS MG5_aMC+Py8 +CMS MG5_aMC+Py8 FxFx +ATLAS Sherpa scale var ME+PS +CMS scale variation ME-only + [GeV] +T +p +-jet +b +Leading +0 +50 +100 +150 +200 +250 +300 +350 +400 +450 +500 +Ratio to Sherpa +0.5 +1 +1.5 + [GeV] +T +p +-jet +b +Leading +0 +50 +100 +150 +200 +250 +300 +350 +400 +450 +500 + [fb/GeV] +0 +b +T +p +d +/ +fid +σ +d +0 +0.5 +1 +1.5 +2 +2.5 +3 +ATLAS + CMS +Generator Level + = 13 TeV, ttW +s + j4 +≥ + +b +2 +≥ + +had +τ +SS 0 +l2 +ATLAS Sherpa 2.2.10 +ATLAS Sherpa 2.2.10 QCD+EW +ATLAS MG5_aMC+Py8 FxFx +ATLAS MG5_aMC+Py8 +CMS MG5_aMC+Py8 FxFx +ATLAS Sherpa scale var ME+PS +CMS scale variation ME-only + [GeV] +T +p +-jet +b +Leading +0 +50 +100 +150 +200 +250 +300 +350 +400 +450 +500 +Ratio to Sherpa +0.5 +1 +1.5 +Figure 12: Distribution of the b-jet multiplicities (top) and the leading b-jet transverse momentum +(bottom), for the Region 1 with Nb−jets=1 (left) and Region 2 with Nb−jets ≥2 (right) selection +requiring four and more jets. All distributions are normalised to the YR4 cross section of 600.8 fb +except Sherpa 2.2.10 QCD+EW which is normalised to 614.7 fb. +28 + + [GeV] +T +p +Leading lepton +0 +50 +100 +150 +200 +250 +300 +350 +400 +450 +500 + [fb/GeV] +l_0 +T +p +d +/ +fid +σ +d +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +ATLAS + CMS +Generator Level + = 13 TeV, ttW +s + j4 +≥ + +b + 1 +had +τ +SS 0 +l2 +ATLAS Sherpa 2.2.10 +ATLAS Sherpa 2.2.10 QCD+EW +ATLAS MG5_aMC+Py8 FxFx +ATLAS MG5_aMC+Py8 +CMS MG5_aMC+Py8 FxFx +ATLAS Sherpa scale var ME+PS +CMS scale variation ME-only + [GeV] +T +p +Leading lepton +0 +50 +100 +150 +200 +250 +300 +350 +400 +450 +500 +Ratio to Sherpa +0.5 +1 +1.5 + [GeV] +T +p +Leading lepton +0 +50 +100 +150 +200 +250 +300 +350 +400 +450 +500 + [fb/GeV] +l_0 +T +p +d +/ +fid +σ +d +0 +0.5 +1 +1.5 +2 +2.5 +ATLAS + CMS +Generator Level + = 13 TeV, ttW +s + j4 +≥ + +b +2 +≥ + +had +τ +SS 0 +l2 +ATLAS Sherpa 2.2.10 +ATLAS Sherpa 2.2.10 QCD+EW +ATLAS MG5_aMC+Py8 FxFx +ATLAS MG5_aMC+Py8 +CMS MG5_aMC+Py8 FxFx +ATLAS Sherpa scale var ME+PS +CMS scale variation ME-only + [GeV] +T +p +Leading lepton +0 +50 +100 +150 +200 +250 +300 +350 +400 +450 +500 +Ratio to Sherpa +0.5 +1 +1.5 +,jet +0l + R +∆ + +min +0 +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +4 +4.5 + [fb] +,jet +0l + R +∆ + +min +d +/ +fid +σ +d +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +ATLAS + CMS +Generator Level + = 13 TeV, ttW +s + j4 +≥ + +b + 1 +had +τ +SS 0 +l2 +ATLAS Sherpa 2.2.10 +ATLAS Sherpa 2.2.10 QCD+EW +ATLAS MG5_aMC+Py8 FxFx +ATLAS MG5_aMC+Py8 +CMS MG5_aMC+Py8 FxFx +ATLAS Sherpa scale var ME+PS +CMS scale variation ME-only +,jet +0l + R +∆ + +min +0 +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +4 +4.5 +Ratio to Sherpa +0.5 +1 +1.5 +,jet +0l + R +∆ + +min +0 +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +4 +4.5 + [fb] +,jet +0l + R +∆ + +min +d +/ +fid +σ +d +0 +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +ATLAS + CMS +Generator Level + = 13 TeV, ttW +s + j4 +≥ + +b +2 +≥ + +had +τ +SS 0 +l2 +ATLAS Sherpa 2.2.10 +ATLAS Sherpa 2.2.10 QCD+EW +ATLAS MG5_aMC+Py8 FxFx +ATLAS MG5_aMC+Py8 +CMS MG5_aMC+Py8 FxFx +ATLAS Sherpa scale var ME+PS +CMS scale variation ME-only +,jet +0l + R +∆ + +min +0 +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +4 +4.5 +Ratio to Sherpa +0.5 +1 +1.5 +Figure 13: Distribution of the leading lepton transverse momentum (top) and the minimum angular +separation between the leading lepton and the nearest jet (bottom), for the Region 1 with Nb−jets=1 +(left) and Region 2 with Nb−jets ≥2 (right) selection requiring four and more jets. All distributions +are normalised to the YR4 cross section of 600.8 fb except Sherpa 2.2.10 QCD+EW which is +normalised to 614.7 fb. +29 + +1 +,l +0l + R +∆ +0 +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +4 +4.5 + [fb] +1 +,l +0l + R +∆ +d +/ +fid +σ +d +0 +0.1 +0.2 +0.3 +0.4 +0.5 +ATLAS + CMS +Generator Level + = 13 TeV, ttW +s + j4 +≥ + +b + 1 +had +τ +SS 0 +l2 +ATLAS Sherpa 2.2.10 +ATLAS Sherpa 2.2.10 QCD+EW +ATLAS MG5_aMC+Py8 FxFx +ATLAS MG5_aMC+Py8 +CMS MG5_aMC+Py8 FxFx +ATLAS Sherpa scale var ME+PS +CMS scale variation ME-only +1 +,l +0l + R +∆ +0 +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +4 +4.5 +Ratio to Sherpa +0.5 +1 +1.5 +1 +,l +0l + R +∆ +0 +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +4 +4.5 + [fb] +1 +,l +0l + R +∆ +d +/ +fid +σ +d +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +1.4 +1.6 +1.8 +2 +2.2 +ATLAS + CMS +Generator Level + = 13 TeV, ttW +s + j4 +≥ + +b +2 +≥ + +had +τ +SS 0 +l2 +ATLAS Sherpa 2.2.10 +ATLAS Sherpa 2.2.10 QCD+EW +ATLAS MG5_aMC+Py8 FxFx +ATLAS MG5_aMC+Py8 +CMS MG5_aMC+Py8 FxFx +ATLAS Sherpa scale var ME+PS +CMS scale variation ME-only +1 +,l +0l + R +∆ +0 +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +4 +4.5 +Ratio to Sherpa +0.5 +1 +1.5 +|l +η + | +max +0 +0.5 +1 +1.5 +2 +2.5 +| [fb] +l +η + | +max +d +/ +fid +σ +d +0 +0.05 +0.1 +0.15 +0.2 +0.25 +0.3 +0.35 +ATLAS + CMS +Generator Level + = 13 TeV, ttW +s + j4 +≥ + +b + 1 +had +τ +SS 0 +l2 +ATLAS Sherpa 2.2.10 +ATLAS Sherpa 2.2.10 QCD+EW +ATLAS MG5_aMC+Py8 FxFx +ATLAS MG5_aMC+Py8 +CMS MG5_aMC+Py8 FxFx +ATLAS Sherpa scale var ME+PS +CMS scale variation ME-only +|l +η + | +max +0 +0.5 +1 +1.5 +2 +2.5 +Ratio to Sherpa +0.5 +1 +1.5 +|l +η + | +max +0 +0.5 +1 +1.5 +2 +2.5 +| [fb] +l +η + | +max +d +/ +fid +σ +d +0.2 +0.4 +0.6 +0.8 +1 +1.2 +1.4 +ATLAS + CMS +Generator Level + = 13 TeV, ttW +s + j4 +≥ + +b +2 +≥ + +had +τ +SS 0 +l2 +ATLAS Sherpa 2.2.10 +ATLAS Sherpa 2.2.10 QCD+EW +ATLAS MG5_aMC+Py8 FxFx +ATLAS MG5_aMC+Py8 +CMS MG5_aMC+Py8 FxFx +ATLAS Sherpa scale var ME+PS +CMS scale variation ME-only +|l +η + | +max +0 +0.5 +1 +1.5 +2 +2.5 +Ratio to Sherpa +0.5 +1 +1.5 +Figure 14: Distribution of the angular distance between the two leptons (top), maximum between +lepton |ηℓ0| and |ηℓ1| (centre), for the Region 1 with Nb−jets=1 (left) and Region 2 with Nb−jets ≥2 +(right) selection requiring four and more jets. All distributions are normalised to the YR4 cross +section of 600.8 fb except Sherpa 2.2.10 QCD+EW which is normalised to 614.7 fb. +30 + +Number of jets +3 +4 +5 +6 +7 +8 +9 + [fb] +jet +N +d +/ +fid +σ +d +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +ATLAS + CMS +Generator Level + = 13 TeV, ttW +s + j3 +≥ + +b +1 +≥ + +had +τ +SS 1 +l2 +ATLAS Sherpa 2.2.10 +ATLAS Sherpa 2.2.10 QCD+EW +ATLAS MG5_aMC+Py8 FxFx +ATLAS MG5_aMC+Py8 +CMS MG5_aMC+Py8 FxFx +ATLAS Sherpa scale var ME+PS +CMS scale variation ME-only +Number of jets +3 +4 +5 +6 +7 +8 +9 +Ratio to Sherpa +0.5 +1 +1.5 +-jets +b +Number of +0.5 +1 +1.5 +2 +2.5 +3 +3.5 + [fb] +-jet +b +N +d +/ +fid +σ +d +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +1.4 +ATLAS + CMS +Generator Level + = 13 TeV, ttW +s + j3 +≥ + +b +1 +≥ + +had +τ +SS 1 +l2 +ATLAS Sherpa 2.2.10 +ATLAS Sherpa 2.2.10 QCD+EW +ATLAS MG5_aMC+Py8 FxFx +ATLAS MG5_aMC+Py8 +CMS MG5_aMC+Py8 FxFx +ATLAS Sherpa scale var ME+PS +CMS scale variation ME-only +-jets +b +Number of +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +Ratio to Sherpa +0.5 +1 +1.5 + [GeV] +T +p +Leading lepton +0 +50 +100 +150 +200 +250 +300 +350 +400 +450 +500 + [fb/GeV] +l_0 +T +p +d +/ +fid +σ +d +0 +0.05 +0.1 +0.15 +0.2 +0.25 +0.3 +0.35 +ATLAS + CMS +Generator Level + = 13 TeV, ttW +s + j3 +≥ + +b +1 +≥ + +had +τ +SS 1 +l2 +ATLAS Sherpa 2.2.10 +ATLAS Sherpa 2.2.10 QCD+EW +ATLAS MG5_aMC+Py8 FxFx +ATLAS MG5_aMC+Py8 +CMS MG5_aMC+Py8 FxFx +ATLAS Sherpa scale var ME+PS +CMS scale variation ME-only + [GeV] +T +p +Leading lepton +0 +50 +100 +150 +200 +250 +300 +350 +400 +450 +500 +Ratio to Sherpa +0.5 +1 +1.5 +1 +,l +0l + R +∆ +0 +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +4 +4.5 + [fb] +1 +,l +0l + R +∆ +d +/ +fid +σ +d +0 +0.05 +0.1 +0.15 +0.2 +0.25 +0.3 +ATLAS + CMS +Generator Level + = 13 TeV, ttW +s + j3 +≥ + +b +1 +≥ + +had +τ +SS 1 +l2 +ATLAS Sherpa 2.2.10 +ATLAS Sherpa 2.2.10 QCD+EW +ATLAS MG5_aMC+Py8 FxFx +ATLAS MG5_aMC+Py8 +CMS MG5_aMC+Py8 FxFx +ATLAS Sherpa scale var ME+PS +CMS scale variation ME-only +1 +,l +0l + R +∆ +0 +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +4 +4.5 +Ratio to Sherpa +0.5 +1 +1.5 +Figure 15: Distribution of the the jet multiplicity, number of b-jets, the leading lepton transverse +momentum and the angular distance between the two leptons ∆Rℓℓ for the Region 5 with 1τhad +selection. All distributions are normalised to the YR4 cross section of 600.8 fb except Sherpa +2.2.10 QCD+EW which is normalised to 614.7 fb. +31 + +3.4 +Conclusions +The t¯tW preditions of Sherpa and MG5 aMC@NLO+Pythia8 with different settings have been +compared with respect to their inclusive ttW cross section predictions and their differential cross +section predictions in regions and observables relevant for the measurement of t¯tH in the multi- +lepton final state. +For the inclusive t¯tW cross section slightly different values are predicted [27, 50] for calculations +with similar theoretical accuracy which is subject to ongoing theoretical studies. Based on the +studies presented in this note, additional studies and discussions in the LHC Higgs Working Group +and the LHC Top Working Group [73], ATLAS and CMS agreed to use the inclusive t¯tW cross +section of 722+70 +−78 (scale) ±7 (PDF) fb [50] as a reference inclusive cross section to allow direct +comparisons between experiments. +The normalised distributions sensitive to shape differences have very small scale uncertainties, +below 10 % in most of the phase space, while these scale uncertainties are significant when the +acceptance effects are included, i.e. the distributions are normalised to the t¯tW cross section. +The inclusion of tree-level EW effects only causes minor shape effects but can lead to up to 20 % +difference in the cross section at high jet multiplicity. As expected, including the FxFx algorithm +into the MG5 aMC@NLO+Pythia8 prediction leads to significant effects in all regions, especially +at low HT. Significant differences between the MG5 aMC@NLO+Pythia8 FxFx predictions of +ATLAS and CMS are observed, especially in the jet multiplicity. Further studies are required to +investigate the origin of these differences. Given that both setups consider the same perturbative +accuracy such differences could be attributed to the choice of merging scale value or Pythia8 tune, +so this could be an area of future study. +For many observables the shape differences between the various model predictions are within the +scale uncertainties of each prediction. Observables relating to jet activity such as the jet multiplicity +and HT are notable exceptions to this. This is especially the case for Region 3 where the differences +in shape between predictions for HT is particlularly large. This region is important to constrain the +interplay between ttW background and backgrounds arising from ttbar production where at least +one lepton is mis-identified. It represents a phase space where one of the jets in the ttW decay is +not reconstructed or is out of acceptance and is not expected to be as sensitive to the additional +jet modelling as Regions 1 and 2. Therefore, it is somewhat surprising that such large differences +are observed between predictions. This should be investigated in future studies. +The inclusion of EW corrections shows only a small shape and normalisation effects for most +observables. One place where a notable effect on the shape of a distribution can be observed is for +the jet multiplicity, however the effect is small enough to be covered by the QCD scale variations. +Future studies could specifically target the sub-leading EW contribution with cuts related to the +rapidity difference between jets which has been shown [51] to be different with respect to the central +ttW QCD process. +These distributions shall be used as a starting point to derive a strategy for the theory uncer- +tainty estimates for a combination of the expected measurement results based on the full Run-2 data +set. Beyond what has been shown in the comparisons included in this document, this strategy is +expected to take into consideration the latest developments on the theoretical models. For example, +the NLO+PS calculations provided in Powheg [48] can act as systematic variation with respect to +the MG5 aMC@NLO+Pythia8 and Sherpa calculations for more inclusive phase-spaces. They +can also be used to understand parton shower, hadronisation and underlying event effects through +interfaces to Pythia and Herwig. In addition, recent off-shell calculations [56, 57, 58, 59] and +in particular the single-resonant contributions could be of importance. In the absence of explicit +parton shower-matched calculations, corrections can be applied through the procedure outlined +32 + +in Ref. [60]. It would also be important to extend existing calculations to additional final states, +such as 2ℓSS. Finally, given the current discrepancy between ATLAS and CMS, the strategy must +address how different model predictions are considered in addition to the scale uncertainties as part +of the theoretical uncertainties on the measurement. +33 + +References +[1] ATLAS Collaboration, “Measurement of Higgs boson decay into b-quarks in associated +production with a top-quark pair in pp collisions at √s = 13 TeV with the ATLAS detector”, +arXiv:2111.06712. Submitted to JHEP (2021). +[2] ATLAS Collaboration, “Search for the standard model Higgs boson produced in association +with top quarks and decaying into a b¯b pair in pp collisions at √s = 13 TeV with the ATLAS +detector”, Phys. Rev. D 97 (2018) 072016, arXiv:1712.08895. +doi:10.1103/PhysRevD.97.072016. +[3] CMS Collaboration, “Measurement of ttH production in the H → bb decay channel in +41.5 fb−1 of proton-proton collision data at √s = 13 TeV”. CMS-PAS-HIG-18-030, 2019. +[4] CMS Collaboration, “Search for ttH production in the H→ bb decay channel with leptonic tt +decays in proton–proton collisions at √s = 13 TeV”, JHEP 03 (2019) 026, +arXiv:1804.03682. doi:10.1007/JHEP03(2019)026. +[5] ATLAS Collaboration, “Analysis of ttH and ttW production in the multilepton final states +with the ATLAS detector”. ATLAS-CONF-2019-045, 2019. +[6] CMS Collaboration, “Measurement of the Higgs boson production rate in association with +top quarks in final states with electrons, muons, and hadronically decaying tau leptons at +√s = 13 TeV”, Eur. Phys. J. C 81 (2021) 378, arXiv:2011.03652. +doi:10.1140/epjc/s10052-021-09014-x. +[7] A. Buckley, J. Butterworth, D. Grellscheid et al., “Rivet user manual”, Comput. Phys. +Commun. 184 (2013) 2803–2819, arXiv:1003.0694. doi:10.1016/j.cpc.2013.05.021. +[8] T. Sj¨ostrand, S. Ask, J. R. Christiansen et al., “An Introduction to PYTHIA 8.2”, Comput. +Phys. Commun. 191 (2015) 159, arXiv:1410.3012. doi:10.1016/j.cpc.2015.01.024. +[9] P. Nason, “A new method for combining NLO QCD with shower Monte Carlo algorithms”, +JHEP 11 (2004) 040, arXiv:hep-ph/0409146. doi:10.1088/1126-6708/2004/11/040. +[10] S. Frixione, P. Nason, and C. Oleari, “Matching NLO QCD computations with parton +shower simulations: the POWHEG method”, JHEP 11 (2007) 070, arXiv:0709.2092. +doi:10.1088/1126-6708/2007/11/070. +[11] S. Frixione, G. Ridolfi, and P. Nason, “A positive-weight next-to-leading-order Monte Carlo +for heavy flavour hadroproduction”, Journal of High Energy Physics 2007 (2007), no. 09, +126–126. doi:10.1088/1126-6708/2007/09/126. +[12] S. Alioli, P. Nason, C. Oleari et al., “A general framework for implementing NLO +calculations in shower Monte Carlo programs: the POWHEG BOX”, JHEP 06 (2010) 043, +arXiv:1002.2581. doi:10.1007/JHEP06(2010)043. +[13] J. M. Campbell, R. K. Ellis, P. Nason et al., “Top-Pair Production and Decay at NLO +Matched with Parton Showers”, JHEP 04 (2015) 114, arXiv:1412.1828. +doi:10.1007/JHEP04(2015)114. +34 + +[14] A. Bredenstein, A. Denner, S. Dittmaier et al., “NLO QCD Corrections to Top Anti-Top +Bottom Anti-Bottom Production at the LHC: 2. full hadronic results”, JHEP 03 (2010) 021, +arXiv:1001.4006. doi:10.1007/JHEP03(2010)021. +[15] A. Bredenstein, A. Denner, S. Dittmaier et al., “NLO QCD corrections to pp → t¯tb¯b + X at +the LHC”, Phys. Rev. Lett. 103 (2009) 012002, arXiv:0905.0110. +doi:10.1103/PhysRevLett.103.012002. +[16] G. Bevilacqua, M. Czakon, C. Papadopoulos et al., “Assault on the NLO wishlist:pp → ttbb”, +Journal of High Energy Physics 2009 (2009), no. 09, 109–109. +doi:10.1088/1126-6708/2009/09/109. +[17] M. V. Garzelli, A. Kardos, and Z. Tr´ocs´anyi, “Hadroproduction of t¯tb¯b final states at LHC: +predictions at NLO accuracy matched with Parton Shower”, JHEP 03 (2015) 083, +arXiv:1408.0266. doi:10.1007/JHEP03(2015)083. +[18] G. Bevilacqua, H.-Y. Bi, H. B. Hartanto et al., “ttbb at the LHC: on the size of corrections +and b-jet definitions”, Journal of High Energy Physics 2021 (2021), no. 8,. +doi:10.1007/jhep08(2021)008. +[19] A. Denner, J.-N. Lang, and M. Pellen, “Full NLO QCD corrections to off-shell ttbb +production”, Physical Review D 104 (2021), no. 5,. doi:10.1103/physrevd.104.056018. +[20] G. Bevilacqua, H.-Y. Bi, H. B. Hartanto et al., “t¯tb¯b at the LHC: On the size of off-shell +effects and prompt b-jet identification”, 2022. doi:10.48550/ARXIV.2202.11186. +[21] T. Jeˇzo, J. M. Lindert, N. Moretti et al., “New NLOPS predictions for t¯t + b -jet production +at the LHC”, Eur. Phys. J. C78 (2018) 502, arXiv:1802.00426. +doi:10.1140/epjc/s10052-018-5956-0. +[22] ATLAS Collaboration, “Studies of Monte Carlo predictions for the t¯tb¯b process”. +ATL-PHYS-PUB-2022-006, 2022. +[23] J. Alwall, R. Frederix, S. Frixione et al., “The automated computation of tree-level and +next-to-leading order differential cross sections, and their matching to parton shower +simulations”, JHEP 07 (2014) 079, arXiv:1405.0301. doi:10.1007/JHEP07(2014)079. +[24] R. Frederix and S. Frixione, “Merging meets matching in MC@NLO”, JHEP 12 (2012) 061, +arXiv:1209.6215. doi:10.1007/JHEP12(2012)061. +[25] J. Bellm, S. Gieseke, D. Grellscheid et al., “Herwig 7.0/Herwig++ 3.0 release note”, The +European Physical Journal C 76 (2016), no. 4,. doi:10.1140/epjc/s10052-016-4018-8. +[26] B. Cooper, J. Katzy, M. L. Mangano et al., “Importance of a consistent choice of alpha(s) in +the matching of AlpGen and Pythia”, Eur. Phys. J. C 72 (2012) 2078, arXiv:1109.5295. +doi:10.1140/epjc/s10052-012-2078-y. +[27] Sherpa Collaboration, “Event Generation with Sherpa 2.2”, SciPost Phys. 7 (2019) 034, +arXiv:1905.09127. doi:10.21468/SciPostPhys.7.3.034. +[28] F. Cascioli, P. Maierh¨ofer, N. Moretti et al., “NLO matching for t¯tb¯b production with +massive b-quarks”, Phys. Lett. B734 (2014) 210–214, arXiv:1309.5912. +doi:10.1016/j.physletb.2014.05.040. +35 + +[29] F. Cascioli, P. Maierhofer, and S. Pozzorini, “Scattering Amplitudes with Open Loops”, +Phys. Rev. Lett. 108 (2012) 111601, arXiv:1111.5206. +doi:10.1103/PhysRevLett.108.111601. +[30] The routine will be released as MC-ttbb routine in the Rivet analysis toolkit [7]. +[31] A. Ryd, D. Lange, N. Kuznetsova et al., “EvtGen: A Monte Carlo Generator for B-Physics”. +EVTGEN-V00-11-07, 2005. +[32] F. Buccioni, S. Kallweit, S. Pozzorini et al., “NLO QCD predictions for t¯tb¯b production in +association with a light jet at the LHC”, JHEP 12 (2019) 015, arXiv:1907.13624. +doi:10.1007/JHEP12(2019)015. +[33] ATLAS Collaboration, “ATLAS Pythia 8 tunes to 7 TeV data”. ATL-PHYS-PUB-2014-021, +2014. +[34] CMS Collaboration, “Extraction and validation of a new set of CMS Pythia 8 tunes from +underlying-event measurements”, Eur. Phys. J. C 80 (2020) 4, arXiv:1903.12179. +doi:10.1140/epjc/s10052-019-7499-4. +[35] T. Gleisberg and S. H¨oche, “Comix, a new matrix element generator”, JHEP 12 (2008) 039, +arXiv:0808.3674. doi:10.1088/1126-6708/2008/12/039. +[36] S. Schumann and F. Krauss, “A Parton shower algorithm based on Catani-Seymour dipole +factorisation”, JHEP 03 (2008) 038, arXiv:0709.1027. +doi:10.1088/1126-6708/2008/03/038. +[37] T. Jeˇzo and P. Nason, “On the Treatment of Resonances in Next-to-Leading Order +Calculations Matched to a Parton Shower”, JHEP 12 (2015) 065, arXiv:1509.09071. +doi:10.1007/JHEP12(2015)065. +[38] M. Cacciari, G. P. Salam, and G. Soyez, “The anti-kt jet clustering algorithm”, JHEP 04 +(2008) 063, arXiv:0802.1189. doi:10.1088/1126-6708/2008/04/063. +[39] M. Cacciari, G. P. Salam, and G. Soyez, “FastJet user manual”, Eur. Phys. J. C 72 (2012) +1896, arXiv:1111.6097. doi:10.1140/epjc/s10052-012-1896-2. +[40] M. Cacciari, G. P. Salam, and G. Soyez, “The Catchment Area of Jets”, JHEP 04 (2008) +005, arXiv:0802.1188. doi:10.1088/1126-6708/2008/04/005. +[41] CMS Collaboration, “Measurement of the cross section of top quark-antiquark pair +production in association with a W boson in proton-proton collisions at √s = 13 TeV”. +CMS-PAS-TOP-21-011, 2022. +[42] F. Maltoni, M. L. Mangano, I. Tsinikos et al., “Top-quark charge asymmetry and +polarization in ttW ± production at the LHC”, Phys. Lett. B 736 (2014) 252–260, +arXiv:1406.3262. doi:10.1016/j.physletb.2014.07.033. +[43] J. M. Campbell and R. K. Ellis, “t¯tW +− production and decay at NLO”, JHEP 07 (2012) +052, arXiv:1204.5678. doi:10.1007/JHEP07(2012)052. +[44] M. V. Garzelli, A. Kardos, C. G. Papadopoulos et al., “t ¯t W +− and t ¯t Z Hadroproduction +at NLO accuracy in QCD with Parton Shower and Hadronization effects”, JHEP 11 (2012) +056, arXiv:1208.2665. doi:10.1007/JHEP11(2012)056. +36 + +[45] F. Maltoni, D. Pagani, and I. Tsinikos, “Associated production of a top-quark pair with +vector bosons at NLO in QCD: impact on ttH searches at the LHC”, JHEP 02 (2016) 113, +arXiv:1507.05640. doi:10.1007/JHEP02(2016)113. +[46] S. Frixione, V. Hirschi, D. Pagani et al., “Electroweak and QCD corrections to top-pair +hadroproduction in association with heavy bosons”, JHEP 06 (2015) 184, +arXiv:1504.03446. doi:10.1007/JHEP06(2015)184. +[47] D. de Florian et al., “Handbook of LHC Higgs Cross Sections: 4. Deciphering the Nature of +the Higgs Sector”, arXiv:1610.07922. (2017). doi:10.23731/CYRM-2017-002. +[48] F. Febres Cordero, M. Kraus, and L. Reina, “Top-quark pair production in association with +a W ± gauge boson in the POWHEG-BOX”, Phys. Rev. D 103 (2021), no. 9, 094014, +arXiv:2101.11808. doi:10.1103/PhysRevD.103.094014. +[49] S. von Buddenbrock, R. Ruiz, and B. Mellado, “Anatomy of inclusive t¯tW production at +hadron colliders”, Phys. Lett. B 811 (2020) 135964, arXiv:2009.00032. +doi:10.1016/j.physletb.2020.135964. +[50] R. Frederix and I. Tsinikos, “On improving NLO merging for ttW production”, JHEP 11 +(2021) 029, arXiv:2108.07826. doi:10.1007/JHEP11(2021)029. +[51] J. A. Dror, M. Farina, E. Salvioni et al., “Strong tW Scattering at the LHC”, JHEP 01 +(2016) 071, arXiv:1511.03674. doi:10.1007/JHEP01(2016)071. +[52] R. Frederix, D. Pagani, and M. Zaro, “Large NLO corrections in t¯tW ± and t¯tt¯t +hadroproduction from supposedly subleading EW contributions”, JHEP 02 (2018) 031, +arXiv:1711.02116. doi:10.1007/JHEP02(2018)031. +[53] A. Kulesza, L. Motyka, D. Schwartl¨ander et al., “Associated production of a top quark pair +with a heavy electroweak gauge boson at NLO+NNLL accuracy”, Eur. Phys. J. C 79 (2019), +no. 3, 249, arXiv:1812.08622. doi:10.1140/epjc/s10052-019-6746-z. +[54] A. Broggio, A. Ferroglia, R. Frederix et al., “Top-quark pair hadroproduction in association +with a heavy boson at NLO+NNLL including EW corrections”, JHEP 08 (2019) 039, +arXiv:1907.04343. doi:10.1007/JHEP08(2019)039. +[55] A. Kulesza, L. Motyka, D. Schwartl¨ander et al., “Associated top quark pair production with +a heavy boson: differential cross sections at NLO+NNLL accuracy”, Eur. Phys. J. C 80 +(2020), no. 5, 428, arXiv:2001.03031. doi:10.1140/epjc/s10052-020-7987-6. +[56] G. Bevilacqua, H.-Y. Bi, H. B. Hartanto et al., “The simplest of them all: t¯tW ± at NLO +accuracy in QCD”, JHEP 08 (2020) 043, arXiv:2005.09427. +doi:10.1007/JHEP08(2020)043. +[57] A. Denner and G. Pelliccioli, “NLO QCD corrections to off-shell t¯tW+ production at the +LHC”, JHEP 11 (2020) 069, arXiv:2007.12089. doi:10.1007/JHEP11(2020)069. +[58] G. Bevilacqua, H.-Y. Bi, H. B. Hartanto et al., “NLO QCD corrections to off-shell t¯tW ± +production at the LHC: correlations and asymmetries”, Eur. Phys. J. C 81 (2021), no. 7, +675, arXiv:2012.01363. doi:10.1140/epjc/s10052-021-09478-x. +37 + +[59] A. Denner and G. Pelliccioli, “Combined NLO EW and QCD corrections to off-shell ttW +production at the LHC”, Eur. Phys. J. C 81 (2021), no. 4, 354, arXiv:2102.03246. +doi:10.1140/epjc/s10052-021-09143-3. +[60] G. Bevilacqua, H. Y. Bi, F. Febres Cordero et al., “Modeling uncertainties of t¯tW ± +multilepton signatures”, Phys. Rev. D 105 (2022), no. 1, 014018, arXiv:2109.15181. +doi:10.1103/PhysRevD.105.014018. +[61] T. Gleisberg, S. H¨oche, F. Krauss et al., “Event generation with SHERPA 1.1”, JHEP 02 +(2009) 007, arXiv:0811.4622. doi:10.1088/1126-6708/2009/02/007. +[62] S. H¨oche, F. Krauss, M. Sch¨onherr et al., “QCD matrix elements + parton showers: The +NLO case”, JHEP 04 (2013) 027, arXiv:1207.5030. doi:10.1007/JHEP04(2013)027. +[63] S. Kallweit, J. M. Lindert, P. Maierhofer et al., “NLO QCD+EW predictions for V + jets +including off-shell vector-boson decays and multijet merging”, JHEP 04 (2016) 021, +arXiv:1511.08692. doi:10.1007/JHEP04(2016)021. +[64] C. G¨utschow, J. M. Lindert, and M. Sch¨onherr, “Multi-jet merged top-pair production +including electroweak corrections”, Eur. Phys. J. C 78 (2018), no. 4, 317, +arXiv:1803.00950. doi:10.1140/epjc/s10052-018-5804-2. +[65] ATLAS Collaboration, “Modelling of rare top quark processes at √s = 13 TeV”. +ATL-PHYS-PUB-2020-024, 2020. +[66] P. Artoisenet, R. Frederix, O. Mattelaer et al., “Automatic spin-entangled decays of heavy +resonances in Monte Carlo simulations”, JHEP 03 (2013) 015, arXiv:1212.3460. +doi:10.1007/JHEP03(2013)015. +[67] S. Catani, F. Krauss, R. Kuhn et al., “QCD matrix elements + parton showers”, JHEP 11 +(2001) 063, arXiv:hep-ph/0109231. doi:10.1088/1126-6708/2001/11/063. +[68] K. Hamilton, P. Nason, and G. Zanderighi, “MINLO: Multi-Scale Improved NLO”, JHEP 10 +(2012) 155, arXiv:1206.3572. doi:10.1007/JHEP10(2012)155. +[69] J. Alwall et al., “A Standard format for Les Houches event files”, Comput. Phys. Commun. +176 (2007) 300–304, arXiv:hep-ph/0609017. doi:10.1016/j.cpc.2006.11.010. +[70] J. Alwall, R. Frederix, S. Frixione et al., “The automated computation of tree-level and +next-to-leading order differential cross sections, and their matching to parton shower +simulations”, Journal of High Energy Physics 2014 (2014). doi:10.1007/jhep07(2014)079. +[71] NNPDF Collaboration, “Parton distributions for the LHC Run II”, JHEP 04 (2015) 040, +arXiv:1410.8849. doi:10.1007/JHEP04(2015)040. +[72] NNPDF Collaboration, “Parton distributions from high-precision collider data”, Eur. Phys. +J. C 77 (2017), no. 10, 663, arXiv:1706.00428. doi:10.1140/epjc/s10052-017-5199-5. +[73] LHC Higgs and Top Working Groups, “ttW modeling in light of ttH measurements”, 2022. +https://indico.cern.ch/event/1219500/. +38 + diff --git a/XdFJT4oBgHgl3EQf5i00/content/tmp_files/load_file.txt b/XdFJT4oBgHgl3EQf5i00/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..76f6edf2cb23f15bff8dcdb87f8514597e394b4e --- /dev/null +++ b/XdFJT4oBgHgl3EQf5i00/content/tmp_files/load_file.txt @@ -0,0 +1,2488 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf,len=2487 +page_content='LHCHWG-2022-003 December 15,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' 2022 LHC Higgs Working Groupa Public Note Study of t¯tb¯b and t¯tW background modelling for t¯tH analyses Lars Ferencz 1,' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='11670v1 [hep-ex] 27 Jan 2023 Abstract This note presents Monte Carlo generator comparisons of the t¯tb¯b and t¯tW processes at particle level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' The aim is to compare the modelling of important backgrounds to t¯tH measurements in multi- lepton final states and in the t¯tH(H → b¯b) decay channel and the treatment of the associated theory uncertainties for a combination of the full Run-2 t¯tH results from ATLAS and CMS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' As a first step, modelling and theory uncertainties as used in ATLAS an CMS are compared in the relevant analysis regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Significant differences in the treatment of systematic uncertainties between the experiments have been observed in t¯tb¯b and t¯tW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' As a first step, ATLAS and CMS agreed on a common reference value of the inclusive t¯tW cross section to allow direct comparisons between experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Contents 1 Introduction 1 2 Comparisons of Monte Carlo predictions for the t¯tb¯b process 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='1 MC generator set-ups .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' 20 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='4 Conclusions .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' 32 1 Introduction The search for Higgs boson production in association with a top quark pair (t¯tH) has been per- formed in the H → b¯b [1, 2, 3, 4] decay channel and in multi-lepton final states [5, 6] which are primarily sensitive to the decays of H→ WW ∗, H→ ττ and H→ ZZ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' These searches are limited by the modelling uncertainties of the main backgrounds, t¯tb¯b and t¯tW, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Examples of tree-level diagrams of the background processes are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' A comparison of Monte Carlo (MC) generators used by ATLAS and CMS is thus performed to compare the background modelling and the estimates of modelling uncertainties in view of future combinations of the experimental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' The goals is to provide input to a discussion between the experiments and between experiments and theorists to define modelling uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Furthermore, the experiments aim to develop a common strategy for combination of the t¯tH(H → b¯b) and t¯tH(multi-lepton) analyses of the full Run-2 data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Comparisons of observables relevant for the analyses are made at stable particle level, in a phase space similar to the reference measurements using the Rivet analysis toolkit [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' The note is structured as follows: comparisons of t¯tb¯b distributions will be presented in Section 2 and comparisons of t¯tW distributions in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Figure 1: Examples of tree-level Feynman diagrams for t¯tb¯b (left) and t¯tW (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' 1 d g 000000 七 d n2 Comparisons of Monte Carlo predictions for the t¯tb¯b process In the following section t¯tb¯b background predictions and variations considered to estimate their uncertainties used by ATLAS and CMS in published and future analyses of t¯tH(H → b¯b) are compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' The first Run-2 t¯tH(H → b¯b) analyses of both experiments [2, 4] based on partial data sets predicted the t¯t + jets background with a t¯t matrix element (ME) calculated at next- to-leading-order (NLO) accuracy in QCD in the five-flavour scheme (5FS) and matched to the Pythia8 parton shower (PS) [8] in the Powheg framework [9, 10, 11, 12, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' In this set-up, b-quarks not originating in the top quark decay chain are produced by Pythia8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' The first predictions using a t¯tb¯b ME at NLO have been performed with stable top quarks in 5FS some time ago [14, 15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' They have been matched subsequently to parton shower programs [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Very recently complete calculations for the t¯tb¯b process in di-lepton top quark decay channel have been carried out in 5FS without matching to PS by two independent groups [18, 19, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Such computations are based on e+νeµ−νµbbbb matrix elements and include all resonant and non- resonant Feynman diagrams, interferences and off-shell effects of the top quark and the W gauge boson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' The first t¯tH(H → b¯b) analysis based on the full Run-2 data set from ATLAS [1] (”first full Run-2 analysis”) used as nominal generator a calculation where the t¯tb¯b ME is calculated at NLO with massive b-quarks1 in the four-flavour scheme (4FS) [21] and matched to Pythia8 in the Powheg-Box-Res framework [21], referred to as t¯tb¯b-Powheg in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' For future analyses both experiments consider to use the calculations of t¯tb¯b-Powheg matched to Pythia8 as nominal generator however with different settings of the renormalisation and fac- torisation scale compared to the original paper [21] and slightly different settings of the internal parameters based on more recent studies [22] as will be discussed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' The estimation of systematic uncertainties differs significantly between the two experiments for the published analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' ATLAS considered uncertainties due the particular choice of matching algorithm and of the parton shower generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' For the analyses based on partial and first full Run-2 data set, these differences were derived from 5FS t¯t sample predicted by MG5 aMC@NLO [23, 24] matched to Pythia8 for the first and a sample where Powheg is matched to Herwig7 [25] for the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Since the nominal generator in the first full Run-2 analysis was based on a t¯tb¯b-Powheg calculation, the relative uncertainties derived from the 5FS t¯t samples were used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Uncertainties due to higher order effects were estimated by varying the renormalisation and factorisation scales in the ME, µR and µF, simultaneously up and down by a factor of two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Correlations between the scale settings in the ME and αs in the PS ISR were considered by simultaneous variation with µR and µF to cover the effects of PS variations in the presence of matching [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' In the first Run-2 analysis, CMS considered the uncertainty due to the choice of generator settings by varying the hdamp parameter in Powheg which controls the transverse momentum (pT) of the first additional emission beyond the leading-order Feynman diagram in the PS and therefore regulates the high-pT emission against which the t¯t system recoils.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Comparisons with Sherpa were done internally but not added to the list of systematic uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' The renormalisation and factorisation scales µR and µF as well as αs in both the PS ISR and FSR were varied independently, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' one parameter was changed at a time while keeping the other parameters at their nominal values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' For future analyses, both experiments consider predictions with varied µR and µF scales and varied PS αs as well as different settings of t¯tb¯b-Powheg internal parameters, however ATLAS studies additional uncertainties due to parton shower and matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' To estimate the dependence on t¯tb¯b-Powheg internal parameters, ATLAS varies the parameter hbzd which regulates the splitting 1“quarks” refers to both quarks and anti-quarks 2 into the finite and the singular part of the real emission in the Powheg framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Variations of the parameter hdamp were studied in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' [22] but no significant differences were found and therefore this variation is not further considered for uncertainty estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Uncertainties due to the particular setting of PS are estimated with set-ups of t¯tb¯b-Powheg matched to Herwig7 and Pythia8 with a dipole recoil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' The dependence on the particular choice of generator and the NLO matching algorithm is studied by comparing to NLO 4FS predictions of t¯tb¯b generated with Sherpa 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='10 [27, 28, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Details of the studies are given in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' In case of CMS, the dependence on t¯tb¯b-Powheg internal parameters is estimated by varying the matching parameter hdamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Both experiments consider PDF uncertainties in the published and future analyses, however they are neglected in the studies presented here due to the smallness of the effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Finally, in order to get comparable results, the scale uncertainties are treated the same way for both experiments in all studies presented here, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' µR and µF, PS ISR and PS FSR are changed individually by a factor 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='5 (2) while keeping the other parameters at their nominal values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' All comparisons are performed using stable final-state particles in a fiducial phase space simi- lar to the experimental measurements implemented in a dedicated routine in the Rivet analysis toolkit [7, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' The chapter is organised as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='1 describes the samples used for the comparison and the technical set-up of their generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='2 describes the observables and the fiducial phase space used for the comparison and finally, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='3 displays the resulting comparisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='1 MC generator set-ups The set-ups used to generate t¯tb¯b predictions with t¯tb¯b-Powheg, Powheg, MG5 aMC@NLO and Sherpa are described in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' The generator configurations and version numbers are summarised in Table 1 and their scale settings are given in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' The systematic uncertainty estimates due to scale and αs variations are summarised in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' The b-quark mass is set to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='75 GeV for CMS samples and for Sherpa, and to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='95 GeV for all other ATLAS samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' The top quark mass is set to 172.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='5 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' The decay of the top quark is calculated by the corresponding generators (Powheg, Sherpa) respecting the spin correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' The PDF sets used in the ME calculation are selected from the NNPDF family for all samples, where ATLAS uses version 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='0 while CMS uses version 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' The ATLAS t¯tb¯b-Powheg, Powheg and MG5 aMC@NLO samples use EvtGen [31] for simulation of the B-hadron decays, while the Sherpa sample and all CMS samples calculate the decays within the corresponding PS codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' All samples were produced for final states with one or two leptons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' t¯tb¯b-Powheg samples: Nominal t¯tb¯b predictions are calculated using the Powheg-Box-Res framework at NLO with massive b-quarks [21] with the “4FS NLO as 0118” PDF sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' The renormalisation scale is set to half of the geometric average of the transverse mass of top- and b-quarks defined as mT,i = � m2 i + p2 T,i, where mi refers to the mass, pT,i to the transverse momentum and i to the top or b-quark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' The factorisation scale is related to the average of the transverse mass of the outgoing partons in the ME calculation, see Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' For ATLAS, it follows Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' [21], while it is set to a factor two smaller in CMS following Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' The t¯tb¯b-Powheg internal parameters differ between the experiments: hbzd is set to 5 for ATLAS and to 2 for CMS, hdamp is set to HT/2 for ATLAS and to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='379 times the top quark mass for CMS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' The Pythia8 parameters for PS and hadronisation modelling are set to the A14 [33] and CP5 [34] 3 tunes for ATLAS and CMS and the samples are referred to as ATLAS and CMS PP8 t¯tb¯b samples, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' To vary t¯tb¯b-Powheg internal parameters, ATLAS sets the parameter hbzd to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' CMS varies in its set-up the hdamp parameter to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='305 times the top quark mass for the “hdamp up” variation and to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='8738 times the top quark mass for the “hdamp down” variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' The ATLAS t¯tb¯b-Powheg calculation was performed using a special option where virtual corrections are switched off and then reweighted with virtual corrections switched on2, while the CMS samples used default calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' For the PS variations, ATLAS uses the set of LHE files which store the results of the ME calculation by t¯tb¯b-Powheg for the PP8 t¯tb¯b sample and matches them to a different PS prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' For the prediction with the Pythia8 dipole shower only the treatment of the recoil of the radiated parton in the shower is changed and all other parameters are kept as the A14 tuned values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Another sample is produced where Herwig7 is used with the default tune provided with this generator version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Sherpa t¯tb¯b samples: A t¯tb¯b sample was generated using Sherpa version 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='10 [27, 28, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' The t¯tb¯b MEs were calculated with massive b-quarks at NLO, using the COMIX [35] and Openloops [29] ME generators, and merged with the Sherpa PS, tuned by the authors [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' The same renormal- isation and factorisation scales and PDFs are used as for the ATLAS PP8 t¯tb¯b prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Inclusive t¯t samples: The inclusive t¯t samples are generated with the Powheg v2 NLO event generator [9, 10, 12, 13, 37] and MG5 aMC@NLO using a 5FS PDF set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' The renormalisation and factorisation scales were set to the average transverse mass of the top quark and antiquark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' For the Powheg samples of both experiments, the PS and hadronisation is modeled by Pythia8 with the same versions and settings as for the PP8 t¯tb¯b samples above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' The hdamp parameter was set to the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='5 times the top quark mass for ATLAS and to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='379 times the top quark mass for CMS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Another ATLAS sample is generated using Herwig7 for the PS and hadronization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' These samples are referred to as ATLAS (CMS) PP8 t¯t and ATLAS PH7 t¯t samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' The inclusive MG5 aMC@NLO t¯t sample uses the same scale settings and the same Pythia8 version as the ATLAS PP8 t¯t sample and is referred to as ATLAS aMC+P8 t¯t sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' 2steered via ”for reweight 1” 4 Table 1: Configurations used for the event generation of the t¯tb¯b process and the predicted total cross section for events with at least one lepton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' name ME Generator ME order Shower Tunea NNPDF PDF set (ME) hdamp hbzd σ≥1lep [pb] ATLAS PP8 t¯tb¯b t¯tb¯b t¯tb¯b-Powheg NLO Pythia 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='224 A14 4FS 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='0 NLO as 0118 HT /2 5 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='72 CMS PP8 t¯tb¯b t¯tb¯b t¯tb¯b-Powheg NLO Pythia 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='230 CP5 4FS 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='1 NLO as 0118 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='379 · mt 2 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='86 ATLAS PP8 t¯tb¯b hbzd 2 t¯tb¯b t¯tb¯b-Powheg NLO Pythia 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='224 A14 4FS 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='0 NLO as 0118 HT/2 2 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='46 ATLAS PP8 t¯tb¯b dipole t¯tb¯b t¯tb¯b-Powheg NLO Pythia 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='224 A14, dipoleRecoilb 4FS 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='0 NLO as 0118 HT/2 2 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='72 ATLAS PH7 t¯tb¯b t¯tb¯b t¯tb¯b-Powheg NLO Herwig 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='6 default 4FS 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='0 NLO as 0118 HT/2 5 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='47 ATLAS Sherpa t¯tb¯b t¯tb¯b Sherpa 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='10 NLO Sherpa default 4FS 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='0 NNLO as 0118 — — 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='24 CMS PP8 t¯tb¯b hdamp up t¯tb¯b t¯tb¯b-Powheg NLO Pythia 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='230 CP5 4FS 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='1 NLO as 0118 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='305 · mt 5 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='86 CMS PP8 t¯tb¯b hdamp down t¯tb¯b t¯tb¯b-Powheg NLO Pythia 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='230 CP5 4FS 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='1 NLO as 0118 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='8738 · mt 5 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='86 ATLAS PP8 t¯t t¯t Powheg v2 NLO Pythia 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='210 A14 5FS 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='0 NLO 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='5 · mt 5 451.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='78c CMS PP8 t¯t t¯t Powheg v2 NLO Pythia 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='230 CP5 5FS 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='1 NLO 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='5 · mt 5 451.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='78c ATLAS PH7 t¯t t¯t Powheg v2 NLO Herwig 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='13 default 5FS 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='0 NLO 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='5 · mt 5 451.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='78c ATLAS aMC+P8 t¯t t¯t MG5 aMC@NLO NLO Pythia 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='210 A14 5FS 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='0 NLO — — 451.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='78c CMS PP8 t¯t hdamp up t¯t Powheg v2 NLO Pythia 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='230 CP5 5FS 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='1 NLO 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='305 · mt 5 451.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='78c CMS PP8 t¯t hdamp down t¯t Powheg v2 NLO Pythia 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='230 CP5 5FS 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='1 NLO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='8738 · mt 5 451.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='78c a“default” refers to the generator’s default tune bcalled by SpaceShower::dipoleRecoil “on” ccross section predicted by NNLO calculation 5 Table 2: Scale choices used in the event generation of t¯tb¯b and t¯t processes for the different generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' ME Generator µR µF ATLAS t¯tb¯b-Powheg t¯tb¯b 1 2 4√mT,t · mT,¯t · mT,b · mT,¯b 1 2(mT,t + mT,¯t + mT,b + mT,¯b + mT,g) CMS t¯tb¯b-Powheg t¯tb¯b 1 2 4√mT,t · mT,¯t · mT,b · mT,¯b 1 4(mT,t + mT,¯t + mT,b + mT,¯b + mT,g) Sherpa 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='10 1 2 4√mT,t · mT,¯t · mT,b · mT,¯b 1 2(mT,t + mT,¯t + mT,b + mT,¯b + mT,g) ATLAS Powheg t¯t � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='5 · (m2 T,t + m2 T,¯t) � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='5 · (m2 T,t + m2 T,¯t) CMS Powheg t¯t � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='5 · (m2 T,t + m2 T,¯t) � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='5 · (m2 T,t + m2 T,¯t) ATLAS aMC t¯t � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='5 · (m2 T,t + m2 T,¯t) � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='5 · (m2 T,t + m2 T,¯t) 6 Table 3: Systematic variations of scales in the ME and PS codes used for all comparisons presented here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Variation Scale variation ME µR × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='5, µF × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' µR × 2, µF × 2 ISR variation (PS) αISR s × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' αsISR × 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='0 FSR variation (PS) αFSR s × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' αsFSR × 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='2 Object reconstruction, fiducial volume and observables The object definition and event selection applied in this comparison study is defined at particle level and is the same for ATLAS and CMS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' All objects are defined using stable final-state particles with a mean lifetime of τ > 3 × 10−11 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Jets are reconstructed from all stable final-state particles (but excluding leptons and neutrinos from the top quark decay chain) using the anti-kt jet algorithm [38, 39] with a radius parameter of R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Jets which contain at least one ghost-associated [40] B- hadron with pT > 5 GeV are defined as b-jets, all other jets are considered “light” jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' The four-momentum of the bare leptons from top quark decay are modified (“dressed”) by adding the four-momenta of all radiated photons within a cone of size ∆R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' All objects are considered within pseudo-rapidity |η| < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='5 and with pT > 27 GeV for leptons and pT > 25 GeV for jets and b-jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Leptons are removed if they are separated from a jet by less than ∆R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='4, where ∆R = � (∆η)2 + (∆φ)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Events are selected with at least four b-jets, and further separated into two analysis regions: events with exactly one lepton and at least six jets (single lepton channel) and events with exactly two leptons and at least four jets (dilepton channel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' A set of observables relevant for the t¯tH(H → b¯b) analysis is studied within this fiducial phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' All observables are studied for both the single lepton and the dilepton channel, however only the variables listed in Table 4 are shown in the following figures, as no significant qualitative difference is observed between the different top quark decay channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Table 4: The list of observables used for the comparison of the generators for the t¯tb¯b process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='Variable ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='Description ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='Channel ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='∆Rmin∆R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='bb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='∆R of the two b-jets in the event which are closest in ∆R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='dilepton ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='mmin∆R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='bb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='Invariant mass of the two b-jets closest in ∆R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='dilepton ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='Njets ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='Number of jets in the event (all jet flavours) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='dilepton ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='Light jet pT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='Transverse momentum of the light jets in the event ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='dilepton ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='Nb-jets ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='Number of b-jets in the event ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='single lepton ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='Hjets ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='Scalar sum of pT of jets in the event (all jet flavours) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='single lepton ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='Leading b-jet pT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='pT of b-jet with largest pT in the event ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='single lepton ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='Fourth b-jet pT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='pT of b-jet with fourth largest pT in the event ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='single lepton ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='3 Results Three sets of generator predictions are compared for the observables given in Table 4 as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' All comparisons are performed with respect to the t¯tb¯b PP8 sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' The PP8 t¯tb¯b sample and the 7 alternative predictions are normalised to an integral of one, after all selections and in each histogram individually for a shape-only comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' The scale uncertainty variations on PP8 t¯tb¯b are derived as listed in Table 3 and the differences are added in quadrature to the statistical uncertainties to form the shaded area displayed in the figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Figure 2 shows the nominal t¯tb¯b predictions from ATLAS and CMS to be used in future analyses compared to the nominal predictions used in the early Run-2 analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' The differences between ATLAS and CMS set-ups cause only minor differences between the predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' However, significant differences between the PP8 t¯tb¯b predictions and the PP8 t¯t predictions are observed in ∆Rmin∆R b¯b , the jet multiplicity and in the number of events with more than four b-jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Furthermore, the uncertainty band is slightly larger in the CMS t¯tb¯b predictions, potentially caused by the lower factorisation scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' 3, the ATLAS nominal PP8 t¯tb¯b prediction is compared to all generator variations potentially considered as modelling uncertainties for future ATLAS t¯tH(H → b¯b) analyses, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' variations in t¯tb¯b-Powheg and Pythia8 parameter settings as well as Sherpa as alternative generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' As already discussed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' [22], the parameter hbzd has only a minor influence on the observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Interestingly, predictions of t¯tb¯b-Powheg matched to Pythia8 using the dipole shower and matched to the Herwig7 PS both show a significant decrease with respect to the nominal PP8 t¯tb¯b in the jet multiplicity and HT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Sherpa differs up to 10–20 % in all distributions with significant differences in shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' 4, the CMS nominal PP8 t¯tb¯b prediction is compared to generator variations potentially considered for the CMS t¯tH(H → b¯b) analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' The scale uncertainties, which include the scale variations in the ME and the PS, are significantly larger than the differences observed for the different hdamp variations, except at very low HT and low leading b-jet pT where the hdamp down variations shows up to 20 % differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Significant statistical fluctuations are observed at regions of low event yields, which are, however, not expected to be relevant for the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Finally, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' 5 shows the distributions used to estimate the systematic modelling uncertainties of the first Run-2 analysis by CMS [4] and of the first full Run-2 analysis by ATLAS [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' In addition to the scale and PS αs variations, the uncertainty on the t¯tb¯b PP8 prediction is estimated in case of ATLAS by assigning the relative difference between PP8 t¯t and alternative t¯t predictions listed in Table 1 to the t¯tb¯b prediction, and in case of CMS by the hdamp variations, where also a cross check with Sherpa t¯t has been made but was not included in the fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Due to displaying purposes, the ATLAS PP8 t¯t prediction, which is very similar to the CMS t¯t prediction as demonstrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' 2, is not shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='4 Conclusions Comparisons of generator predictions used by ATLAS and CMS in a typical phase space of the t¯tH(H → b¯b) measurement were presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Two sets are used for comparison: the generators used in the most recent published analyses involving t¯t inclusive predictions based on 5FS PDFs to estimate uncertainties and the set of generators in the future effort using t¯tb¯b calculations at NLO based on 4FS PDFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' The difference between the predictions exceeds the uncertainties from the scale variations both for the uncertainties considered in the published t¯tH(H → b¯b) analysis and for the future analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' The uncertainties due to the choice of PS and NLO generator are reduced when estimating them based on t¯tb¯b ME predictions compared to the previously used t¯t ME matched predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Despite differences in the set-ups between the experiments for the nominal PP8 t¯tb¯b generator, only small differences are observed in the predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' However, the considerations of the mod- elling uncertainties differs significantly: CMS considers inherent variations of the chosen model 8 as uncertainty, while ATLAS studied inherent variations and differences obtained with alternative generator choices and the latter dominates the uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Scale variations are applied by both experiments, however the details of the estimates differ between ATLAS and CMS in the published analysis but the effect of the different treatment are not yet studied for the future analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' The presented studies shall be used as input to discussions between the experiments and theo- rists to define theory uncertainties for future combinations of ATLAS and CMS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='25 Arbitrary Units b b t ATLAS PP8 t b b t CMS PP8 t t ATLAS PP8 t t CMS PP8 t ATLAS + 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hdamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' All predictions are normalised to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' The ratio shows the different curves divided by PP8 t¯tb¯b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' The error band contains the statistical uncertainty and the scale variations (ME and PS) for the CMS PP8 t¯tb¯b sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Statistical uncertainties are indicated by vertical lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' 12 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='2 b +b t ATLAS t Ratio to PP8 Figure 5: Comparison of predictions used for the systematic uncertainties of the first Run-2 analysis by CMS [4] and of the first full Run-2 analysis by ATLAS [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' All distributions are normalised to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' The ratio shows the different curves divided by ATLAS PP8 t¯tb¯b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' The error bands are constructed from the statistical uncertainties and the scale variations (ME and PS) for the ATLAS PP8 t¯tb¯b (blue) and the CMS PP8 t¯tb¯b (red) samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Statistical uncertainties are indicated by vertical lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' 13 3 Comparisons of Monte Carlo predictions for the t¯tW process The ATLAS [5] and CMS [6] experiments measured the t¯tH production cross section in multi-lepton final states, which are primarily sensitive to the decays of H → WW ∗, H → ττ and H → ZZ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' The dominant background in these measurements stems from t¯tW production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' These measurements along with the recent CMS measurement of t¯tW production [41] show some tension with the SM t¯tW predictions which were used to calculate the inclusive cross section and the acceptance in the analysis phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Different nominal MC predictions were used by the experiments for these measurements, AT- LAS used Sherpa 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='1 [27] and CMS used MG5 aMC@NLO 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='2 matched to Pythia8 using the FxFx merging scheme [24] and including sub-leading electroweak (EW) corrections of the order αsα3 where α (αs) refers to the EW (QCD) coupling constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' The experiments applied different corrections to predict the theoretical inclusive t¯tW cross section that entered the calculation of the scale factor to data, resulting in a value of 727 fb for ATLAS [5] and 650 fb for CMS [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Both experiments estimate the uncertainty of the MC prediction related to missing higher order correc- tions by varying the renormalisation and factorisation scales in the ME.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' However, ATLAS considers additionally uncertainties associated with the modelling of additional QCD radiation by comparing the nominal t¯tW prediction with that of MG5 aMC@NLO+Pythia8 as alternative MC generator differing in particular in the number of additional partons in the ME calculation, the parton shower and merging algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' In recent times there have been significant theoretical developments in t¯tW modelling despite the challenges associated with calculations of t¯tW with higher order corrections in the QCD, αs, and EWK, α, couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Even at LO in αs, complications arise because t¯tW is a qq-initiated process in which the radiation of the W-boson from one of the initial state quarks polarises the incoming quark, making spin correlations all the more important [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Initial calculations of t¯tW production at next-to-leading order (NLO) in QCD at fixed order [43] and later matched to a parton shower [44, 45] were later augmented with NLO EWK corrections (of order α2α2 s) [46] to provide the higher order cross sections used across the LHC programme for a number of years [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Furthermore, full NLO calculations including fixed-order corrections matched to parton shower in the POWHEG-BOX framework and accounting for LO spin-correlation of decay product have recently been provided in [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Since then there has been significant theoretical progress in calculating more complex and precise predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Higher order QCD corrections including t¯tW production with additional partons open gluon-initiated production modes with significant contributions to the total cross section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Recent studies show that these contributions also have large next-to-leading order (NLO) corrections [23] and that t¯tW jj can be large [49], both of which require NLO-merged calculations [50] for such effects to be properly included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Furthermore, beyond the traditionally “leading” NLO EWK cor- rections (of order α2α2 s) there are even larger contributions from traditionally “sub-leading” NLO corrections (of order α3αs) [51, 52, 48] due to the existence of tW scattering contributions embedded in to the t¯tWj process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Calculations at NLO in QCD accounting for next-to-next-to-leading loga- rithmic effects (NNLL) are also available [53] as well as recent predictions at NLO+NNLL in QCD also with NLO EWK corrections [54, 55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Full off-shell calculations at NLO in QCD [56, 57, 58] are also now available and more recently the NLO EWK corrections have also been incorporated [59] into these calculations, along with the development of procedures to apply the off-shell corrections to NLO+PS setups [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' A first attempt to formulate an uncertainty estimate in view of these theoretical predictions has been made in [48] where different generator codes at NLO QCD are compared with fixed order calculations to demonstrate that a robust theoretical prediction of hadronic t¯tW production cannot 14 be expressed as a simple recipe covering the specifics of all experimental observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Therefore the value of comparing several well tested tools is emphasised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' For future analyses, updated MC models will be used and the estimate of systematic uncertainty is under development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' In particular, ATLAS is considering Sherpa predictions including several higher order EW corrections in addition to the predictions at NLO in the strong coupling, namely of the order α3, α2α2 s and α3αs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Furthermore, calculations of MG5 aMC@NLO+Pythia8 employing the FxFx merging scheme will be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' For inclusive predictions, Powheg predictions [48] are also considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' CMS will continue to use MG5 aMC@NLO+Pythia8 with the FxFx merging scheme including subleading EW corrections however the EW corrections are not included in the present document in order to facilitate the comparison between the setups used by each experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' The samples will be described in the following and an overview with detailed information on the samples is given in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' The use of other theoretical developments, already outlined, will also be considered in future but are beyond the scope of this document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Comparisons are performed using stable final-state particles in a fiducial phase space similar to the experimental measurements in the two same-sign leptons (2lSS) channel as implemented in a dedicated routine in the Rivet analysis toolkit [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Two sets of distributions are presented, one where the histograms are normalised to unit area to asses shape differences in the differential distributions and another set where the generator cross sections are set to 600.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='8 fb the value reported in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' This allows to study differences in acceptance for the different generator predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' The chapter is organised as follows: Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='1 gives the detailed set-up for the generator samples, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='2 describes the object reconstruction and event selection, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='3 gives the two sets of results and finally conclusions are drawn in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='1 MC generator set-ups This chapter describes in detail the set-up of the MC generator set-ups used for the ATLAS and CMS samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' ATLAS setup The nominal sample for the comparison of this note was generated using the Sherpa 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='10 [27, 61] generator with the NNPDF3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='0 NLO PDF set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' The t¯tW matrix element was calculated for up to one additional parton at NLO and up to two partons at leading order (LO) accuracy using Comix [35] and OpenLoops [29], and merged with the Sherpa parton shower [36] using the MEPs@NLO prescription [62] with a merging scale of 30 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' The choice of renormalisation and factorisation scales of the core process is µR = µF = HT/2, where HT is defined as the scalar sum of the transverse masses � p2 T + m2 of all final state particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Systematic uncertainties due to missing higher-order QCD corrections are estimated in the nominal sample by varying the factorisation and renormalisation scales together with αs in the parton shower by a factor of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='5 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='0) with respect to the central value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' In addition to this nominal prediction at NLO in the strong coupling, a separate sample is produced which contains also higher order corrections relating to EW contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' These are added in two ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' First, event-by-event correction factors are applied that provide virtual NLO EW corrections of the order α2α2 s derived using the formalism described in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' [63] along with LO corrections of order α3, both are implemented using the prescription outlined in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' [27, 64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Second, sub-leading EW corrections at order α3αs [52] are partially accounted for (only the real emission contribution) via the addition of an independent Sherpa 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='10 sample produced at LO in QCD for this final state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' This sample is marked as “QCD+EW” in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' 15 Alternative t¯tW predictions are produced using the MG5 aMC@NLO 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='3 program to gen- erate t¯tW production with up to one additional parton in the final state at NLO accuracy in the strong coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' The renormalisation and factorisation scales are the same as in the nominal sam- ple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Another sample is generated using MG5 aMC@NLO 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='3 for up to one additional parton at NLO accuracy and up to two additional partons at LO accuracy in the ME and merging the different jet multiplicities using the FxFx NLO matrix-element and parton-shower merging pre- scription [24], see detailed description in [65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' As part of the FxFx merging algorithm, scales are dynamically chosen and set to the characteristic scale of the hard process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' In both samples, spin correlation effects between the ME decay products are accounted for by Madspin [66] and the showering and subsequent hadronization is performed using Pythia 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='210 and Pythia 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='245 [8], respectively, with the A14 tune [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' These samples are referred to as “ATLAS MG5 aMC+Py8” and “ATLAS MG5 aMC+Py8 FxFx” in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' CMS setup CMS simulates proton-proton to t¯tℓν processes at NLO accuracy in the matrix element calcula- tion using MG5 aMC@NLO 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Spin correlation effects between the ME decay products are accounted for by Madspin [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' The ME calculation includes diagrams with up to one additional parton at NLO and any further partons are generated by the parton shower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' The renormalisation and factorisation scales are set to the characteristic scale of the hard process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' They are chosen dynamically and are dependent kinematics of the event after the FxFx merging prescription3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Theoretical uncertainties associated with missing higher-order QCD corrections from the ME calculation are estimated by varying the renormalisation and factorisation scale by a factor of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='5 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' All possible combinations of these variations, implemented using a dedicated set of per-event weights, are then used to construct the uncertainty envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' The parton shower, hadronization processes and decays of τ leptons (including polarisation effects) are modelled using Pythia 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='226 with the CP5 tune.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' The samples is called “CMS MG5 aMC+Py8 FxFx” in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' 3see in particular section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='3 of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' [24] where elements of Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' [67, 68] are taken into account 16 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='2 Object reconstruction, fiducial volume and observables Object and event selection is defined at stable particle-level that closely matches the detector-level described in reference [5] (ATLAS) and [6] (CMS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Jets are reconstructed from all stable final state particles with a mean lifetime of τ > 3 × 10−11 s (but excluding leptons and neutrinos from the top quark decay chain), using the anti-kt algorithm with a radius parameter of R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Jets are required to satisfy pT > 25 GeV and |η| < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Jets that are matched to a b-hadron4 by ghost matching [40] are referred to as b-jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Electrons and muons, referred to as light leptons ℓ, are required to be separated from selected jets by ∆R > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='4 and are otherwise removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Hadronically decaying τ leptons are required to satisfy pT > 25 GeV and |η| < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Events are selected with exactly two light leptons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' The four-momentum of the bare leptons from top quark decay are modified (”dressed”) by adding the four-momenta of all radiated photons within a cone of size ∆R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Leptons are required to have |η| < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='5 and pT > 25(20) GeV for leading ℓ0 (subleading ℓ1) lepton (pT ordered).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Leptons are required to have same charge, targeting the semi-leptonic t¯t decay and leptonic W decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Events with at least 3 jets and at least one of them being a b-jet are considered in the fiducial volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' The object definition and event selection is summarised in Tables 6 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' These are then split into five regions, categorized by the number of jets of any flavour (three or ≥4), Nb−jets (one or ≥2) as well as the presence of hadronically decaying τ lepton, as summarised in Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' The definitions of the regions are motivated by the t¯tH multi-lepton analysis strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Regions 1 and 2 corresponds to the signal regions5 and Regions 3 and 4 are used as control regions in the 2ℓ same-sign 0-τhad t¯tH channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Definition of Region 5 is closely followed6 by the selections in the 2ℓ same-sign 1-τhad t¯tH channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' The list of variables for the comparison of the t¯tW generators presented in this note are sum- marised in Table 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' 4no pT cut is applied 5slightly different then in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' [5], in order to define a common selection with the CMS Collaboration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' 6requirement on jet multiplicity is relaxed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' 17 Table 5: The configurations used for the event generation of the t¯tW processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Scale settings given in terms of HT = �N i=0 � p2 T,i + m2 i , where N corresponds to the number of final state particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Label ATLAS Sherpa 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='10 ATLAS Sherpa 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='10 ATLAS MG5 aMC+Py8 FxFx ATLAS MG5 aMC+Py8 CMS MG5 aMC+Py8 FxFx QCD+EW Process t¯tW inclusive t¯tW inclusive t¯tW inclusive t¯tW inclusive t¯tℓν (t¯tW inclusive) Generator Sherpa 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='10 [27] Sherpa 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='10 [27] MG5 aMC@NLO 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='3 [69] MG5 aMC@NLO 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='3 [70] MG5 aMC@NLO 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='2 order of QCD ME 0,1 j@NLOa 0,1 j@NLOa 0,1 j@NLO NLO 0,1 j@NLO ME or core scale µR = µF = HT/2 µR = µF = HT/2 dynamic scale choice [24, 67, 68] µR = µF = HT/2 dynamic scale choice [24, 67, 68] order of EW corr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' α3, α2α2 s, α3αs Parton Shower Sherpa 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='10 Sherpa 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='10 Pythia 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='245 [8] Pythia 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='210 [8] Pythia 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='226 Merging Scheme MEPs@NLO [62] MEPs@NLO [62] FxFx [24] FxFx Merging Scale 30 GeV 30 GeV 30 GeV 42 GeV PDF NNPDF3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='0 NNLO [71] NNPDF3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='0 NNLO NNPDF3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='0 NLO NNPDF3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='0 NLO NNPDF3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='1 NLO [72] Tune Sherpa default Sherpa default A14 [33] A14 CP5 [34] Cross sectionb 597 fb 615 fb 613 fb 548 fb 220 fb (666 fbc) aIn addition to the implicit 2j@LO contribution from the real emission part of the 1j@NLO calculation, Sherpa adds the 2j@LO as an explicit separate process within the merging such that the ME is supplemented with higher-order improvements such as the CKKW scale choice and Sudakov factors.” bσtot=600.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='8 fb from YR4 is used for all samples in the generator comparisons in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='2 except for Sherpa QCD+EW ccalculated from t¯tℓν as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='2198 x (1/ (3 x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='11) ) 18 Table 6: The object reconstruction used in the Rivet analysis of the t¯tW processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Leptons are ordered in pT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Object reconstruction and selection jets stable final state particles with anti-kt algorithm, radius R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='4 prompt ”dressed” leptons and neutrinos are vetoed from jet pT > 25 GeV and |η| < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='5 b-jets jets ghost matched to B-hadrons pT > 25 GeV and |η| < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='5 light leptons (electrons and muons) dressed with photons within ∆R < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='1 |η| < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='5 and pT > 25(20) GeV for leading (subleading) lepton overlap removal remove light lepton if ∆R(jet, lepton) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='4 hadronicaly decaying τ leptons (before decay) pT > 25 GeV and |η| < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='5 Table 7: The event selection used in the Rivet analysis for the t¯tW processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Njets refers to all jets independent of jet flavour, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' b-jets are included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Event selection for 2ℓSS exactly 2 leptons with same charge Njets ≥3 Nb−jets ≥1 Table 8: The region definitions used in the Rivet analysis for the t¯tW processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Region Selection 1 Nb−jets =1, Njets ≥4, 0-τhad 2 Nb−jets ≥2, Njets ≥4, 0-τhad 3 Nb−jets =1, Njets=3, 0-τhad 4 Nb−jets ≥2, Njets=3, 0-τhad 5 Nb−jets ≥1, Njets ≥3, 1-τhad Table 9: List of the observables for the comparison of t¯tW predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Leptons and b-jets are ordered in pT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Variable Description Regions Njets Jet multiplicity 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='5 Nb−jets Number of b-jets 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='5 Hjets T Scalar sum of transverse momentum of all jets in the event 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='4 pb0 T Leading b-jet transverse momentum 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='2 pℓ0 T Leading lepton transverse momentum 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='5 ∆Rℓ0jets Minimum angular separation between the leading lepton ℓ0 and the nearest jet 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='2 ∆Rℓ0ℓ1 Angular distance between the two leptons 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='5 max|ηℓ| Value of the highest lepton’s pseudorapidity in the event 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='2 19 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='3 Results The samples described in Table 5 are compared in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' The ratio plots show the ratios of the all MC samples with respect to ATLAS Sherpa 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='10, the shaded band represents scale variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' The same set of distributions are presented twice with different focus: in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='1 shapes are compared and in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='2 acceptance effects are studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='1 Shape comparison In the following, shape comparisons between nominal and alternative generators will be presented, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' the distributions are normalised to unit area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' The modelling of jet based distributions are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' 6 for the regions without hadronic τ leptons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Sizeable discrepancies in the mod- elling of high jet multiplicities can be observed between the ATLAS and CMS MG5 aMC@NLO FxFx predictions which are in opposite direction compared to Sherpa t¯tW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' All predictions except ATLAS MG5 aMC@NLO+Pythia8 agree well on HT in regions with at least four jets, but larger discrepancies are observed for the three jet regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' The distributions of b-jet pT differ more in the regions with one b-jet, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Only ATLAS MG5 aMC@NLO+Pythia8 shows significant differences for the angular distance between the two leptons and the value of lepton’s pseudo-rapidity as demonstrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' The lepton pT distributions are similar, but their distance to the closest jet vary at as seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Distributions of the jet multiplicity, number of b-jets, the leading lepton transverse momentum and the angular distance between the two leptons ∆Rℓ0ℓ1 for the Region 5 with Nτhad = 1 selection are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' 10.' metadata={'source': 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scale variation ME-only [GeV] jets T H 0 200 400 600 800 1000 1200 1400 Ratio to Sherpa 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='5 Figure 6: Distribution of the jet multiplicities (top) and the scalar sum of jets transverse momentum, Hjets T (middle), for the Region 1 with Nb−jets =1 (left) and Region 2 with Nb−jets ≥2 (right) selection requiring four and more jets, and for the Region 3 Nb−jets = 1 (bottom, left) and Region 4 with Nb−jets ≥2 (bottom, right) selection requiring exactly three jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' 21 jets b Number of 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' 25 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='2 Comparisons of predictions including acceptance effects In the following section, a comparison of the generators will be given in the fiducial phase space, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' the predicted distributions include acceptance effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' For this comparison, all distributions are normalised to a common total cross section value of σYR4 tot = 600.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='8 fb as given in the Yellow Report 4 [47], except the distributions of Sherpa 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='10 QCD+EW which is normalised to its generator cross section of 614.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='7 fb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' The same set of distributions as discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='1 are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' In all distributions, a significant increase of scale uncertainties is observed, reaching up to 50 % at high jet multiplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' The observables related to jet multiplicity and HT show similar trends as in the shape comparisons, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Only the discrepancy of the jet multiplicity prediction in MG5 aMC+Py8 FxFx is significantly enhanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' 26 Number of jets 3 4 5 6 7 8 9 [fb] jet N d / fid σ d 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='8 1 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='35 ATLAS + CMS Generator Level = 13 TeV, ttW s j3 ≥ b 1 ≥ had τ SS 1 l2 ATLAS Sherpa 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='10 ATLAS Sherpa 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='10 QCD+EW ATLAS MG5_aMC+Py8 FxFx ATLAS MG5_aMC+Py8 CMS MG5_aMC+Py8 FxFx ATLAS Sherpa scale var ME+PS CMS scale variation ME-only [GeV] T p Leading lepton 0 50 100 150 200 250 300 350 400 450 500 Ratio to Sherpa 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='5 1 ,l 0l R ∆ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='5 1 1.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='3 ATLAS + CMS Generator Level = 13 TeV, ttW s j3 ≥ b 1 ≥ had τ SS 1 l2 ATLAS Sherpa 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='10 ATLAS Sherpa 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='10 QCD+EW ATLAS MG5_aMC+Py8 FxFx ATLAS MG5_aMC+Py8 CMS MG5_aMC+Py8 FxFx ATLAS Sherpa scale var ME+PS CMS scale variation ME-only 1 ,l 0l R ∆ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='5 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='5 Ratio to Sherpa 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='5 Figure 15: Distribution of the the jet multiplicity, number of b-jets, the leading lepton transverse momentum and the angular distance between the two leptons ∆Rℓℓ for the Region 5 with 1τhad selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' All distributions are normalised to the YR4 cross section of 600.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='8 fb except Sherpa 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='10 QCD+EW which is normalised to 614.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='7 fb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' 31 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='4 Conclusions The t¯tW preditions of Sherpa and MG5 aMC@NLO+Pythia8 with different settings have been compared with respect to their inclusive ttW cross section predictions and their differential cross section predictions in regions and observables relevant for the measurement of t¯tH in the multi- lepton final state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' For the inclusive t¯tW cross section slightly different values are predicted [27, 50] for calculations with similar theoretical accuracy which is subject to ongoing theoretical studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Based on the studies presented in this note, additional studies and discussions in the LHC Higgs Working Group and the LHC Top Working Group [73], ATLAS and CMS agreed to use the inclusive t¯tW cross section of 722+70 −78 (scale) ±7 (PDF) fb [50] as a reference inclusive cross section to allow direct comparisons between experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' The normalised distributions sensitive to shape differences have very small scale uncertainties, below 10 % in most of the phase space, while these scale uncertainties are significant when the acceptance effects are included, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' the distributions are normalised to the t¯tW cross section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' The inclusion of tree-level EW effects only causes minor shape effects but can lead to up to 20 % difference in the cross section at high jet multiplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' As expected, including the FxFx algorithm into the MG5 aMC@NLO+Pythia8 prediction leads to significant effects in all regions, especially at low HT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Significant differences between the MG5 aMC@NLO+Pythia8 FxFx predictions of ATLAS and CMS are observed, especially in the jet multiplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Further studies are required to investigate the origin of these differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Given that both setups consider the same perturbative accuracy such differences could be attributed to the choice of merging scale value or Pythia8 tune, so this could be an area of future study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' For many observables the shape differences between the various model predictions are within the scale uncertainties of each prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Observables relating to jet activity such as the jet multiplicity and HT are notable exceptions to this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' This is especially the case for Region 3 where the differences in shape between predictions for HT is particlularly large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' This region is important to constrain the interplay between ttW background and backgrounds arising from ttbar production where at least one lepton is mis-identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' It represents a phase space where one of the jets in the ttW decay is not reconstructed or is out of acceptance and is not expected to be as sensitive to the additional jet modelling as Regions 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Therefore, it is somewhat surprising that such large differences are observed between predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' This should be investigated in future studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' The inclusion of EW corrections shows only a small shape and normalisation effects for most observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' One place where a notable effect on the shape of a distribution can be observed is for the jet multiplicity, however the effect is small enough to be covered by the QCD scale variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Future studies could specifically target the sub-leading EW contribution with cuts related to the rapidity difference between jets which has been shown [51] to be different with respect to the central ttW QCD process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' These distributions shall be used as a starting point to derive a strategy for the theory uncer- tainty estimates for a combination of the expected measurement results based on the full Run-2 data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Beyond what has been shown in the comparisons included in this document, this strategy is expected to take into consideration the latest developments on the theoretical models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' For example, the NLO+PS calculations provided in Powheg [48] can act as systematic variation with respect to the MG5 aMC@NLO+Pythia8 and Sherpa calculations for more inclusive phase-spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' They can also be used to understand parton shower, hadronisation and underlying event effects through interfaces to Pythia and Herwig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' In addition, recent off-shell calculations [56, 57, 58, 59] and in particular the single-resonant contributions could be of importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' In the absence of explicit parton shower-matched calculations, corrections can be applied through the procedure outlined 32 in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' It would also be important to extend existing calculations to additional final states, such as 2ℓSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Finally, given the current discrepancy between ATLAS and CMS, the strategy must address how different model predictions are considered in addition to the scale uncertainties as part of the theoretical uncertainties on the measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' 33 References [1] ATLAS Collaboration, “Measurement of Higgs boson decay into b-quarks in associated production with a top-quark pair in pp collisions at √s = 13 TeV with the ATLAS detector”, arXiv:2111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='06712.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Submitted to JHEP (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' [2] ATLAS Collaboration, “Search for the standard model Higgs boson produced in association with top quarks and decaying into a b¯b pair in pp collisions at √s = 13 TeV with the ATLAS detector”, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' D 97 (2018) 072016, arXiv:1712.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='08895.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='072016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' [3] CMS Collaboration, “Measurement of ttH production in the H → bb decay channel in 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='5 fb−1 of proton-proton collision data at √s = 13 TeV”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' CMS-PAS-HIG-18-030, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' [4] CMS Collaboration, “Search for ttH production in the H→ bb decay channel with leptonic tt decays in proton–proton collisions at √s = 13 TeV”, JHEP 03 (2019) 026, arXiv:1804.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='03682.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='1007/JHEP03(2019)026.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' [5] ATLAS Collaboration, “Analysis of ttH and ttW production in the multilepton final states with the ATLAS detector”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' ATLAS-CONF-2019-045, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' [6] CMS Collaboration, “Measurement of the Higgs boson production rate in association with top quarks in final states with electrons, muons, and hadronically decaying tau leptons at √s = 13 TeV”, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' C 81 (2021) 378, arXiv:2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='03652.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='1140/epjc/s10052-021-09014-x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' [7] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Buckley, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Butterworth, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Grellscheid et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=', “Rivet user manual”, Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' 184 (2013) 2803–2819, arXiv:1003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='0694.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='cpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' [8] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Sj¨ostrand, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Ask, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Christiansen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=', “An Introduction to PYTHIA 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='2”, Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' 191 (2015) 159, arXiv:1410.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='3012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='cpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='024.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' [9] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Nason, “A new method for combining NLO QCD with shower Monte Carlo algorithms”, JHEP 11 (2004) 040, arXiv:hep-ph/0409146.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='1088/1126-6708/2004/11/040.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' [10] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Frixione, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Nason, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Oleari, “Matching NLO QCD computations with parton shower simulations: the POWHEG method”, JHEP 11 (2007) 070, arXiv:0709.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='2092.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='1088/1126-6708/2007/11/070.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' [11] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Frixione, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Ridolfi, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Nason, “A positive-weight next-to-leading-order Monte Carlo for heavy flavour hadroproduction”, Journal of High Energy Physics 2007 (2007), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' 09, 126–126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='1088/1126-6708/2007/09/126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' [12] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Alioli, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Nason, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Oleari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=', “A general framework for implementing NLO calculations in shower Monte Carlo programs: the POWHEG BOX”, JHEP 06 (2010) 043, arXiv:1002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='2581.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='1007/JHEP06(2010)043.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' [13] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Campbell, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Ellis, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Nason et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=', “Top-Pair Production and Decay at NLO Matched with Parton Showers”, JHEP 04 (2015) 114, arXiv:1412.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='1828.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='1007/JHEP04(2015)114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' 34 [14] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Bredenstein, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Denner, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Dittmaier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=', “NLO QCD Corrections to Top Anti-Top Bottom Anti-Bottom Production at the LHC: 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' full hadronic results”, JHEP 03 (2010) 021, arXiv:1001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='4006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='1007/JHEP03(2010)021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' [15] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Bredenstein, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Denner, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Dittmaier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=', “NLO QCD corrections to pp → t¯tb¯b + X at the LHC”, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' 103 (2009) 012002, arXiv:0905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='0110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='1103/PhysRevLett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='012002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' [16] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Bevilacqua, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Czakon, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Papadopoulos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=', “Assault on the NLO wishlist:pp → ttbb”, Journal of High Energy Physics 2009 (2009), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' 09, 109–109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='1088/1126-6708/2009/09/109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' [17] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Garzelli, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Kardos, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Tr´ocs´anyi, “Hadroproduction of t¯tb¯b final states at LHC: predictions at NLO accuracy matched with Parton Shower”, JHEP 03 (2015) 083, arXiv:1408.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='0266.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='1007/JHEP03(2015)083.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' [18] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Bevilacqua, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Bi, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Hartanto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=', “ttbb at the LHC: on the size of corrections and b-jet definitions”, Journal of High Energy Physics 2021 (2021), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' 8,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='1007/jhep08(2021)008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' [19] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Denner, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='-N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Lang, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Pellen, “Full NLO QCD corrections to off-shell ttbb production”, Physical Review D 104 (2021), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' 5,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='1103/physrevd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='056018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' [20] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Bevilacqua, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Bi, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Hartanto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=', “t¯tb¯b at the LHC: On the size of off-shell effects and prompt b-jet identification”, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='48550/ARXIV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='2202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='11186.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' [21] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Jeˇzo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Lindert, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Moretti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=', “New NLOPS predictions for t¯t + b -jet production at the LHC”, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' C78 (2018) 502, arXiv:1802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='00426.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='1140/epjc/s10052-018-5956-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' [22] ATLAS Collaboration, “Studies of Monte Carlo predictions for the t¯tb¯b process”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' ATL-PHYS-PUB-2022-006, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' [23] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Alwall, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Frederix, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Frixione et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=', “The automated computation of tree-level and next-to-leading order differential cross sections, and their matching to parton shower simulations”, JHEP 07 (2014) 079, arXiv:1405.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='0301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='1007/JHEP07(2014)079.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' [24] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Frederix and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Frixione, “Merging meets matching in MC@NLO”, JHEP 12 (2012) 061, arXiv:1209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='6215.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='1007/JHEP12(2012)061.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' [25] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Bellm, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Gieseke, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Grellscheid et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=', “Herwig 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='0/Herwig++ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='0 release note”, The European Physical Journal C 76 (2016), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' 4,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='1140/epjc/s10052-016-4018-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' [26] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Cooper, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Katzy, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Mangano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=', “Importance of a consistent choice of alpha(s) in the matching of AlpGen and Pythia”, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' C 72 (2012) 2078, arXiv:1109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='5295.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='1140/epjc/s10052-012-2078-y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' [27] Sherpa Collaboration, “Event Generation with Sherpa 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='2”, SciPost Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' 7 (2019) 034, arXiv:1905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='09127.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='21468/SciPostPhys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='034.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' [28] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Cascioli, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Maierh¨ofer, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Moretti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=', “NLO matching for t¯tb¯b production with massive b-quarks”, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' B734 (2014) 210–214, arXiv:1309.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='5912.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='physletb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='040.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' 35 [29] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Cascioli, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Maierhofer, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Pozzorini, “Scattering Amplitudes with Open Loops”, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' 108 (2012) 111601, arXiv:1111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='5206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='1103/PhysRevLett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='111601.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' [30] The routine will be released as MC-ttbb routine in the Rivet analysis toolkit [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' [31] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Ryd, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Lange, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Kuznetsova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=', “EvtGen: A Monte Carlo Generator for B-Physics”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' EVTGEN-V00-11-07, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' [32] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Buccioni, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Kallweit, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Pozzorini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=', “NLO QCD predictions for t¯tb¯b production in association with a light jet at the LHC”, JHEP 12 (2019) 015, arXiv:1907.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='13624.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='1007/JHEP12(2019)015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' [33] ATLAS Collaboration, “ATLAS Pythia 8 tunes to 7 TeV data”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' ATL-PHYS-PUB-2014-021, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' [34] CMS Collaboration, “Extraction and validation of a new set of CMS Pythia 8 tunes from underlying-event measurements”, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' C 80 (2020) 4, arXiv:1903.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='12179.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='1140/epjc/s10052-019-7499-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' [35] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Gleisberg and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' H¨oche, “Comix, a new matrix element generator”, JHEP 12 (2008) 039, arXiv:0808.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='3674.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='1088/1126-6708/2008/12/039.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' [36] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Schumann and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Krauss, “A Parton shower algorithm based on Catani-Seymour dipole factorisation”, JHEP 03 (2008) 038, arXiv:0709.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='1027.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='1088/1126-6708/2008/03/038.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' [37] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Jeˇzo and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Nason, “On the Treatment of Resonances in Next-to-Leading Order Calculations Matched to a Parton Shower”, JHEP 12 (2015) 065, arXiv:1509.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='09071.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='1007/JHEP12(2015)065.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' [38] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Cacciari, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Salam, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Soyez, “The anti-kt jet clustering algorithm”, JHEP 04 (2008) 063, arXiv:0802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='1189.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='1088/1126-6708/2008/04/063.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' [39] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Cacciari, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Salam, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Soyez, “FastJet user manual”, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' C 72 (2012) 1896, arXiv:1111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='6097.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='1140/epjc/s10052-012-1896-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' [40] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Cacciari, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Salam, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Soyez, “The Catchment Area of Jets”, JHEP 04 (2008) 005, arXiv:0802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='1188.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='1088/1126-6708/2008/04/005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' [41] CMS Collaboration, “Measurement of the cross section of top quark-antiquark pair production in association with a W boson in proton-proton collisions at √s = 13 TeV”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' CMS-PAS-TOP-21-011, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' [42] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Maltoni, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Mangano, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Tsinikos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=', “Top-quark charge asymmetry and polarization in ttW ± production at the LHC”, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' B 736 (2014) 252–260, arXiv:1406.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='3262.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='physletb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='033.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' [43] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Campbell and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Ellis, “t¯tW +− production and decay at NLO”, JHEP 07 (2012) 052, arXiv:1204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='5678.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='1007/JHEP07(2012)052.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' [44] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Garzelli, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Kardos, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Papadopoulos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=', “t ¯t W +− and t ¯t Z Hadroproduction at NLO accuracy in QCD with Parton Shower and Hadronization effects”, JHEP 11 (2012) 056, arXiv:1208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='2665.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='1007/JHEP11(2012)056.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' 36 [45] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Maltoni, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Pagani, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Tsinikos, “Associated production of a top-quark pair with vector bosons at NLO in QCD: impact on ttH searches at the LHC”, JHEP 02 (2016) 113, arXiv:1507.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='05640.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='1007/JHEP02(2016)113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' [46] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Frixione, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Hirschi, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Pagani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=', “Electroweak and QCD corrections to top-pair hadroproduction in association with heavy bosons”, JHEP 06 (2015) 184, arXiv:1504.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='03446.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='1007/JHEP06(2015)184.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' [47] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' de Florian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=', “Handbook of LHC Higgs Cross Sections: 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Deciphering the Nature of the Higgs Sector”, arXiv:1610.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='07922.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='23731/CYRM-2017-002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' [48] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Febres Cordero, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Kraus, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Reina, “Top-quark pair production in association with a W ± gauge boson in the POWHEG-BOX”, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' D 103 (2021), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' 9, 094014, arXiv:2101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='11808.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='094014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' [49] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' von Buddenbrock, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Ruiz, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Mellado, “Anatomy of inclusive t¯tW production at hadron colliders”, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' B 811 (2020) 135964, arXiv:2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='00032.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='physletb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='135964.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' [50] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Frederix and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Tsinikos, “On improving NLO merging for ttW production”, JHEP 11 (2021) 029, arXiv:2108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='07826.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='1007/JHEP11(2021)029.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' [51] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Dror, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Farina, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Salvioni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=', “Strong tW Scattering at the LHC”, JHEP 01 (2016) 071, arXiv:1511.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='03674.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='1007/JHEP01(2016)071.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' [52] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Frederix, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Pagani, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Zaro, “Large NLO corrections in t¯tW ± and t¯tt¯t hadroproduction from supposedly subleading EW contributions”, JHEP 02 (2018) 031, arXiv:1711.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='02116.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='1007/JHEP02(2018)031.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' [53] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Kulesza, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Motyka, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Schwartl¨ander et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=', “Associated production of a top quark pair with a heavy electroweak gauge boson at NLO+NNLL accuracy”, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' C 79 (2019), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' 3, 249, arXiv:1812.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='08622.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='1140/epjc/s10052-019-6746-z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' [54] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Broggio, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Ferroglia, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Frederix et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=', “Top-quark pair hadroproduction in association with a heavy boson at NLO+NNLL including EW corrections”, JHEP 08 (2019) 039, arXiv:1907.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='04343.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='1007/JHEP08(2019)039.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' [55] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Kulesza, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Motyka, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Schwartl¨ander et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=', “Associated top quark pair production with a heavy boson: differential cross sections at NLO+NNLL accuracy”, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' C 80 (2020), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' 5, 428, arXiv:2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='03031.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='1140/epjc/s10052-020-7987-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' [56] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Bevilacqua, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Bi, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Hartanto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=', “The simplest of them all: t¯tW ± at NLO accuracy in QCD”, JHEP 08 (2020) 043, arXiv:2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='09427.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='1007/JHEP08(2020)043.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' [57] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Denner and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Pelliccioli, “NLO QCD corrections to off-shell t¯tW+ production at the LHC”, JHEP 11 (2020) 069, arXiv:2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='12089.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='1007/JHEP11(2020)069.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' [58] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Bevilacqua, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Bi, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Hartanto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=', “NLO QCD corrections to off-shell t¯tW ± production at the LHC: correlations and asymmetries”, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' C 81 (2021), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' 7, 675, arXiv:2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='01363.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='1140/epjc/s10052-021-09478-x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' 37 [59] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Denner and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Pelliccioli, “Combined NLO EW and QCD corrections to off-shell ttW production at the LHC”, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' C 81 (2021), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' 4, 354, arXiv:2102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='03246.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='1140/epjc/s10052-021-09143-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' [60] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Bevilacqua, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Bi, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Febres Cordero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=', “Modeling uncertainties of t¯tW ± multilepton signatures”, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' D 105 (2022), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' 1, 014018, arXiv:2109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='15181.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='014018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' [61] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Gleisberg, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' H¨oche, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Krauss et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=', “Event generation with SHERPA 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='1”, JHEP 02 (2009) 007, arXiv:0811.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='4622.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='1088/1126-6708/2009/02/007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' [62] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' H¨oche, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Krauss, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Sch¨onherr et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=', “QCD matrix elements + parton showers: The NLO case”, JHEP 04 (2013) 027, arXiv:1207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='5030.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='1007/JHEP04(2013)027.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' [63] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Kallweit, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Lindert, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Maierhofer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=', “NLO QCD+EW predictions for V + jets including off-shell vector-boson decays and multijet merging”, JHEP 04 (2016) 021, arXiv:1511.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='08692.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='1007/JHEP04(2016)021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' [64] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' G¨utschow, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Lindert, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Sch¨onherr, “Multi-jet merged top-pair production including electroweak corrections”, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' C 78 (2018), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' 4, 317, arXiv:1803.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='00950.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='1140/epjc/s10052-018-5804-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' [65] ATLAS Collaboration, “Modelling of rare top quark processes at √s = 13 TeV”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' ATL-PHYS-PUB-2020-024, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' [66] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Artoisenet, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Frederix, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Mattelaer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=', “Automatic spin-entangled decays of heavy resonances in Monte Carlo simulations”, JHEP 03 (2013) 015, arXiv:1212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='3460.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='1007/JHEP03(2013)015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' [67] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Catani, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Krauss, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Kuhn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=', “QCD matrix elements + parton showers”, JHEP 11 (2001) 063, arXiv:hep-ph/0109231.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='1088/1126-6708/2001/11/063.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' [68] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Hamilton, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Nason, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Zanderighi, “MINLO: Multi-Scale Improved NLO”, JHEP 10 (2012) 155, arXiv:1206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='3572.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='1007/JHEP10(2012)155.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' [69] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Alwall et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=', “A Standard format for Les Houches event files”, Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' 176 (2007) 300–304, arXiv:hep-ph/0609017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='cpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' [70] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Alwall, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Frederix, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Frixione et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=', “The automated computation of tree-level and next-to-leading order differential cross sections, and their matching to parton shower simulations”, Journal of High Energy Physics 2014 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='1007/jhep07(2014)079.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' [71] NNPDF Collaboration, “Parton distributions for the LHC Run II”, JHEP 04 (2015) 040, arXiv:1410.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='8849.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='1007/JHEP04(2015)040.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' [72] NNPDF Collaboration, “Parton distributions from high-precision collider data”, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' C 77 (2017), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' 10, 663, arXiv:1706.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='00428.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='1140/epjc/s10052-017-5199-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' [73] LHC Higgs and Top Working Groups, “ttW modeling in light of ttH measurements”, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' https://indico.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='cern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content='ch/event/1219500/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} +page_content=' 38' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFJT4oBgHgl3EQf5i00/content/2301.11670v1.pdf'} diff --git a/ZtAzT4oBgHgl3EQfK_te/content/tmp_files/2301.01106v1.pdf.txt b/ZtAzT4oBgHgl3EQfK_te/content/tmp_files/2301.01106v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..227cf2f42e512ba4c80891048ddaf11ce55b21ad --- /dev/null +++ b/ZtAzT4oBgHgl3EQfK_te/content/tmp_files/2301.01106v1.pdf.txt @@ -0,0 +1,1721 @@ +Towards retrospective motion correction and +reconstruction for clinical 3D brain MRI +protocols with a reference contrast +Gabrio Rizzuti1,2, Tim Schakel2, Niek R. F. Huttinga2, Jan Willem Dankbaar2, +Tristan van Leeuwen1,3, and Alessandro Sbrizzi2 +1Utrecht University, Utrecht, 3584 CS, The Netherlands +2Universitair Medisch Centrum Utrecht, Utrecht, 3584 CX, The Netherlands +3Centrum Wiskunde & Informatica, Amsterdam, 1098 XG, The Netherlands +Abstract +Motion artifacts often spoil the radiological interpretation of MR im- +ages, and in the most severe cases the scan needs be repeated, with ad- +ditional costs for the provider. +We discuss the application of a novel +3D retrospective rigid motion correction and reconstruction scheme for +MRI, which leverages multiple scans contained in a MR session. Typi- +cally, in a multi-contrast MR session, motion does not equally affect all +the scans, and some motion-free scans are generally available, so that we +can exploit their anatomic similarity. The uncorrupted scan is used as a +reference in a generalized rigid-motion registration problem to remove the +motion artifacts affecting the corrupted scans. We discuss the potential +of the proposed algorithm with a prospective in-vivo study and clinical +3D brain protocols. This framework can be easily incorporated into the +existing clinical practice with no disruption to the conventional workflow. +Keywords +Image Reconstruction, Motion Correction, Multi-Contrast, Brain, Total varia- +tion +1 +Introduction +Magnetic resonance imaging (MRI) is fundamentally prone to motion artifacts, +since the data acquisition process usually lasts several minutes for each acquired +contrast, and the MR exam can be an uncomfortable experience for the patient. +Motion corruption impedes a correct radiological assessment, which then may +require a scan repetition, leading to considerable waste of resources for the +hospital [Andre et al., 2015]. +1 +arXiv:2301.01106v1 [eess.IV] 3 Jan 2023 + +Motion reduction strategies are broadly classified as preventive, prospective, +or retrospective techniques [Zaitsev et al., 2015, Godenschweger et al., 2016]. +Preventive strategies include physical devices to limit the motion (e.g. head +holders) or sedation, but their application is limited by ethical or health consid- +erations, and are often ineffective in eliminating patient movement. Prospective +and retrospective strategies, on the other hand, directly or indirectly estimate +the motion that the object of interest undergoes inside the scanner, and remove +its effect from the data or in the reconstruction phase. This correction step is +said to be applied “prospectively” [Maclaren et al., 2013] when the position of +the patient is tracked in real time and the scan settings are adjusted accord- +ingly on-the-fly. For example, the relative change of position can be estimated +by acquiring additional k-space or image-space navigators [Ehman and Felm- +lee, 1989b, Welch et al., 2002], or with “self-navigating” sequences [Pipe, 1999, +Welch et al., 2004, Bookwalter et al., 2010]. Alternatively, camera devices or +markers [Zaitsev et al., 2006, Forman et al., 2011] can be used to estimate the +imaging object position. However, most tracking modalities are often defective +in terms of either precision, patient interaction, or sequence independence [Ma- +claren et al., 2013]. Therefore, although effective in many respects, prospective +methods have somewhat limited range of application. +Retrospective algorithms are characterized by the removal of motion artifacts +in the final reconstruction phase, after the data acquisition. The main advan- +tage of retrospective schemes is in their flexibility, since they do not necessarily +require additional hardware, scanner modifications, MR navigators, markers, +and so on. Note, however, that they may benefit from using prior information +about the target imaging object and motion pattern. One main challenge for +this class of methods is the need for time-intensive computations. The scientific +literature on retrospective motion correction is quite rich: examples of retro- +spective techniques for rigid motion using navigators or markers can be found +in Ehman and Felmlee [1989a], Korin et al. [1995], Mendes et al. [2009], Book- +walter et al. [2010], Vaillant et al. [2014], while examples of “blind” techniques +(in this context, meaning that they are not using navigators or markers) are +presented in Atkinson et al. [1997, 1999], Manduca et al. [2000], Lin and Song +[2006], Loktyushin et al. [2013]. +Retrospective correction schemes are typically formulated as a bi-level opti- +mization problem, where two types of unknown are jointly estimated: the recon- +structed (2D/3D) image and the motion parameters. Due to the ill-posedness of +the problem here considered, the choice of the regularization method is crucial: +see, for example, gradient-entropy regularization in Manduca et al. [2000], Lin +and Song [2006], Loktyushin et al. [2013], sparsity regularization in M¨oller et al. +[2015], or iteratively re-weighted least-squares regularization in Cordero-Grande +et al. [2020]. Another strategy to ease the ill-posedness is to resort to special ac- +quisition patterns in k-space that are more robust in terms of motion correction, +as described in the DISORDER method in Cordero-Grande et al. [2020]. Al- +ternatively, many machine-learning approaches have been recently proposed for +retrospective motion correction [Pawar et al., 2018, K¨ustner et al., 2019, Haskell +et al., 2019, Lee et al., 2020, Ghaffari et al., 2021, Lee et al., 2021, Hossbach +2 + +et al., 2022]. +Some previous work in Rizzuti et al. [2022] introduced a retrospective motion +correction scheme, whose novel aspect is the use of a contrast free of motion +artifacts that can be leveraged as a reference to remove motion effects from +any other contrast from the same patient, akin to a generalized rigid motion +registration. The chief assumption of this work is the following: in a multi- +contrast MR session, motion does not typically affect all the scans and some +motion-free scans are generally available, so that we can exploit their anatomic +similarity. Structural similarity is technically achieved via structure-guided total +variation (TV), as originally proposed in Ehrhardt and Betcke [2016] and further +developed in Bungert and Ehrhardt [2020] (see also Ehrhardt et al. [2015]). +The goal of this paper is to extend the scope of Rizzuti et al. [2022], limited to +2D synthetic results, to general 3D randomized acquisitions and 3D rigid-motion +correction. We experimentally verify that a 3D extension is indeed feasible for +brain imaging. We do not assume data-driven priors (so that machine learning +is not available), any additional navigator data, nor consider motion-resilient +acquisition schemes, in order to conform to more broadly available clinical pro- +tocols. Note that the proposed method can employ any acquisition scheme, in +principle, but we stick to Cartesian acquisition, which are the standard encoding +strategies of clinical protocols. Since we focus on brain imaging, rigid motion can +be effectively assumed for our scope. The reference and the corrupted contrast +do not need be co-registered or acquired with the same resolution. +We thoroughly validate the method with a prospective in-vivo study based +on three volunteers and several motion types. +The strength and limitations +of the method are highlighted with the comparison of correction quality with +varying degrees of motion artifacts and contrast type as a reference prior. +2 +Theory +In this section, we present the basic mathematical formulation underpinning +the proposed motion correction method (further details can be found in Rizzuti +et al. [2022]). +The contrast volume, in the remainder of this section, will be denoted by +u ∈ Cnx, where nx is the number of voxels contained in a rectangular field of +view. The 3D image undergoes a time-dependent rigid motion +ut = Tθθθtu, +(1) +where t is a time-related label. In practice, t corresponds to the index of the k- +space readout line in the phase-encoding plane. The corresponding rigid trans- +formation is given by Tθθθt, and is parameterized by a time-dependent motion +parameter θθθt ∈ R6, which includes translations and rotations in 3D: +θθθ = (τττ,ϕϕϕ), +τττ = (τx, τy, τz), +ϕϕϕ = (ϕxy, ϕxz, ϕyz). +(2) +The rigid motion consists of a 3D rotation (defined by the 2D rotation angles +3 + +ϕxy, ϕxz, ϕyz, performed in this order in the corresponding planes) followed by +a translation (governed by the translation parameters τx, τy, τz). +Without loss of generality, we are assuming a Cartesian acquisition. At each +given time t, the MR acquisition process corresponds to the evaluation of the +Fourier transform F of ut in a particular subset Kt of the k-space. In practice, +the acquisition is structured in such a way that all the subsets Kt consist of +parallel lines in the k-space (the common direction being the readout direction). +We refer to the Fourier transform of a rigidly moving object uθθθ := Tθθθu as the +perturbed Fourier transform Fθθθu := Fuθθθ, and can be directly characterized as +Fθθθu (k) := exp (−i k · τττ)Fu (R−1 +ϕϕϕ k), +(3) +where the rotational operator with respect to the 3D angle ϕϕϕ is indicated by +Rϕϕϕ. This definition is motivated by classical Fourier identities that describe the +action of rigid motion under the Fourier transform. Due to rotational effects, +one must resort to the non-uniform discrete Fourier transform (NUFFT) to +evaluate equation (3) [Barnett et al., 2019, Barnett, 2021]. +Note that we implicitly assumed that no motion occurs while sampling the +elements of Kt, since the state of the object at the time t is associated to a +single motion parameter θθθt. +The assumption is motivated by the fact that +Kt will correspond, in practice, to a single Cartesian readout line, which lasts +few milliseconds. Hence, the data acquisition at time t is symbolized by the +application of the selection operator St to the Fourier-transformed volume: +dt = StFθθθtu = (Fθθθtu (k1), . . . , Fθθθtu (knr)), +k1, . . . , knr ∈ Kt. +(4) +Here, nr is the number of k-space samples in a single readout. +The resulting inverse problem can be cast as an optimization problem over +the reconstruction unknowns u and the motion parameters θθθt, that is: +min +u,θθθ1:nt +f(u,θθθ1:nt) + λgu(u) + µgθ(θθθ1:nt), +(5) +where θθθ1:nt = (θθθ1, . . . ,θθθnt), and nt is the number of time steps. The weighting +parameters λ, µ (both positive numbers) set the strength of the corresponding +regularization terms. The first term of the objective functional in equation (5) +corresponds to the data misfit: +f(u,θθθ1:nt) = +nt +� +t=1 +1 +2 ∥Fθθθtu − dt∥2 . +(6) +The least-squares norm is indicated here by ∥·∥. The regularization terms gu and +gθ are crucial in ensuring the well-posedness of the problem. Indeed, the objec- +tive in equation (6) will be sensitive to the relatively high signal-to-noise ratios +(SNR) of the high-frequency components of the data. Moreover, the objective is +highly non-convex as a function of θθθ1:nt. The motion-parameter regularization +is designed to ensure some form of regularity in time (e.g. smoothness), this +4 + +can be achieved for example by setting +gθ(θθθ1:nt) = +nt−1 +� +t=1 +1 +2 ∥θθθt+1 − θθθt∥2 . +(7) +Alternatively, higher-order derivatives may be used. Another strategy, adopted +in this paper, is to impose smoothness by setting hard constraints for the motion +parameters, rather than via an additive penalty term as in equation (7) [Rizzuti +et al., 2022]. +2.1 +Reference-guided total variation regularization +The crux of the proposed method is related to the choice of the regularization +term gu in equation (5). We adopt the structure-guided total variation scheme +proposed in Ehrhardt and Betcke [2016] in the context of multi-contrast imaging, +that is: +gu(u) = +� +x +∥Πv|x∇u|x∥ , +Πv|x = I3 − ξv|xξv|x +H, +(8) +where I3 is the 3 × 3 identity matrix, ∇·|x is the discretized gradient operator +evaluated at the voxel with center x, and Πv|x is the projection operator on the +linear space that is orthogonal to the vector ξv|x ∈ C3. The symbol H indicates +the adjoint operation. The vector ξv|x corresponds to the normalized gradient of +a given motion-free contrast v, e.g. ξv|x ≈ ∇v|x/ ∥∇v|x∥. The actual definition +is +ξv|x = +∇v|x +� +∥∇v|x∥2 + η2 +, +(9) +for some constant η > 0. The regularization term in equation (8) enforces the +gradient structure of v onto u, when v and u are anatomically compatible. It +is important to observe that v is not required to be registered with the target +contrast u, since the estimation of the motion parameters in equation (5) will +automatically compensate for the initial misalignment [see also Bungert and +Ehrhardt, 2020]. +In this work, we actually adopt a constrained formulation +of equation (8), meaning that structural similarity is imposed by forcing the +solution to belong to the constraint set Cu = {u : gu(u) ≤ ε}, where ε > 0 is a +prescribed regularization level [see Rizzuti et al., 2022, Peters et al., 2018, for +more details]. +2.2 +Optimization +In order to solve equation (5), we adopt an alternating update scheme based +on the proximal alternating minimization algorithm (PALM) described in Bolte +et al. [2014]. The algorithm of the optimization strategy is exemplified in Algo- +rithm 1. Each update requires the linearization of the smooth objective f and +the application of the proximal operators associated to gu and gθ. As it is com- +monly noted in the image registration literature, we will make use of multi-scale +5 + +methods to ease the ill-posedness of the problem. Two types of scale are con- +sidered, here. One is traditionally associated to the reconstruction grid size, by +considering a sequence of optimization problems defined on progressively finer +grids. Note that spatial coarsening of the reconstructed image u is intertwined +with the temporal coarsening of the motion parameters θθθ1:nt, since they are +associated to sample locations in the k-space. The other scale is related to the +regularization strength λ, as defined in equation (5). Hence, strongly weighted +problems are solved first, and the regularization is gradually relaxed as in a +continuation strategy. Overall, two nested sequences of optimization problems +are considered here. +Algorithm 1 Joint motion correction and reconstruction with alternating prox- +imal operator evaluation +Input: d, u(=0), θθθ(=0), αu, αθ, N ▷ Data, starting guesses, steplengths, iters +Output: u, θθθ +for scale = coarse,. . ., fine do +Downscaling of d, u, θθθ +for λ = high,. . ., low do +for n = 1 : N do +u ← proxαugu(u − αu∇uf(·,θθθ)) +▷ Reconstruction proxy +θθθ ← proxαθgθ(θθθ − αθ∇θθθf(u, ·)) +▷ Motion parameter proxy +end for +end for +Upscaling of u, θθθ +end for +3 +Experiments +In this section, we set up several experiments that demonstrate the capabilities +of the retrospective motion correction algorithm detailed in Section 2, whose +main novel aspect and strength is the use of a reference contrast to guide the cor- +rection. Our objective is to tackle motion correction for brain imaging, and we +focus on acquisition protocols that are relevant for the clinical practice. All the +imaging sequences considered in this study were taken from actual clinical brain +protocols of the Radiology and Radiotherapy departments of the UMC Utrecht. +The data considered in this section is based on 3D Cartesian acquisition. The +sampling pattern used in these acquisitions typically utilizes pseudo-random +undersampling. +The main assumptions underlying the proposed method are +related to the availability of a motion-free reference contrast and the motion +artifacts being produced by rigid motion. +We consider several studies with volunteer data (three volunteers in total1), +where motion artifacts are prospectively generated by instructing the volunteer +1We have informed written consent from the volunteers. The experiments were approved +by the ethical review board of the UMC Utrecht. +6 + +Experiment +Contrast +Sequence +Resolution +FOV +TR +TE +Flip angle +Phase-encoding pattern +Duration +Section 3.1 +T2-FLAIR +3D TSE +1.2 mm3 +230×230×237.6 mm3 +4800 ms +320 ms +90◦ +Randomized +350 s +T1∗ +3D TFE +1 mm3 +230×230×238 mm3 +7.8 ms +3.6 ms +8◦ +Randomized +180 s +Section 3.2 +T1 +3D TFE +1 mm3 +230×230×238 mm3 +7.9 ms +3.6 ms +8◦ +Randomized +180 s +T2∗ +3D TSE +1.1 mm3 +250×250×190.3 mm3 +3000 ms +260 ms +90◦ +Randomized +180 s +Section 3.3 +T2 +3D TSE +1.3 mm3 +250×250×183.26 mm3 +2000 ms +318 ms +90◦ +Regular (acc. 2×2) +300 s +T1∗ +3D TFE +1 mm3 +250×250×183 mm3 +7.7 ms +3.6 ms +8◦ +Randomized +150 s +Section 3.3 +T2-FLAIR +3D TSE +1.2 mm3 +230×230×238 mm3 +4800 ms +291 ms +90◦ +Randomized +350 s +T1∗ +3D TFE +1 mm3 +230×230×238 mm3 +7.5 ms +3.4 ms +8◦ +Randomized +180 s +Table 1: Specification of the acquisition sequences utilized in the experiments +in Section 3. We use a 1.5 T Philips Ingenia scanner with a 15-channel head +coil. For each experiment, the asterisk indicates the reference contrast. The +“randomized” sampling pattern indicated in this table more specifically refers +to variable density Cartesian randomized undersampling, while the “regular” +pattern refers to classical accelerated linear filling undersampling. +to actively move during the scan (a certain number of times). While we did +not track the type of rigid motion produced by the volunteers, we prompted +them to maintain the same position in between our instructions. In this way, +we have some fair qualitative expectations about the motion estimated by the +correction algorithm (that is, a stepwise behavior, see also Appendix C for the +estimated motion unknowns associated to each of the experiments described in +this section). The ‘ground-truth’ acquisition and reconstruction is obtained by +simply asking the volunteers not to move. +The volunteer studies aim at investigating several relevant questions related +to the application of the proposed retrospective motion correction technique. +The first study in Section 3.1 is a qualitative assessment of the robustness of +the motion correction with respect to motion complexity, here equated to the +number of volunteer poses during the scan. +In Section 3.2, we demonstrate +that many combinations of corrupted-contrast and reference-contrast types are +possible for adequate correction. In the experiment in Section 3.3, we ascertain +whether using the scanner reconstruction in the DICOM format, as opposed +to the raw k-space data, is suited as input data for the algorithm after apply- +ing the Fourier transform (e.g., d in equation 6). We note that the proposed +method assumes coil-resolved data as input for computational reasons, therefore +it is sensitive to how the raw k-space data is post-processed, and, in particular, +to the degree of which the post-processed data can be adequately corrected by +rigid-motion estimation. Finally, further experimentation is deferred to the sup- +plemental section in Appendix B, where we demonstrate the effectiveness of the +reference-based motion correction against a “blind” motion correction method, +which does not use a reference contrast to eliminate the motion artifacts. +All the following investigations use a 1.5 T Philips Ingenia scanner with a +15-channel head coil. We considered several contrast acquisition sequences with +the specifications highlighted in Table 1. For all the experiments except the +one described in Section 3.3, the raw k-space data (pertaining to corrupted or +ground-truth scans) was exported for off-line processing. A pre-processing step +is dedicated to remove coil dependency, by performing a SENSE reconstruction +and then applying the Fourier transform. +7 + +3.1 +Experiment 1: robustness with respect to motion com- +plexity +In order to test the robustness of the proposed motion correction scheme in terms +of motion complexity, we instruct volunteer 1 to move multiple times during +acquisition. With “motion complexity” we specifically refer to the number of +position changes performed by the volunteer within one prospectively corrupted +scan. The goal of this in-vivo study is to provide a qualitative assessment of the +degradation of the reconstruction quality as a function of motion complexity. +We consider three levels of motion corruption: (i) the volunteer moves once, +(ii) the volunteer moves twice, and (iii) the volunteer moves five times. The +volunteer is instructed to change its head position every time it is prompted to +do so, and maintain that position in between instructions. We use T2-FLAIR- +weighted contrasts as corrupted scans, with T1-weighted contrast as a reference +(see Table 1 for further details). The corrupted acquisition employs randomized +sampling. +The results of this experiment are collected in Section 4.1. Note that, in +Appendix B, we use the same settings detailed in this experiment to compare +the proposed algorithm with a baseline method without a reference guide. +3.2 +Experiment 2: on the choice of the reference contrast +This in-vivo experiment tests the proposed correction scheme with respect to a +different combination of corrupted and reference contrast, namely a T1-weighted +corrupted contrast with a T2-weighted reference contrast (see Table 1). For this +experiment, we prompt volunteer 2 to move five times during the acquisition. +The corrupted acquisition employs randomized sampling. +In Section 4.2, we gather the results for this experiment. +3.3 +Experiment 3: +scanner reconstruction vs processed +raw k-space data as input for retrospective motion +correction +With the in-vivo studies presented in this section, we investigate a question +related to the nature of the input data d (equation 6) required by the algo- +rithm. Due to the formulation of the problem directly in k-space (by means +of the NUFFT), the method assumes coil-resolved data. One must then assess +whether the scanner reconstruction (available in the DICOM format) is suit- +able for this purpose, since many different reconstruction methods are available +depending on the acquisition protocol. In particular, the default reconstruc- +tion method for linear-filling patterns in k-space employs the SENSE frame- +work [Pruessmann et al., 1999], while compressed-sensing reconstruction (via +the wavelet transform) is used for randomized acquisitions [Lustig et al., 2007]. +Note that our experimentation suggests that without the phase map of the scan- +ner reconstruction our motion correction scheme does not perform adequately. +8 + +Therefore, with “scanner reconstruction”, we will always refer to the complex- +valued scanner reconstruction (comprising both the respective amplitude and +phase). +In the first experiment, we asked volunteer 3 to change position once during +the prospectively-corrupted acquisition. We consider a corrupted T2-weighted +contrast and a reference T1-weighted contrast (see Table 1). One important as- +pect of this experiment is related to the acquisition protocol of the T2-weighted +contrast, based on a linear-filling pattern in k-space. The corrupted data used +as input for the proposed motion-correction algorithm is obtained by export- +ing the reconstructed volume directly from the scanner, followed by a simple +Fourier transform. Note that this 3D image has been obtained by a SENSE +reconstruction. +The second experiment is set up similarly to the previous one. We asked +volunteer 3 to change position only once during the acquisition phase. +We +consider, now, a corrupted T2-FLAIR-weighted contrast with a reference T1- +weighted contrast (see Table 1). The most important difference with the previ- +ous experiment, besides the type of contrast pair considered, is related to the +randomized acquisition protocol. In this case, the scanner reconstruction em- +ploys a compressed-sensing reconstruction, and is not suited as input for the +proposed motion-correction algorithm (see Appendix A). Therefore, for ade- +quate motion correction, we must set up an intermediate step for processing the +raw k-space data via the SENSE reconstruction. +We further discuss the results of this experiment in Section 4.3. +4 +Results +In this section, we display and briefly analyze the results of the experiments +presented in the previous section. For ease of exposition, we organized the power +signal-to-noise ratio (PSNR) and structural similarity index (SSIM) values of +the reconstructions (with respect to a known ground truth) in Table 2. +The motion-corrected full-volume scans were analyzed by a neuroradiologist +with 16 years of experience. These were generally deemed of good radiological +quality. The motion-related artifacts have been completely removed, and the +results are quite close to the ground truth. In Table 3, we organized a more +detailed qualitative analysis of the 3D results, geared toward a radiological as- +sessment of the corrected scans. +4.1 +Experiment 1: robustness test +We gather the results for the robustness test described in Section 3.1 (volunteer +1) in Figures 1, 2, and 3 for motion corruption mechanisms associated to one, +two, and five changes of position, respectively. Furthermore, we juxtapose the +corrected images with varying degrees of corruption in Figure 4. We observe that +the proposed method consistently ameliorates the corrupted scan. The quality +9 + +Experiment +Slice orientation +PSNR (↑) +SSIM (↑) +Corrupted +Corrected +Corrupted +Corrected +Section 3.1, Figure 1 +Sagittal +23.94 +27.95 +0.7068 +0.7936 +Coronal +26.66 +29.82 +0.7653 +0.8332 +Axial +25.40 +30.16 +0.7616 +0.8490 +Section 3.1, Figure 2 +Sagittal +25.78 +27.76 +0.7263 +0.7816 +Coronal +28.19 +29.73 +0.7847 +0.8244 +Axial +27.79 +29.70 +0.8104 +0.8362 +Section 3.1, Figure 3 +Sagittal +22.45 +25.28 +0.6116 +0.7661 +Coronal +24.54 +27.40 +0.6734 +0.8060 +Axial +24.15 +27.66 +0.7086 +0.8298 +Section 3.2, Figure 5 +Sagittal +25.84 +28.07 +0.7032 +0.8093 +Coronal +26.35 +30.40 +0.7851 +0.9021 +Axial +28.11 +30.54 +0.8248 +0.9012 +Section 3.3, Figure 6 +Sagittal +22.26 +27.54 +0.6963 +0.8409 +Coronal +23.46 +31.65 +0.7321 +0.8370 +Axial +24.55 +32.33 +0.7895 +0.8144 +Section 3.3, Figure 7 +Sagittal +24.72 +28.76 +0.6762 +0.7818 +Coronal +25.95 +29.54 +0.7238 +0.8107 +Axial +25.08 +29.59 +0.7263 +0.8407 +Table 2: Summary of the motion-correction results shown in Section 4 in terms +of PSNR and SSIM +Experiment +Contrast +Motion resolution +Blurring +Artifacts +Additional comments +Section 3.1, Figure 1 +T2-FLAIR +Completely corrected +Some blurring +No additional artifacts +Good grey white matter differentiation +Section 3.1, Figure 2 +T2-FLAIR +Completely corrected +Some blurring +No additional artifacts +Good grey white matter differentiation +Section 3.1, Figure 3 +T2-FLAIR +Completely corrected +Some blurring +Darker areas within the white matter +Good grey white matter differentiation +Section 3.2, Figure 5 +T1 +Completely corrected +Some blurring +No additional artifacts +Good grey white matter differentiation, +some loss of grey matter low signal +Section 3.3, Figure 6 +T2 +Completely corrected +No blurring +No additional artifacts +Section 3.3, Figure 7 +T2-FLAIR +Completely corrected +Some blurring +No additional artifacts +Good grey white matter differentiation +Table 3: Qualitative radiological analysis of the motion-corrected results shown +in Section 4. The corrected scans are radiologically equivalent to the ground +truth. +indexes based on PSNR and SSIM show only a modest decrease in correction +quality as a function of motion complexity (Figure 4). +4.2 +Experiment 2: choice of the reference contrast +With the experiment described in Section 3.2, we demonstrate the flexibility +of the correction scheme with respect to the choice of the reference contrast. +The results are shown in Figure 5. Contrary to the experiments detailed in the +previous section, we are now considering a T2-weighted reference contrast to +guide the correction of a T1-weighted corrupted contrast. The quality of the +correction indicates that the proposed technique is rather flexible in terms of +reference contrast. +10 + +4.3 +Experiment 3: scanner reconstruction vs raw k-space +data +The results of the two experiments described in Section 3.3 are depicted in +Figures 6 and 7. The main difference between the two experiments is related to +the input data for the proposed motion-correction algorithm. +In the first experiment, the corrupted contrast has been acquired with a +protocol based on a linear filling pattern in k-space. Note that, in this particu- +lar case, the scanner reconstruction implements the SENSE method. We then +extracted the DICOM of both amplitude and phase produced by the scanner, +and used it as input data (after a Fourier transform) for the algorithm. The +proposed scheme is able to successfully remove the motion artifacts in Figure 6. +In the case of randomized sampling, the scanner reconstruction is not ad- +equate as input data for the proposed motion-correction algorithm, because it +employs a compressed-sensing algorithm. We speculate that compressed-sensing +reconstructions degrade the information contained in the corrupted volume, +and the corrected contrast cannot be effectively recovered by simply removing +rigid-motion artifacts (we defer the degraded results when using scanner re- +construction data in Appendix A). However, when the input data is obtained +by directly processing the raw k-space data via the SENSE reconstruction, the +motion-correction scheme is able to successfully remove the motion artifacts +(Figure 7). +5 +Discussion +Reference-guided TV regularization substantially improves the motion correc- +tion quality, both visually and in terms of quality metrics based on PSNR and +SSIM, when compared to basic Fourier reconstruction without motion correc- +tion. The comparison is also substantially favorable with standard “blind” mo- +tion correction techniques, for example based on conventional regularization +such as TV, which do not employ a reference to guide the correction (see Ap- +pendix B). In fact, for randomized sampling patterns that are now common in +the clinical practice, we verified that blind retrospective techniques are wholly +inadequate for motion correction of radiological quality (cf. the comparison in +Appendix B, Figure 9). +Our experimentation based on volunteer data aimed at assessing the ro- +bustness of the correction quality with respect to motion artifacts of increasing +complexity. In this study, we equated this complexity to the number of volun- +teer changes of pose during the acquisition phase. Clearly, this does not fully +describe the complexity of motion encountered in practice in the clinic, but it +only constitutes a preliminary step in that direction. Nevertheless, the results +described in Section 4.1 support the indication that the retrospective motion +correction of T2-FLAIR weighted images based on a T1 reference contrast is +quite robust in terms of reconstruction quality, with only minor degradations in +terms of contrast and resolution. +11 + +Furthermore, the flexibility of the proposed motion-correction method is +demonstrated with different combinations of motion-corrupted and reference +contrasts (Section 4.2). Our experience suggests that an important factor in +assessing the effectiveness of the reference contrast as a guide for motion cor- +rection lies in the similarity of the k-space distribution of the two contrasts. +Good reconstruction quality can be expected when the reference contrast has +similar or higher frequency content when compared to the corrupted contrast, +regardless of the type of contrast considered. +A significant part of our experimentation was devoted to assess whether the +scanner reconstruction (available as DICOM format) can be directly used as in- +put data for the proposed correction method (Section 4.3). We established that +the scanner reconstruction is not suitable for this purpose when it is obtained +via compressed-sensing algorithms (Appendix A), which is the case for random- +ized sampling on the 1.5 T Philips Ingenia scanner utilized in this work. In +this case, we must resort to the raw k-space data and perform an intermediate +SENSE reconstruction for effective motion correction. +The computational times of the motion correction are, generally speaking, +problem dependent, since complex motion artifacts require an increasing num- +ber of iterations as a function of motion complexity (Section 2.2). The examples +illustrated in this study, where a fixed number of iterations was consider irre- +spectively of motion complexity, are completed within 1 h 30 min for 3D images +of approximately 256×256×256 voxels. The current CPU implementation was +run on a consumer-grade laptop with the following processor specifications: Intel +Core i7-10750H CPU@2.60GHz×12. An effective implementation in a clinical +scenario for on-line reconstructions will likely require GPUs. +The basic assumption of the proposed retrospective correction method is +related to the availability of a motion-free contrast. While we believe that it is +a realistic possibility within an MR session, we note that the reference contrast +may come from previous MR sessions (or even different imaging modalities +altogether, such as CT). In this particular case, the bias introduced by the +structural prior may have an adverse effect in case of an evolving pathology. +However, when structural changes involve a limited pathological region, the +adverse bias can be easily mitigated by masking the affected zone. +Note that the motion-free reference can be exploited differently than the +reference-guided TV regularization introduced in Ehrhardt and Betcke [2016], +and adopted in this work. For example, one may consider several competing +techniques advanced for multi-contrast MRI, such as Bayesian compressed sens- +ing [Bilgic et al., 2011], group sparsity [Huang et al., 2014], reference-based MRI +[Weizman et al., 2015], or multi-contrast graph-based sparsity [Lai et al., 2017, +2018]. +The method here presented is limited to rigid motion. Indeed, some de- +crease in correction quality is noticeable in Figure 6 in the neck region (which +is not supposed to behave rigidly). However, our technique may be extended to +non-rigid motion and, hence, different body regions other than the brain [see, +for example, Huttinga et al., 2020]. A major challenge for such extension is a +computationally effective parameterization of the motion effects, and the result- +12 + +ing ill-posedness of the inverse problem. Note that a significant computational +advantage of rigid motion over non-rigid motion is related to the direct imple- +mentation of the rigid motion in k-space, via equation (3), which results in a +data model that requires a single NUFFT evaluation, regardless of the number +of time samples considered. Other interesting extensions of the method are re- +lated to the integration of specialized motion-resilient acquisition patterns, e.g. +as described in Cordero-Grande et al. [2020]. +6 +Conclusions +We assessed the performance of the proposed retrospective motion correction +method based on a reference contrast not affected by motion artifacts. The +current prospective in-vivo study targets 3D clinical protocols conventionally +used in brain imaging. +The method is tested with several degrees of motion artifacts, by instructing +the volunteers to change position during the scan multiple times. While, we +observe that the corrupted images are severely degraded as a function of motion +complexity, the corrected images are generally robustly estimated. +We also +verified that the proposed technique is agnostic with respect to the choice of the +reference contrast, as long as the frequency content of the reference and target +contrasts is comparable. +Further assessment of the proposed method will be devoted to patient data. +Data +The 3D results of the experiment described in Sections 3, 4 are freely available +online in the DICOM format at the following link: +github.com/grizzuti/ReferenceGuidedMotionCorrection Supplementary DICOM. +Acknowledgments +This publication is part of the project “Reducing re-scans in clinical MRI ex- +ams” (with project numbers 0104022007, 01040222210001 of the research pro- +gram “IMDI, Technologie voor bemensbare zorg: Doorbraakprojecten”) which +is financed by The Netherlands Organization for Health Research and Devel- +opment (ZonMW). The project is also supported by Philips Medical Systems +Netherlands BV. +13 + +Sagittal +Corrupted +PSNR: 22.93 +SSIM: 0.6655 +Corrected +PSNR: 27.95 +SSIM: 0.7936 +Ground truth +Reference +Coronal +PSNR: 25.91 +SSIM: 0.7461 +PSNR: 29.82 +SSIM: 0.8332 +Axial +PSNR: 24.46 +SSIM: 0.7371 +PSNR: 30.16 +SSIM: 0.8490 +Axial detail +Figure 1: Reconstruction results for volunteer 1. The volunteer is instructed +to move once during the scan. The corrupted contrast is T2-FLAIR-weighted, +while the reference contrast is T1-weighted. Compare these results with the one +obtained with different motion complexity in Figures 2, 3. +14 + +SSagittal +Corrupted +PSNR: 21.82 +SSIM: 0.6071 +Corrected +PSNR: 27.76 +SSIM: 0.7816 +Ground truth +Reference +Coronal +PSNR: 24.08 +SSIM: 0.6751 +PSNR: 29.73 +SSIM: 0.8244 +Axial +PSNR: 22.72 +SSIM: 0.6440 +PSNR: 29.70 +SSIM: 0.8362 +Axial detail +Figure 2: Reconstruction results for volunteer 1. The volunteer is instructed +to move twice during the scan. The corrupted contrast is T2-FLAIR-weighted, +while the reference contrast is T1-weighted. Compare these results with the one +obtained with different motion complexity in Figures 1, 3. +15 + +SSagittal +Corrupted +PSNR: 20.65 +SSIM: 0.5141 +Corrected +PSNR: 25.28 +SSIM: 0.7661 +Ground truth +Reference +Coronal +PSNR: 22.77 +SSIM: 0.5998 +PSNR: 27.40 +SSIM: 0.8060 +Axial +PSNR: 21.23 +SSIM: 0.5727 +PSNR: 27.66 +SSIM: 0.8298 +Axial detail +Figure 3: Reconstruction results for volunteer 1. The volunteer is instructed to +move five times during the scan. The corrupted contrast is T2-FLAIR-weighted, +while the reference contrast is T1-weighted. Compare these results with the one +obtained with different motion complexity in Figures 1, 2. +16 + +SMove once +Corrupted +PSNR: 24.46 +SSIM: 0.7371 +Corrected +PSNR: 30.16 +SSIM: 0.8490 +Ground truth +Reference +Move twice +PSNR: 22.72 +SSIM: 0.6440 +PSNR: 29.70 +SSIM: 0.8362 +Move five times +PSNR: 21.23 +SSIM: 0.5727 +PSNR: 27.66 +SSIM: 0.8298 +Figure 4: Summary of the reconstruction results for volunteer 1 (see Figures 1, +2, 3). The volunteer is instructed to move a variable number of times during +the scan in order to test the robustness of the proposed correction scheme with +respect the motion complexity. The corrupted images are increasingly affected +by motion artifacts, however only modest decrease in reconstruction quality can +be observed for the corrected images (here axial slices). +17 + +Sagittal +Corrupted +PSNR: 25.84 +SSIM: 0.7032 +Corrected +PSNR: 28.07 +SSIM: 0.8093 +Ground truth +Reference +Coronal +PSNR: 26.35 +SSIM: 0.7851 +PSNR: 30.40 +SSIM: 0.9021 +Axial +PSNR: 28.11 +SSIM: 0.8248 +PSNR: 30.55 +SSIM: 0.9012 +Coronal detail +Figure 5: Reconstruction results for volunteer 2. The volunteer is instructed +to move five times during the scan. The corrupted contrast is T1-weighted, +while the reference contrast is T2-weighted. The proposed correction scheme +is agnostic about the choice of corrupted/reference contrast combinations with +similar spectral content. +18 + +Sagittal +Corrupted +PSNR: 22.26 +SSIM: 0.6963 +Corrected +PSNR: 27.54 +SSIM: 0.8409 +Ground truth +Reference +Coronal +PSNR: 23.46 +SSIM: 0.7321 +PSNR: 31.65 +SSIM: 0.8370 +Axial +PSNR: 24.55 +SSIM: 0.7895 +PSNR: 32.33 +SSIM: 0.8144 +Axial detail +Figure 6: Reconstruction results for volunteer 3. The volunteer is instructed +to move once, halfway through the scan (the two overlapping positions are +clearly visible in the corrupted slices). The corrupted contrast is T2-weighted, +while the reference contrast is T1-weighted. In this case, the input data for +the correction algorithm is directly extracted from the scanner reconstruction in +DICOM format (comprising both amplitude and phase). The acquisition scheme +for the T2-weighted contrast follows a linear filling pattern in k-space. +The +proposed method successfully removes the motion artifacts because the scanner +reconstruction is obtained through a conventional SENSE reconstruction (cf. +Figure 7). +19 + +Sagittal +Corrupted +PSNR: 24.72 +SSIM: 0.6762 +Corrected +PSNR: 28.76 +SSIM: 0.7818 +Ground truth +Reference +Coronal +PSNR: 25.95 +SSIM: 0.7238 +PSNR: 29.54 +SSIM: 0.8107 +Axial +PSNR: 25.08 +SSIM: 0.7263 +PSNR: 29.59 +SSIM: 0.8407 +Axial detail +Figure 7: Reconstruction results for volunteer 3. The volunteer is instructed to +move once, halfway through the scan (the two overlapping positions are clearly +visible in the corrupted slices). The corrupted contrast is T2-FLAIR-weighted, +while the reference contrast is T1-weighted. Unlike Figure 6, the input data +for the correction algorithm is not extracted from the scanner, but is obtained +via SENSE reconstruction of the raw k-space data. The acquisition scheme for +the T2-FLAIR-weighted contrast is randomized in k-space. When the scanner +reconstruction is directly used as input data, the motion correction is highly +suboptimal (cf. Figure 8 in Appendix A). +20 + +CeTTYUReferences +Jalal B. Andre, Brian W. Bresnahan, Mahmud Mossa-Basha, Michael N. Hoff, +C. Patrick Smith, Yoshimi Anzai, and Wendy A. Cohen. Toward quantify- +ing the prevalence, severity, and cost associated with patient motion during +clinical MR examinations. Journal of the American College of Radiology, 12 +(7):689–695, jul 2015. ISSN 1558349X. doi: 10.1016/j.jacr.2015.03.007. URL +https://pubmed.ncbi.nlm.nih.gov/25963225/. +David Atkinson, Derek L.G. Hill, Peter N.R. Stoyle, Paul E. Summers, and +Stephen F. Keevil. +Automatic correction of motion artifacts in magnetic +resonance images using an entropy focus criterion. IEEE Transactions on +Medical Imaging, 16(6):903–910, 1997. doi: 10.1109/42.650886. +David Atkinson, Derek L.G. Hill, Peter N.R. Stoyle, Paul E. Summers, Stuart +Clare, Richard Bowtell, and Stephen F. Keevil. Automatic compensation of +motion artifacts in MRI. Magnetic Resonance in Medicine, 41(1):163–170, +1999. ISSN 07403194. doi: 10.1002/mrm.1910310505. +Alex H. Barnett. +Aliasing error of the exp(β1-z2) kernel in the nonuniform +fast Fourier transform. Applied and Computational Harmonic Analysis, 51 +(2):1–16, 2021. ISSN 1096603X. doi: 10.1016/j.acha.2020.10.002. +Alexander H. Barnett, Jeremy Magland, and Ludvig A.F. Klinteberg. A par- +allel nonuniform fast Fourier transform library based on an “exponential of +semicircle” kernel. SIAM Journal on Scientific Computing, 41(5):C479–C504, +2019. ISSN 10957197. doi: 10.1137/18M120885X. +Berkin Bilgic, Vivek K Goyal, and Elfar Adalsteinsson. Multi-contrast recon- +struction with bayesian compressed sensing. Magnetic resonance in medicine, +66(6):1601–1615, 2011. +J´erˆome Bolte, Shoham Sabach, and Marc Teboulle. Proximal alternating lin- +earized minimization for nonconvex and nonsmooth problems. Mathematical +Programming, 146(1):459–494, 2014. +Candice A Bookwalter, Mark A Griswold, and Duerk Jeffrey L. +Multiple +Overlapping k-Space Junctions for Investigating Translating Objects (MO- +JITO). IEEE Transactions on Medical Imaging, 29(2):339–349, 2010. doi: +10.1109/TMI.2009.2029854.Multiple. +Leon Bungert and Matthias J Ehrhardt. Robust Image Reconstruction With +Misaligned Structural Information. IEEE Access, 8:222944–222955, 2020. +Lucilio Cordero-Grande, Giulio Ferrazzi, Rui Pedro A.G. Teixeira, Jonathan +O’Muircheartaigh, Anthony N. Price, and Joseph V. Hajnal. Motion-corrected +MRI with DISORDER: Distributed and incoherent sample orders for re- +construction deblurring using encoding redundancy. Magnetic Resonance in +Medicine, 84(2):713–726, 2020. ISSN 15222594. doi: 10.1002/mrm.28157. +21 + +R. L. Ehman and J. P. Felmlee. Adaptive technique for high-definition MR imag- +ing of moving structures. Radiology, 173(1):255–263, 1989a. ISSN 00338419. +doi: 10.1148/radiology.173.1.2781017. +Richard L Ehman and Joel P Felmlee. Adaptive technique for high-definition +MR imaging of moving structures. Radiology, 173(1):255–263, 1989b. +Matthias J. Ehrhardt and Marta M. Betcke. Multicontrast MRI reconstruction +with structure-guided total variation. SIAM Journal on Imaging Sciences, 9 +(3):1084–1106, 2016. ISSN 19364954. doi: 10.1137/15M1047325. +Matthias J. Ehrhardt, Kris Thielemans, Luis Pizarro, David Atkinson, S´ebastien +Ourselin, Brian F. Hutton, and Simon R. Arridge. Joint reconstruction of +PET-MRI by exploiting structural similarity. Inverse Problems, 31(1), 2015. +doi: 10.1088/0266-5611/31/1/015001. +Christoph Forman, Murat Aksoy, Joachim Hornegger, and Roland Bammer. +Self-encoded marker for optical prospective head motion correction in mri. +Medical image analysis, 15(5):708–719, 2011. +Mina Ghaffari, Kamlesh Pawar, and Ruth Oliver. +Brain MRI motion arti- +fact reduction using 3D conditional generative adversarial networks on simu- +lated motion. In 2021 Digital Image Computing: Techniques and Applications +(DICTA), pages 1–7, 2021. doi: 10.1109/DICTA52665.2021.9647370. +Frank Godenschweger, Urte K¨agebein, Daniel Stucht, Uten Yarach, Alessandro +Sciarra, Renat Yakupov, Falk L¨usebrink, Peter Schulze, and Oliver Speck. +Motion correction in MRI of the brain. Physics in Medicine & Biology, 61 +(5):R32–R56, 2016. +Melissa W. Haskell, Stephen F. Cauley, Berkin Bilgic, Julian Hossbach, +Daniel N. Splitthoff, Josef Pfeuffer, Kawin Setsompop, and Lawrence L. +Wald. Network Accelerated Motion Estimation and Reduction (NAMER): +Convolutional neural network guided retrospective motion correction us- +ing a separable motion model. +Magnetic Resonance in Medicine, 82(4): +1452–1461, 2019. +doi: https://doi.org/10.1002/mrm.27771. +URL https: +//onlinelibrary.wiley.com/doi/abs/10.1002/mrm.27771. +Julian Hossbach, Daniel Nicolas Splitthoff, Stephen Cauley, Bryan Clifford, +Daniel Polak, Wei-Ching Lo, Heiko Meyer, and Andreas Maier. Deep learning- +based motion quantification from k-space for fast model-based mri motion cor- +rection. Medical Physics, 2022. doi: https://doi.org/10.1002/mp.16119. URL +https://aapm.onlinelibrary.wiley.com/doi/abs/10.1002/mp.16119. +Junzhou Huang, Chen Chen, and Leon Axel. Fast multi-contrast mri recon- +struction. Magnetic resonance imaging, 32(10):1344–1352, 2014. +Niek RF Huttinga, Cornelis AT Van den Berg, Peter R Luijten, and Alessandro +Sbrizzi. Mr-motus: model-based non-rigid motion estimation for mr-guided +22 + +radiotherapy using a reference image and minimal k-space data. Physics in +Medicine & Biology, 65(1):015004, 2020. +Hope W Korin, Joel P Felmlee, Stephen J Riederer, and Richard L Ehman. +Spatial-frequency-tuned markers and adaptive correction for rotational mo- +tion. Magnetic resonance in medicine, 33(5):663–669, 1995. +Thomas K¨ustner, Karim Armanious, Jiahuan Yang, Bin Yang, Fritz Schick, +and Sergios Gatidis. Retrospective correction of motion-affected MR images +using deep learning frameworks. +Magnetic Resonance in Medicine, 82(4): +1527–1540, 2019. +doi: https://doi.org/10.1002/mrm.27783. +URL https: +//onlinelibrary.wiley.com/doi/abs/10.1002/mrm.27783. +Zongying Lai, Xiaobo Qu, Hengfa Lu, Xi Peng, Di Guo, Yu Yang, Gang Guo, +and Zhong Chen. Sparse mri reconstruction using multi-contrast image guided +graph representation. Magnetic resonance imaging, 43:95–104, 2017. +Zongying Lai, Xinlin Zhang, Di Guo, Xiaofeng Du, Yonggui Yang, Gang Guo, +Zhong Chen, and Xiaobo Qu. Joint sparse reconstruction of multi-contrast +mri images with graph based redundant wavelet transform. BMC medical +imaging, 18(1):1–16, 2018. +Jongyeon Lee, Byungjai Kim, and HyunWook Park. MC2-Net: motion correc- +tion network for multi-contrast brain MRI. Magnetic Resonance in Medicine, +86(2):1077–1092, 2021. +doi: +https://doi.org/10.1002/mrm.28719. +URL +https://onlinelibrary.wiley.com/doi/abs/10.1002/mrm.28719. +Seul Lee, Soozy Jung, Kyu-Jin Jung, and Dong-Hyun Kim. Deep Learning in +MR Motion Correction: a Brief Review and a New Motion Simulation Tool +(view2Dmotion). Investigative Magnetic Resonance Imaging, 24(4):196–206, +2020. +Wei Lin and Hee Kwon Song. Improved optimization strategies for autofocusing +motion compensation in MRI via the analysis of image metric maps. Magnetic +Resonance Imaging, 24(6):751–760, 2006. ISSN 0730725X. doi: 10.1016/j.mri. +2006.02.003. +Alexander Loktyushin, +Hannes Nickisch, +Rolf Pohmann, +and Bernhard +Sch¨olkopf. +Blind retrospective motion correction of MR images. +Mag- +netic Resonance in Medicine, 70(6):1608–1618, 2013. ISSN 15222594. doi: +10.1002/mrm.24615. +Michael Lustig, David Donoho, and John M. Pauly. Sparse MRI: The appli- +cation of compressed sensing for rapid MR imaging. Magnetic Resonance in +Medicine, 58(6):1182–1195, 2007. ISSN 07403194. doi: 10.1002/mrm.21391. +Julian Maclaren, Michael Herbst, Oliver Speck, and Maxim Zaitsev. Prospec- +tive motion correction in brain imaging: a review. Magnetic resonance in +medicine, 69(3):621–636, 2013. +23 + +Armando Manduca, Kiaran P. McGee, E. Brian Welch, Joel P. Felmlee, Roger C. +Grimm, and Richard L. Ehman. Autocorrection in MR Imaging: Adaptive +Motion Correction without Navigator Echoes. Radiology, 215(3):904–909, jun +2000. doi: 10.1148/radiology.215.3.r00jn19904. +Jason Mendes, Eugene Kholmovski, and Dennis L Parker. Rigid-Body Motion +Correction with Self-Navigation MRI. Magnetic Resonance in Medicine, 61 +(3):739–747, 2009. doi: 10.1002/mrm.21883.Rigid-Body. +Anita M¨oller, Marco Maaß, and Alfred Mertins. Blind sparse motion mri with +linear subpixel interpolation. In Bildverarbeitung f¨ur die Medizin 2015, pages +510–515. Springer, 2015. +Kamlesh Pawar, Zhaolin Chen, N Jon Shah, and Gary F Egan. Motion Correc- +tion in MRI using Deep Convolutional Neural Network. In Proceedings of the +27th Annual Meeting ISMRM, page 1174, 2018. +Bas Peters, Brendan R. Smithyman, and Felix J. Herrmann. Projection meth- +ods and applications for seismic nonlinear inverse problems with multiple con- +straints. Geophysics, 84(2):R251–R269, 02 2018. doi: 10.1190/geo2018-0192. +1. +James G Pipe. Motion correction with PROPELLER MRI: application to head +motion and free-breathing cardiac imaging. Magnetic Resonance in Medicine, +42(5):963–969, 1999. +Klaas P. Pruessmann, Markus Weiger, Markus B. Scheidegger, and Pe- +ter +Boesiger. +SENSE: +Sensitivity +encoding +for +fast +MRI. +Mag- +netic Resonance in Medicine, +42(5):952–962, +nov 1999. +ISSN 0740- +3194. +doi: +10.1002/(SICI)1522-2594(199911)42:5⟨952::AID-MRM16⟩3.0. +CO;2-S. +URL https://onlinelibrary.wiley.com/doi/10.1002/(SICI) +1522-2594(199911)42:5%3C952::AID-MRM16%3E3.0.CO;2-S. +Gabrio Rizzuti, Alessandro Sbrizzi, and Tristan van Leeuwen. Joint retrospec- +tive motion correction and reconstruction for brain mri with a reference con- +trast. IEEE Transactions on Computational Imaging, 8:490–504, 2022. ISSN +2333-9403. doi: 10.1109/TCI.2022.3183383. +Ghislain Vaillant, Claudia Prieto, Christoph Kolbitsch, Graeme Penney, and +Tobias Schaeffter. Retrospective rigid motion correction in k-space for seg- +mented radial MRI. IEEE Transactions on Medical Imaging, 33(1):1–10, 2014. +ISSN 02780062. doi: 10.1109/TMI.2013.2268898. +Lior Weizman, Yonina C Eldar, Alon Eilam, S Londner, Moran Artzi, and D Ben +Bashat. Fast reference based mri. In 2015 37th Annual International Con- +ference of the IEEE Engineering in Medicine and Biology Society (EMBC), +pages 7486–7489. IEEE, 2015. +24 + +Edward Brian Welch, Armando Manduca, Roger C Grimm, Heidi A Ward, and +Clifford R Jack Jr. Spherical navigator echoes for full 3D rigid body motion +measurement in MRI. Magnetic Resonance in Medicine, 47(1):32–41, 2002. +Edward Brian Welch, Phillip J Rossman, Joel P Felmlee, and Armando Man- +duca. Self-navigated motion correction using moments of spatial projections +in radial MRI. Magnetic Resonance in Medicine, 52(2):337–345, 2004. +Maxim Zaitsev, Christian Dold, Georgios Sakas, J¨urgen Hennig, and Oliver +Speck. +Magnetic resonance imaging of freely moving objects: prospective +real-time motion correction using an external optical motion tracking system. +Neuroimage, 31(3):1038–1050, 2006. +Maxim Zaitsev, Julian Maclaren, and Michael Herbst. +Motion artifacts in +MRI: A complex problem with many partial solutions. +Journal of Mag- +netic Resonance Imaging, 42(4):887–901, oct 2015. +ISSN 10531807. +doi: +10.1002/jmri.24850. URL http://doi.wiley.com/10.1002/jmri.24850. +25 + +A +Inadequate motion correction with scanner +reconstruction as input data +As anticipated in Section 3.3, directly using the scanner reconstruction (ex- +tracted as DICOM files of both the amplitude and phase of the reconstruction) +as input data for the proposed motion correction scheme may degrade the per- +formance when compressed-sensing reconstruction tools have been employed in +the reconstruction process. To motivate this conclusion, we setup an experiment +with the same setting as described in the second experiment in Section 3.3, the +only difference being in how the input data is generated. In this case, the input +data consist of the Fourier transform of the extracted scanner reconstruction. +The related suboptimal correction is quite evident when comparing Figure 8 +with Figure 7. +Axial +Corrupted +Corrected +Ground truth +Reference +Figure 8: Reconstruction results for volunteer 3. The volunteer is instructed to +move once, halfway through the scan. The corrupted contrast is T2-FLAIR- +weighted, while the reference contrast is T1-weighted. +In this experiment, +the proposed motion correction scheme processes the scanner reconstruction +directly. +Since the reconstruction algorithm implemented in the scanner de- +stroys the coherence of the rigid motion artifact, the proposed method cannot +properly recover the correct reconstruction by simply estimating the motion +parameters. With contrasts obtained by randomized acquisitions, we advise to +use raw k-space data instead (cf. Figure 7). +B +Comparison of motion correction with and +without a reference guide +The reference-guided motion-correction algorithm described in Section 2 is com- +pared with a standard retrospective motion-correction algorithm based on the +TV regularization. +We note that most retrospective motion-correction methods follows the basic +mathematical framework detailed in Section 2 (see, for example, Loktyushin +et al. [2013] or Cordero-Grande et al. [2020]), where the main mathematical +difference consists in the choice of the regularization term gu, in equation (5). +Hence, in order to assess the effect of the reference contrast, we adopt the +26 + +same formulation described in Section 2 with a simple TV regularization term +gu(u) = � +x ∥∇u|x∥ (cf. equation 8 for the reference-guided version of TV). +For the comparison with the baseline method, we use the same experimen- +tal settings in Section 3.1. Once again, the motion artifacts are prospectively +induced by prompting the volunteer to move during the scan. The results are +summarized in Figure 9 and Table 4. +Move once +Corrupted +PSNR: 25.40 +SSIM: 0.7616 +Corrected (ours) +PSNR: 30.16 +SSIM: 0.8490 +Corrected (baseline) +PSNR: 27.54 +SSIM: 0.8007 +Ground truth +Move twice +PSNR: 27.79 +SSIM: 0.8104 +PSNR: 29.70 +SSIM: 0.8362 +PSNR: 26.57 +SSIM: 0.7708 +Move five times +PSNR: 24.15 +SSIM: 0.7086 +PSNR: 27.66 +SSIM: 0.8298 +PSNR: 24.96 +SSIM: 0.7562 +Figure 9: +Comparison of the reconstruction results for volunteer 1 with a +reference-guided (ours) and a baseline motion-correction method (e.g., not +guided by a reference contrast). The volunteer is instructed to move a variable +number of times during the scan in order to test the robustness of the proposed +correction schemes with respect the motion complexity. The corrupted images +are increasingly affected by motion artifacts. The decrease in reconstruction +quality for the baseline method is substantially more pronounced than the re- +sults obtained with our reference-guided correction (see also Figures 1–3). +We note that the difference in performance between the reference-guided +and blind motion correction is even more pronounced in this example than what +was previously shown in Rizzuti et al. [2022] (which was limited to 2D synthetic +data). This might depend on the fact that the problem is substantially more +ill-posed in 3D with randomized sampling than the 2D full-acquisition setup +considered in Rizzuti et al. [2022]. It is also worth noting that, in our experience, +the results for blind motion correction depend more sensibly on the choice of the +27 + +Experiment +Slice orientation +PSNR (↑) +SSIM (↑) +Corrupted +Corrected +Corrupted +Corrected +Ours +Baseline +Ours +Baseline +Move once +Sagittal +23.94 +27.95 +25.27 +0.7068 +0.7936 +0.7529 +Coronal +26.66 +29.82 +27.61 +0.7653 +0.8332 +0.7818 +Axial +25.40 +30.16 +27.54 +0.7616 +0.8490 +0.8007 +Move twice +Sagittal +25.78 +27.76 +24.13 +0.7263 +0.7816 +0.6925 +Coronal +28.19 +29.73 +26.68 +0.7847 +0.8244 +0.7448 +Axial +27.79 +29.70 +26.57 +0.8104 +0.8362 +0.7708 +Move five times +Sagittal +22.45 +25.28 +22.10 +0.6116 +0.7661 +0.6719 +Coronal +24.54 +27.40 +24.72 +0.6734 +0.8060 +0.7327 +Axial +24.15 +27.66 +24.96 +0.7086 +0.8298 +0.7562 +Table 4: +Comparison of the motion-correction results for the baseline and +reference-guided methods in terms of PSNR and SSIM. The experiment setup +is described in Section 3.1. +hyper-parameters in equation (5) than the proposed reference-based version. +C +Motion parameter estimation +The proposed motion correction algorithm described in Section 2 estimates the +rigid motion that the object of interest undergoes during the scan, in order +to undo its effect on the reconstructed 3D image. In 3D, the rigid motion is +performed by: a plane rotation θxy in the corresponding plane xy, a plane +rotation θxz in the xz plane, a plane rotation θyz in the yz plane, a translation +τx in the x direction, a translation τy in the y direction, and a translation +τz in the z direction (in this order). We adopt the following convention: the +x direction corresponds to the left-right direction, y to the posterior-anterior +direction, and z to the inferior-superior direction, the xy plane corresponds to +the axial plane, xz to the coronal plane, and yz to the sagittal plane. Left/right, +anterior/posterior, and inferior/superior are meant from the patient perspective. +The orientation of the rotation planes is determined by the right-hand rule. +By design, the prospectively-induced motion for all the experiments detailed +in Section 3 follows a step-wise behavior (each step corresponding to a change +of pose). In this appendix, we gather the estimated rigid motion parameters for +the results shown in Section 4, as a function of time. As noted in the main body +of the paper, time is equated to the phase-encoding plane coordinate index, +ordered by the corresponding acquisition ordering. We display the estimated +motion parameters in Figure 10 (see Sections 3.1, 4.1, Figure 1), Figure 11 (see +Sections 3.1, 4.1, Figure 2), Figure 12 (see Sections 3.1, 4.1, Figure 3), Figure +13 (see Sections 3.2, 4.2, Figure 5), Figure 14 (see Sections 3.3, 4.3, Figure 6), +and Figure 15 (see Sections 3.3, 4.3, Figure 7). +28 + +−2.5 +0.0 +τ +x + (mm) +Estimated +0 +2 +τ +y + (mm) +0.0 +2.5 +τ +z + (mm) +0 +5 +θ +xy + ( +∘ +) +−7.5 +−5.0 +−2.5 +θ +xz + ( +∘ +) +0 +1000 +2000 +3000 +4000 +5000 +6000 +t = phase encoding index +0 +5 +θ +yz + ( +∘ +) +Figure 10: Estimated rigid motion parameters for the experiment described in +Sections 3.1, 4.1, with motion-correction results in Figure 1. The volunteer was +asked to move once during the scan. +−2.5 +0.0 +2.5 +τ +x + (mm) +Estimated +2.5 +5.0 +7.5 +τ +y + (mm) +0 +5 +τ +z + (mm) +−5 +0 +5 +θ +xy + ( +∘ +) +−10 +0 +θ +xz + ( +∘ +) +0 +1000 +2000 +3000 +4000 +5000 +6000 +t = phase encoding index +0 +10 +θ +yz + ( +∘ +) +Figure 11: Estimated rigid motion parameters for the experiment described in +Sections 3.1, 4.1, with motion-correction results in Figure 2. The volunteer was +asked to move twice during the scan. +29 + +−5 +0 +5 +τ +x + (mm) +Estimated +0 +10 +τ +y + (mm) +0 +10 +τ +z + (mm) +−10 +0 +θ +xy + ( +∘ +) +−10 +0 +10 +θ +xz + ( +∘ +) +0 +1000 +2000 +3000 +4000 +5000 +6000 +t = phase encoding index +−10 +0 +10 +θ +yz + ( +∘ +) +Figure 12: Estimated rigid motion parameters for the experiment described in +Sections 3.1, 4.1, with motion-correction results in Figure 3. The volunteer was +asked to move five times during the scan. +7.5 +10.0 +τ +x + (mm) +Estimated +−1 +0 +1 +τ +y + (mm) +17.5 +20.0 +τ +z + (mm) +−1 +0 +1 +θ +xy + ( +∘ +) +−1 +0 +1 +θ +xz + ( +∘ +) +0 +2500 +5000 +7500 +10000 +12500 +15000 +17500 +20000 +t = phase encoding index +−3 +−2 +−1 +θ +yz + ( +∘ +) +Figure 13: Estimated rigid motion parameters for the experiment described in +Sections 3.2, 4.2, with motion-correction results in Figure 5 +30 + +−5 +0 +τ +x + (mm) +Estimated +−5 +0 +5 +τ +y + (mm) +−5 +0 +τ +z + (mm) +0 +5 +θ +xy + ( +∘ +) +−5 +0 +θ +xz + ( +∘ +) +0 +10000 +20000 +30000 +40000 +50000 +60000 +t = phase encoding index +−5 +0 +5 +θ +yz + ( +∘ +) +Figure 14: Estimated rigid motion parameters for the experiment described in +Sections 3.3, 4.3, with motion-correction results in Figure 6 +0 +5 +τ +x + (mm) +Estimated +−5 +0 +τ +y + (mm) +0 +5 +τ +z + (mm) +0 +10 +θ +xy + ( +∘ +) +0 +10 +θ +xz + ( +∘ +) +0 +2000 +4000 +6000 +8000 +10000 +12000 +t = phase encoding index +−10 +0 +θ +yz + ( +∘ +) +Figure 15: Estimated rigid motion parameters for the experiment described in +Sections 3.3, 4.3, with motion-correction results in Figure 7 +31 + diff --git a/ZtAzT4oBgHgl3EQfK_te/content/tmp_files/load_file.txt b/ZtAzT4oBgHgl3EQfK_te/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9e9a1f45d7b4c615070bc4f31ecf46a09b4974f2 --- /dev/null +++ b/ZtAzT4oBgHgl3EQfK_te/content/tmp_files/load_file.txt @@ -0,0 +1,1025 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf,len=1024 +page_content='Towards retrospective motion correction and reconstruction for clinical 3D brain MRI protocols with a reference contrast Gabrio Rizzuti1,2, Tim Schakel2, Niek R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Huttinga2, Jan Willem Dankbaar2, Tristan van Leeuwen1,3, and Alessandro Sbrizzi2 1Utrecht University, Utrecht, 3584 CS, The Netherlands 2Universitair Medisch Centrum Utrecht, Utrecht, 3584 CX, The Netherlands 3Centrum Wiskunde & Informatica, Amsterdam, 1098 XG, The Netherlands Abstract Motion artifacts often spoil the radiological interpretation of MR im- ages, and in the most severe cases the scan needs be repeated, with ad- ditional costs for the provider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' We discuss the application of a novel 3D retrospective rigid motion correction and reconstruction scheme for MRI, which leverages multiple scans contained in a MR session.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Typi- cally, in a multi-contrast MR session, motion does not equally affect all the scans, and some motion-free scans are generally available, so that we can exploit their anatomic similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' The uncorrupted scan is used as a reference in a generalized rigid-motion registration problem to remove the motion artifacts affecting the corrupted scans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' We discuss the potential of the proposed algorithm with a prospective in-vivo study and clinical 3D brain protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' This framework can be easily incorporated into the existing clinical practice with no disruption to the conventional workflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Keywords Image Reconstruction, Motion Correction, Multi-Contrast, Brain, Total varia- tion 1 Introduction Magnetic resonance imaging (MRI) is fundamentally prone to motion artifacts, since the data acquisition process usually lasts several minutes for each acquired contrast, and the MR exam can be an uncomfortable experience for the patient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Motion corruption impedes a correct radiological assessment, which then may require a scan repetition, leading to considerable waste of resources for the hospital [Andre et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=', 2015].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='01106v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='IV] 3 Jan 2023 Motion reduction strategies are broadly classified as preventive, prospective, or retrospective techniques [Zaitsev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=', 2015, Godenschweger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=', 2016].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Preventive strategies include physical devices to limit the motion (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' head holders) or sedation, but their application is limited by ethical or health consid- erations, and are often ineffective in eliminating patient movement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Prospective and retrospective strategies, on the other hand, directly or indirectly estimate the motion that the object of interest undergoes inside the scanner, and remove its effect from the data or in the reconstruction phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' This correction step is said to be applied “prospectively” [Maclaren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=', 2013] when the position of the patient is tracked in real time and the scan settings are adjusted accord- ingly on-the-fly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' For example, the relative change of position can be estimated by acquiring additional k-space or image-space navigators [Ehman and Felm- lee, 1989b, Welch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=', 2002], or with “self-navigating” sequences [Pipe, 1999, Welch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=', 2004, Bookwalter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=', 2010].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Alternatively, camera devices or markers [Zaitsev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=', 2006, Forman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=', 2011] can be used to estimate the imaging object position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' However, most tracking modalities are often defective in terms of either precision, patient interaction, or sequence independence [Ma- claren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=', 2013].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Therefore, although effective in many respects, prospective methods have somewhat limited range of application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Retrospective algorithms are characterized by the removal of motion artifacts in the final reconstruction phase, after the data acquisition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' The main advan- tage of retrospective schemes is in their flexibility, since they do not necessarily require additional hardware, scanner modifications, MR navigators, markers, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Note, however, that they may benefit from using prior information about the target imaging object and motion pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' One main challenge for this class of methods is the need for time-intensive computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' The scientific literature on retrospective motion correction is quite rich: examples of retro- spective techniques for rigid motion using navigators or markers can be found in Ehman and Felmlee [1989a], Korin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' [1995], Mendes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' [2009], Book- walter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' [2010], Vaillant et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' [2014], while examples of “blind” techniques (in this context, meaning that they are not using navigators or markers) are presented in Atkinson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' [1997, 1999], Manduca et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' [2000], Lin and Song [2006], Loktyushin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' [2013].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Retrospective correction schemes are typically formulated as a bi-level opti- mization problem, where two types of unknown are jointly estimated: the recon- structed (2D/3D) image and the motion parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Due to the ill-posedness of the problem here considered, the choice of the regularization method is crucial: see, for example, gradient-entropy regularization in Manduca et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' [2000], Lin and Song [2006], Loktyushin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' [2013], sparsity regularization in M¨oller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' [2015], or iteratively re-weighted least-squares regularization in Cordero-Grande et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' [2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Another strategy to ease the ill-posedness is to resort to special ac- quisition patterns in k-space that are more robust in terms of motion correction, as described in the DISORDER method in Cordero-Grande et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' [2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Al- ternatively, many machine-learning approaches have been recently proposed for retrospective motion correction [Pawar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=', 2018, K¨ustner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=', 2019, Haskell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=', 2019, Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=', 2020, Ghaffari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=', 2021, Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=', 2021, Hossbach 2 et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=', 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Some previous work in Rizzuti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' [2022] introduced a retrospective motion correction scheme, whose novel aspect is the use of a contrast free of motion artifacts that can be leveraged as a reference to remove motion effects from any other contrast from the same patient, akin to a generalized rigid motion registration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' The chief assumption of this work is the following: in a multi- contrast MR session, motion does not typically affect all the scans and some motion-free scans are generally available, so that we can exploit their anatomic similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Structural similarity is technically achieved via structure-guided total variation (TV), as originally proposed in Ehrhardt and Betcke [2016] and further developed in Bungert and Ehrhardt [2020] (see also Ehrhardt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' [2015]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' The goal of this paper is to extend the scope of Rizzuti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' [2022], limited to 2D synthetic results, to general 3D randomized acquisitions and 3D rigid-motion correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' We experimentally verify that a 3D extension is indeed feasible for brain imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' We do not assume data-driven priors (so that machine learning is not available), any additional navigator data, nor consider motion-resilient acquisition schemes, in order to conform to more broadly available clinical pro- tocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Note that the proposed method can employ any acquisition scheme, in principle, but we stick to Cartesian acquisition, which are the standard encoding strategies of clinical protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Since we focus on brain imaging, rigid motion can be effectively assumed for our scope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' The reference and the corrupted contrast do not need be co-registered or acquired with the same resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' We thoroughly validate the method with a prospective in-vivo study based on three volunteers and several motion types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' The strength and limitations of the method are highlighted with the comparison of correction quality with varying degrees of motion artifacts and contrast type as a reference prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' 2 Theory In this section, we present the basic mathematical formulation underpinning the proposed motion correction method (further details can be found in Rizzuti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' [2022]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' The contrast volume, in the remainder of this section, will be denoted by u ∈ Cnx, where nx is the number of voxels contained in a rectangular field of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' The 3D image undergoes a time-dependent rigid motion ut = Tθθθtu, (1) where t is a time-related label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' In practice, t corresponds to the index of the k- space readout line in the phase-encoding plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' The corresponding rigid trans- formation is given by Tθθθt, and is parameterized by a time-dependent motion parameter θθθt ∈ R6, which includes translations and rotations in 3D: θθθ = (τττ,ϕϕϕ), τττ = (τx, τy, τz), ϕϕϕ = (ϕxy, ϕxz, ϕyz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' (2) The rigid motion consists of a 3D rotation (defined by the 2D rotation angles 3 ϕxy, ϕxz, ϕyz, performed in this order in the corresponding planes) followed by a translation (governed by the translation parameters τx, τy, τz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Without loss of generality, we are assuming a Cartesian acquisition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' At each given time t, the MR acquisition process corresponds to the evaluation of the Fourier transform F of ut in a particular subset Kt of the k-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' In practice, the acquisition is structured in such a way that all the subsets Kt consist of parallel lines in the k-space (the common direction being the readout direction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' We refer to the Fourier transform of a rigidly moving object uθθθ := Tθθθu as the perturbed Fourier transform Fθθθu := Fuθθθ, and can be directly characterized as Fθθθu (k) := exp (−i k · τττ)Fu (R−1 ϕϕϕ k), (3) where the rotational operator with respect to the 3D angle ϕϕϕ is indicated by Rϕϕϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' This definition is motivated by classical Fourier identities that describe the action of rigid motion under the Fourier transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Due to rotational effects, one must resort to the non-uniform discrete Fourier transform (NUFFT) to evaluate equation (3) [Barnett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=', 2019, Barnett, 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Note that we implicitly assumed that no motion occurs while sampling the elements of Kt, since the state of the object at the time t is associated to a single motion parameter θθθt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' The assumption is motivated by the fact that Kt will correspond, in practice, to a single Cartesian readout line, which lasts few milliseconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Hence, the data acquisition at time t is symbolized by the application of the selection operator St to the Fourier-transformed volume: dt = StFθθθtu = (Fθθθtu (k1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' , Fθθθtu (knr)), k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' , knr ∈ Kt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' (4) Here, nr is the number of k-space samples in a single readout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' The resulting inverse problem can be cast as an optimization problem over the reconstruction unknowns u and the motion parameters θθθt, that is: min u,θθθ1:nt f(u,θθθ1:nt) + λgu(u) + µgθ(θθθ1:nt), (5) where θθθ1:nt = (θθθ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' ,θθθnt), and nt is the number of time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' The weighting parameters λ, µ (both positive numbers) set the strength of the corresponding regularization terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' The first term of the objective functional in equation (5) corresponds to the data misfit: f(u,θθθ1:nt) = nt � t=1 1 2 ∥Fθθθtu − dt∥2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' (6) The least-squares norm is indicated here by ∥·∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' The regularization terms gu and gθ are crucial in ensuring the well-posedness of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Indeed, the objec- tive in equation (6) will be sensitive to the relatively high signal-to-noise ratios (SNR) of the high-frequency components of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Moreover, the objective is highly non-convex as a function of θθθ1:nt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' The motion-parameter regularization is designed to ensure some form of regularity in time (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' smoothness), this 4 can be achieved for example by setting gθ(θθθ1:nt) = nt−1 � t=1 1 2 ∥θθθt+1 − θθθt∥2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' (7) Alternatively, higher-order derivatives may be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Another strategy, adopted in this paper, is to impose smoothness by setting hard constraints for the motion parameters, rather than via an additive penalty term as in equation (7) [Rizzuti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=', 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='1 Reference-guided total variation regularization The crux of the proposed method is related to the choice of the regularization term gu in equation (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' We adopt the structure-guided total variation scheme proposed in Ehrhardt and Betcke [2016] in the context of multi-contrast imaging, that is: gu(u) = � x ∥Πv|x∇u|x∥ , Πv|x = I3 − ξv|xξv|x H, (8) where I3 is the 3 × 3 identity matrix, ∇·|x is the discretized gradient operator evaluated at the voxel with center x, and Πv|x is the projection operator on the linear space that is orthogonal to the vector ξv|x ∈ C3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' The symbol H indicates the adjoint operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' The vector ξv|x corresponds to the normalized gradient of a given motion-free contrast v, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' ξv|x ≈ ∇v|x/ ∥∇v|x∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' The actual definition is ξv|x = ∇v|x � ∥∇v|x∥2 + η2 , (9) for some constant η > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' The regularization term in equation (8) enforces the gradient structure of v onto u, when v and u are anatomically compatible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' It is important to observe that v is not required to be registered with the target contrast u, since the estimation of the motion parameters in equation (5) will automatically compensate for the initial misalignment [see also Bungert and Ehrhardt, 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' In this work, we actually adopt a constrained formulation of equation (8), meaning that structural similarity is imposed by forcing the solution to belong to the constraint set Cu = {u : gu(u) ≤ ε}, where ε > 0 is a prescribed regularization level [see Rizzuti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=', 2022, Peters et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=', 2018, for more details].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='2 Optimization In order to solve equation (5), we adopt an alternating update scheme based on the proximal alternating minimization algorithm (PALM) described in Bolte et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' [2014].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' The algorithm of the optimization strategy is exemplified in Algo- rithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Each update requires the linearization of the smooth objective f and the application of the proximal operators associated to gu and gθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' As it is com- monly noted in the image registration literature, we will make use of multi-scale 5 methods to ease the ill-posedness of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Two types of scale are con- sidered, here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' One is traditionally associated to the reconstruction grid size, by considering a sequence of optimization problems defined on progressively finer grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Note that spatial coarsening of the reconstructed image u is intertwined with the temporal coarsening of the motion parameters θθθ1:nt, since they are associated to sample locations in the k-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' The other scale is related to the regularization strength λ, as defined in equation (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Hence, strongly weighted problems are solved first, and the regularization is gradually relaxed as in a continuation strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Overall, two nested sequences of optimization problems are considered here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Algorithm 1 Joint motion correction and reconstruction with alternating prox- imal operator evaluation Input: d, u(=0), θθθ(=0), αu, αθ, N ▷ Data, starting guesses, steplengths, iters Output: u, θθθ for scale = coarse,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=', fine do Downscaling of d, u, θθθ for λ = high,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=', low do for n = 1 : N do u ← proxαugu(u − αu∇uf(·,θθθ)) ▷ Reconstruction proxy θθθ ← proxαθgθ(θθθ − αθ∇θθθf(u, ·)) ▷ Motion parameter proxy end for end for Upscaling of u, θθθ end for 3 Experiments In this section, we set up several experiments that demonstrate the capabilities of the retrospective motion correction algorithm detailed in Section 2, whose main novel aspect and strength is the use of a reference contrast to guide the cor- rection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Our objective is to tackle motion correction for brain imaging, and we focus on acquisition protocols that are relevant for the clinical practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' All the imaging sequences considered in this study were taken from actual clinical brain protocols of the Radiology and Radiotherapy departments of the UMC Utrecht.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' The data considered in this section is based on 3D Cartesian acquisition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' The sampling pattern used in these acquisitions typically utilizes pseudo-random undersampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' The main assumptions underlying the proposed method are related to the availability of a motion-free reference contrast and the motion artifacts being produced by rigid motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' We consider several studies with volunteer data (three volunteers in total1), where motion artifacts are prospectively generated by instructing the volunteer 1We have informed written consent from the volunteers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' The experiments were approved by the ethical review board of the UMC Utrecht.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' 6 Experiment Contrast Sequence Resolution FOV TR TE Flip angle Phase-encoding pattern Duration Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='1 T2-FLAIR 3D TSE 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='2 mm3 230×230×237.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='6 mm3 4800 ms 320 ms 90◦ Randomized 350 s T1∗ 3D TFE 1 mm3 230×230×238 mm3 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='8 ms 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='6 ms 8◦ Randomized 180 s Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='2 T1 3D TFE 1 mm3 230×230×238 mm3 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='9 ms 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='6 ms 8◦ Randomized 180 s T2∗ 3D TSE 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='1 mm3 250×250×190.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='3 mm3 3000 ms 260 ms 90◦ Randomized 180 s Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='3 T2 3D TSE 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='3 mm3 250×250×183.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='26 mm3 2000 ms 318 ms 90◦ Regular (acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' 2×2) 300 s T1∗ 3D TFE 1 mm3 250×250×183 mm3 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='7 ms 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='6 ms 8◦ Randomized 150 s Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='3 T2-FLAIR 3D TSE 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='2 mm3 230×230×238 mm3 4800 ms 291 ms 90◦ Randomized 350 s T1∗ 3D TFE 1 mm3 230×230×238 mm3 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='5 ms 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='4 ms 8◦ Randomized 180 s Table 1: Specification of the acquisition sequences utilized in the experiments in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' We use a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='5 T Philips Ingenia scanner with a 15-channel head coil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' For each experiment, the asterisk indicates the reference contrast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' The “randomized” sampling pattern indicated in this table more specifically refers to variable density Cartesian randomized undersampling, while the “regular” pattern refers to classical accelerated linear filling undersampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' to actively move during the scan (a certain number of times).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' While we did not track the type of rigid motion produced by the volunteers, we prompted them to maintain the same position in between our instructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' In this way, we have some fair qualitative expectations about the motion estimated by the correction algorithm (that is, a stepwise behavior, see also Appendix C for the estimated motion unknowns associated to each of the experiments described in this section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' The ‘ground-truth’ acquisition and reconstruction is obtained by simply asking the volunteers not to move.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' The volunteer studies aim at investigating several relevant questions related to the application of the proposed retrospective motion correction technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' The first study in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='1 is a qualitative assessment of the robustness of the motion correction with respect to motion complexity, here equated to the number of volunteer poses during the scan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' In Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='2, we demonstrate that many combinations of corrupted-contrast and reference-contrast types are possible for adequate correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' In the experiment in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='3, we ascertain whether using the scanner reconstruction in the DICOM format, as opposed to the raw k-space data, is suited as input data for the algorithm after apply- ing the Fourier transform (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=', d in equation 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' We note that the proposed method assumes coil-resolved data as input for computational reasons, therefore it is sensitive to how the raw k-space data is post-processed, and, in particular, to the degree of which the post-processed data can be adequately corrected by rigid-motion estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Finally, further experimentation is deferred to the sup- plemental section in Appendix B, where we demonstrate the effectiveness of the reference-based motion correction against a “blind” motion correction method, which does not use a reference contrast to eliminate the motion artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' All the following investigations use a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='5 T Philips Ingenia scanner with a 15-channel head coil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' We considered several contrast acquisition sequences with the specifications highlighted in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' For all the experiments except the one described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='3, the raw k-space data (pertaining to corrupted or ground-truth scans) was exported for off-line processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' A pre-processing step is dedicated to remove coil dependency, by performing a SENSE reconstruction and then applying the Fourier transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' 7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='1 Experiment 1: robustness with respect to motion com- plexity In order to test the robustness of the proposed motion correction scheme in terms of motion complexity, we instruct volunteer 1 to move multiple times during acquisition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' With “motion complexity” we specifically refer to the number of position changes performed by the volunteer within one prospectively corrupted scan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' The goal of this in-vivo study is to provide a qualitative assessment of the degradation of the reconstruction quality as a function of motion complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' We consider three levels of motion corruption: (i) the volunteer moves once, (ii) the volunteer moves twice, and (iii) the volunteer moves five times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' The volunteer is instructed to change its head position every time it is prompted to do so, and maintain that position in between instructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' We use T2-FLAIR- weighted contrasts as corrupted scans, with T1-weighted contrast as a reference (see Table 1 for further details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' The corrupted acquisition employs randomized sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' The results of this experiment are collected in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Note that, in Appendix B, we use the same settings detailed in this experiment to compare the proposed algorithm with a baseline method without a reference guide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='2 Experiment 2: on the choice of the reference contrast This in-vivo experiment tests the proposed correction scheme with respect to a different combination of corrupted and reference contrast, namely a T1-weighted corrupted contrast with a T2-weighted reference contrast (see Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' For this experiment, we prompt volunteer 2 to move five times during the acquisition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' The corrupted acquisition employs randomized sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='2, we gather the results for this experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='3 Experiment 3: scanner reconstruction vs processed raw k-space data as input for retrospective motion correction With the in-vivo studies presented in this section, we investigate a question related to the nature of the input data d (equation 6) required by the algo- rithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Due to the formulation of the problem directly in k-space (by means of the NUFFT), the method assumes coil-resolved data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' One must then assess whether the scanner reconstruction (available in the DICOM format) is suit- able for this purpose, since many different reconstruction methods are available depending on the acquisition protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' In particular, the default reconstruc- tion method for linear-filling patterns in k-space employs the SENSE frame- work [Pruessmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=', 1999], while compressed-sensing reconstruction (via the wavelet transform) is used for randomized acquisitions [Lustig et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=', 2007].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Note that our experimentation suggests that without the phase map of the scan- ner reconstruction our motion correction scheme does not perform adequately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' 8 Therefore, with “scanner reconstruction”, we will always refer to the complex- valued scanner reconstruction (comprising both the respective amplitude and phase).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' In the first experiment, we asked volunteer 3 to change position once during the prospectively-corrupted acquisition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' We consider a corrupted T2-weighted contrast and a reference T1-weighted contrast (see Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' One important as- pect of this experiment is related to the acquisition protocol of the T2-weighted contrast, based on a linear-filling pattern in k-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' The corrupted data used as input for the proposed motion-correction algorithm is obtained by export- ing the reconstructed volume directly from the scanner, followed by a simple Fourier transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Note that this 3D image has been obtained by a SENSE reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' The second experiment is set up similarly to the previous one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' We asked volunteer 3 to change position only once during the acquisition phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' We consider, now, a corrupted T2-FLAIR-weighted contrast with a reference T1- weighted contrast (see Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' The most important difference with the previ- ous experiment, besides the type of contrast pair considered, is related to the randomized acquisition protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' In this case, the scanner reconstruction em- ploys a compressed-sensing reconstruction, and is not suited as input for the proposed motion-correction algorithm (see Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Therefore, for ade- quate motion correction, we must set up an intermediate step for processing the raw k-space data via the SENSE reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' We further discuss the results of this experiment in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' 4 Results In this section, we display and briefly analyze the results of the experiments presented in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' For ease of exposition, we organized the power signal-to-noise ratio (PSNR) and structural similarity index (SSIM) values of the reconstructions (with respect to a known ground truth) in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' The motion-corrected full-volume scans were analyzed by a neuroradiologist with 16 years of experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' These were generally deemed of good radiological quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' The motion-related artifacts have been completely removed, and the results are quite close to the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' In Table 3, we organized a more detailed qualitative analysis of the 3D results, geared toward a radiological as- sessment of the corrected scans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='1 Experiment 1: robustness test We gather the results for the robustness test described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='1 (volunteer 1) in Figures 1, 2, and 3 for motion corruption mechanisms associated to one, two, and five changes of position, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Furthermore, we juxtapose the corrected images with varying degrees of corruption in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' We observe that the proposed method consistently ameliorates the corrupted scan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' The quality 9 Experiment Slice orientation PSNR (↑) SSIM (↑) Corrupted Corrected Corrupted Corrected Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='1, Figure 1 Sagittal 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='94 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='7068 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='7936 Coronal 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='66 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='82 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='7653 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='8332 Axial 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='40 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='7616 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='8490 Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='1, Figure 2 Sagittal 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='78 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='7263 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='7816 Coronal 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='19 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='73 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='7847 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='8244 Axial 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='79 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='8104 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='8362 Section 3.' 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+page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='6734 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='8060 Axial 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='15 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='7086 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='8298 Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} 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+page_content='8107 Axial 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='08 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='59 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='7263 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='8407 Table 2: Summary of the motion-correction results shown in Section 4 in terms of PSNR and SSIM Experiment Contrast Motion resolution Blurring Artifacts Additional comments Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='1, Figure 1 T2-FLAIR Completely corrected Some blurring No additional artifacts Good grey white matter differentiation Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='1, Figure 2 T2-FLAIR Completely corrected Some blurring No additional artifacts Good grey white matter differentiation Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='1, Figure 3 T2-FLAIR Completely corrected Some blurring Darker areas within the white matter Good grey white matter differentiation Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='2, Figure 5 T1 Completely corrected Some blurring No additional artifacts Good grey white matter differentiation, some loss of grey matter low signal Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='3, Figure 6 T2 Completely corrected No blurring No additional artifacts Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='3, Figure 7 T2-FLAIR Completely corrected Some blurring No additional artifacts Good grey white matter differentiation Table 3: Qualitative radiological analysis of the motion-corrected results shown in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' The corrected scans are radiologically equivalent to the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' indexes based on PSNR and SSIM show only a modest decrease in correction quality as a function of motion complexity (Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='2 Experiment 2: choice of the reference contrast With the experiment described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='2, we demonstrate the flexibility of the correction scheme with respect to the choice of the reference contrast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' The results are shown in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Contrary to the experiments detailed in the previous section, we are now considering a T2-weighted reference contrast to guide the correction of a T1-weighted corrupted contrast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' The quality of the correction indicates that the proposed technique is rather flexible in terms of reference contrast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' 10 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='3 Experiment 3: scanner reconstruction vs raw k-space data The results of the two experiments described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='3 are depicted in Figures 6 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' The main difference between the two experiments is related to the input data for the proposed motion-correction algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' In the first experiment, the corrupted contrast has been acquired with a protocol based on a linear filling pattern in k-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Note that, in this particu- lar case, the scanner reconstruction implements the SENSE method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' We then extracted the DICOM of both amplitude and phase produced by the scanner, and used it as input data (after a Fourier transform) for the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' The proposed scheme is able to successfully remove the motion artifacts in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' In the case of randomized sampling, the scanner reconstruction is not ad- equate as input data for the proposed motion-correction algorithm, because it employs a compressed-sensing algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' We speculate that compressed-sensing reconstructions degrade the information contained in the corrupted volume, and the corrected contrast cannot be effectively recovered by simply removing rigid-motion artifacts (we defer the degraded results when using scanner re- construction data in Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' However, when the input data is obtained by directly processing the raw k-space data via the SENSE reconstruction, the motion-correction scheme is able to successfully remove the motion artifacts (Figure 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' 5 Discussion Reference-guided TV regularization substantially improves the motion correc- tion quality, both visually and in terms of quality metrics based on PSNR and SSIM, when compared to basic Fourier reconstruction without motion correc- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' The comparison is also substantially favorable with standard “blind” mo- tion correction techniques, for example based on conventional regularization such as TV, which do not employ a reference to guide the correction (see Ap- pendix B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' In fact, for randomized sampling patterns that are now common in the clinical practice, we verified that blind retrospective techniques are wholly inadequate for motion correction of radiological quality (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' the comparison in Appendix B, Figure 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Our experimentation based on volunteer data aimed at assessing the ro- bustness of the correction quality with respect to motion artifacts of increasing complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' In this study, we equated this complexity to the number of volun- teer changes of pose during the acquisition phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Clearly, this does not fully describe the complexity of motion encountered in practice in the clinic, but it only constitutes a preliminary step in that direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Nevertheless, the results described in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='1 support the indication that the retrospective motion correction of T2-FLAIR weighted images based on a T1 reference contrast is quite robust in terms of reconstruction quality, with only minor degradations in terms of contrast and resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' 11 Furthermore, the flexibility of the proposed motion-correction method is demonstrated with different combinations of motion-corrupted and reference contrasts (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Our experience suggests that an important factor in assessing the effectiveness of the reference contrast as a guide for motion cor- rection lies in the similarity of the k-space distribution of the two contrasts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Good reconstruction quality can be expected when the reference contrast has similar or higher frequency content when compared to the corrupted contrast, regardless of the type of contrast considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' A significant part of our experimentation was devoted to assess whether the scanner reconstruction (available as DICOM format) can be directly used as in- put data for the proposed correction method (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' We established that the scanner reconstruction is not suitable for this purpose when it is obtained via compressed-sensing algorithms (Appendix A), which is the case for random- ized sampling on the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='5 T Philips Ingenia scanner utilized in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' In this case, we must resort to the raw k-space data and perform an intermediate SENSE reconstruction for effective motion correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' The computational times of the motion correction are, generally speaking, problem dependent, since complex motion artifacts require an increasing num- ber of iterations as a function of motion complexity (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' The examples illustrated in this study, where a fixed number of iterations was consider irre- spectively of motion complexity, are completed within 1 h 30 min for 3D images of approximately 256×256×256 voxels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' The current CPU implementation was run on a consumer-grade laptop with the following processor specifications: Intel Core i7-10750H CPU@2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='60GHz×12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' An effective implementation in a clinical scenario for on-line reconstructions will likely require GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' The basic assumption of the proposed retrospective correction method is related to the availability of a motion-free contrast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' While we believe that it is a realistic possibility within an MR session, we note that the reference contrast may come from previous MR sessions (or even different imaging modalities altogether, such as CT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' In this particular case, the bias introduced by the structural prior may have an adverse effect in case of an evolving pathology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' However, when structural changes involve a limited pathological region, the adverse bias can be easily mitigated by masking the affected zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Note that the motion-free reference can be exploited differently than the reference-guided TV regularization introduced in Ehrhardt and Betcke [2016], and adopted in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' For example, one may consider several competing techniques advanced for multi-contrast MRI, such as Bayesian compressed sens- ing [Bilgic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=', 2011], group sparsity [Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=', 2014], reference-based MRI [Weizman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=', 2015], or multi-contrast graph-based sparsity [Lai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=', 2017, 2018].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' The method here presented is limited to rigid motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Indeed, some de- crease in correction quality is noticeable in Figure 6 in the neck region (which is not supposed to behave rigidly).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' However, our technique may be extended to non-rigid motion and, hence, different body regions other than the brain [see, for example, Huttinga et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=', 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' A major challenge for such extension is a computationally effective parameterization of the motion effects, and the result- 12 ing ill-posedness of the inverse problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Note that a significant computational advantage of rigid motion over non-rigid motion is related to the direct imple- mentation of the rigid motion in k-space, via equation (3), which results in a data model that requires a single NUFFT evaluation, regardless of the number of time samples considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Other interesting extensions of the method are re- lated to the integration of specialized motion-resilient acquisition patterns, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' as described in Cordero-Grande et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' [2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' 6 Conclusions We assessed the performance of the proposed retrospective motion correction method based on a reference contrast not affected by motion artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' The current prospective in-vivo study targets 3D clinical protocols conventionally used in brain imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' The method is tested with several degrees of motion artifacts, by instructing the volunteers to change position during the scan multiple times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' While, we observe that the corrupted images are severely degraded as a function of motion complexity, the corrected images are generally robustly estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' We also verified that the proposed technique is agnostic with respect to the choice of the reference contrast, as long as the frequency content of the reference and target contrasts is comparable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Further assessment of the proposed method will be devoted to patient data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Data The 3D results of the experiment described in Sections 3, 4 are freely available online in the DICOM format at the following link: github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='com/grizzuti/ReferenceGuidedMotionCorrection Supplementary DICOM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Acknowledgments This publication is part of the project “Reducing re-scans in clinical MRI ex- ams” (with project numbers 0104022007, 01040222210001 of the research pro- gram “IMDI, Technologie voor bemensbare zorg: Doorbraakprojecten”) which is financed by The Netherlands Organization for Health Research and Devel- opment (ZonMW).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' The project is also supported by Philips Medical Systems Netherlands BV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' 13 Sagittal Corrupted PSNR: 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='93 SSIM: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='6655 Corrected PSNR: 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='95 SSIM: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='7936 Ground truth Reference Coronal PSNR: 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='91 SSIM: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='7461 PSNR: 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='82 SSIM: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='8332 Axial PSNR: 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='46 SSIM: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='7371 PSNR: 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='16 SSIM: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='8490 Axial detail Figure 1: Reconstruction results for volunteer 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' The volunteer is instructed to move once during the scan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' The corrupted contrast is T2-FLAIR-weighted, while the reference contrast is T1-weighted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Compare these results with the one obtained with different motion complexity in Figures 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' 14 SSagittal Corrupted PSNR: 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='82 SSIM: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='6071 Corrected PSNR: 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='76 SSIM: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='7816 Ground truth Reference Coronal PSNR: 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='08 SSIM: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='6751 PSNR: 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='73 SSIM: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='8244 Axial PSNR: 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='72 SSIM: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='6440 PSNR: 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='70 SSIM: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='8362 Axial detail Figure 2: Reconstruction results for volunteer 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' The volunteer is instructed to move twice during the scan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' The corrupted contrast is T2-FLAIR-weighted, while the reference contrast is T1-weighted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Compare these results with the one obtained with different motion complexity in Figures 1, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' 15 SSagittal Corrupted PSNR: 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='65 SSIM: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='5141 Corrected PSNR: 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='28 SSIM: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='7661 Ground truth Reference Coronal PSNR: 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='77 SSIM: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='5998 PSNR: 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='40 SSIM: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='8060 Axial PSNR: 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='23 SSIM: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='5727 PSNR: 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='66 SSIM: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='8298 Axial detail Figure 3: Reconstruction results for volunteer 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' The volunteer is instructed to move five times during the scan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' The corrupted contrast is T2-FLAIR-weighted, while the reference contrast is T1-weighted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Compare these results with the one obtained with different motion complexity in Figures 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' 16 SMove once Corrupted PSNR: 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='46 SSIM: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='7371 Corrected PSNR: 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='16 SSIM: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='8490 Ground truth Reference Move twice PSNR: 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='72 SSIM: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='6440 PSNR: 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='70 SSIM: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='8362 Move five times PSNR: 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='23 SSIM: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='5727 PSNR: 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='66 SSIM: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='8298 Figure 4: Summary of the reconstruction results for volunteer 1 (see Figures 1, 2, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' The volunteer is instructed to move a variable number of times during the scan in order to test the robustness of the proposed correction scheme with respect the motion complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' The corrupted images are increasingly affected by motion artifacts, however only modest decrease in reconstruction quality can be observed for the corrected images (here axial slices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' 17 Sagittal Corrupted PSNR: 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='84 SSIM: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='7032 Corrected PSNR: 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='07 SSIM: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='8093 Ground truth Reference Coronal PSNR: 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='35 SSIM: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='7851 PSNR: 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='40 SSIM: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='9021 Axial PSNR: 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='11 SSIM: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='8248 PSNR: 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='55 SSIM: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='9012 Coronal detail Figure 5: Reconstruction results for volunteer 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' The volunteer is instructed to move five times during the scan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' The corrupted contrast is T1-weighted, while the reference contrast is T2-weighted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' The proposed correction scheme is agnostic about the choice of corrupted/reference contrast combinations with similar spectral content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' 18 Sagittal Corrupted PSNR: 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='26 SSIM: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='6963 Corrected PSNR: 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='54 SSIM: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='8409 Ground truth Reference Coronal PSNR: 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='46 SSIM: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='7321 PSNR: 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='65 SSIM: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='8370 Axial PSNR: 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='55 SSIM: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='7895 PSNR: 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='33 SSIM: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='8144 Axial detail Figure 6: Reconstruction results for volunteer 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' The volunteer is instructed to move once, halfway through the scan (the two overlapping positions are clearly visible in the corrupted slices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' The corrupted contrast is T2-weighted, while the reference contrast is T1-weighted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' In this case, the input data for the correction algorithm is directly extracted from the scanner reconstruction in DICOM format (comprising both amplitude and phase).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' The acquisition scheme for the T2-weighted contrast follows a linear filling pattern in k-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' The proposed method successfully removes the motion artifacts because the scanner reconstruction is obtained through a conventional SENSE reconstruction (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Figure 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' 19 Sagittal Corrupted PSNR: 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='72 SSIM: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='6762 Corrected PSNR: 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='76 SSIM: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='7818 Ground truth Reference Coronal PSNR: 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='95 SSIM: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='7238 PSNR: 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='54 SSIM: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='8107 Axial PSNR: 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='08 SSIM: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='7263 PSNR: 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='59 SSIM: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='8407 Axial detail Figure 7: Reconstruction results for volunteer 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' The volunteer is instructed to move once, halfway through the scan (the two overlapping positions are clearly visible in the corrupted slices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' The corrupted contrast is T2-FLAIR-weighted, while the reference contrast is T1-weighted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Unlike Figure 6, the input data for the correction algorithm is not extracted from the scanner, but is obtained via SENSE reconstruction of the raw k-space data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' The acquisition scheme for the T2-FLAIR-weighted contrast is randomized in k-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' When the scanner reconstruction is directly used as input data, the motion correction is highly suboptimal (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Figure 8 in Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' 20 CeTTYUReferences Jalal B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Andre, Brian W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Bresnahan, Mahmud Mossa-Basha, Michael N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Hoff, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Patrick Smith, Yoshimi Anzai, and Wendy A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Cohen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Toward quantify- ing the prevalence, severity, and cost associated with patient motion during clinical MR examinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Journal of the American College of Radiology, 12 (7):689–695, jul 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' ISSN 1558349X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='jacr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' URL https://pubmed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='ncbi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='nlm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='nih.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='gov/25963225/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' David Atkinson, Derek L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Hill, Peter N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Stoyle, Paul E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Summers, and Stephen F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Keevil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Automatic correction of motion artifacts in magnetic resonance images using an entropy focus criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' IEEE Transactions on Medical Imaging, 16(6):903–910, 1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='1109/42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='650886.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Klinteberg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' A par- allel nonuniform fast Fourier transform library based on an “exponential of semicircle” kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' SIAM Journal on Scientific Computing, 41(5):C479–C504, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' ISSN 10957197.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='1137/18M120885X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Berkin Bilgic, Vivek K Goyal, and Elfar Adalsteinsson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Multi-contrast recon- struction with bayesian compressed sensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Magnetic resonance in medicine, 66(6):1601–1615, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' J´erˆome Bolte, Shoham Sabach, and Marc Teboulle.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='1109/TMI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='2029854.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='Multiple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Leon Bungert and Matthias J Ehrhardt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Mina Ghaffari, Kamlesh Pawar, and Ruth Oliver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Brain MRI motion arti- fact reduction using 3D conditional generative adversarial networks on simu- lated motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' In 2021 Digital Image Computing: Techniques and Applications (DICTA), pages 1–7, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='1109/DICTA52665.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='9647370.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Frank Godenschweger, Urte K¨agebein, Daniel Stucht, Uten Yarach, Alessandro Sciarra, Renat Yakupov, Falk L¨usebrink, Peter Schulze, and Oliver Speck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Motion correction in MRI of the brain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Physics in Medicine & Biology, 61 (5):R32–R56, 2016.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Bas Peters, Brendan R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Smithyman, and Felix J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Herrmann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Projection meth- ods and applications for seismic nonlinear inverse problems with multiple con- straints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Geophysics, 84(2):R251–R269, 02 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' doi: 10.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Joint retrospec- tive motion correction and reconstruction for brain mri with a reference con- trast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' IEEE Transactions on Computational Imaging, 8:490–504, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' ISSN 2333-9403.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='1109/TCI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='3183383.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Ghislain Vaillant, Claudia Prieto, Christoph Kolbitsch, Graeme Penney, and Tobias Schaeffter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Retrospective rigid motion correction in k-space for seg- mented radial MRI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' IEEE Transactions on Medical Imaging, 33(1):1–10, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' ISSN 02780062.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='1109/TMI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='2268898.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Lior Weizman, Yonina C Eldar, Alon Eilam, S Londner, Moran Artzi, and D Ben Bashat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Fast reference based mri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' In 2015 37th Annual International Con- ference of the IEEE Engineering in Medicine and Biology Society (EMBC), pages 7486–7489.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' IEEE, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' 24 Edward Brian Welch, Armando Manduca, Roger C Grimm, Heidi A Ward, and Clifford R Jack Jr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Spherical navigator echoes for full 3D rigid body motion measurement in MRI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Magnetic Resonance in Medicine, 47(1):32–41, 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Edward Brian Welch, Phillip J Rossman, Joel P Felmlee, and Armando Man- duca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Self-navigated motion correction using moments of spatial projections in radial MRI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Magnetic Resonance in Medicine, 52(2):337–345, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Maxim Zaitsev, Christian Dold, Georgios Sakas, J¨urgen Hennig, and Oliver Speck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Magnetic resonance imaging of freely moving objects: prospective real-time motion correction using an external optical motion tracking system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Neuroimage, 31(3):1038–1050, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Maxim Zaitsev, Julian Maclaren, and Michael Herbst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Motion artifacts in MRI: A complex problem with many partial solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Journal of Mag- netic Resonance Imaging, 42(4):887–901, oct 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' ISSN 10531807.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='1002/jmri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='24850.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' URL http://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='wiley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='com/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='1002/jmri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='24850.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' 25 A Inadequate motion correction with scanner reconstruction as input data As anticipated in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='3, directly using the scanner reconstruction (ex- tracted as DICOM files of both the amplitude and phase of the reconstruction) as input data for the proposed motion correction scheme may degrade the per- formance when compressed-sensing reconstruction tools have been employed in the reconstruction process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' To motivate this conclusion, we setup an experiment with the same setting as described in the second experiment in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='3, the only difference being in how the input data is generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' In this case, the input data consist of the Fourier transform of the extracted scanner reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' The related suboptimal correction is quite evident when comparing Figure 8 with Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Axial Corrupted Corrected Ground truth Reference Figure 8: Reconstruction results for volunteer 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' The volunteer is instructed to move once, halfway through the scan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' The corrupted contrast is T2-FLAIR- weighted, while the reference contrast is T1-weighted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' In this experiment, the proposed motion correction scheme processes the scanner reconstruction directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Since the reconstruction algorithm implemented in the scanner de- stroys the coherence of the rigid motion artifact, the proposed method cannot properly recover the correct reconstruction by simply estimating the motion parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' With contrasts obtained by randomized acquisitions, we advise to use raw k-space data instead (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Figure 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' B Comparison of motion correction with and without a reference guide The reference-guided motion-correction algorithm described in Section 2 is com- pared with a standard retrospective motion-correction algorithm based on the TV regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' We note that most retrospective motion-correction methods follows the basic mathematical framework detailed in Section 2 (see, for example, Loktyushin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' [2013] or Cordero-Grande et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' [2020]), where the main mathematical difference consists in the choice of the regularization term gu, in equation (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Hence, in order to assess the effect of the reference contrast, we adopt the 26 same formulation described in Section 2 with a simple TV regularization term gu(u) = � x ∥∇u|x∥ (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' equation 8 for the reference-guided version of TV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' For the comparison with the baseline method, we use the same experimen- tal settings in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Once again, the motion artifacts are prospectively induced by prompting the volunteer to move during the scan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' The results are summarized in Figure 9 and Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Move once Corrupted PSNR: 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='40 SSIM: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='7616 Corrected (ours) PSNR: 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='16 SSIM: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='8490 Corrected (baseline) PSNR: 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='54 SSIM: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='8007 Ground truth Move twice PSNR: 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='79 SSIM: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='8104 PSNR: 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='70 SSIM: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='8362 PSNR: 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='57 SSIM: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='7708 Move five times PSNR: 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='15 SSIM: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='7086 PSNR: 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='66 SSIM: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='8298 PSNR: 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='96 SSIM: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='7562 Figure 9: Comparison of the reconstruction results for volunteer 1 with a reference-guided (ours) and a baseline motion-correction method (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=', not guided by a reference contrast).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' The volunteer is instructed to move a variable number of times during the scan in order to test the robustness of the proposed correction schemes with respect the motion complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' The corrupted images are increasingly affected by motion artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' The decrease in reconstruction quality for the baseline method is substantially more pronounced than the re- sults obtained with our reference-guided correction (see also Figures 1–3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' We note that the difference in performance between the reference-guided and blind motion correction is even more pronounced in this example than what was previously shown in Rizzuti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' [2022] (which was limited to 2D synthetic data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' This might depend on the fact that the problem is substantially more ill-posed in 3D with randomized sampling than the 2D full-acquisition setup considered in Rizzuti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' [2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' It is also worth noting that, in our experience, the results for blind motion correction depend more sensibly on the choice of the 27 Experiment Slice orientation PSNR (↑) SSIM (↑) Corrupted Corrected Corrupted Corrected Ours Baseline Ours Baseline Move once Sagittal 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='94 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='95 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='7068 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='7936 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='7529 Coronal 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='66 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='82 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='61 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='8490 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='8007 Move twice Sagittal 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='78 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='76 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='7263 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='7816 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='6925 Coronal 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='19 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='73 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='68 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='7847 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='8244 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='7448 Axial 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='79 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='70 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='57 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='8104 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='8362 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='7708 Move five times Sagittal 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='45 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='28 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='6116 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='7661 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='6719 Coronal 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='54 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='40 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='6734 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='8060 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='7327 Axial 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='15 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='66 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='7086 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='8298 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='7562 Table 4: Comparison of the motion-correction results for the baseline and reference-guided methods in terms of PSNR and SSIM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' The experiment setup is described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' hyper-parameters in equation (5) than the proposed reference-based version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' C Motion parameter estimation The proposed motion correction algorithm described in Section 2 estimates the rigid motion that the object of interest undergoes during the scan, in order to undo its effect on the reconstructed 3D image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' In 3D, the rigid motion is performed by: a plane rotation θxy in the corresponding plane xy, a plane rotation θxz in the xz plane, a plane rotation θyz in the yz plane, a translation τx in the x direction, a translation τy in the y direction, and a translation τz in the z direction (in this order).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' We adopt the following convention: the x direction corresponds to the left-right direction, y to the posterior-anterior direction, and z to the inferior-superior direction, the xy plane corresponds to the axial plane, xz to the coronal plane, and yz to the sagittal plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' Left/right, anterior/posterior, and inferior/superior are meant from the patient perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' The orientation of the rotation planes is determined by the right-hand rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' By design, the prospectively-induced motion for all the experiments detailed in Section 3 follows a step-wise behavior (each step corresponding to a change of pose).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' In this appendix, we gather the estimated rigid motion parameters for the results shown in Section 4, as a function of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' As noted in the main body of the paper, time is equated to the phase-encoding plane coordinate index, ordered by the corresponding acquisition ordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' We display the estimated motion parameters in Figure 10 (see Sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='1, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='1, Figure 1), Figure 11 (see Sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='1, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='1, Figure 2), Figure 12 (see Sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='1, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='1, Figure 3), Figure 13 (see Sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='2, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='2, Figure 5), Figure 14 (see Sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='3, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='3, Figure 6), and Figure 15 (see Sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='3, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='3, Figure 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' 28 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='0 τ x (mm) Estimated 0 2 τ y (mm) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='5 τ z (mm) 0 5 θ xy ( ∘ ) −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='5 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='0 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='5 θ xz ( ∘ ) 0 1000 2000 3000 4000 5000 6000 t = phase encoding index 0 5 θ yz ( ∘ ) Figure 10: Estimated rigid motion parameters for the experiment described in Sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='1, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='1, with motion-correction results in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' The volunteer was asked to move once during the scan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='5 τ x (mm) Estimated 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='5 τ y (mm) 0 5 τ z (mm) −5 0 5 θ xy ( ∘ ) −10 0 θ xz ( ∘ ) 0 1000 2000 3000 4000 5000 6000 t = phase encoding index 0 10 θ yz ( ∘ ) Figure 11: Estimated rigid motion parameters for the experiment described in Sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='1, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='1, with motion-correction results in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' The volunteer was asked to move twice during the scan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' 29 −5 0 5 τ x (mm) Estimated 0 10 τ y (mm) 0 10 τ z (mm) −10 0 θ xy ( ∘ ) −10 0 10 θ xz ( ∘ ) 0 1000 2000 3000 4000 5000 6000 t = phase encoding index −10 0 10 θ yz ( ∘ ) Figure 12: Estimated rigid motion parameters for the experiment described in Sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='1, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='1, with motion-correction results in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' The volunteer was asked to move five times during the scan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='0 τ x (mm) Estimated −1 0 1 τ y (mm) 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='5 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='0 τ z (mm) −1 0 1 θ xy ( ∘ ) −1 0 1 θ xz ( ∘ ) 0 2500 5000 7500 10000 12500 15000 17500 20000 t = phase encoding index −3 −2 −1 θ yz ( ∘ ) Figure 13: Estimated rigid motion parameters for the experiment described in Sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='2, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='2, with motion-correction results in Figure 5 30 −5 0 τ x (mm) Estimated −5 0 5 τ y (mm) −5 0 τ z (mm) 0 5 θ xy ( ∘ ) −5 0 θ xz ( ∘ ) 0 10000 20000 30000 40000 50000 60000 t = phase encoding index −5 0 5 θ yz ( ∘ ) Figure 14: Estimated rigid motion parameters for the experiment described in Sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='3, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='3, with motion-correction results in Figure 6 0 5 τ x (mm) Estimated −5 0 τ y (mm) 0 5 τ z (mm) 0 10 θ xy ( ∘ ) 0 10 θ xz ( ∘ ) 0 2000 4000 6000 8000 10000 12000 t = phase encoding index −10 0 θ yz ( ∘ ) Figure 15: Estimated rigid motion parameters for the experiment described in Sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='3, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAzT4oBgHgl3EQfK_te/content/2301.01106v1.pdf'} +page_content='3, with motion-correction results in Figure 7 31' metadata={'source': 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a/a9AyT4oBgHgl3EQfiviR/content/tmp_files/2301.00402v1.pdf.txt b/a9AyT4oBgHgl3EQfiviR/content/tmp_files/2301.00402v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..5ae9a947620613438a4c903636bc84f98e6fbed3 --- /dev/null +++ b/a9AyT4oBgHgl3EQfiviR/content/tmp_files/2301.00402v1.pdf.txt @@ -0,0 +1,749 @@ + +xxxx-xxxx/xx/xxxxxx + +1 +© xxxx IOP Publishing Ltd + +Edge States of α-Bismuthene +Nanostructures +Sara Salehitaleghani1, Tobias Maerkl1, Pawel J Kowalczyk2, Maxime Le +Ster2, Xiaoxiong Wang3, Guang Bian4, Tai-Chang Chiang5, and Simon A +Brown1* +1 The MacDiarmid Institute for Advanced Materials and Nanotechnology, School of +Physical and Chemical Sciences, University of Canterbury, Christchurch, New Zealand +2 Department of Solid-State Physics, Faculty of Physics and Applied Informatics, +University of Lodz, Lodz, Poland +3 College of Science, Nanjing University of Science and Technology, +Nanjing 210094, People’s Republic of China +4 Department of Physics and Astronomy, University of Missouri, Columbia, +MO 65211, United States of America +5 Department of Physics, University of Illinois at Urbana-Champaign, +1110 West Green Street, Urbana, IL 61801-3080, United States of America + +E-mail: simon.brown@canterbury.ac.nz +Received xxxxxx +Accepted for publication xxxxxx +Published xxxxxx +Abstract +We present a systematic investigation of the edge states of two-dimensional α- +bismuthene (α-Bi) structures self-assembled on HOPG substrates, using +scanning tunnelling microscopy and scanning tunnelling spectroscopy. The +measurements are carried out for 3ML, 5ML and 7ML thick Bi structures. Our +spectroscopy studies reveal clear features at the edges of the 5ML and 7ML thick +structures, and the positions of the edge states (ESs) coincide with the +topographical step edges. In contrast, in 3ML structures the ESs appear to be +absent and instead new states are sometimes observed, far from the +topographical edge. These states are associated with a moiré pattern and result +from strain-induced modulation of the topology. Our observations demonstrate +the impact on the edge states of coupling to adjacent structures. + +Keywords: 2D Materials, α-Bismuthene, Topological Edge States. + + +IOP Publishing + Journal Title +Journal XX (XXXX) XXXXXX https://doi.org/XXXX/XXXX + +Xxxx xxxx/xx/xxxxxx +2 +© xxxx IOP Publishing Ltd + +1. Introduction +α-bismuthene (α-Bi) is a two-dimensional black-phosphorus-like allotrope of bismuth that is +expected to have robust quantum spin Hall states [1], allowing spintronic applications such as long- +distance spin transport and spin-to-charge conversion [2]. The SOC-induced bulk band gap of α-Bi +[3] is large enough to make it a candidate for topotronic devices that operate efficiently at room +temperature [2]. These properties originate from the band structure, which has a complex dependence +on thickness that leads to interesting quantum size effects [4], and which derives from the non- +symmorphic symmetry of the lattice (see Figure 1 (a, b)) [5]. +Both α-Bi and the other (hexagonal) 2D allotrope of bismuth (β-Bi) are theoretically expected to +have spin-polarized 1D edge states (ESs) [6-8]. α-Bi is energetically favoured at small thicknesses but +β-Bi is generally observed at large thicknesses [9]. β-Bi structures have been thoroughly investigated +[10-18], and it is found the nature of the ESs varies depending on the substrate on which it is grown. +α-Bi structures were grown on several substrates (Si(111) [19], Ge(111) [20], the superconductor +NbSe2 [21], graphene supported by SiC [22, 23], MoS2 [24], TiSe2 [25] and HOPG [3, 26]), but less +is known about their topological properties and ESs. Calculations suggest a strong dependence of the +topology on the atomic buckling at the α-Bi surface i.e. the difference in height, h, between the surface +atoms (compare Figure 1 (c) and (d)) [3]. It was reported that for flat surfaces (h = 0, Figure 1 (d)) the +ESs are topologically protected and that a nontrivial-trivial transition occurs as the buckling is +increased to h ~ 0.1 Å [3]. +Here we report a comprehensive characterization of the edge states of α-Bi structures grown on +HOPG, with emphasis on how the edges are affected by adjacent structures (either other Bi structures +or the HOPG substrate). ESs for 5ML and 7ML structures are similar to those reported in ref. [3], +including a characteristic peak at energies ~0.1 eV. However, we find no evidence for ESs on 3ML +structures. Instead we observe states with similar characteristics to the previously reported ESs in +along lines parallel to the fringes of a moiré pattern, and localised regions that are most likely +associated with defects. These results indicate that the buckling of the α-Bi islands is modified locally, +consistent with a topological transition. + +IOP Publishing + Journal Title +Journal XX (XXXX) XXXXXX https://doi.org/XXXX/XXXX + +Xxxx xxxx/xx/xxxxxx +3 +© xxxx IOP Publishing Ltd + +2. Methods +1.2ML 5N purity Bi from a Knudsen cell was thermally deposited on freshly cleaved highly +oriented pyrolytic graphite (HOPG) substrates at room temperature and under ultrahigh vacuum +conditions (UHV, <1 × 10-9 mbar). The deposition rate was kept at ~0.01 Å s-1 and black-phosphorus- +like α-Bi structures are always observed for mean thicknesses less than ~12ML [27]. Scanning +tunnelling microscopy (STM) and spectroscopy (STS) measurements were carried out using a +variable-temperature UHV scanning tunnelling microscope (Scienta Omicron VT-AFM XA) system. +The STS data was acquired in various bias voltage ranges (typically in the range ±1V with a setpoint +current of 300 pA) and using a lock-in amplifier; modulation voltage and frequency were 20 mV and + +Figure 1 Black phosphorus structure and experimentally observed morphology for α-Bi. (a) +Perspective, (b) top view, (c) side view along 〈001〉 for the buckled structure (h ~ 0.1 Å [3]), (d) side +view along 〈1̅10〉 for the structure with zero buckling, and (e) side view along 〈001〉 directions. (f) +Large area STM image showing typical α-Bi island morphology when grown on HOPG. The islands +typically comprise 3ML thick bases (darker orange) topped by additional 2ML stripes forming 5ML +Bi and 7ML Bi (yellow and white respectively). Isolated 5ML and 7ML structures i.e. the 5ML and +7ML islands which sit directly on the HOPG substrate can occasionally occur (an example of a 5ML +island is arrowed). (g) Schematic illustration of the basic morphology. Left: 3ML island. Centre: 5ML +stripe (edge adjacent to 3ML island). Right: 5ML island (edge adjacent to the HOPG substrate). +Colours are chosen to match the image in (f). Note that island edges are always adjacent to the +substrate, whereas edges of stripes are adjacent to other α-Bi layers. Note also that the schematic +includes an inert wetting layer which means that the observed density of states for each of the 3ML, +5ML and 7ML islands correspond to calculated band structures for 2ML, 4ML and 6ML α-Bi +respectively [4]. + +(a) +(b) +(d) +(c) +〈 +〉 +〈 +〉 +(g) +(e) +(f) + +150 nm5ML +[3ML] +SubstrateIOP Publishing + Journal Title +Journal XX (XXXX) XXXXXX https://doi.org/XXXX/XXXX + +Xxxx xxxx/xx/xxxxxx +4 +© xxxx IOP Publishing Ltd + +2.7 kHz. All spectra for bulk α-Bi structures are in reasonable agreement with those measured +previously using standard DC bias techniques [4]. All data reported here was obtained at low +temperatures ~50 K. +3. Experimental Results +α-Bi typically forms [4, 5, 24, 26, 28-32] atomically flat structures with layered “wedding cake” +profiles. Figure 1 (a-e) show the atomic structure and Figure 1 (f, g) show the typical morphology of +the islands. Detailed STM measurements [28] show that the islands comprise 3ML thick bases (see +Figure 1 and caption) topped by additional 2ML stripes, so that the stripes have total thicknesses of +5ML and 7ML. It is also occasionally possible to find isolated 5ML and 7ML islands which are not +surrounded by other Bi structures. This allows us to investigate the impact of the interactions with +adjacent structures on the nature of the ESs i.e. to compare results from edges adjacent to another Bi +structure (e.g. 5ML stripe next to a 3ML base) and edges adjacent to HOPG (e.g. 5ML island). +Previous calculations show that the band structures change with the thickness of the material [4, +5] and so it is expected that the edge states and topology could vary as well. 3ML, 5ML and 7 ML +structures exhibit very different STS spectra, corresponding to the results of calculations for +freestanding 2ML, 4ML and 6ML -Bi. (The difference in measured height is believed to be due to +the presence of an inert wetting layer [4]). Therefore, we present data for structures of each thickness +separately, starting by considering examples of ESs for 7ML structures (both islands and stripes); we +then move on to a discussion of 5ML and 3ML structures. +3.1. 7ML Structures +A typical isolated 7ML Bi island is shown in Figure 2 (a). STS measurements were performed on +the different edges of the island with magnified views shown in Figure 2 (b) to (e) and corresponding +maps of the local density of states (LDOS) at +100 mV in Figure 2 (f) to (i). The edge states are +immediately visible as prominent bright features (high-LDOS near the Fermi level; see also Figure +S1). ESs are observed over the entire perimeter of the island, including the rounded ends. +This particular 7ML island exhibits some unusual additional complexity that provides further +information. Firstly, the grain boundary arrowed in Figure 2 (g) separates two grains of -Bi that are +rotated by ~90° with respect to each other [29, 30] and therefore have different orientations with +respect to the HOPG substrate. The observed ESs are very similar, which is a strong indication that +the ESs are not sensitive to the orientation of Bi on HOPG. Secondly, this island happened to grow +over an HOPG step edge and so the interaction of lower part of the island (Figure 2 (d)) with the +substrate is expected to be different to that of the upper part of the island (Figure 2 (b,c)): the signature + +IOP Publishing + Journal Title +Journal XX (XXXX) XXXXXX https://doi.org/XXXX/XXXX + +Xxxx xxxx/xx/xxxxxx +5 +© xxxx IOP Publishing Ltd + +of the edge state remains unchanged, again suggesting the ESs are not affected by interactions with +the substrate. The lack of dependence on the crystallographic orientation of the edge or interaction +with the substrate indicates the ESs are inherent to the edge, are robust, and therefore are likely have +a topological origin. [3] +Figure 3 (a) shows a 7ML Bi stripe that is surrounded by a 5ML Bi island and Figure 3 (b) shows +a magnified view of the edge. The ES is consistently observed in the LDOS maps at energies ~+100 +mV (see Figure 3 (c) and discussion of STS spectra below) and it is not dependent on the +crystallographic orientation of the edges (see Figure S1). Despite the fact that the ES is adjacent to a +material with a very different band structure (5ML Bi versus HOPG) the results for the ESs are +qualitatively similar to those of the isolated 7ML Bi islands. Figure 3 (d) shows an LDOS intensity +profile perpendicular to the edge of the stripe (along the white line in Figure 3 (c)). The ES is clearly +resolved as an additional light blue feature at ~0.1 eV (between the red dashed lines that mark the +position of the topographic edge). The ES decays completely into the bulk of the 7ML structure. +Hence the ES is aligned with the topographical edge shown in the height profile in Figure 3 (e). To +illustrate this more clearly Figure 3 (f) shows a mapping of the LDOS at +100 mV (Figure 3 (c)) onto +the 3D topography image (Figure 3 (b)). +The intrinsic width of the ESs in our experiments cannot be exactly determined due to the tip +broadening effect (see Figure S2). While the measured widths are close to those calculated for 2ML +α-Bi nanoribbons (i.e. ~2 nm) [3], we believe that this agreement is accidental as the width is also +found to correspond to the measured (i.e. tip-broadened) width of the topographical edge. The width +is found to be slightly smaller for the stripes than for islands, simply because tip broadening is more +significant for taller structures. Within the range that it can be observed, the position and width of the +ES do not change as a function of bias (Figure S3). +The fact that the width of the ESs always corresponds to the measured width of the topographic +step suggests that the ESs are physically located right at the topographic edge. Importantly, no +enhancement of the DOS is observed on the flat part of the island – Figure 3 (f) shows clearly that the +position of the ES corresponds to that shown schematically in Figure S2 (c). This is in strong contrast +with previous calculations for α-Bi (see especially figure 4 in ref. [3]) which predicted that the ESs +should extend from the topographic edge into the bulk, as illustrated schematically in Figure S2 (b). +We emphasise that similar data were obtained for many islands and stripes of different thicknesses, +and so the position of the ES is significant. Similar positioning of the ES was previously reported in +calculations of the edge state band structure for monolayer -Bi on α-Bi [11]: there the ES +wavefunction clearly extends beyond the final atom in the overlayer in some cases and the differences +in coupling lead to a modulation of the topology of the ESs. + +IOP Publishing + Journal Title +Journal XX (XXXX) XXXXXX https://doi.org/XXXX/XXXX + +Xxxx xxxx/xx/xxxxxx +6 +© xxxx IOP Publishing Ltd + + +Figure 2 (a) Large-scale STM image of an isolated 7ML Bi island. Imaging conditions: Vt = +1 V, It = 20 +pA and T = 50 K. (b)-(e) Magnified view of the regions inside the red squares in (a) showing various edges +of the island with different crystallographic directions. The coloured rectangles in (e) correspond to spectra +shown in Figure 4 (a). (f)-(i) Corresponding dI/dV(V) maps at +140 mV. STS parameters: ±1 V, 300 pA +and 50 K. Note (i) a feature with similar characteristics to the ESs is observed at a grain boundary (arrowed +in g)), (ii) even though the island grew over an HOPG step edge the signature of the edge state remains +unchanged in (h) indicating that the interaction of the ES with the substrate is the same. DOS colour scale: +red is high, blue is low. +(e) +10 nm +(b) +(f) +10 nm +(c) +(g) +10 nm +(i) +10 nm +(h) +b +(e) +7ML Bi +50 nm +e +(a) +d +c +min +max +(d) +10 nm + +IOP Publishing + Journal Title +Journal XX (XXXX) XXXXXX https://doi.org/XXXX/XXXX + +Xxxx xxxx/xx/xxxxxx +7 +© xxxx IOP Publishing Ltd + + + +Figure 3 (a) STM image of a 7ML Bi stripe (adjacent to a 5ML Bi island). Imaging conditions: Vt = +1 +V, It = 20 pA and T = 50 K. (b) Magnified view of the region inside the purple rectangle in (a) showing +the edge. The coloured rectangles correspond to spectra shown in Figure 4 (b). (c) Corresponding LDOS +map at +100 mV. (d) LDOS intensity profile drawn along the white arrow in (c), perpendicular to the +edge of the 7ML Bi stripe. (e) Corresponding height profile. (f) Mapping of the +100 mV LDOS shown +in (c) onto the STM 3D-topograph of (b); red regions exhibit the highest DOS and highlight that the ES +is clearly located at the sloping edge of the stripe. STS parameters: ±1 V, 300 pA and 50 K. + + ++1 +max +(a) ++0.8 +(d) ++0.6 ++0.4 +国 +5MLBi +M ++0.2 +7MLBi +Bias +0 +-0.2 +-0.4 +-0.6 +100 nm +-0.8 +-1 +0 +2 +4 +6 +8 +10min +Distance +(nm) +(b) +(e) +P1 +5MLBi +0.8 +Height (nm) +0.6 +0.4 +7MLBi +0.2 +5nm +0 +0 +2 +4 +6 +8 +10 +Distance(nm) +max +(c) +(f) +0.8nm +18 nm +20nm +minIOP Publishing + Journal Title +Journal XX (XXXX) XXXXXX https://doi.org/XXXX/XXXX + +Xxxx xxxx/xx/xxxxxx +8 +© xxxx IOP Publishing Ltd + +The dI/dV(V) spectra obtained from a 7ML Bi island and from a 7ML stripe are shown in Figure +4 (a) and (b) respectively. The spectra were recorded at positions marked by the colour-coded +rectangles in Figure 2 (e) and Figure 3 (b). The cyan curves in Figure 4 (a) and (b) are spectra obtained +from the bulk (i.e. interior) of the stripe and the island respectively, and share strong similarities such +as the shape of their LDOS valleys and the number and position of the main peaks [4]. Spectra from +the edges of the two structures (pink curves) also have very similar shapes. They exhibit LDOS peaks +at ~+100 mV that are absent in the spectra of bulk and thus are clearly associated with the ESs. It is +important to note that the enhancement of the LDOS can be observed over an energy window between +-200 mV and +250 mV. Hence these observations are consistent with the previous calculations [3] +suggesting that the ES dispersion spans an energy range corresponding to the energy gap of the Bi. +Note that near EF the LDOS at the edges is always higher than the corresponding bulk. The higher +energy peaks in the spectra (particularly those at +0.3 eV and -0.4 eV) observed in the bulk of 7ML +Bi appear to be suppressed at the edges, suggesting that those states are involved in the formation of +the ESs. + +Figure 4 Experimental dI/dV(V) spectra from (a) an isolated 7ML Bi island and (b) a 7ML Bi stripe +adjacent to a 5ML Bi structure. [Cartoons above the spectra illustrate the specific edges that are under +investigation; colours correspond to the experimental image in Figure 1(g).] The spectra are extracted +from the regions in the bulk (cyan) and at the edges (pink) of the structures inside the colour-coded +rectangles in Figure 2 (e) and Figure 3 (b). (c) Comparison with spectra from a 3ML Bi island, which +has a very different band structure but exhibits a similar peak at ~+150 mV – see Section 3.3. The +spectra were recorded at different positions of the island marked by the colour-coded rectangles in +Figure 6 (b). STS parameters: ±1 V, 300 pA and 50 K + +0 +0.5 +1 +-1 +-0.5 +0 +0.5 +1 +dI/dV (a.u.) +Bias (V) +0 +0.5 +1 +-1 +-0.5 +0 +0.5 +1 +dI/dV (a.u.) +Bias (V) +(a) +(b) +0 +0.5 +1 +-1 +-0.5 +0 +0.5 +1 +dI/dV (a.u.) +Bias (V) +(c) +tip +tip +tip + +SIMIl +SubstrateMI. +SIMII +Substrate/IMI +SubstrateIOP Publishing + Journal Title +Journal XX (XXXX) XXXXXX https://doi.org/XXXX/XXXX + +Xxxx xxxx/xx/xxxxxx +9 +© xxxx IOP Publishing Ltd + +3.2. 5ML Structures +Results obtained for 5ML Bi stripes are similar to those of the 7ML Bi stripes i.e. continuous ESs +are visible at ~+100 mV, which coincide with the topographical edge (Figure S4). The ESs have a +width of ~3nm, as shown in Figure S6 (c) and (d). For isolated 5ML Bi islands, however, the ESs +appear rather differently. Figure 5 (a) and (b) show a large-scale image and a magnified view +respectively, while Figure 5 (c) and (c) show the corresponding LDOS map at +150 mV and a LDOS +intensity profile drawn parallel to the edge (d). The key feature in Figure 5(c) is that the ES near EF +obviously has a modulated intensity. The modulation of the ES is clearly seen in spectra (Figure S5) +but the modulation does not seem to be periodic even in wider-scale images (not shown). This +aperiodic behaviour may be partially explained by previously published [23] atomic resolution images +which show that the reconstruction of Bi edges is not fully periodic. It is possible that the ESs could + +Figure 5 (a) STM image of an isolated 5ML Bi island. (b) Magnified view of the region inside the +red rectangle in (a) showing the edge. The coloured rectangles correspond to spectra shown in Figure +S5 (a). (c) +150 mV LDOS map corresponding to (b). (d) LDOS intensity profile drawn parallel to +the edge along the white arrow in (c). Imaging conditions: Vt = +1 V, It = 20 pA and T = 50K. STS +parameters: ±1 V, 300 pA and 50 K. +Distance (nm) +Bias (V) +5ML Bi +5 nm +(c) +(b) +200 nm +(a) +0 2 4 6 8 10 12 +Bias (V) +min +max +min +max +(d) + ++1 ++0.8 ++0.6 ++0.4 ++0.2 +0 +-0.2 +-0.4 +-0.6 +-0.8 +-1IOP Publishing + Journal Title +Journal XX (XXXX) XXXXXX https://doi.org/XXXX/XXXX + +Xxxx xxxx/xx/xxxxxx +10 +© xxxx IOP Publishing Ltd + +be better understood through an investigation of the reconstruction of dangling bonds at the edges, +which might be expected to be different for the edges of islands (where five atomic layers terminate) +and edges of stripes (where only two atomic layers terminate). However, we were unable to obtain +reliable atomic scale images of the edges; our experimental conditions, including the tips, are +optimised for the long scans required for spectroscopy, and typically the tips that provide the best +spectra are not sharp enough to obtain good atomic resolutions images. To our knowledge only one +example of an atomic resolution image of the edge reconstruction in -Bi has been reported in the +literature [23]. We note however that discontinuous ESs have previously been observed for WTe2 +systems due to structural inhomogeneities [33-35]. Another possibility is that the edge states couple +differently to the different adjacent structures (HOPG or Bi). +3.3. 3ML Structures +An STM image of a 3ML Bi island is shown in Figure 6 (a) along with magnified views of its +edges in Figure 6 (b-d). The corresponding LDOS maps at +150 mV are shown in Figure 6 (e-g). No +obvious electronic state is observed at the topographic edges, and in fact we were unable to observe +ESs for any 3ML islands, despite making measurements under a range of imaging conditions using +both lock-in and standard DC bias techniques. This is very much in contrast with Ref. [3] where ESs +were observed for 3ML structures. +Interestingly, linear features with a high LDOS at ~+150 mV are observed ~5 nm away from the +edge in Figure 6 (e) and (f) (white arrows). Isolated regions with high LDOS are also observed at +various positions (red arrows in Figure 6 (e)) and turn out to have similar spectral characteristics. The +spectra measured on the high LDOS regions (pink and red curves in Figure 4 (c), corresponding to +the colour-coded rectangles in Figure 6 (b)) exhibit a clear peak at ~+150 mV, which is completely +absent in spectra taken from other positions of the island (purple and blue curves in Figure 4 (c)). +Figure 6 (g) provides an important indication as to the origin of these linear features int eh 3ML +islands: it shows a periodic modulation of the LDOS intensity perpendicular to the edge. This +modulation is consistent with the presence of a moiré pattern (MP) [36]. A similar modulation is +shown in Figure S7 (c) over a larger spatial region for a different island. The orientation of the MP +(parallel to the edges of the island) is unusual for Bi on HOPG [37], but detailed analysis (Figure S8 +(a)) reveals that this type of the MP is in fact expected when the Bi 〈1̅10〉 direction is rotated~28° +from the HOPG 〈101̅0〉 direction. + +IOP Publishing + Journal Title +Journal XX (XXXX) XXXXXX https://doi.org/XXXX/XXXX + +Xxxx xxxx/xx/xxxxxx +11 +© xxxx IOP Publishing Ltd + +The MP causes a modulation of the strain in the -Bi structure [30] and we believe that this +induces different amounts of buckling, and hence different topologies, in each region. The position of +the peak in the LDOS at ~+150 mV agrees well with that of the topologically protected edge bands +calculated previously in Ref. [3] for freestanding 2ML α-Bi nanoribbon, and with the results presented +above for 5ML and 7ML structures. Hence it is possible that the additional features (both the high +LDOS 1D stripes and localised patches arrowed in Figure 6 (e-g)) might be associated with +topological states. Modulation of topology by a moiré pattern has previously been predicted only in a +very few special cases [38, 39] and experimentally observed in -Bi/-Bi heterostructures [11]. In + +Figure 6 (a) STM image of a 3ML Bi island on HOPG. (b-d) Magnified views of the regions inside +the white rectangles in (a) showing different edges. The coloured rectangles in (b) correspond to +spectra shown in Figure 4 (c). Imaging conditions: Vt = +1 V, It = 20 pA and T = 50 K. (e-g) +150mV +LDOS maps corresponding to (b-d) showing that no clear edge state is observed on 3ML structures; +white arrows indicate an additional extended bright LDOS feature near the edges and red arrows (in +(e)) indicate localised bright regions inside the bulk and at the edges. Pink arrows in (g) indicate +LDOS features associated with moiré fringes in the bulk of the island. STS parameters: ±0.5 V, 300 +pA and 50 K. + +50 nm +b +c +(a) +(b) +5 nm +(b) +3ML Bi +HOPG +5 nm +(c) +3ML Bi +HOPG +(e) +min +max +(f) +min +max +5 nm +min +max +(d) +(g) +d +3ML Bi +HOPG + +IOP Publishing + Journal Title +Journal XX (XXXX) XXXXXX https://doi.org/XXXX/XXXX + +Xxxx xxxx/xx/xxxxxx +12 +© xxxx IOP Publishing Ltd + +this interpretation the crests and troughs of the MP would each correspond to different topological +states, as predicted in [38]. Other interpretations are however possible, and for example it is possible +that the observed stripes correspond to the edges of linear topological regions created by the MP [39]. +In that interpretation, the MP must have twice the period observed in Figure 6 (g); Figure S8 (b) shows +that such a MP is in fact possible if the Bi lattice is allowed to expand fractionally. +4. Summary and Discussion +We used scanning tunnelling spectroscopy at ~50 K to reveal edge states for 7ML and 5ML Bi +nanostructures on HOPG substrate. ESs are observed for both islands (i.e. 5ML and 7ML Bi adjacent +to bare HOPG substrates) and stripes (i.e. 5ML Bi structures adjacent to 3ML, and 7ML structures +adjacent to 5ML). High LDOS features are only visible near the Fermi level, and are always located +at the topographic edges, and not on the flat part of the islands as expected from previous calculations +[3] ESs are found on the whole perimeter of the 7ML and 5ML Bi nanostructures including their +round ends. This insensitivity to the different crystallographic directions of the edges provides a strong +argument that the ES do not originate simply from a reconstruction (which would be expected to be +different on edges with different crystallographic facets), and might be indicative of the topological +origin of ESs, as previously reported by Lu et al. [3]. In the case of 5ML islands, the intensity of the +ESs is modulated, whereas no modulation was observed for the 5ML stripes. This appears to be +because the topological states couple differently to the adjacent structures and/or that the edge +reconstruction is different for stripes and islands. +We found no clear signature of ESs for 3ML Bi islands. Instead, we observed localised states +(high conductance regions at ~+150 mV) in small regions near the edges, and in extended regions +along the fringes of a moiré pattern. We suggest that these states most likely result from a modulation +of the local strain (by defects and by the moiré pattern) [40]. The localised states certainly do not +originate from the reconstruction of dangling bonds since they are found far from the edges. Such a +modulation of the topology could be valuable for engineering devices [38, 39]. +The absence of previously reported [3] ESs for 3ML Bi islands and the presence of localised +states with ES-like spectra can more generally be understood in terms of the sensitivity of the topology +to the level of buckling in the -Bi structure [5]. The buckling is likely to be controlled by local +interaction with the underlying HOPG, the Bi/HOPG orientation, electronic coupling to the substrate, +and the presence of grain boundaries (as well as strain and defects, as mentioned in the previous +paragraph; note also the appearance of an ES-like feature in Figure 2 (c) at a grain boundary, which +can be regarded as a chain of defects). Previous µLEED patterns for the 3ML structures exhibit weak +spots that are consistent with a small amount of buckling (or a buckling that is modulated across the + +IOP Publishing + Journal Title +Journal XX (XXXX) XXXXXX https://doi.org/XXXX/XXXX + +Xxxx xxxx/xx/xxxxxx +13 +© xxxx IOP Publishing Ltd + +island), but these spots are completely absent in thicker structures, consistent with an absence of +buckling and non-symmorphic symmetry of the lattice [4]. It also seems likely that electronic states +in the 3ML structures are more strongly influenced than thicker structures by coupling to the substrate +and charge transfer, and this might account for a higher sensitivity of 3ML ESs to perturbations. New +calculations are required to fully understand the topological states in the hybrid -Bi/HOPG system, +however such calculations are very challenging due to the lack of commensuration between the -Bi +and HOPG lattices (very large supercells are required) and the inter-relationships between the effects +of buckling, strain, substrate coupling, and the long-range periodicity of the moiré pattern. + +5. Acknowledgement +This work was supported by the MacDiarmid Institute for Advanced Materials and Nanotechnology +and the Marsden Fund (SS, TM, MLS, SAB), the National Science Centre, Poland (DEC-2015/17/B/ +ST3/02362, PJK), the National Natural Science Foundation of China (11204133, XXW) and the U.S. +National Science Foundation (NSF-DMR-1809160) and Department of Energy, Office of Science, +Office of Basic Energy Sciences, Division of Materials Science and Engineering (DE-FG02- +07ER46383) + + +xxxx-xxxx/xx/xxxxxx + +14 +© xxxx IOP Publishing Ltd + + + + + + +| 15 + + +Xxxx xxxx/xx/xxxxxx +15 +© xxxx IOP Publishing Ltd + +6. References +[1] S.-S. Li, W.-X. Ji, P. Li, S.-J. Hu, L. Cai, C.-W. Zhang, S.-S. Yan, Tunability of the Quantum Spin +Hall Effect in Bi (110) Films: Effects of Electric Field and Strain Engineering, ACS Applied Materials +& Interfaces, 9 (2017) 21515-21523. +[2] X. Liu, S. Zhang, S. Guo, B. Cai, S.A. Yang, F. Shan, M. Pumera, H. Zeng, Advances of 2D +bismuth in energy sciences, Chemical Society Reviews, 49 (2020) 263-285. +[3] Y. Lu, W. Xu, M. Zeng, G. Yao, L. Shen, M. Yang, Z. Luo, F. Pan, K. Wu, T. Das, P. He, J. Jiang, +J. Martin, Y.P. Feng, H. Lin, X.-S. Wang, Topological properties determined by atomic buckling in +self-assembled ultrathin Bi (110), Nano Letters, 15 (2015) 80-87. +[4] P.J. Kowalczyk, O. Mahapatra, S. Brown, G. Bian, X. Wang, T.-C. Chiang, Electronic size effects +in three-dimensional nanostructures, Nano Letters, 13 (2013) 43-47. +[5] P.J. Kowalczyk, S.A. Brown, T. Maerkl, Q. Lu, C.-K. Chiu, Y. Liu, S.A. Yang, X. Wang, I. +Zasada, F. Genuzio, T.O. Menteş, A. Locatelli, T.-C. Chiang, G. Bian, Realization of symmetry- +enforced two-dimensional dirac fermions in nonsymmorphic α-bismuthene, ACS Nano, 14 (2020) +1888-1894. +[6] L. Fu, C.L. Kane, E.J. Mele, Topological insulators with inversion symmetry, Physical Review B, +76 (2007) 045302. +[7] M. Wada, S. Murakami, F. Freimuth, G. Bihlmayer, Localized edge states in two-dimensional +topological insulators: Ultrathin Bi films, Physical Review B, 83 (2011) 121310. +[8] C. Niu, G. Bihlmayer, H. Zhang, D. Wortmann, S. Blügel, Y. Mokrousov, Functionalized bismuth +films: Giant gap quantum spin Hall and valley-polarized quantum anomalous Hall states, Physical +Review B, 91 (2015) 041303. +[9] T. Nagao, J. Sadowski, M. Saito, S. Yaginuma, Y. Fujikawa, T. Kogure, T. Ohno, Y. Hasegawa, +S. Hasegawa, T. Sakurai, Nanofilm Allotrope and Phase Transformation of Ultrathin Bi Film on +Si(111)−7×7, Physical Review Letters, 93 (2004) 105501. +[10] I.K. Drozdov, A. Alexandradinata, S. Jeon, S. Nadj-Perge, H. Ji, R. Cava, B.A. Bernevig, A. +Yazdani, One-dimensional topological edge states of bismuth bilayers, Nature Physics, 10 (2014) +664-669. +[11] J. Gou, L. Kong, X. He, Y.L. Huang, J. Sun, S. Meng, K. Wu, L. Chen, A.T.S. Wee, The effect +of moiré superstructures on topological edge states in twisted bismuthene homojunctions, Science +Advances, 6 (2020) eaba2773. +[12] T. Hirahara, G. Bihlmayer, Y. Sakamoto, M. Yamada, H. Miyazaki, S.-i. Kimura, S. Blügel, S. +Hasegawa, Interfacing 2D and 3D topological insulators: Bi (111) bilayer on Bi2Te3, Physical Review +Letters, 107 (2011) 166801. +[13] N. Kawakami, C.-L. Lin, M. Kawai, R. Arafune, N. Takagi, One-dimensional edge state of Bi +thin film grown on Si (111), Applied Physics Letters, 107 (2015) 031602. + + + +| 16 + + +Xxxx xxxx/xx/xxxxxx +16 +© xxxx IOP Publishing Ltd + +[14] S.H. Kim, K.-H. Jin, J. Park, J.S. Kim, S.-H. Jhi, T.-H. Kim, H.W. Yeom, Edge and interfacial +states in a two-dimensional topological insulator: Bi (111) bilayer on Bi2Te2 Se, Physical Review B, +89 (2014) 155436. +[15] F. Reis, G. Li, L. Dudy, M. Bauernfeind, S. Glass, W. Hanke, R. Thomale, J. Schäfer, R. +Claessen, Bismuthene on a SiC substrate: A candidate for a high-temperature quantum spin Hall +material, Science, 357 (2017) 287-290. +[16] A. Takayama, T. Sato, S. Souma, T. Oguchi, T. Takahashi, One-dimensional edge states with +giant spin splitting in a bismuth thin film, Physical Review Letters, 114 (2015) 066402. +[17] Z. Wang, M.-Y. Yao, W. Ming, L. Miao, F. Zhu, C. Liu, C. Gao, D. Qian, J.-F. Jia, F. Liu, +Creation of helical Dirac fermions by interfacing two gapped systems of ordinary fermions, Nature +Communications, 4 (2013) 1-6. +[18] F. Yang, L. Miao, Z. Wang, M.-Y. Yao, F. Zhu, Y. Song, M.-X. Wang, J.-P. Xu, A.V. Fedorov, +Z. Sun, G.B. Zhang, A. Liu, F. Liu, D. Qian, C.L. Gao, J.-F. Jia, Spatial and energy distribution of +topological edge states in single Bi (111) bilayer, Physical Review Letters, 109 (2012) 016801. +[19] I. Kokubo, Y. Yoshiike, K. Nakatsuji, H. Hirayama, Ultrathin Bi (110) films on Si (111) 3×3−B +substrates, Physical Review B, 91 (2015) 075429. +[20] S. Hatta, Y. Ohtsubo, S. Miyamoto, H. Okuyama, T. Aruga, Epitaxial growth of Bi thin films on +Ge (111), Applied Surface Science, 256 (2009) 1252-1256. +[21] L. Peng, J. Qiao, J.-J. Xian, Y. Pan, W. Ji, W. Zhang, Y.-S. Fu, Unusual electronic states and +superconducting proximity effect of Bi films modulated by a NbSe2 substrate, ACS Nano, 13 (2019) +1885-1892. +[22] T. Hu, X. Hui, X. Zhang, X. Liu, D. Ma, R. Wei, K. Xu, F. Ma, Nanostructured Bi grown on +epitaxial graphene/SiC, The Journal of Physical Chemistry Letters, 9 (2018) 5679-5684. +[23] J.-T. Sun, H. Huang, S.L. Wong, H.-J. Gao, Y.P. Feng, A.T.S. Wee, Energy-gap opening in a Bi +(110) nanoribbon induced by edge reconstruction, Physical Review Letters, 109 (2012) 246804. +[24] T. Märkl, P.J. Kowalczyk, M. Le Ster, I.V. Mahajan, H. Pirie, Z. Ahmed, G. Bian, X. Wang, T.- +C. Chiang, S.A. Brown, Engineering multiple topological phases in nanoscale Van der Waals +heterostructures: realisation of α-antimonene, 2D Materials, 5 (2017) 011002. +[25] X. Dong, Y. Li, J. Li, X. Peng, L. Qiao, D. Chen, H. Yang, X. Xiong, Q. Wang, X. Li, J. Duan, +J. Han, W. Xiao, Epitaxial growth and structural properties of Bi (110) thin films on TiSe2 substrates, +The Journal of Physical Chemistry C, 123 (2019) 13637-13641. +[26] P. Kowalczyk, D. Belic, O. Mahapatra, S. Brown, E. Kadantsev, T. Woo, B. Ingham, W. +Kozlowski, Anisotropic oxidation of bismuth nanostructures: Evidence for a thin film allotrope of +bismuth, Applied Physics Letters, 100 (2012) 151904. +[27] S. Scott, M. Kral, S. Brown, A crystallographic orientation transition and early stage growth +characteristics of thin Bi films on HOPG, Surface Science, 587 (2005) 175-184. + + + +| 17 + + +Xxxx xxxx/xx/xxxxxx +17 +© xxxx IOP Publishing Ltd + +[28] P. Kowalczyk, O. Mahapatra, S. Brown, G. Bian, T.-C. Chiang, STM driven modification of +bismuth nanostructures, Surface Science, 621 (2014) 140-145. +[29] P.J. Kowalczyk, D. Belić, O. Mahapatra, S. Brown, Grain boundaries between bismuth +nanocrystals, Acta Materialia, 60 (2012) 674-681. +[30] P.J. Kowalczyk, O. Mahapatra, D. Belić, S. Brown, G. Bian, T.-C. Chiang, Origin of the moiré +pattern in thin Bi films deposited on HOPG, Physical Review B, 91 (2015) 045434. +[31] P.J. Kowalczyk, O. Mahapatra, D. McCarthy, W. Kozlowski, Z. Klusek, S. Brown, STM and +XPS investigations of bismuth islands on HOPG, Surface Science, 605 (2011) 659-667. +[32] T. Märkl, S. Salehitaleghani, M. Le Ster, P.J. Kowalczyk, X. Wang, P. Wang, M. Snyder, G. +Bian, T.-C. Chiang, S.A. Brown, Antimony oxide nanostructures in the monolayer limit: self- +assembly of van der Waals-bonded molecular building blocks, Nanotechnology, 32 (2020) 125701. +[33] P. Chen, W.W. Pai, Y.-H. Chan, W.-L. Sun, C.-Z. Xu, D.-S. Lin, M. Chou, A.-V. Fedorov, T.- +C. Chiang, Large quantum-spin-Hall gap in single-layer 1T′ WSe2, Nature communications, 9 (2018) +1-7. +[34] L. Peng, Y. Yuan, G. Li, X. Yang, J.-J. Xian, C.-J. Yi, Y.-G. Shi, Y.-S. Fu, Observation of +topological states residing at step edges of WTe2, Nature communications, 8 (2017) 1-7. +[35] M.M. Ugeda, A. Pulkin, S. Tang, H. Ryu, Q. Wu, Y. Zhang, D. Wong, Z. Pedramrazi, A. Martín- +Recio, Y. Chen, F. Wang, Z.-X. Shen, S.-K. Mo, O.V. Yazyev, M.F. Crommie, Observation of +topologically protected states at crystalline phase boundaries in single-layer WSe2, Nature +communications, 9 (2018) 1-7. +[36] M. Le Ster, T. Maerkl, P.J. Kowalczyk, S.A. Brown, Moiré patterns in van der Waals +heterostructures, Physical Review B, 99 (2019) 075422. +[37] M. Le Ster, T. Märkl, S. Brown, Moiré patterns: a simple analytical model, 2D Materials, 7 +(2019) 011005. +[38] C. Hu, V. Michaud-Rioux, W. Yao, H. Guo, Moiré valleytronics: Realizing dense arrays of +topological helical channels, Physical Review Letters, 121 (2018) 186403. +[39] Q. Tong, H. Yu, Q. Zhu, Y. Wang, X. Xu, W. Yao, Topological mosaics in moiré superlattices +of van der Waals heterobilayers, Nature Physics, 13 (2017) 356-362. +[40] J. Cook, D. Halbertal, Q. Lu, M. Snyder, Y.S. Hor, D.N. Basov, G. Bian, Moiré Modulated +Lattice Strain and Thickness-Dependent Lattice Expansion in Epitaxial Ultrathin Films of PdTe2, +arXiv preprint arXiv:2206.07035, (2022). + + diff --git a/a9AyT4oBgHgl3EQfiviR/content/tmp_files/load_file.txt b/a9AyT4oBgHgl3EQfiviR/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..bb88e9b88dbe003b169d6dfc5c73f81d0cb9e28d --- /dev/null +++ b/a9AyT4oBgHgl3EQfiviR/content/tmp_files/load_file.txt @@ -0,0 +1,729 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf,len=728 +page_content='xxxx xxxx/xx/xxxxxx 1 © xxxx IOP Publishing Ltd Edge States of α-Bismuthene Nanostructures Sara Salehitaleghani1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Tobias Maerkl1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Pawel J Kowalczyk2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Maxime Le Ster2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Xiaoxiong Wang3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Guang Bian4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Tai-Chang Chiang5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' and Simon A Brown1* 1 The MacDiarmid Institute for Advanced Materials and Nanotechnology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' School of Physical and Chemical Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' University of Canterbury,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Christchurch,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' New Zealand 2 Department of Solid-State Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Faculty of Physics and Applied Informatics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' University of Lodz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Lodz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Poland 3 College of Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Nanjing University of Science and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Nanjing 210094,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' People’s Republic of China 4 Department of Physics and Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' University of Missouri,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Columbia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' MO 65211,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' United States of America 5 Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' University of Illinois at Urbana-Champaign,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' 1110 West Green Street,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Urbana,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' IL 61801-3080,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' United States of America E-mail: simon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='brown@canterbury.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='nz Received xxxxxx Accepted for publication xxxxxx Published xxxxxx Abstract We present a systematic investigation of the edge states of two-dimensional α- bismuthene (α-Bi) structures self-assembled on HOPG substrates, using scanning tunnelling microscopy and scanning tunnelling spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' The measurements are carried out for 3ML, 5ML and 7ML thick Bi structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Our spectroscopy studies reveal clear features at the edges of the 5ML and 7ML thick structures, and the positions of the edge states (ESs) coincide with the topographical step edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' In contrast, in 3ML structures the ESs appear to be absent and instead new states are sometimes observed, far from the topographical edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' These states are associated with a moiré pattern and result from strain-induced modulation of the topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Our observations demonstrate the impact on the edge states of coupling to adjacent structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Keywords: 2D Materials, α Bismuthene, Topological Edge States.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' IOP Publishing Journal Title Journal XX (XXXX) XXXXXX https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='org/XXXX/XXXX Xxxx xxxx/xx/xxxxxx 2 © xxxx IOP Publishing Ltd 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Introduction α-bismuthene (α-Bi) is a two-dimensional black-phosphorus-like allotrope of bismuth that is expected to have robust quantum spin Hall states [1], allowing spintronic applications such as long- distance spin transport and spin-to-charge conversion [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' The SOC-induced bulk band gap of α-Bi [3] is large enough to make it a candidate for topotronic devices that operate efficiently at room temperature [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' These properties originate from the band structure, which has a complex dependence on thickness that leads to interesting quantum size effects [4], and which derives from the non- symmorphic symmetry of the lattice (see Figure 1 (a, b)) [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Both α-Bi and the other (hexagonal) 2D allotrope of bismuth (β-Bi) are theoretically expected to have spin-polarized 1D edge states (ESs) [6-8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' α-Bi is energetically favoured at small thicknesses but β-Bi is generally observed at large thicknesses [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' β-Bi structures have been thoroughly investigated [10-18], and it is found the nature of the ESs varies depending on the substrate on which it is grown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' α-Bi structures were grown on several substrates (Si(111) [19], Ge(111) [20], the superconductor NbSe2 [21], graphene supported by SiC [22, 23], MoS2 [24], TiSe2 [25] and HOPG [3, 26]), but less is known about their topological properties and ESs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Calculations suggest a strong dependence of the topology on the atomic buckling at the α-Bi surface i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' the difference in height, h, between the surface atoms (compare Figure 1 (c) and (d)) [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' It was reported that for flat surfaces (h = 0, Figure 1 (d)) the ESs are topologically protected and that a nontrivial-trivial transition occurs as the buckling is increased to h ~ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='1 Å [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Here we report a comprehensive characterization of the edge states of α-Bi structures grown on HOPG, with emphasis on how the edges are affected by adjacent structures (either other Bi structures or the HOPG substrate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' ESs for 5ML and 7ML structures are similar to those reported in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' [3], including a characteristic peak at energies ~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='1 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' However, we find no evidence for ESs on 3ML structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Instead we observe states with similar characteristics to the previously reported ESs in along lines parallel to the fringes of a moiré pattern, and localised regions that are most likely associated with defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' These results indicate that the buckling of the α-Bi islands is modified locally, consistent with a topological transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' IOP Publishing Journal Title Journal XX (XXXX) XXXXXX https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='org/XXXX/XXXX Xxxx xxxx/xx/xxxxxx 3 © xxxx IOP Publishing Ltd 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Methods 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='2ML 5N purity Bi from a Knudsen cell was thermally deposited on freshly cleaved highly oriented pyrolytic graphite (HOPG) substrates at room temperature and under ultrahigh vacuum conditions (UHV, <1 × 10-9 mbar).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' The deposition rate was kept at ~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='01 Å s-1 and black-phosphorus- like α-Bi structures are always observed for mean thicknesses less than ~12ML [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Scanning tunnelling microscopy (STM) and spectroscopy (STS) measurements were carried out using a variable-temperature UHV scanning tunnelling microscope (Scienta Omicron VT-AFM XA) system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' The STS data was acquired in various bias voltage ranges (typically in the range ±1V with a setpoint current of 300 pA) and using a lock-in amplifier;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' modulation voltage and frequency were 20 mV and Figure 1 Black phosphorus structure and experimentally observed morphology for α-Bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' (a) Perspective, (b) top view, (c) side view along 〈001〉 for the buckled structure (h ~ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='1 Å [3]), (d) side view along 〈1̅10〉 for the structure with zero buckling, and (e) side view along 〈001〉 directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' (f) Large area STM image showing typical α-Bi island morphology when grown on HOPG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' The islands typically comprise 3ML thick bases (darker orange) topped by additional 2ML stripes forming 5ML Bi and 7ML Bi (yellow and white respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Isolated 5ML and 7ML structures i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' the 5ML and 7ML islands which sit directly on the HOPG substrate can occasionally occur (an example of a 5ML island is arrowed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' (g) Schematic illustration of the basic morphology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Left: 3ML island.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Centre: 5ML stripe (edge adjacent to 3ML island).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Right: 5ML island (edge adjacent to the HOPG substrate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Colours are chosen to match the image in (f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Note that island edges are always adjacent to the substrate, whereas edges of stripes are adjacent to other α-Bi layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Note also that the schematic includes an inert wetting layer which means that the observed density of states for each of the 3ML, 5ML and 7ML islands correspond to calculated band structures for 2ML, 4ML and 6ML α-Bi respectively [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' (a) (b) (d) (c) 〈 〉 〈 〉 (g) (e) (f) 150 nm5ML [3ML] SubstrateIOP Publishing Journal Title Journal XX (XXXX) XXXXXX https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='org/XXXX/XXXX Xxxx xxxx/xx/xxxxxx 4 © xxxx IOP Publishing Ltd 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='7 kHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' All spectra for bulk α-Bi structures are in reasonable agreement with those measured previously using standard DC bias techniques [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' All data reported here was obtained at low temperatures ~50 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Experimental Results α-Bi typically forms [4, 5, 24, 26, 28-32] atomically flat structures with layered “wedding cake” profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Figure 1 (a-e) show the atomic structure and Figure 1 (f, g) show the typical morphology of the islands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Detailed STM measurements [28] show that the islands comprise 3ML thick bases (see Figure 1 and caption) topped by additional 2ML stripes, so that the stripes have total thicknesses of 5ML and 7ML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' It is also occasionally possible to find isolated 5ML and 7ML islands which are not surrounded by other Bi structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' This allows us to investigate the impact of the interactions with adjacent structures on the nature of the ESs i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' to compare results from edges adjacent to another Bi structure (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' 5ML stripe next to a 3ML base) and edges adjacent to HOPG (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' 5ML island).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Previous calculations show that the band structures change with the thickness of the material [4, 5] and so it is expected that the edge states and topology could vary as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' 3ML, 5ML and 7 ML structures exhibit very different STS spectra, corresponding to the results of calculations for freestanding 2ML, 4ML and 6ML \uf061-Bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' (The difference in measured height is believed to be due to the presence of an inert wetting layer [4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Therefore, we present data for structures of each thickness separately, starting by considering examples of ESs for 7ML structures (both islands and stripes);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' we then move on to a discussion of 5ML and 3ML structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' 7ML Structures A typical isolated 7ML Bi island is shown in Figure 2 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' STS measurements were performed on the different edges of the island with magnified views shown in Figure 2 (b) to (e) and corresponding maps of the local density of states (LDOS) at +100 mV in Figure 2 (f) to (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' The edge states are immediately visible as prominent bright features (high-LDOS near the Fermi level;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' see also Figure S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' ESs are observed over the entire perimeter of the island, including the rounded ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' This particular 7ML island exhibits some unusual additional complexity that provides further information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Firstly, the grain boundary arrowed in Figure 2 (g) separates two grains of \uf061-Bi that are rotated by ~90° with respect to each other [29, 30] and therefore have different orientations with respect to the HOPG substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' The observed ESs are very similar, which is a strong indication that the ESs are not sensitive to the orientation of Bi on HOPG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Secondly, this island happened to grow over an HOPG step edge and so the interaction of lower part of the island (Figure 2 (d)) with the substrate is expected to be different to that of the upper part of the island (Figure 2 (b,c)): the signature IOP Publishing Journal Title Journal XX (XXXX) XXXXXX https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='org/XXXX/XXXX Xxxx xxxx/xx/xxxxxx 5 © xxxx IOP Publishing Ltd of the edge state remains unchanged, again suggesting the ESs are not affected by interactions with the substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' The lack of dependence on the crystallographic orientation of the edge or interaction with the substrate indicates the ESs are inherent to the edge, are robust, and therefore are likely have a topological origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' [3] Figure 3 (a) shows a 7ML Bi stripe that is surrounded by a 5ML Bi island and Figure 3 (b) shows a magnified view of the edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' The ES is consistently observed in the LDOS maps at energies ~+100 mV (see Figure 3 (c) and discussion of STS spectra below) and it is not dependent on the crystallographic orientation of the edges (see Figure S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Despite the fact that the ES is adjacent to a material with a very different band structure (5ML Bi versus HOPG) the results for the ESs are qualitatively similar to those of the isolated 7ML Bi islands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Figure 3 (d) shows an LDOS intensity profile perpendicular to the edge of the stripe (along the white line in Figure 3 (c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' The ES is clearly resolved as an additional light blue feature at ~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='1 eV (between the red dashed lines that mark the position of the topographic edge).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' The ES decays completely into the bulk of the 7ML structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Hence the ES is aligned with the topographical edge shown in the height profile in Figure 3 (e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' To illustrate this more clearly Figure 3 (f) shows a mapping of the LDOS at +100 mV (Figure 3 (c)) onto the 3D topography image (Figure 3 (b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' The intrinsic width of the ESs in our experiments cannot be exactly determined due to the tip broadening effect (see Figure S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' While the measured widths are close to those calculated for 2ML α-Bi nanoribbons (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' ~2 nm) [3], we believe that this agreement is accidental as the width is also found to correspond to the measured (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' tip-broadened) width of the topographical edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' The width is found to be slightly smaller for the stripes than for islands, simply because tip broadening is more significant for taller structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Within the range that it can be observed, the position and width of the ES do not change as a function of bias (Figure S3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' The fact that the width of the ESs always corresponds to the measured width of the topographic step suggests that the ESs are physically located right at the topographic edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Importantly, no enhancement of the DOS is observed on the flat part of the island – Figure 3 (f) shows clearly that the position of the ES corresponds to that shown schematically in Figure S2 (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' This is in strong contrast with previous calculations for α-Bi (see especially figure 4 in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' [3]) which predicted that the ESs should extend from the topographic edge into the bulk, as illustrated schematically in Figure S2 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' We emphasise that similar data were obtained for many islands and stripes of different thicknesses, and so the position of the ES is significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Similar positioning of the ES was previously reported in calculations of the edge state band structure for monolayer \uf062-Bi on α-Bi [11]: there the ES wavefunction clearly extends beyond the final atom in the overlayer in some cases and the differences in coupling lead to a modulation of the topology of the ESs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' IOP Publishing Journal Title Journal XX (XXXX) XXXXXX https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='org/XXXX/XXXX Xxxx xxxx/xx/xxxxxx 6 © xxxx IOP Publishing Ltd Figure 2 (a) Large-scale STM image of an isolated 7ML Bi island.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Imaging conditions: Vt = +1 V, It = 20 pA and T = 50 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' (b)-(e) Magnified view of the regions inside the red squares in (a) showing various edges of the island with different crystallographic directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' The coloured rectangles in (e) correspond to spectra shown in Figure 4 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' (f)-(i) Corresponding dI/dV(V) maps at +140 mV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' STS parameters: ±1 V, 300 pA and 50 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Note (i) a feature with similar characteristics to the ESs is observed at a grain boundary (arrowed in g)), (ii) even though the island grew over an HOPG step edge the signature of the edge state remains unchanged in (h) indicating that the interaction of the ES with the substrate is the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' DOS colour scale: red is high, blue is low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' (e) 10 nm (b) (f) 10 nm (c) (g) 10 nm (i) 10 nm (h) b (e) 7ML Bi 50 nm e (a) d c min max (d) 10 nm IOP Publishing Journal Title Journal XX (XXXX) XXXXXX https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='org/XXXX/XXXX Xxxx xxxx/xx/xxxxxx 7 © xxxx IOP Publishing Ltd Figure 3 (a) STM image of a 7ML Bi stripe (adjacent to a 5ML Bi island).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Imaging conditions: Vt = +1 V, It = 20 pA and T = 50 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' (b) Magnified view of the region inside the purple rectangle in (a) showing the edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' The coloured rectangles correspond to spectra shown in Figure 4 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' (c) Corresponding LDOS map at +100 mV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' (d) LDOS intensity profile drawn along the white arrow in (c), perpendicular to the edge of the 7ML Bi stripe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' (e) Corresponding height profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' (f) Mapping of the +100 mV LDOS shown in (c) onto the STM 3D-topograph of (b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' red regions exhibit the highest DOS and highlight that the ES is clearly located at the sloping edge of the stripe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' STS parameters: ±1 V, 300 pA and 50 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' +1 max (a) +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='8 (d) +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='6 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='4 国 5MLBi M +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='2 7MLBi Bias 0 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='2 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='4 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='6 100 nm -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='8 -1 0 2 4 6 8 10min Distance (nm) (b) (e) P1 5MLBi 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='8 Height (nm) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='4 7MLBi 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='2 5nm 0 0 2 4 6 8 10 Distance(nm) max (c) (f) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='8nm 18 nm 20nm minIOP Publishing Journal Title Journal XX (XXXX) XXXXXX https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='org/XXXX/XXXX Xxxx xxxx/xx/xxxxxx 8 © xxxx IOP Publishing Ltd The dI/dV(V) spectra obtained from a 7ML Bi island and from a 7ML stripe are shown in Figure 4 (a) and (b) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' The spectra were recorded at positions marked by the colour-coded rectangles in Figure 2 (e) and Figure 3 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' The cyan curves in Figure 4 (a) and (b) are spectra obtained from the bulk (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' interior) of the stripe and the island respectively, and share strong similarities such as the shape of their LDOS valleys and the number and position of the main peaks [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Spectra from the edges of the two structures (pink curves) also have very similar shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' They exhibit LDOS peaks at ~+100 mV that are absent in the spectra of bulk and thus are clearly associated with the ESs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' It is important to note that the enhancement of the LDOS can be observed over an energy window between -200 mV and +250 mV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Hence these observations are consistent with the previous calculations [3] suggesting that the ES dispersion spans an energy range corresponding to the energy gap of the Bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Note that near EF the LDOS at the edges is always higher than the corresponding bulk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' The higher energy peaks in the spectra (particularly those at +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='3 eV and -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='4 eV) observed in the bulk of 7ML Bi appear to be suppressed at the edges, suggesting that those states are involved in the formation of the ESs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Figure 4 Experimental dI/dV(V) spectra from (a) an isolated 7ML Bi island and (b) a 7ML Bi stripe adjacent to a 5ML Bi structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' [Cartoons above the spectra illustrate the specific edges that are under investigation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' colours correspond to the experimental image in Figure 1(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='] The spectra are extracted from the regions in the bulk (cyan) and at the edges (pink) of the structures inside the colour-coded rectangles in Figure 2 (e) and Figure 3 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' (c) Comparison with spectra from a 3ML Bi island, which has a very different band structure but exhibits a similar peak at ~+150 mV – see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' The spectra were recorded at different positions of the island marked by the colour-coded rectangles in Figure 6 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' STS parameters: ±1 V, 300 pA and 50 K 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='5 1 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='5 1 dI/dV (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=') Bias (V) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='5 1 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='5 1 dI/dV (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=') Bias (V) (a) (b) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='5 1 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='5 1 dI/dV (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=') Bias (V) (c) tip tip tip SIMIl SubstrateMI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' SIMII Substrate/IMI SubstrateIOP Publishing Journal Title Journal XX (XXXX) XXXXXX https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='org/XXXX/XXXX Xxxx xxxx/xx/xxxxxx 9 © xxxx IOP Publishing Ltd 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' 5ML Structures Results obtained for 5ML Bi stripes are similar to those of the 7ML Bi stripes i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' continuous ESs are visible at ~+100 mV, which coincide with the topographical edge (Figure S4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' The ESs have a width of ~3nm, as shown in Figure S6 (c) and (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' For isolated 5ML Bi islands, however, the ESs appear rather differently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Figure 5 (a) and (b) show a large-scale image and a magnified view respectively, while Figure 5 (c) and (c) show the corresponding LDOS map at +150 mV and a LDOS intensity profile drawn parallel to the edge (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' The key feature in Figure 5(c) is that the ES near EF obviously has a modulated intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' The modulation of the ES is clearly seen in spectra (Figure S5) but the modulation does not seem to be periodic even in wider-scale images (not shown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' This aperiodic behaviour may be partially explained by previously published [23] atomic resolution images which show that the reconstruction of Bi edges is not fully periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' It is possible that the ESs could Figure 5 (a) STM image of an isolated 5ML Bi island.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' (b) Magnified view of the region inside the red rectangle in (a) showing the edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' The coloured rectangles correspond to spectra shown in Figure S5 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' (c) +150 mV LDOS map corresponding to (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' (d) LDOS intensity profile drawn parallel to the edge along the white arrow in (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Imaging conditions: Vt = +1 V, It = 20 pA and T = 50K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' STS parameters: ±1 V, 300 pA and 50 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Distance (nm) Bias (V) 5ML Bi 5 nm (c) (b) 200 nm (a) 0 2 4 6 8 10 12 Bias (V) min max min max (d) +1 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='8 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='6 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='4 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='2 0 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='2 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='4 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='6 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='8 -1IOP Publishing Journal Title Journal XX (XXXX) XXXXXX https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='org/XXXX/XXXX Xxxx xxxx/xx/xxxxxx 10 © xxxx IOP Publishing Ltd be better understood through an investigation of the reconstruction of dangling bonds at the edges, which might be expected to be different for the edges of islands (where five atomic layers terminate) and edges of stripes (where only two atomic layers terminate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' However, we were unable to obtain reliable atomic scale images of the edges;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' our experimental conditions, including the tips, are optimised for the long scans required for spectroscopy, and typically the tips that provide the best spectra are not sharp enough to obtain good atomic resolutions images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' To our knowledge only one example of an atomic resolution image of the edge reconstruction in \uf061-Bi has been reported in the literature [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' We note however that discontinuous ESs have previously been observed for WTe2 systems due to structural inhomogeneities [33-35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Another possibility is that the edge states couple differently to the different adjacent structures (HOPG or Bi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' 3ML Structures An STM image of a 3ML Bi island is shown in Figure 6 (a) along with magnified views of its edges in Figure 6 (b-d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' The corresponding LDOS maps at +150 mV are shown in Figure 6 (e-g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' No obvious electronic state is observed at the topographic edges, and in fact we were unable to observe ESs for any 3ML islands, despite making measurements under a range of imaging conditions using both lock-in and standard DC bias techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' This is very much in contrast with Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' [3] where ESs were observed for 3ML structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Interestingly, linear features with a high LDOS at ~+150 mV are observed ~5 nm away from the edge in Figure 6 (e) and (f) (white arrows).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Isolated regions with high LDOS are also observed at various positions (red arrows in Figure 6 (e)) and turn out to have similar spectral characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' The spectra measured on the high LDOS regions (pink and red curves in Figure 4 (c), corresponding to the colour-coded rectangles in Figure 6 (b)) exhibit a clear peak at ~+150 mV, which is completely absent in spectra taken from other positions of the island (purple and blue curves in Figure 4 (c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Figure 6 (g) provides an important indication as to the origin of these linear features int eh 3ML islands: it shows a periodic modulation of the LDOS intensity perpendicular to the edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' This modulation is consistent with the presence of a moiré pattern (MP) [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' A similar modulation is shown in Figure S7 (c) over a larger spatial region for a different island.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' The orientation of the MP (parallel to the edges of the island) is unusual for Bi on HOPG [37], but detailed analysis (Figure S8 (a)) reveals that this type of the MP is in fact expected when the Bi 〈1̅10〉 direction is rotated~28° from the HOPG 〈101̅0〉 direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' IOP Publishing Journal Title Journal XX (XXXX) XXXXXX https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='org/XXXX/XXXX Xxxx xxxx/xx/xxxxxx 11 © xxxx IOP Publishing Ltd The MP causes a modulation of the strain in the \uf061-Bi structure [30] and we believe that this induces different amounts of buckling, and hence different topologies, in each region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' The position of the peak in the LDOS at ~+150 mV agrees well with that of the topologically protected edge bands calculated previously in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' [3] for freestanding 2ML α-Bi nanoribbon, and with the results presented above for 5ML and 7ML structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Hence it is possible that the additional features (both the high LDOS 1D stripes and localised patches arrowed in Figure 6 (e-g)) might be associated with topological states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Modulation of topology by a moiré pattern has previously been predicted only in a very few special cases [38, 39] and experimentally observed in \uf062-Bi/\uf061-Bi heterostructures [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' In Figure 6 (a) STM image of a 3ML Bi island on HOPG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' (b-d) Magnified views of the regions inside the white rectangles in (a) showing different edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' The coloured rectangles in (b) correspond to spectra shown in Figure 4 (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Imaging conditions: Vt = +1 V, It = 20 pA and T = 50 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' (e-g) +150mV LDOS maps corresponding to (b-d) showing that no clear edge state is observed on 3ML structures;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' white arrows indicate an additional extended bright LDOS feature near the edges and red arrows (in (e)) indicate localised bright regions inside the bulk and at the edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Pink arrows in (g) indicate LDOS features associated with moiré fringes in the bulk of the island.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' STS parameters: ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='5 V, 300 pA and 50 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' 50 nm b c (a) (b) 5 nm (b) 3ML Bi HOPG 5 nm (c) 3ML Bi HOPG (e) min max (f) min max 5 nm min max (d) (g) d 3ML Bi HOPG IOP Publishing Journal Title Journal XX (XXXX) XXXXXX https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='org/XXXX/XXXX Xxxx xxxx/xx/xxxxxx 12 © xxxx IOP Publishing Ltd this interpretation the crests and troughs of the MP would each correspond to different topological states, as predicted in [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Other interpretations are however possible, and for example it is possible that the observed stripes correspond to the edges of linear topological regions created by the MP [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' In that interpretation, the MP must have twice the period observed in Figure 6 (g);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Figure S8 (b) shows that such a MP is in fact possible if the Bi lattice is allowed to expand fractionally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Summary and Discussion We used scanning tunnelling spectroscopy at ~50 K to reveal edge states for 7ML and 5ML Bi nanostructures on HOPG substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' ESs are observed for both islands (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' 5ML and 7ML Bi adjacent to bare HOPG substrates) and stripes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' 5ML Bi structures adjacent to 3ML, and 7ML structures adjacent to 5ML).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' High LDOS features are only visible near the Fermi level, and are always located at the topographic edges, and not on the flat part of the islands as expected from previous calculations [3] ESs are found on the whole perimeter of the 7ML and 5ML Bi nanostructures including their round ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' This insensitivity to the different crystallographic directions of the edges provides a strong argument that the ES do not originate simply from a reconstruction (which would be expected to be different on edges with different crystallographic facets), and might be indicative of the topological origin of ESs, as previously reported by Lu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' In the case of 5ML islands, the intensity of the ESs is modulated, whereas no modulation was observed for the 5ML stripes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' This appears to be because the topological states couple differently to the adjacent structures and/or that the edge reconstruction is different for stripes and islands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' We found no clear signature of ESs for 3ML Bi islands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Instead, we observed localised states (high conductance regions at ~+150 mV) in small regions near the edges, and in extended regions along the fringes of a moiré pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' We suggest that these states most likely result from a modulation of the local strain (by defects and by the moiré pattern) [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' The localised states certainly do not originate from the reconstruction of dangling bonds since they are found far from the edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Such a modulation of the topology could be valuable for engineering devices [38, 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' The absence of previously reported [3] ESs for 3ML Bi islands and the presence of localised states with ES-like spectra can more generally be understood in terms of the sensitivity of the topology to the level of buckling in the \uf061-Bi structure [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' The buckling is likely to be controlled by local interaction with the underlying HOPG, the Bi/HOPG orientation, electronic coupling to the substrate, and the presence of grain boundaries (as well as strain and defects, as mentioned in the previous paragraph;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' note also the appearance of an ES-like feature in Figure 2 (c) at a grain boundary, which can be regarded as a chain of defects).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Previous µLEED patterns for the 3ML structures exhibit weak spots that are consistent with a small amount of buckling (or a buckling that is modulated across the IOP Publishing Journal Title Journal XX (XXXX) XXXXXX https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='org/XXXX/XXXX Xxxx xxxx/xx/xxxxxx 13 © xxxx IOP Publishing Ltd island), but these spots are completely absent in thicker structures, consistent with an absence of buckling and non-symmorphic symmetry of the lattice [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' It also seems likely that electronic states in the 3ML structures are more strongly influenced than thicker structures by coupling to the substrate and charge transfer, and this might account for a higher sensitivity of 3ML ESs to perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' New calculations are required to fully understand the topological states in the hybrid \uf061-Bi/HOPG system, however such calculations are very challenging due to the lack of commensuration between the \uf061-Bi and HOPG lattices (very large supercells are required) and the inter-relationships between the effects of buckling, strain, substrate coupling, and the long-range periodicity of the moiré pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Acknowledgement This work was supported by the MacDiarmid Institute for Advanced Materials and Nanotechnology and the Marsden Fund (SS, TM, MLS, SAB), the National Science Centre, Poland (DEC-2015/17/B/ ST3/02362, PJK), the National Natural Science Foundation of China (11204133, XXW) and the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' National Science Foundation (NSF-DMR-1809160) and Department of Energy, Office of Science, Office of Basic Energy Sciences, Division of Materials Science and Engineering (DE-FG02- 07ER46383) xxxx xxxx/xx/xxxxxx 14 © xxxx IOP Publishing Ltd | 15 Xxxx xxxx/xx/xxxxxx 15 © xxxx IOP Publishing Ltd 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' References [1] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Li, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Ji, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Li, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Hu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Cai, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Zhang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Yan, Tunability of the Quantum Spin Hall Effect in Bi (110) Films: Effects of Electric Field and Strain Engineering, ACS Applied Materials & Interfaces, 9 (2017) 21515-21523.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' [2] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Liu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Zhang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Guo, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Cai, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Yang, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Shan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Pumera, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Zeng, Advances of 2D bismuth in energy sciences, Chemical Society Reviews, 49 (2020) 263-285.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' [3] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Lu, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Xu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Zeng, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Yao, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Shen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Yang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Luo, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Pan, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Wu, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Das, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' He, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Jiang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Martin, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Feng, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Lin, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Wang, Topological properties determined by atomic buckling in self-assembled ultrathin Bi (110), Nano Letters, 15 (2015) 80-87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' [4] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Kowalczyk, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Mahapatra, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Brown, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Bian, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Wang, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Chiang, Electronic size effects in three-dimensional nanostructures, Nano Letters, 13 (2013) 43-47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' [5] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Kowalczyk, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Brown, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Maerkl, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Lu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='-K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Chiu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Liu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Yang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Wang, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Zasada, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Genuzio, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Menteş, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Locatelli, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Chiang, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Bian, Realization of symmetry- enforced two-dimensional dirac fermions in nonsymmorphic α-bismuthene, ACS Nano, 14 (2020) 1888-1894.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' [6] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Fu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Kane, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Mele, Topological insulators with inversion symmetry, Physical Review B, 76 (2007) 045302.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' [7] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Wada, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Murakami, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Freimuth, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Bihlmayer, Localized edge states in two-dimensional topological insulators: Ultrathin Bi films, Physical Review B, 83 (2011) 121310.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' [8] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Niu, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Bihlmayer, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Zhang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Wortmann, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Blügel, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Mokrousov, Functionalized bismuth films: Giant gap quantum spin Hall and valley-polarized quantum anomalous Hall states, Physical Review B, 91 (2015) 041303.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' [9] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Nagao, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Sadowski, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Saito, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Yaginuma, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Fujikawa, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Kogure, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Ohno, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Hasegawa, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Hasegawa, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Sakurai, Nanofilm Allotrope and Phase Transformation of Ultrathin Bi Film on Si(111)−7×7, Physical Review Letters, 93 (2004) 105501.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' [10] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Drozdov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Alexandradinata, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Jeon, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Nadj-Perge, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Ji, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Cava, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Bernevig, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Yazdani, One-dimensional topological edge states of bismuth bilayers, Nature Physics, 10 (2014) 664-669.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' [11] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Gou, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Kong, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' He, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Huang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Sun, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Meng, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Wu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Chen, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Wee, The effect of moiré superstructures on topological edge states in twisted bismuthene homojunctions, Science Advances, 6 (2020) eaba2773.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' [12] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Hirahara, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Bihlmayer, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Sakamoto, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Yamada, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Miyazaki, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='-i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Kimura, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Blügel, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Hasegawa, Interfacing 2D and 3D topological insulators: Bi (111) bilayer on Bi2Te3, Physical Review Letters, 107 (2011) 166801.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' [13] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Kawakami, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Lin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Kawai, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Arafune, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Takagi, One-dimensional edge state of Bi thin film grown on Si (111), Applied Physics Letters, 107 (2015) 031602.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' | 16 Xxxx xxxx/xx/xxxxxx 16 © xxxx IOP Publishing Ltd [14] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Kim, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Jin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Park, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Kim, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Jhi, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Kim, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Yeom, Edge and interfacial states in a two-dimensional topological insulator: Bi (111) bilayer on Bi2Te2 Se, Physical Review B, 89 (2014) 155436.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' [15] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Reis, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Li, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Dudy, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Bauernfeind, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Glass, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Hanke, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Thomale, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Schäfer, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Claessen, Bismuthene on a SiC substrate: A candidate for a high-temperature quantum spin Hall material, Science, 357 (2017) 287-290.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' [16] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Takayama, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Sato, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Souma, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Oguchi, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Takahashi, One-dimensional edge states with giant spin splitting in a bismuth thin film, Physical Review Letters, 114 (2015) 066402.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' [17] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Wang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Yao, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Ming, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Miao, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Zhu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Liu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Gao, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Qian, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Jia, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Liu, Creation of helical Dirac fermions by interfacing two gapped systems of ordinary fermions, Nature Communications, 4 (2013) 1-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' [18] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Yang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Miao, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Wang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Yao, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Zhu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Song, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Xu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Fedorov, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Sun, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Zhang, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Liu, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Liu, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Qian, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Gao, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Jia, Spatial and energy distribution of topological edge states in single Bi (111) bilayer, Physical Review Letters, 109 (2012) 016801.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' [19] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Kokubo, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Yoshiike, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Nakatsuji, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Hirayama, Ultrathin Bi (110) films on Si (111) 3×3−B substrates, Physical Review B, 91 (2015) 075429.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' [20] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Hatta, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Ohtsubo, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Miyamoto, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Okuyama, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Aruga, Epitaxial growth of Bi thin films on Ge (111), Applied Surface Science, 256 (2009) 1252-1256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' [21] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Peng, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Qiao, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Xian, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Pan, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Ji, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Zhang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Fu, Unusual electronic states and superconducting proximity effect of Bi films modulated by a NbSe2 substrate, ACS Nano, 13 (2019) 1885-1892.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' [22] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Hu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Hui, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Zhang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Liu, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Ma, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Wei, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Xu, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Ma, Nanostructured Bi grown on epitaxial graphene/SiC, The Journal of Physical Chemistry Letters, 9 (2018) 5679-5684.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' [23] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Sun, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Huang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Wong, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Gao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Feng, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Wee, Energy-gap opening in a Bi (110) nanoribbon induced by edge reconstruction, Physical Review Letters, 109 (2012) 246804.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' [24] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Märkl, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Kowalczyk, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Le Ster, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Mahajan, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Pirie, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Ahmed, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Bian, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Wang, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='- C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Chiang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Brown, Engineering multiple topological phases in nanoscale Van der Waals heterostructures: realisation of α-antimonene, 2D Materials, 5 (2017) 011002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' [25] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Dong, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Li, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Li, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Peng, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Qiao, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Chen, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Yang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Xiong, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Wang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Li, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Duan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Han, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Xiao, Epitaxial growth and structural properties of Bi (110) thin films on TiSe2 substrates, The Journal of Physical Chemistry C, 123 (2019) 13637-13641.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' [26] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Kowalczyk, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Belic, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Mahapatra, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Brown, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Kadantsev, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Woo, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Ingham, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Kozlowski, Anisotropic oxidation of bismuth nanostructures: Evidence for a thin film allotrope of bismuth, Applied Physics Letters, 100 (2012) 151904.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' [27] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Scott, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Kral, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Brown, A crystallographic orientation transition and early stage growth characteristics of thin Bi films on HOPG, Surface Science, 587 (2005) 175-184.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' | 17 Xxxx xxxx/xx/xxxxxx 17 © xxxx IOP Publishing Ltd [28] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Kowalczyk, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Mahapatra, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Brown, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Bian, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Chiang, STM driven modification of bismuth nanostructures, Surface Science, 621 (2014) 140-145.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' [29] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Kowalczyk, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Belić, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Mahapatra, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Brown, Grain boundaries between bismuth nanocrystals, Acta Materialia, 60 (2012) 674-681.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' [30] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Kowalczyk, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Mahapatra, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Belić, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Brown, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Bian, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Chiang, Origin of the moiré pattern in thin Bi films deposited on HOPG, Physical Review B, 91 (2015) 045434.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' [31] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Kowalczyk, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Mahapatra, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' McCarthy, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Kozlowski, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Klusek, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Brown, STM and XPS investigations of bismuth islands on HOPG, Surface Science, 605 (2011) 659-667.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' [32] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Märkl, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Salehitaleghani, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Le Ster, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Kowalczyk, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Wang, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Wang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Snyder, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Bian, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Chiang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Brown, Antimony oxide nanostructures in the monolayer limit: self- assembly of van der Waals-bonded molecular building blocks, Nanotechnology, 32 (2020) 125701.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' [33] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Chen, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Pai, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Chan, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Sun, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='-Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Xu, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Lin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Chou, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='-V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Fedorov, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='- C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Chiang, Large quantum-spin-Hall gap in single-layer 1T′ WSe2, Nature communications, 9 (2018) 1-7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' [34] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Peng, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Yuan, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Li, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Yang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Xian, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Yi, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Shi, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Fu, Observation of topological states residing at step edges of WTe2, Nature communications, 8 (2017) 1-7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' [35] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Ugeda, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Pulkin, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Tang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Ryu, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Wu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Zhang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Wong, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Pedramrazi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Martín- Recio, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Chen, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Wang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Shen, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='-K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Mo, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Yazyev, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Crommie, Observation of topologically protected states at crystalline phase boundaries in single-layer WSe2, Nature communications, 9 (2018) 1-7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' [36] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Le Ster, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Maerkl, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Kowalczyk, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Brown, Moiré patterns in van der Waals heterostructures, Physical Review B, 99 (2019) 075422.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' [37] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Le Ster, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Märkl, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Brown, Moiré patterns: a simple analytical model, 2D Materials, 7 (2019) 011005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' [38] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Hu, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Michaud-Rioux, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Yao, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Guo, Moiré valleytronics: Realizing dense arrays of topological helical channels, Physical Review Letters, 121 (2018) 186403.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' [39] Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Tong, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Yu, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Zhu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Wang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Xu, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Yao, Topological mosaics in moiré superlattices of van der Waals heterobilayers, Nature Physics, 13 (2017) 356-362.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' [40] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Cook, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Halbertal, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Lu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Snyder, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Hor, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Basov, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content=' Bian, Moiré Modulated Lattice Strain and Thickness-Dependent Lattice Expansion in Epitaxial Ultrathin Films of PdTe2, arXiv preprint arXiv:2206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} +page_content='07035, (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQfiviR/content/2301.00402v1.pdf'} diff --git a/b9E1T4oBgHgl3EQfxQXl/content/2301.03421v1.pdf b/b9E1T4oBgHgl3EQfxQXl/content/2301.03421v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..73dd01898fbc9a3a9a3f529d9c5d07661b261d61 --- /dev/null +++ b/b9E1T4oBgHgl3EQfxQXl/content/2301.03421v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:bf0979f2aa4a412463084bff7d2079f53833400acbf7f6a994a07a8a19cc9c1b +size 803896 diff --git 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we identify a new concept associated with the risk +of assessing the financial exposure by a measure that is not adequate to the ac- +tual time horizon of the position. +This will be called horizon risk. +We clarify +that dynamic risk measures are subject to horizon risk, so we propose to use +the fully-dynamic version. To quantify horizon risk, we introduce h-longevity as +an indicator. We investigate these notions together with other properties of risk +measures as normalization, restriction property, and different formulations of time- +consistency. We also consider these concepts for fully-dynamic risk measures gen- +erated by backward stochastic differential equations (BSDEs), backward stochastic +Volterra integral equations (BSVIEs), and families of these. Within this study, we +provide new results for BSVIEs such as a converse comparison theorem and the +dual representation of the associated risk measures. +Keywords: Fully-dynamic risk measures, time-consistency, BSDEs, BSVIEs, con- +verse comparison theorem for BSVIEs, dual representation, horizon risk, h-longevity +MSC2020: 60H10, 60H20, 91B70, 91G70 +1 +Introduction +Monetary risk assessment spans across time horizons with different length, from the +very short ones for trading operations to the decades-long ones typical of sovereign +wealth or pension funds. Hence the use of adequate risk measures across time as well +as a time-consistent evaluation of risk are important factors in risk quantification and +management. +We address the problem of using an appropriate risk evaluation for the given time +horizon. To explain, it is reasonable to think that it is not correct to use a risk measure +∗Department of Mathematics, University of Oslo, P.O. Box 1053 Blindern, N-0316 Oslo Norway. +Email: giulian@math.uio.no +†NHH - Norwegian School of Economics, Helleveien 30, N-5045 Bergen, Norway. +‡Department of Statistics and Quantitative Methods, University of Milano Bicocca, via Bicocca +degli Arcimboldi 8, 20126 Milano Italy. Email: emanuela.rosazza1@unimib.it +1 + +designed for long term positions to evaluate risks that occur in the short term. This is +considered a “common error” in risk quantification by some quants, see e.g. [12]. Indeed, +bearing in mind that the purpose of risk assessment and management is not avoiding +risk, but giving the rightful possibility to invest in risky assets with a reasonable control +on what is acceptable to the investor, then it is clear that this form of “horizon risk” +should be tackled. +Indeed, in this work, we investigate horizon risk and we propose a way to quantify +this via the here called horizon longevity, or h-longevity in short. For this we work with +the general framework of fully-dynamic risk measures that embeds the classical dy- +namic risk measures. Indeed, a fully-dynamic risk measure is a family of risk measures +(ρsu)0≤s≤u≤T (with T < ∞) indexed by two time parameters, the first represents the +evaluation time, the second the horizon to which the risk pertains. Instead, a dynamic +risk measure is a family of risk measures (ρs)0≤s≤T indexed by only one time parameter +representing the time of risk evaluation. So we have that +ρs = ρsT. +(1) +Whenever we consider a risk X occurring at time t ∈ [0, T] (hence Ft-measurable for +some given information flow), we can consider different risk evaluations at s, either +ρst(X) or ρsu(X), for any other horizon u > t (even much later than t), since Ft ⊆ Fu. +According to the above, it is natural to penalize the use of the “wrong” risk measure, +that is the one that does not pertain to the right time horizon. We propose h-longevity +to quantify this penalization. +The impact of the length of the time horizon on risk evaluation has already been +detected in the work [15] on epistemic uncertainty. There, it comes out that ambiguity +on the model choice is growing with the time horizon. While [15] is not related with +our considerations, it shows anyhow that the time horizon effects the precision on risk +quantification. +Another concept under investigation in this paper is time-consistency, which has +different formulations and it plays an important role in dynamic risk evaluation. See +e.g. [1, 4, 6, 7]. +In our study we highlight that both h-longevity and time-consistency are linked to +the restriction property (see [7]), that is +ρst(X) = ρsu(X), +for Ft-measurable X, +u ≥ t. +(2) +Indeed, the presence of restriction induces that the different formulations of time- +consistency are equivalent, while the absence of restriction provides the very possibility +to introduce h-longevity. The restriction property is naturally satisfied by dynamic risk +measures (ρs)s (see (1)) and this may be a reason for which the horizon risk has not been +earlier identified in its own being and hence quantified. Similarly, also normalization, +i.e. +ρsu(0) = 0, +(3) +is discovered to play a crucial role in the relationships among the concepts above. Note +that normalization is often standardly assumed in many financial risk evaluations. +2 + +In the first part of this paper, we work in full generality with an axiomatic setting. +In the second part we concentrate on risk measures induced by backward dynamics. It +is well-known the relationship between dynamic risk measures and backward stochastic +differential equations (BSDEs), see e.g. [3, 17, 21, 22, 24, 25]. Here we focus on the +Brownian framework and use a BSDE with driver g to generate a fully-dynamic risk +measure (ρsu)s,u. Then we study how the properties of the driver are connected to the +concepts of time-consistency, h-longevity, and restriction. +Since a fully-dynamic risk measure (ρsu)s,u depends on the horizon u, then we can +also generate it from a family of BSDEs with drivers G = (gu)u. +In this way it is +emphasized that each period [s, u] is associated to the BSDE with driver gu providing +the risk evaluation ρsu. In this framework we investigate again the relationships among +the concepts of interest with the family of drivers. +Recently, backward stochastic Volterra integral equations (BSVIEs) have been sug- +gested to generate dynamic risk measures, see [26, 2]. However, at that stage, some +important results on BSVIEs were still lacking. Here we provide a converse comparison +theorem for BSVIEs and a dual representation of risk measures induced by BSVIEs. +With these two results we have a full picture of the relationship between the properties +of the driver and those of the corresponding risk measures. +We recall that a BSVIE is induced by a family of BSDEs, see [27]. This has been +a further motivation to study fully-dynamic risk measures induced by a BSVIE with +Volterra driver g = g(t, s, ·), s ≥ t. As a final stage we have also considered families +of BSVIEs with Volterra drivers G = (gu)u again to emphasize the role of the time +horizon in the risk evaluation. +Working with families of backward equations has made evident the surprisingly +crucial role of the restriction property in risk measurement across time. For example, +it turns out that the only way to have a strong (recursive) time-consistency is to work +with fully-dynamic risk measures induced by a single standard BSDE. +Summarizing, in this work we have first identified horizon risk in its own being +and proposed one way to quantify it. We have hence considered fully-dynamic risk +measures and studied the different forms of time-consistency and h-longevity, showing +how the restriction property is actually playing a crucial role in these matters. Our +work has covered both the general axiomatic approach and the risk measures generated +by backward dynamics, both single and in families, both of standard and of Volterra +type. In this we have seen how the properties of the drivers reflect the properties of the +fully-dynamic risk measures in respect of the risk evaluations across time. In between +we have also provided results on BSVIEs of independent interest. +2 +Fully-dynamic risk measures, time-consistency, and hori- +zon risk +In the sequel, we will focus on fully-dynamic risk measures that have been recently +introduced by Bion-Nadal and Di Nunno [7]. +See also [5], where this concept was +actually simply called dynamic risk measure. +3 + +Definition 1 A fully-dynamic risk measure is a family (ρst)0≤s≤t≤T of risk measures +indexed by two time parameters +ρst : Lp(Ft) → Lp(Fs), with p ∈ [1, +∞], +that are monotone, convex, Fs-translation invariant, and, for p = ∞, continuous from +below. +In the definition above, continuity from below is assumed only in the case where p = ∞. +In fact, for any p ∈ [1, ∞), it is implied by the other assumptions, see [7, Remark 2.5]. +Furthermore, fully-dynamic risk measures satisfy weak Fs-homogeneity, i.e. +1Aρst(X) = 1Aρst(1AX), +for any X ∈ Lp(Ft), A ∈ Fs +(see [7, Remark 2.6]) and have the following dual representations +ρst(X) += +ess.max +Q∈Qst +{EQ[−X|Fs] − αst(Q)} +(4) += +ess.max +Q≪P : +EP [αst(Q)]<∞ +{EQ[−X|Fs] − αst(Q)} +(5) +where +Qst ≜ {Q on Ft : Q ≪ P and Q|Fs ≡ P|Fs} +(6) +and +αst(Q) ≜ ess.sup +X∈Lp(Ft) +{EQ[−X|Fs] − ρst(X)} +(7) +is the minimal penalty functional. See [7, Proposition 2.8] for details. +We stress that, in general, the risk measures in the family of the fully dynamic +risk measure are not normalized, i.e. ρst(0) ̸= 0, and the family does not satisfy the +restriction property, that is ρrt(Y ) ̸= ρrs(Y ) for Y ∈ Lp(Fs), for s ≤ t. +In view of the dynamic nature of the risk evaluation, we consider the relationships +among the inter-temporal evaluations given by the fully-dynamic risk measure. For +this, we recall here below the notions of time-consistency mostly used in the literature +and their connections expressed in the present setting. See, e.g., [1, 4, 6], among others. +• Strong time-consistency (or recursivity): for any s, t, u ∈ [0, T] with s ≤ t ≤ u, +ρst(−ρtu(X)) = ρsu(X) +for any X ∈ Lp(Fu). +(8) +Observe that this notion is equivalent to the cocycle condition on the minimal penalties: +αsu(Q) = αst(Q|Ft) + EQ[αtu(Q)|Fs], +∀s, t, u, Q ∈ Qsu, +(9) +together with the m-stability of the Radon-Nykodym derivatives associated to the cor- +responding sets of measures (Qst)s,t, i.e. the probability measure S defined by +dS +dP = dQ +dP +dR +dP , +for all Q ∈ Qst, R ∈ Qtu +(10) +4 + +belongs to Qsu. +See [5, Theorem 2.5]. +The operation in (10) is called pasting of +probability measures, also shortly denoted Q · R. Naturally, any measure S ∈ Qsu +admits a representation in the form (10). In fact, we have +dS +dP = dS|Ft +dP +dR +dP +with dR +dP = +dS +dP +EP +� dS +dP |Ft +�1A + 1Ac +where A ≜ +� +ω ∈ Ω : EP +� dS +dP |Ft +� +(ω) > 0 +� +. +• Order time-consistency: for any s, t, u ∈ [0, T] with s ≤ t ≤ u, +ρtu(X) = ρtu(Y ), +X, Y ∈ Lp(Fu) =⇒ ρsu(X) = ρsu(Y ). +(11) +We recall from Proposition 2.3 and Corollary 2.4 of Bion-Nadal and Di Nunno [7] +that, for fully-dynamic risk measures, strong time-consistency implies order time- +consistency; furthermore, for normalized fully-dynamic risk measures strong time- +consistency is equivalent to the restriction property plus order time-consistency. +Motivated by the discussion on time-consistency done by Bion-Nadal and Di Nunno [7], +where it was shown that risk indifference pricing does not satisfy strong time-consistency, +but only a weaker version, we have decided to investigate this new axiom introduced +in the work above mentioned, and that for the (non-normalized) fully-dynamic risk +measures appears in the form: +• Weak time-consistency: for any s, t, u ∈ [0, T] with s ≤ t ≤ u, +ρsu(ρtu(0) − ρtu(X)) = ρsu(X) +for any X ∈ Lp(Fu). +(12) +It is worth emphasizing that, differently from strong time-consistency, the formulation +of weak time-consistency is in terms of risk measures with the same time horizon u. +Notice that under the additional assumption of normalization (3), weak time- +consistency (12) becomes +ρsu(−ρtu(X)) = ρsu(X) +for any X ∈ Lp(Fu), +(13) +while, under both normalization and the restriction property, weak time-consistency (12) +reduces to the classical strong time-consistency (8). +Again from [7] and from the arguments above, it emerges that the restriction prop- +erty has an extremely important role on time-consistency and on risk measures. Once +fully-dynamic risk measures are considered, hence the restriction property is dropped +in general, it could be reasonable to impose the following axiom in order to take into +account the increasing cost of riskiness for longer time horizons: +• Horizon longevity or h-longevity, for short: for any fixed s ∈ [0, T], +ρst(X) ≤ ρsu(X) +for any 0 ≤ t ≤ u, X ∈ Lp(Ft). +(14) +5 + +Differently from weak time-consistency (12) that has an impact on the first time pa- +rameter, horizon longevity (14) focuses on the behavior of the second time parameter, +that is the time horizon. +Our aim is to investigate weak time-consistency and h-longevity on fully-dynamic +risk measures (ρst)0≤s≤t≤T from an axiomatic point of view as well as their impact on +the relation between fully-dynamic risk measures and BSDEs or BSVIEs. +2.1 +Time-consistency +The next results focus on weak time-consistency and on its characterizations. Also we +introduce the new concept of sub time-consistency, see Proposition 3, which will turn +out to interplay with h-longevity. +Proposition 2 A fully-dynamic risk measure (ρst)0≤s≤t≤T is weakly time-consistent if +and only if it satisfies order time-consistency. +Proof. Assume order time-consistency. Let X ∈ Lp(Fu) and t ≤ u be arbitrary. By +Ft-translation invariance of ρtu it follows +ρtu(X) = ρtu(0 − ρtu(X)) − ρtu(0) = ρtu(ρtu(0) − ρtu(X)). +(15) +Since Y ≜ ρtu(0) − ρtu(X) ∈ Lp(Ft), (15) and order time-consistency imply that +ρsu(X) = ρsu(Y ) for any s ≤ t. The thesis is therefore proved. +Conversely, suppose that ρtu(X) = ρtu(Y ) holds for some X, Y ∈ Lp(Fu) and t ≤ u. +By weak time-consistency it follows that +ρsu(X) = ρsu(ρtu(0) − ρtu(X)) = ρsu(ρtu(0) − ρtu(Y )) = ρsu(Y ) +for any s ≤ t. +Naturally, since strong time-consistency implies order time-consistency, the result above +gives that strong time-consistency implies also weak time-consistency. +Proposition 3 Let (αst)0≤s≤t≤T be the minimal penalty terms of (ρst)0≤s≤t≤T . +a) If (αru)0≤r≤u≤T satisfies, for all s ≤ t ≤ u, +αsu(S) = αsu(Q) + EQ[αtu(R) − ess.inf ¯R∈Qtu αtu( ¯R)|Fs], +(16) +for all Q ∈ Qsu, R ∈ Qtu, where S = Q|Ft · R ∈ Qsu is obtained by pasting Q on [s, t] +and R on [t, u], then (ρst)0≤s≤t≤T is weakly time-consistent. +b) Weak time-consistency of (ρst)0≤s≤t≤T implies that, for all s ≤ t ≤ u, +αsu(S) ≤ αsu(Q) + EQ[αtu(R) − ess.inf ¯R∈Qtu αtu( ¯R)|Fs], +(17) +for all Q ∈ Qsu, R ∈ Qtu, where S = Q|Ft · R ∈ Qsu is obtained by pasting Q on [s, t] +and R on [t, u]. In (17) equality holds at least for the optimal scenarios in the dual +representation (4)-(5). +6 + +It is worth emphasizing that (16) is different from the usual cocycle condition (9) +for two reasons. First, it depends on the additional term ess.inf ¯R∈Qtu αtu( ¯R), which +is due to the non-normalization of the risk measure. Second, both αsu(S) and αsu(Q) +refer to the time horizon u. Note that ess.inf ¯R∈Qtu αtu( ¯R) ∈ Lp(Ft) because of ρtu(0) = +− ess.inf ¯R∈Qtu αtu( ¯R) and, by assumption, ρtu(0) ∈ Lp(Ft). +In terms of the normalized risk measure +�ρst(X) ≜ ρst(X) − ρst(0) +and of its minimal penalty function +�αst(Q) ≜ αst(Q) − ess.inf ¯R∈Qst αst( ¯R), +condition (16) becomes +�αsu(S) = �αsu(Q) + EQ[�αtu(R)|Fs], +∀s, t, u, Q ∈ Qsu, R ∈ Qtu. +(18) +Notice also that while weak time-consistency of a fully-dynamic risk measure (ρst)0≤s≤t≤T +is equivalent to that of the corresponding normalized (�ρst)0≤s≤t≤T , the same is no more +true for h-longevity. The reason is that the normalization terms ρst(0) and ρsu(0) are +different in general. As pointed out by Bion-Nadal and Di Nunno [7], Remark 2.12, +also strong time-consistency of a fully-dynamic risk measure is not transferred to the +normalized version. +Proof of Proposition 3. +In view of the above comments, for simplicity, we prove +the result in terms of the normalized �ρ. +a) Assume that (18) holds true. By the dual representation (4) of �ρ, +�ρsu(−�ρtu(X)) += +ess.max +Q∈Qsu {EQ[�ρtu(X)|Fs] − �αsu(Q)} += +ess.max +Q∈Qsu {EQ[ess.max +R∈Qtu {ER[−X|Ft] − �αtu(R)}|Fs] − �αsu(Q)} += +ess.max +Q∈Qsu,R∈Qtu {EQ[ER[−X|Ft]|Fs] − EQ[�αtu(R)|Fs] − �αsu(Q)} +(19) += +ess.max +S=Q|Ft·R: +Q∈Qsu,R∈Qtu +{ES[−X|Fs] − �αsu(S)} = �ρsu(X), +(20) +where S is obtained by pasting Q on [s, t] and R on [t, u]. Here above, (19) follows from +the same arguments as in [6] and [11], while the first equality in (20) is due to (18) and +to m-stability (10). +b) Assume that weak time-consistency (13) holds for (�ρst)0≤s≤t≤T . On the one hand, +again by the dual representation (4), we have +�ρsu(−�ρtu(X)) += +ess.max +Q∈Qsu,R∈Qtu {EQ[ER[−X|Ft]|Fs] − EQ[�αtu(R)|Fs] − �αsu(Q)}. +(21) +7 + +On the other hand, by weak time-consistency, +�ρsu(−�ρtu(X)) = �ρsu(X) = ess.max +S∈Qsu {ES[−X|Fs] − �αsu(S)} += +ess.max +S=Q|Ft·R: +Q∈Qsu,R∈Qtu +{EQ[ER[−X|Ft]|Fs] − �αsu(S)}, +(22) +which is due to the fact that, for any S ∈ Qsu, there exist Q ∈ Qsu, R ∈ Qtu such that +S = Q|Ft · R and, vice versa, given Q ∈ Qsu, R ∈ Qtu, the pasting S = Q|Ft · R ∈ Qsu. +Since �αsu(S) is the minimal penalty function on [s, u], (21) and (22) imply that +�αsu(S) ≤ �αsu(Q) + EQ[�αtu(R)|Fs] +(23) +for any Q ∈ Qsu, R ∈ Qtu and the pasting S = Q|Ft · R. +It remains to prove that (17) holds with an equality at least for the optimal scenarios +in the dual representation. By the arguments above, it is then enough to prove the +reverse inequality in (23). Since in the dual representation (4) the ess.max is attained, +it follows that �ρsu(−�ρtu(X)) = �ρsu(X) becomes +EQ[ER[−X|Ft] − �αtu(R)|Fs] − �αsu(Q) = �ρsu(X) +(24) +for some R ∈ Qtu, Q ∈ Qsu (depending on X). By defining S as the pasting Q · R of Q +on [s, t] and R on [t, u], (24) reduces to +ES[−X|Fs] − EQ[�αtu(R)|Fs] − �αsu(Q) = �ρsu(X). +(25) +From �ρsu(X) ≥ ES[−X|Fs] − �αsu(S) and (25), it follows that �αsu(S) ≥ �αsu(Q) + +EQ[�αtu(R)|Fs]. +Remark 4 If (ρst)0≤s≤t≤T satisfies weak time-consistency, h-longevity, and ρtu(0) ≤ 0 +for any t, u, then +ρst(−ρtu(X)) ≤ ρsu(X) +for any X ∈ Lp(Fu), +(26) +called sub (strong) time-consistency in the following. Notice that ρtu(0) ≤ 0 is equiva- +lent to ess.infQ∈Qtu αtu(Q) ≥ 0. +On the other hand, sub (strong) time-consistency (26) together with ρtu(0) ≥ 0, for +any t, u, imply h-longevity. For any ¯X ∈ Lp(Ft), indeed, +ρst( ¯X) ≤ ρst( ¯X − ρtu(0)) = ρst(−ρtu( ¯X)) ≤ ρsu( ¯X), +where the first inequality is due to ρtu(0) ≥ 0 and monotonicity, the last to sub time- +consistency, while the equality follows by Ft-translation invariance. +Proceeding similarly to Acciaio and Penner [1], it is easy to prove the relation +between sub time-consistency and acceptance sets for fully-dynamic risk measures, +here formulated as follows. +8 + +Proposition 5 For any fully-dynamic risk measure (ρst)0≤s≤t≤T the following are +equivalent: +(i) Sub (strong) time-consistency, that is ρst(−ρtu(X)) ≤ ρsu(X) for any X ∈ Lp(Fu), +s ≤ t ≤ u; +(ii) Asu ⊆ Ast + Atu where Asu ≜ {Z ∈ Lp(Fu) : ρsu(Z) ≤ 0, P-a.s.} is the acceptance +set associated to ρsu; +(iii) αsu(Q) ≤ αst(Q|Ft) + EQ[αtu(Q)|Fs] for any Q ∈ Qsu, s ≤ t ≤ u. +To summarize, the following implications among the different notions of time- +consistency investigated above and h-longevity hold: +strong TC +⇕ ++ normalization and restriction +(see [7]) +order TC +⇐⇒ +weakly TC +(see Prop. 2) +⇓ ++ h-longevity and ρtu(0) ≤ 0 +(see Remark 4) +sub TC +⇓ ++ ρtu(0) ≥ 0 +(see Remark 4) +h-longevity +2.2 +Horizon risk and h-longevity +We recall that h-longevity (14) has been formulated as +ρst(X) ≤ ρsu(X) +for any 0 ≤ s ≤ t ≤ u, X ∈ Lp(Ft). +From the formulation of h-longevity it is clear that the risk measure with longer +horizon ρsu is relevant only restricted on Lp(Ft). In other words, longevity aims to +penalize the risk measurement of a position done with the wrong risk measure, i.e. +with the risk measure suitable for a longer horizon. +For this reason, in the sequel, we denote the restriction of ρsu on Lp(Ft) by ¯ρt +su, +while ¯αt +su is the corresponding minimal penalty function, i.e. +¯αt +su(Q) = ess.sup +X∈Lp(Ft) +{EQ[−X|Fs] − ¯ρt +su(X)}, +Q ∈ Qst. +It then follows that +¯αt +su(Q|Ft) ≤ αsu(Q), +Q ∈ Qsu. +9 + +In fact, +αsu(Q) += +ess.sup +X∈Lp(Fu) +{EQ[−|XFs] − ρsu(X)} +≥ +ess.sup +X∈Lp(Ft) +{EQ[−X|Fs] − ¯ρt +su(X)} = ¯αt +su(Q|Ft). +Using the notation above, h-longevity naturally leads to +¯ρt +su(X) = ρsu(X) = ρst(X) + γ(s, t, u, X) +(27) +for any 0 ≤ s ≤ t ≤ u, X ∈ Lp(Ft), and for a suitable Fs-measurable γ(s, t, u, X) ≥ 0 +that may depend on the position X, on the time of evaluation s, on the time parameter +t referring to the measurability of X, and on the “wrong” time horizon u used for +evaluating X. The term γ can be then seen as an indicator quantifying the horizon +risk or, roughly speaking, as an additive term of adjustment/calibration. It follows that, +for any s ≤ t ≤ u and X ∈ Lp(Ft), γ(s, t, t, X) = 0 and, by translation invariance of the +fully-dynamic risk measure, that γ(s, t, u, X + cs) = γ(s, t, u, X) for any cs ∈ Lp(Fs). +The restriction property (2) is then equivalent to γ(s, t, u, X) = 0 for all 0 ≤ s ≤ t ≤ u, +X ∈ Lp(Ft). +The following result characterizes longevity in terms of acceptance sets and of +penalty functions. +Proposition 6 For a fully dynamic risk measure (ρst)s,t: +(a) longevity is equivalent to Asu ∩ Lp(Ft) ⊆ Ast for any s ≤ t ≤ u. +(b) restriction is equivalent to Asu ∩ Lp(Ft) = Ast for any s ≤ t ≤ u. +(c) longevity implies that αst(Q) ≥ ¯αt +su(Q) and also +¯αt +su(Q) ≥ αst(Q) − ess.supX∈Lp(Ft) γ(s, t, u, X), +for any s ≤ t ≤ u and Q ∈ Qst. +(d) restriction implies that αst(Q) = ¯αt +su(Q) for any s ≤ t ≤ u and Q ∈ Qst. +Proof. (a) For any X ∈ Asu ∩ Lp(Ft) it follows that ρsu(X) ≤ 0 and, by longevity, +also that ρst(X) ≤ 0. Hence Asu ∩ Lp(Ft) ⊆ Ast. Conversely, from the representation +of risk measures in terms of acceptance sets and from the condition on acceptance sets, +it follows that, for any X ∈ Lp(Ft), +ρsu(X) += +ess.inf{ms ∈ Lp(Fs)| ms + X ∈ Asu} +≥ +ess.inf{ms ∈ Lp(Fs)| ms + X ∈ Ast} = ρst(X). +(b) Assume now restriction. From (27), we see that restriction can be interpreted as +h-longevity with γ(s, t, u, X) = 0. Hence, from (a), it remains to prove the reverse +inclusion for acceptance sets. +For any X ∈ Ast (hence X ∈ Lp(Ft)) we have that +10 + +ρst(X) ≤ 0. By restriction, ρsu(X) = ρst(X) ≤ 0, hence Asu ∩ Lp(Ft) ⊇ Ast. Con- +versely, assume Asu ∩ Lp(Ft) = Ast. From the representation of risk measures in terms +of acceptance sets, we obtain that, for any X ∈ Lp(Ft), +ρsu(X) += +ess.inf{ms ∈ Lp(Fs)| ms + X ∈ Asu} += +ess.inf{ms ∈ Lp(Fs)| ms + X ∈ Ast} = ρst(X). +(c) Proceeding as in F¨ollmer and Schied [14], Thm. 4.16, by translation invariance of +ρsu, we have +αst(Q) = ess.sup +X∈Ast +EQ[−X|Fs] +for any s ≤ t and Q ∈ Qst. By (a), h-longevity and translation invariance imply that, +for any Q ∈ Qst, +αst(Q) += +ess.sup +X∈Ast +EQ[−X|Fs] +≥ +ess.sup +X∈Asu∩Lp(Ft) +EQ[−X|Fs] += +ess.sup +X∈Lp(Ft): +ρsu(X)≤0 +EQ[−X|Fs] = ¯αt +su(Q). +Moreover, in terms of the indicator γ in (27), we have that +¯αt +su(Q) += +ess.sup +X∈Lp(Ft) +{EQ[−X|Fs] − ¯ρt +su(X)} += +ess.sup +X∈Lp(Ft) +{EQ[−X|Fs] − ρst(X) − γ(s, t, u, X)} +≥ +ess.sup +X∈Lp(Ft) +{EQ[−X|Fs] − ρst(X)} − ess.sup +X∈Lp(Ft) +γ(s, t, u, X) += +αst(Q) − ess.sup +X∈Lp(Ft) +γ(s, t, u, X). +(d) By the equivalence between restriction and the condition on acceptance sets in (b), +it follows that for any Q ∈ Qst +αst(Q) += +ess.sup +X∈Ast +EQ[−X|Fs] += +ess.sup +X∈Asu∩Lp(Ft) +EQ[−X|Fs] = ¯αt +su(Q). +The proof is then complete. +With (c), we see that the greater is the term ess.supX∈Lp(Ft) γ(s, t, u, X) for positions +that are Ft-measurable, the lower is the lower bound αst(Q|Ft)−ess.supX∈Lp(Ft) γ(s, t, u, X) +of αsu(Q). The term ess.supX∈Lp(Ft) γ(s, t, u, X) can be then interpreted as the maxi- +mal error (at the level of the penalty function) for a wrong use of ρsu for positions X +belonging to Lp(Ft), that is, ρs· with a wrong time horizon. +11 + +From a motivational point of view, one can imagine γ dependent on the time hori- +zon, that is, γ(s, t, u, X) = γs,t(h, X) with h = u − t or, even, γs,t(h) independent from +X. In particular, h = u − t can be interpreted as the length of the time interval over +which there is an uncorrect use of the risk measure (ρsu versus ρst). See Example 10 +and Example 24. +In what follows, we investigate whether BSDEs or BSVIEs can provide families of +fully-dynamic risk measures satisfying sub or weak time-consistency and h-longevity. +In particular, we study the cases of a single driver g and of a family of drivers (gt)t∈[0,T] +depending on the time horizon t considered in ρst. The former case will be shortened +by BSDE (g) (or BSVIE (g)), the latter by BSDE (gt) (or BSVIE (gt)). +3 +Relation with BSDEs +Hereafter, we consider a fully-dynamic risk measure induced by a single BSDE with +driver g and also measures generated by a family of BSDEs associated to the drivers +G = (gt)t∈[0,T] where the index t refers to the time horizon. +We will restrict our +attention to L2 spaces. +On the probability space (Ω, F, P) we consider a d-dimensional Brownian motion +(Bt)t∈[0,T] and the P-augmented natural filtration (Ft)t∈[0,T] of (Bt)t∈[0,T]. According +to Peng [20], the solution (Yt, Zt)t∈[0,T] of the BSDE +Yt = X + +� T +t +g(s, Ys, Zs) ds − +� T +t +Zs dBs +(28) +can be seen as an operator depending on the driver g and evaluated at the final condition +X ∈ L2(FT ). In our work, we consider a driver +g : Ω × [0, T] × R × Rd → R +satisfying the standard assumptions: +• adapted, +• uniformly Lipschitz, i.e. there exists a constant C > 0 such that, dP × dt-a.e., +|g(ω, t, y1, z1) − g(ω, t, y2, z2)| ≤ C(|y1 − y2| + |z1 − z2|), +for any y1, y2 ∈ R, z1, z2 ∈ Rd, where | · | denotes the Euclidean norm in Rk for +whatever k is relevant; +• E +�� T +0 |g(s, 0, 0)|2 ds +� +< +∞. +Under these conditions, equation (28) admits a unique solution (Yt, Zt)t∈[0,T], with +(Yt)t∈[0,T] ∈ H2 +[0,T](R) and (Zt)t∈[0,T] ∈ H2 +[0,T](Rd), where we have set +H2 +[a,b](Rk)≜ +� +adapted Rk-valued processes (ηs)s∈[a,b] :E +�� b +a +|ηs|2 ds +� +<∞ +� +. +12 + +For further details see, e.g., El Karoui et al. [13]. +In Peng [20, 21] and Rosazza Gianin [24] the relationship among BSDEs, nonlinear +expectations and dynamic risk measures is detailed clarifying that the properties of g +reflect the properties of the risk measures associated. To summarize, in the financial +context of monetary risk measures, if g in (28) does not depend on y, then the risk +measure is translation invariant. Also g convex produces a convex risk measure. See +Barrieu and El Karoui [3], Jiang [17], and Rosazza Gianin [24]. +Beyond the Lipschitz condition on the driver, the connection between BSDEs and +risk measures has been studied in terms of maximal solutions of BSDEs (see, e.g., Bar- +rieu and El Karoui [3] and Kobylanski [18]). When it comes to generalizations beyond +the Brownian framework we can refer, e.g., to Royer [25], Quenez and Sulem [22], and +Laeven and Stadje [19]. +3.1 +Risk measures generated by a single BSDE +Consider the following BSDE with driver g, not depending on y, and terminal condition +X ∈ L2(Fu): +Yt = X + +� u +t +g(s, Zs)ds − +� u +t +ZsdBs, +0 ≤ t ≤ u, +(29) +where u ∈ [0, T]. +To simplify the notation, we will occasionally adopt the Peng’s +notation where Eg (X|Ft) denotes the conditional g-expectation of X at time t, that is +the Y -component of the solution (Y, Z) at time t of the BSDE above.Then, we focus +on risk measures of the following form: +ρtu(X) = Eg (−X|Ft) , +X ∈ L2(Fu). +(30) +The condition g(t, 0) = 0, guarantees the restriction property as well as the nor- +malization (3). Hence, in our case we cannot assume g(t, 0) = 0. +Proposition 7 The following statements are equivalent: +(a) g(t, 0) = 0 for any t ∈ [0, T]; +(b) ρ is normalized, see (3); +(c) ρ has the restriction property (2). +Proof. (a) ⇒ (b). This implication follows immediately by remarking that (Yt, Zt) +with Yt = Zt = 0 for any t is the unique solution when the terminal condition is 0. See +also Peng [20]. +(b) ⇒ (a). Assume that ρtu(0) = 0 for any 0 ≤ t ≤ u ≤ T. It follows that +0 = Yt = +� u +t +g(s, Zs)ds − +� u +t +ZsdBs +holds for any 0 ≤ t ≤ u ≤ T. Hence +� u +t +g(s, Zs)ds = +� u +t +ZsdBs, +for any 0 ≤ t ≤ u ≤ T. +13 + +Since a continuous martingale and a process of finite variation can be equal only if the +martingale is constant (see Prop. 1.2 in Chapter 4 of Revuz and Yor [23]), then for +any t the martingale Mt(u) = +� u +t ZsdBs, u ≥ t, starting from 0 is identically equal to +0. Hence Zs ≡ 0 for any s ≥ t and +� u +t g(s, Zs)ds ≡ 0 for any u ≥ t. By replacing Zs +and deriving this last equation with respect to u, it follows that g(u, 0) = 0 for any u. +(a) ⇒ (c). This implication was proved in Peng [20]. Assume that X ∈ L2(Fu) and +consider ρtu(X) and ρtv(X) for any v ≥ u. +Let us denote by (Y X,u +r +, ZX,u +r +) (resp. +(Y X,v +r +, ZX,v +r +)) the solution corresponding to ρtu(X) (resp. ρtv(X)) at time r ≤ u. Since +(Y X,v +r +, ZX,v +r +) with +Y X,v +r += +� +Y X,u +r +; +r ≤ u +−X; +u < r ≤ v +ZX,v(r, s) = +� +ZX,u +r +; +s ≤ u +0; +u < s ≤ v +is a solution of ρtv(X) when g(t, 0) = 0 for any t ∈ [0, T], the restriction property +follows. +(c) ⇒ (a). By translation invariance and restriction, it holds that for any X ∈ L2(Ft) ⊆ +L2(Fu) +−X + ρtT (0) = ρtT (X) = ρtu(X) = −X + ρtu(0). +Hence ρtT (0) = ρtu(0) for any t ≤ u ≤ T. Since ρtu is induced by the BSDE (29) +via (30), it follows that ρtt(0) = 0. Consequently, ρtu(0) = 0 for any t ≤ u ≤ T. By +this the proof is complete. +Note that while normalization refers to a single risk measure ρtu, restriction involves +the whole family (ρtu)t,u. It is already known that any BSDE satisfies both weak and +strong time-consistency (see Barrieu and El Karoui [3], El Karoui et al. [13], Bion +Nadal and Di Nunno [7]). +From the arguments above, any BSDE with a driver g +such that g(t, 0) = 0 for any t ∈ [0, T] satisfies normalization, restriction and strong +time-consistency. +Here below, we provide some examples where the driver is not normalized and +(ρtu)t,u does not satisfies restriction, but only h-longevity. +Example 8 a) Consider the driver g(t, z) = a for any t ∈ [0, T], z ∈ Rd, with a ∈ +R \ {0}. It can be checked easily that, for any t ∈ [0, T] and X ∈ L2(FT ), +ρtT (X) = EP +� +−X + +� T +t +a ds +���� Ft +� += EP [−X| Ft] + (T − t)a. +This means that, for a ̸= 0, (ρtu)t,u is not normalized and does not satisfy the restriction +property. Instead, it satisfies h-longevity whenever a > 0. +b) Consider now the following driver: g(t, z) = bz + a for any t ∈ [0, T], z ∈ Rd, with +a, b ∈ R \ {0}. By Girsanov Theorem, +−dYt += (bZt + a)dt − Zt dBt += a dt − Zt dBQ +t +14 + +where E +� +dQ +dP +���Ft +� += exp +� +− 1 +2b2t + b · Bt +� +. It then follows easily that +ρtT (X) = EQ [−X| Ft] + (T − t) a. +As before, for a ̸= 0, (ρtu)t,u is not normalized and does not satisfy the restriction +property. Instead, it satisfies h-longevity whenever a = g(t, 0) > 0. +The next result provides a sufficient condition on the driver g for h-longevity of +(ρtu)t,u. +Proposition 9 If g(v, 0) ≥ 0 for any v ∈ [0, T], then h-longevity holds. Furthermore, +for all s, u ∈ [0, T], s ≤ u, γ(s, t, u, X) = E � +QX +�� u +t g(v, 0)dv|Fs +� +, s ≤ t ≤ u, X ∈ +Lp(Ft), where �QX is a probability measure on Qsu depending on X equivalent to P, +with density +d �QX +dP += exp +� +−1 +2 +� u +s +|∆zg(v)|2dv + +� u +s +∆zg(v)dBv +� +. +Here above ∆zg(v) = (∆i +zg(v))i=1,...,d and +∆i +zg(v) ≜ g(v, Zu +v ) − g(v, ¯Zt +v) +d(Zu,i +v +− ¯Zt,i +v ) +1{Zu,i +v +̸= ¯Zt,i +v }. +The probability measure �QX here above can be interpreted as an h-longevity premium +measure. +Proof. Let s ≤ t ≤ u and X ∈ Lp(Ft). The risk measures ρst(X) and ρsu(X) satisfy +the following BSDEs: +ρst(X) += +−X + +� t +s +g(v, Zt +v)dv − +� t +s +Zt +vdBv +ρsu(X) += +−X + +� u +s +g(v, Zu +v )dv − +� u +s +Zu +v dBv, +respectively. Set now the Rd-valued process +¯Zt +v = +� Zt +v; +v ≤ t +0; +t < v ≤ u ; +�Zv = Zu +v − ¯Zt +v. +Then +ρsu(X) − ρst(X) += +� u +s +[g(v, Zu +v ) − g(v, ¯Zt +v)]dv + +� u +t +g(v, ¯Zt +v)dv − +� u +s +[Zu +v − ¯Zt +v]dBv − +� u +t +¯Zt +vdBv += +� u +s +[g(v, Zu +v ) − g(v, ¯Zt +v)]dv − +� u +s +�ZvdBv + +� u +t +g(v, 0)dv += +� u +s +∆zg(v) · �Zvdv − +� u +s +�ZvdBv + +� u +t +g(v, 0)dv. +(31) +15 + +Furthermore, (31) can be rewritten also as +δρs = Γt,u + +� u +s +∆zg(v) · �Zvdv − +� u +s +�ZvdBv, +(32) +where δρs ≜ ρsu(X) − ρst(X) and Γt,u ≜ +� u +t g(v, 0)dv represents the final condition at +time u (which depends on t but not on s) of the linear BSDE (32). +Since Γt,u ≥ 0 for any t by hypothesis and ∆zg(v) ∈ H2 +[s,u](Rd) by the assumption +of g Lipschitz in z, by Prop. 2.2 of El Karoui, Peng and Quenez [13] it follows that +δρs ≥ 0 for any s ≤ t. +By applying Girsanov Theorem, (31) becomes +ρsu(X) − ρst(X) += +� u +s +∆zg(v) · �Zvdv − +� u +s +�ZvdBv + +� u +t +g(v, 0)dv += +− +� u +s +�ZvdB +� +QX +v ++ +� u +t +g(v, 0)dv, +where B +�QX +v +≜ Bv − Bs − +� v +s ∆zg(r) dr, v ∈ [s, u], is a �QX-Brownian motion. Hence, by +taking the conditional expectation with respect to �QX, +ρsu(X) − ρst(X) += +E � +QX +� +− +� u +s +�ZvdB +� +QX +v ++ +� u +t +g(v, 0)dv +���Fs +� += +E � +QX +�� u +t +g(v, 0)dv +���Fs +� +. +By assumption on g(·, 0), it follows that ρsu(X) − ρst(X) ≥ 0 and that γ(s, t, u, X) = +ρsu(X) − ρst(X) = E � +QX +�� u +t g(v, 0)dv|Fs +� +. +As discussed in Section 2.2, γ may depend on the time horizon, that is, γ(s, t, u, X) = +γs,t(h, X) with h = u − t or, even, γs,t(h) independent from X. The following example +provides some cases covering the situation above. +Example 10 Let g(v, 0) ≥ 0 for any v ∈ [0, T]. Hence, by the result above, h-longevity +holds and γ(s, t, u, X) = E � +QX +�� u +t g(v, 0)dv|Fs +� +for any X ∈ L2(Ft). +a) If g(v, 0) = c for any v ∈ [0, T], with c ≥ 0, then c is necessarily deterministic (since +it should be measurable for any v ≥ 0) and, consequently, +γ(s, t, u, X) = E � +QX +�� u +t +g(v, 0)dv +���Fs +� += (u − t)c. +Hence γ only depends on h = u − t, that is, roughly speaking, on the length of the time +interval over which there is an uncorrect use of the risk measure (ρsu versus ρst). +b) If g(v, 0) = exp(−r v) for any v ∈ [0, T], with r ≥ 0, then r is necessarily determin- +istic (for the same arguments as above) and, consequently, +γ(s, t, u, X) = e−rt � +1 − e−r(u−t)� +r +In other words, γ only depends on the “right” time horizon t (referring to the measur- +ability of X) and on the length of the time interval [t, u]. +16 + +3.2 +Risk measures generated by a family of BSDEs +Now we consider general risk measures induced by a family of BSDEs of type (28) with +drivers G = (gu)u∈[0,T] depending on the time horizon u of ρtu. +For later use, by increasing family G = (gu)u∈[0,T] it is meant, for any t ≤ u, +gt(v, y, z) ≤ gu(v, y, z) for any v ∈ [0, t], y ∈ R, z ∈ Rd. +Suppose that, for any t ≤ u, the risk measure ρtu comes from a gu-expectation with +a driver depending on the maturity u. This means that +ρtu(X) = ρG +tu(X) = Egu(−X|Ft), for any X ∈ L2(Fu). +(33) +Assume now that (gu)u∈[0,T] is a family of drivers depending on the maturity u, +independent of y, Lipschitz, and convex in z. Then the risk measure (ρG +tu)t,u defined +by (29), (33) satisfies monotonicity, convexity, continuity from above/below, and trans- +lation invariance. +Furthermore, for u ∈ [0, T], if gu(v, 0) = 0 for any v ≤ u, then +ρG +tu(0) = 0 for any t ≤ u. In general, however, gu(v, 0) = 0 for any v ≤ u does not +imply the restriction property. +In a Brownian setting, ρG +tu can be represented as +ρtu(X) = ρG +tu(X) = ess.sup +Q∈Qtu +� +EQ [−X|Ft] − αG +tu(Q) +� +, +X ∈ L2(Fu), +where Qtu is defined in (6) and the penalty term αG +tu associated to ρG +tu is given by +αG +tu(Q) = EQ +�� u +t +g∗ +u(v, qv)dv +���� Ft +� +, +where g∗ +u denotes the convex conjugate of the function gu. See Delbaen et al. [10]. +The following result shows that, for gu(r, 0) = 0 for any r, u, the restriction property +is satisfied only for risk measures induced by a BSDE with a single driver g (that is, +constant with respect to the maturity time). +Proposition 11 For any u ∈ [0, T], let gu(v, 0) = 0 for any v ≤ u and let gu(v, ·) be +continuous in v. The restriction property (2) holds if and only if gu is constant in u. +Proof. If gu is constant in u, the restriction property follows directly by Proposition 7. +Conversely, assume that the restriction property holds, i.e. ρtu(X) = ρtv(X) for any +t ≤ u ≤ v and X ∈ L2(Fu). Proceeding as in the proof of the Converse Comparison +Theorem of Briand et al. [8], Thm. 4.1, +gu(s, z) = lim +ε→0 +ρsu(−z · (Bs+ε − Bε)) +ε +gv(s, z) = lim +ε→0 +ρsv(−z · (Bs+ε − Bε)) +ε +17 + +with convergence in L2, for any z ∈ Rd, u ≤ v and s ∈ [0, u]. +By extracting a +subsequence to obtain convergence P-a.s. and passing to the limit as ε → 0, it holds +that +ρsv(−z(Bs+ε − Bε)) +ε += ρsu(−z(Bs+ε − Bε)) +ε +−→ gu(s, z), +ǫ → 0, +P − a.s. +where the equality is due to restriction. The thesis then follows because +ρsv(−z(Bs+ε − Bε)) +ε +−→ gv(s, z), +ǫ → 0, +P − a.s. +By this we end the proof. +As done in Example 8 for a single driver, we provide here below some examples +where restriction fails. +Example 12 a) Consider the driver gu(t, z) = au for any t ∈ [0, u], z ∈ Rd, with +au ∈ R \ {0} depending on the maturity u. +It can be checked easily that, for any +t ∈ [0, T] and X ∈ L2(FT ), +ρtu(X) = EP [−X| Ft] + (u − t)au. +This means that, for au ̸= 0, (ρtu)t,u is not normalized and does not satisfy the re- +striction property. Instead, it satisfies h-longevity whenever au > 0 is increasing in u. +b) Consider now the following driver: gu(t, z) = bz + au for any t ∈ [0, u], z ∈ Rd, with +au, b ∈ R \ {0} and au depending on the maturity u. Similarly to Example 8, it follows +that +ρtu(X) = EQ [−X| Ft] + (u − t)au, +where E +� +dQ +dP +��� Ft +� += exp +� +− 1 +2b2t + b · Bt +� +. As before, therefore, for au ̸= 0, (ρtu)t,u +is not normalized and does not satisfy the restriction property. +Instead, it satisfies +h-longevity whenever au > 0 is increasing in u. +3.2.1 +Time-consistency +Thanks to the Comparison Theorem (see El Karoui et al. [13]) and to the Converse +Comparison Theorem (see Briand et al. [8]), the following result establishes that mono- +tonicity of the family G = (gt)t∈[0,T] is equivalent to sub time-consistency. +Theorem 13 Let (ρtu)t,u be induced by a BSDE with a family of drivers G = (gt)t∈[0,T] +as in (33). +a) The family G is increasing if and only if (ρtu)t,u satisfies sub time-consistency. +b) G = {g} if and only if (ρtu)t,u satisfies strong time-consistency. +18 + +Proof. a) By definition (33), sub time-consistency of (ρtu)t,u can be written as +Egt(Egu(−X|Ft)|Fs) ≤ Egu(−X|Fs) +(34) +for any s ≤ t ≤ u and X ∈ L2(Fu). +By definition of Egt, the left-hand and right-hand sides of the previous equation can +be rewritten as follows: +Egt(Egu(−X|Ft)|Fs) += +Egu(−X|Ft) + +� t +s +gt(v, Zv) dv − +� t +s +Zv dBv += +−X + +� u +t +gu(v, ˜Zv) dv − +� u +t +˜Zv dBv + +� t +s +gt(v, Zv) dv − +� t +s +Zv dBv +and +Egu(−X|Fs) = −X + +� u +s +gu(v, ˆZv) dv − +� u +s +ˆZv dBv. +(35) +Set now +¯g(v, z) = +� gt(v, z); +v ∈ [0, t] +gu(v, z); +v ∈ (t, u] +¯Zv = +� Zv; +v ∈ [0, t] +˜Zv; +v ∈ (t, u] +By the arguments above, the left-hand side of (34) becomes +Egt(Egu(−X|Ft)|Fs) = −X + +� u +s +¯g(v, ¯Zv) dv − +� u +s +¯Zv dBv. +(36) +On the one hand, if ¯g ≤ gu on [0, u] (or, equivalently, gt ≤ gu on [0, t]), then by the +Comparison theorem of BSDE (see El Karoui et al. [13]) sub time-consistency (34) is +satisfied. +On the other hand, by (35) and (36) and Converse Comparison Theorem of Briand +et al. [8], sub time-consistency (34) implies that ¯g ≤ gu on [0, u], hence gt ≤ gu on [0, t]. +b) The case of time-consistency can be obtained by replacing inequalities with equalities +in the proof above. +In other words, the result above guarantees that strong time-consistency is fulfilled if +and only if the family of drivers reduces to a singleton, that is, the drivers gt do not +depend on the maturity t. +Remark 14 In the previous result, the proof of item a) can be done also by means of +the penalty term characterization. In this case, one could use +αG +su(Q) = EQ +�� u +s +g∗ +u(v, qv)dv +���� Fs +� +and the fact that, by the Fenchel-Moreau biconjugate Theorem, the condition g∗ +t ≥ g∗ +u +is equivalent to gt ≤ gu. Indeed, if g∗ +t ≥ g∗ +u then +gt(s, z) = sup +q∈Rd{q · z − g∗ +t (s, q)} ≤ sup +q∈Rd{q · z − g∗ +u(s, q)} = gu(s, z). +The converse implication can be checked similarly. +19 + +3.2.2 +H-longevity +In order to investigate h-longevity, we need to compare BSDEs with different horizons. +Hereafter, we suggest a comparison theorem for this. Note that in this result we consider +more general BSDEs (28) with driver depending on y, as the result is of interest well +beyond the use of BSDEs in risk measures modelling. +Let T1, T2 > 0 be two different time horizon with T2 > T1. +Let (Y Ti,ξi +t +, ZTi,ξi +t +)t∈[0,T1] (for i = 1, 2) be the solution of a BSDE with final condition +ξi ∈ L2(FTi) and driver gTi satisfying the usual assumptions. More precisely, +Y Ti +t += ξi + +� Ti +t +gTi(s, Y Ti +s , ZTi +s )ds − +� Ti +t +ZTi +s dBs. +(37) +In the terminology of Peng [20], Y Ti +t += EgTi(ξi +��Ft). +Theorem 15 (Horizon Comparison Theorem) If gT2(s, y, z) ≥ gT1(s, y, z) for any s ∈ +[0, T1], y ∈ R, z ∈ Rd and gT2(s, y, z) ≥ 0 for any s ∈ [T1, T2], y ∈ R, z ∈ Rd and ξ2 ≥ ξ1, +then Y T2 +t +≥ Y T1 +t +for any t ∈ [0, T1] and Y T2 +t +≥ ξ1 for any t ∈ [T1, T2]. +Proof. By (37), it follows that +Y T2 +t +− Y T1 +t += +ξ2 − ξ1 + +� T2 +t +gT2(s, Y T2 +s , ZT2 +s )ds − +� T1 +t +gT1(s, Y T1 +s , ZT1 +s )ds +− +� T2 +t +ZT2 +s dBs + +� T1 +t +ZT1 +s dBs. +(38) +Let us now extend gT1(s, y, z), Y T1 +s +and ZT1 +s +also to [T1, T2] as follows: +g(T1)(s, y, z) = +� gT1(s, y, z); +s ∈ [0, T1] +0; +s ∈ (T1; T2] +Y +T1 +s = +� Y T1 +s ; +s ∈ [0, T1] +ξ1; +s ∈ (T1; T2] ; +Z +T1 +s = +� ZT1 +s ; +s ∈ [0, T1] +0; +s ∈ (T1; T2] +For t ∈ [0, T2], equation (38) becomes then +Y T2 +t +− Y +T1 +t += ξ2 − ξ1 + +� T2 +t +� +gT2(s, Y T2 +s , ZT2 +s ) − gT1(s, Y +T1 +s , Z +T1 +s ) +� +ds − +� T2 +t +(ZT2 +s − Z +T1 +s )dBs +(39) +Set now �Ys ≜ Y T2 +s +− Y +T1 +s , �ξ ≜ ξ2 − ξ1, �Zi +s ≜ ZT2,i +s +− Z +T1,i +s +for i = 1, ..., d, and +∆sg += +gT2(s, Y +T1 +s , Z +T1 +s ) − gT1(s, Y +T1 +s , Z +T1 +s ) +∆sy += +gT2(s, Y T2 +s , ZT2 +s ) − gT2(s, Y +T1 +s , ZT2 +s ) +Y T2 +s +− Y +T1 +s +1{Y T2 +s +̸=Y +T1 +s } +(∆sz)i += +gT2(s, Y +T1 +s , ZT2 +s ) − gT2(s, Y +T1 +s , Z +T1 +s ) +d(ZT2,i +s +− Z +T1,i +s +) +1{ZT2,i +s +̸=Z +T1,i +s +} +20 + +for i = 1, ..., d. Hence (39) can be rewritten as +� +−d�Yt += +� +∆ty �Yt + ∆tz · �Zt + ∆t g +� +dt − �ZtdBt +�YT2 += �ξ +(40) +Since ∆ty ∈ H2 +[0,T](R), ∆tz ∈ H2 +[0,T](Rd) and ∆t g is in H2 +[0,T](R) by the assumption +of gTi Lipschitz, the solution of (40) exists uniquely by Prop. 2.2 of El Karoui, Peng +and Quenez [13]. Moreover, since gT2(s, y, z) ≥ gT1(s, y, z) for any s ∈ [0, T2], y, z and +�ξ ≥ 0, then �Yt ≥ 0 for any t ∈ [0, T2] which concludes the proof. +The previous result guarantees that for fully-dynamic risk measures induced by +g-expectations as in (33), h-longevity holds when gu is increasing in u and gu ≥ 0. +Proposition 16 a) If G is an increasing family of drivers and ρtu(0) ≥ 0 for any +t ≤ u, then (ρtu)t,u satisfies h-longevity. +b) If G is an increasing family of drivers and gt ≥ 0 for any t ∈ [0, T], then (ρtu)t,u +satisfies h-longevity. +Proof. a) follows by Remark 4 and Proposition 13. +b) By Theorem 15, increasing monotonicity of the family G and gt ≥ 0 for any t ∈ [0, T] +imply h-longevity. +Notice that requiring gu ≥ 0 implies that ρtu(X) ≥ EP[−X|Ft] for any X ∈ L2(Fu). +Indeed, by definition of ρtu(X) as in (33) and from gu ≥ 0, it follows that +ρtu(X) += +EP +� +−X + +� u +t +gu(v, Zv) dv − +� u +t +Zv dBv +���� Ft +� += +EP [−X|Ft] + EP +�� u +t +gu(v, Zv) dv +���� Ft +� +≥ +EP [−X|Ft] . +3.2.3 +Beyond the Lipschitz driver: the entropic case +Next, we consider the entropic risk measure, which is generated by a quadratic BSDE. +Even though the Lipschitz condition is clearly not satisfied in this case, the solution of +the BSDE is still unique. This discussion on the uniqueness of solutions is developed +in [3]. +The entropic risk measure is particularly interesting because of its explicit +connection with utility functions as investment preference criteria. In the following +example, we propose a variation to the classical entropic risk measure able to capture +h-longevity. +Example 17 In the one-dimensional case, for every u ∈ [0, T], consider the driver +gu(t, z) = bu +2 z2 + au(t) for any t ∈ [0, T], z ∈ R, with bu > 0 and au positive function. +We recall that bu is the reciprocal of the risk aversion parameter in the corresponding +21 + +utility function. By using the explicit solution of the BSDE with driver ¯gu(t, z) = bu +2 z2, +it can be checked easily that, for any t ∈ [0, T] and X ∈ L2(Fu), +ρsu(X) = 1 +bu +ln (EP [exp(−buX)| Fs]) + +� u +s +au(t)dt. +Hence, (ρsu)s,u is not normalized and does not satisfy the restriction property. Instead, +h-longevity holds whenever (bu)u and (au)u are increasing in u. See Proposition 16. +4 +Relation with BSVIEs +We have come now to consider a fully-dynamic risk measure induced by a single BSVIE +with a driver g and also measures generated by a family of BSVIEs associated to the +drivers G = (gt)t∈[0,T] where the index t refers to the time horizon. +4.1 +Risk measures generated by a single BSVIE +In the same spirit of risk measures generated by BSDEs, we can consider risk measures +induced by BSVIEs as Agram [2] and Yong [26] suggested. Indeed, this can also be +seen in connection with families of BSDEs as Yong [27] has pinpointed. +Let us consider a BSVIE of type +Yt = ξt + +� T +t +g(t, s, Z(t, s)) ds − +� T +t +Z(t, s) dBs, +(41) +where ξt ∈ L2(FT ), for all t ∈ [0, T]. The driver g : Ω × ∆ × Rd → R, with ∆[0,T] ≜ +{(t, s) ∈ [0, T] × [0, T] : s ≥ t}, is independent of y (so to guarantee translation +invariance in the application to monetary risk measures) and it is such that +• for any t ∈ [0, T], z ∈ Rd, g(t, ·, z) is adapted; +• is uniformly Lipschitz, i.e. there exists a constant C > 0 such that +|g(t, s, z1) − g(t, s, z2)| ≤ C|z1 − z2|, for any (t, s) ∈ ∆, z1, z2 ∈ Rd, P − a.s.; +• E +�� T +0 +�� T +t g(t, s, 0) ds +�2 +dt +� +< +∞. +Under there assumptions, we have a unique solution (Y·, Z(·, ·)) ∈ H2 +[0,T](R)×Z2 +[0,T](Rd), +where we have set +Z2 +[a,b](Rd) ≜ +� +Z : Ω × ∆[a,b] → Rd : +Z(t, ·) is adapted for all t and +E +�� b +a +� b +a |Z(t, s)|2 ds dt +� +< +∞ +� +. +See Yong [27], Thm. 2.2, and [26]. Also Yong [27] introduced the following family of +BSDEs parameterized by t and related to the BSVIE (41): +η(r; t, ξt) = ξt + +� T +r +¯g(v, ζ(v; t); t)dv − +� T +r +ζ(v; t)dBv, +r ∈ [t, T] +22 + +where +ζ(v; t) = Z(t, v); +¯g(v, ζ(v; t); t) = g(t, v, Z(t, v)) +and +Yt = η(t; t, ξt). +Although η(·; t, X) and Yt are closely related, Yt, solution of the BSVIE (41) may satisfy +some specific properties, since it corresponds not to the whole family η(·; t, X), but only +to η(t; t, X). +As in Section 3, we assume convex drivers that guarantee convex risk measures +satisfying monotonicity and translation invariance. See Yong [26] and Agram [2], who +also makes an extension to jump dynamics. Then we focus on monetary risk measures +of the type +ρtu(X) = Eg,V (−X|Ft) , +X ∈ L2(Fu), +(42) +where Eg,V (−X|Ft) denotes the Y -component of the solution of the BSVIE of type (41) +with terminal condition ξt = −X, for all t. +As shown in the following result, for a driver independent on y the condition +g(t, u, 0) = 0, for any t ≤ u, guarantees the restriction property as well as the nor- +malization. Hence, in our case we cannot assume g(t, u, 0) = 0. +Proposition 18 The following statements are equivalent: +(a) g(t, u, 0) = 0 for any 0 ≤ t ≤ u ≤ T; +(b) ρ is normalized, see (3): +(c) ρ satisfies the restriction property (2). +Proof. (a) ⇒ (b) The implication is true because (Y·, , Z(·, ·)) with Yt = Z(t, s) = 0, +s ≥ t is the unique solution corresponding to X = 0. +(b) ⇒ (a). Assume that ρtu(0) = 0 for any 0 ≤ t ≤ u ≤ T. we have +0 = Yt = +� u +t +g(t, s, Z(t, s))ds − +� u +t +Z(t, s)dBs +holds for any 0 ≤ t ≤ u ≤ T. Hence +� u +t +g(t, s, Z(t, s))ds = +� u +t +Z(t, s)dBs, +for any 0 ≤ t ≤ u ≤ T. +As in the proof of Proposition 7, it follows that for any fixed t the martingale Mt(u) = +� u +t Z(t, s)dBs, with u ≥ t, starting from 0 should be identically to 0. Hence Z(t, s) ≡ 0 +for any s ≥ t and +� u +t g(t, s, 0)ds ≡ 0 for any u ≥ t. By deriving this last equation with +respect to u, it follows that g(t, u, 0) = 0 for any u ≥ t. +(a) ⇒ (c). Assume that X ∈ L2(Fu) and consider ρtu(X) and ρtv(X) for any v ≥ u. Let +us denote by (Y X,u +r +, ZX,u(r, s)) (resp. (Y X,v +r +, ZX,v(r, s))) the solution corresponding to +ρtu(X) (resp. ρtv(X)) at time r ≤ u. Since (Y X,v +r +, ZX,v(r, s)) with +Y X,v +r += +� +Y X,u +r +; +r ≤ u +−X; +u < r ≤ v +ZX,v(r, s) = +� ZX,u(r, s); +s ≤ u +0; +u < s ≤ v +23 + +is a solution of ρtv(X) whether g(t, u, 0) = 0 for any 0 ≤ t ≤ u ≤ T, the restriction +property follows. +(c) ⇒ (a) The argument is similar to the proof of the corresponding implication in +Proposition 7. +Note that, for all t, the set of probability measures Qt on Fu such that +dQt +dP ≜ exp +�1 +2 +� u +t +|q(t, s)|2ds − +� u +t +q(t, s)dBs +� +, +(43) +for some stochastic process (q(t, s))t≤s≤u ∈ H2 +[t,u](Rd) coincides with Qtu in a Brownian +setting (see Revuz and Yor [23], Chapter VIII, Prop. 1.6). +Theorem 19 (Dual representation) For any t ≤ u fixed, the risk measure ρtu induced +by a BSVIE as in (42) has the following dual representation +ρtu(X) = ess.sup +Qt∈Qtu +{EQt [−X|Ft] − αtu(Qt)} , +(44) +where Qt corresponds to (q(t, s))t≤s≤u via (43), g∗(t, s, ·) denotes the convex conjugate +of g(t, s, ·) and the penalty functional is given by +αtu(Qt) = EQt +�� u +t +g∗(t, s, q(t, s))ds +���� Ft +� +. +Proof. The present proof extends a similar one of Barrieu and El Karoui [3] to the +case of BSVIEs. Let t, u ∈ [0, T] with t ≤ u be fixed arbitrarily. The evaluation of +risk (42) can be written as +Yt = −X + +� u +t +[g(t, s, Z(t, s)) − q(t, s) · Z(t, s)]ds − +� u +t +Z(t, s)(dBs − q(t, s)ds) += −X + +� u +t +[g(t, s, Z(t, s)) − q(t, s) · Z(t, s)]ds − +� u +t +Z(t, s)dBQt +s +and, by Girsanov Theorem, BQt +u +≜ Bu − Bt − +� u +t q(t, s)ds, u ≥ t, is a Brownian motion +with respect to Qt (see (43)) with initial value BQt +t += 0. Set now +g∗(t, s, q) ≜ ess.sup +z∈Rd {q · z − g(t, s, z)}, q ∈ Rd, +for s, t such that s ≥ t. +We obtain that, for any fixed t, g∗(t, ·, q(t, ·)) is adapted. +Furthermore, +Yt += +−X − +� u +t +[q(t, s) · Z(t, s) − g(t, s, Z(t, s))]ds − +� u +t +Z(t, s)dBQt +s +≥ +−X − +� u +t +g∗(t, s, q(t, s))ds − +� u +t +Z(t, s)dBQt +s , +24 + +where the last inequality follows by the definition of g∗. By taking the conditional +expectation with respect to Qt, it follows +Yt ≥ EQt [−X|Ft] − EQt +�� u +t +g∗(t, s, q(t, s))ds +���� Ft +� +, +since EQt[ +� u +t Z(t, s)dBQt +s |Ft] = 0 because (BQt +s )s≥t is a Brownian motion with respect +to Qt. Consequently, +Yt ≥ ess.sup +Qt∈Qtu +� +EQt +� +− X|Ft +� +− EQt +� � u +t +g∗(t, s, q(t, s))ds +��Ft +�� +. +(45) +It remains to prove the converse inequality in (45). In a similar way as in Theorem +7.4 (i) and Lemma 7.5 of Barrieu and El Karoui [3], for t fixed and any Z(t, s), s ≥ t, +there exist some progressively measurable qZ(t, s) (associated to a Qt,Z via (43)) such +that g(t, s, Z(t, s)) = qZ(t, s) · Z(t, s) − g∗(t, s, qZ(t, s)). Hence, +Yt += +−X + +� u +t +g(t, s, Z(t, s))ds − +� u +t +Z(t, s)dBs += +−X + +� u +t +[qZ(t, s) · Z(t, s) − g∗(t, s, qZ(t, s))]ds − +� u +t +Z(t, s)dBs += +−X − +� u +t +g∗(t, s, qZ(t, s))ds − +� u +t +Z(t, s)dBQt,Z +s +, +where BQt,Z +s += Bs−Bt− +� s +t qZ(t, r)dr, t ≤ s ≤ u. By taking the conditional expectation +with respect to Qt,Z, it holds that +Yt += +EQt,Z +� +−X − +� u +t +g∗(t, s, qZ(t, s))ds +���� Ft +� +≤ +ess.sup +Qt∈Qtu +� +EQt [−X|Ft] − EQt +�� u +t +g∗(t, s, q(t, s))ds +���� Ft +�� +. +(46) +The thesis follows then by (45) and (46). +4.1.1 +Time-consistency +The result below provides a necessary and sufficient condition on g for sub time- +consistency of (ρtu)t,u induced by a BSVIE (g). +Theorem 20 Let (ρtu)t,u be induced by a BSVIE with driver g as in (42). +a) The driver g(t, v, z) is decreasing in t for any v ∈ [t, u], z ∈ Rd (meaning that, for +any s ≤ t, g(t, v, z) ≤ g(s, v, z)) if and only if (ρtu)t,u satisfies sub time-consistency. +b) If the driver g(t, ·, ·) is constant in t, then (ρtu)t,u satisfies time-consistency. +25 + +Proof. +a) Assume that sub time-consistency holds. By the dual representation of +(ρtu)t,u in Proposition 19, the penalty term of ρtu is given by +αtu(Qt) = EQt +�� u +t +g∗(t, v, q(t, v))dv +���� Ft +� +for any Qt ∈ Qtu. Let s, t, u ∈ [0, T] with s ≤ t ≤ u and let Qs ∈ Qst, Qt ∈ Qtu be fixed +arbitrarily. Set now ¯Q = Qs · Qt the pasting of Qs on [s, t] and of Qt on [t, u], hence +¯Q ∈ Qsu. Denote by q(s, v), q(t, v) and ¯q(s, v) the corresponding processes as in (43). +By applying the characterization of penalty term of sub time-consistent risk measures +(see Proposition 5(iii)), we obtain +E ¯Q +� � u +s +g∗(s, v, ¯q(s, v))dv +���Fs +� +≤ E ¯Q +� � t +s +g∗(s, v, ¯q(s, v))dv +���Fs +� ++ E ¯Q +� +E ¯Q +� � u +t +g∗(t, v, ¯q(t, v))dv +���Ft +����Fs +� +, +hence +E ¯Q +� � u +t +g∗(s, v, ¯q(s, v))dv +���Fs +� +≤ E ¯Q +� +E ¯Q +� � u +t +g∗(t, v, ¯q(t, v))dv +���Ft +����Fs +� +EQs +� +EQt +� � u +t +g∗(s, v, ¯q(s, v))dv +���Ft +����Fs +� +≤ EQs +� +EQt +� � u +t +g∗(t, v, ¯q(t, v))dv +���Ft +����Fs +� +EQs +� +EQt +� � u +t +� +g∗(t, v, ¯q(t, v)) − g∗(s, v, ¯q(s, v)) +� +dv +���Ft +����Fs +� +≥ 0. +(47) +Since (47) should hold for any s ≤ t ≤ u and any Qs ∈ Qst, Qt ∈ Qtu, it follows that +g∗(t, v, ¯q(t, v)) ≥ g∗(s, v, ¯q(s, v)) +for any s ≤ t, v ∈ [t, u], ¯q ∈ Rd. +Hence, g(t, ·, ·) ≤ g(s, ·, ·) for any s ≤ t. +Conversely, assume that g(t, ·, ·) is decreasing in t, in the meaning specified above. +We are going to prove that +ρst(−ρtu(X)) = Eg,V (Eg,V (−X|Ft)|Fs) ≤ Eg,V (−X|Fs) = ρsu(X) +(48) +for any s ≤ t ≤ u and X ∈ L2(Fu). By definition of Eg,V , the left-hand and right-hand +sides of the previous equation can be rewritten as follows: +Eg,V (Eg,V (−X|Ft)|Fs) += +Eg,V (−X|Ft) + +� t +s +g(s, v, Z(s, v)) dv − +� t +s +Z(s, v) dBv += +−X + +� u +t +g(t, v, ˜Z(t, v) dv − +� u +t +˜Z(t, v) dBv + +� t +s +g(s, v, Z(s, v)) dv − +� t +s +Z(s, v) dBv += +−X + +� u +s +� +g(s, v, Z(s, v))1[s,t](v) + g(t, v, ˜Z(t, v))1(t,u](v) +� +dv +− +� u +s +� +Z(s, v)1[s,t](v) + ˜Z(t, v)1(t,u](v) +� +dBv +(49) +26 + +and +Eg,V (−X|Fs) = −X + +� u +s +g(s, v, ˆZ(s, v) dv − +� u +s +ˆZ(s, v) dBv. +(50) +Furthermore, by decreasing monotonicity of g(t, ·, ·) in t and by (49) it follows that +Eg,V (Eg,V (−X|Ft)|Fs) +≤ +−X + +� u +s +� +g(s, v, Z(s, v))1[s,t](v) + g(s, v, ˜Z(t, v))1(t,u](v) +� +dv +− +� u +s +� +Z(s, v)1[s,t](v) + ˜Z(t, v)1(t,u](v) +� +dBv. +Set now +¯Zt(s, v) = Z(s, v)1[s,t](v) + ˜Z(t, v)1(t,u](v), +v ≥ s. +By the arguments above, +Eg,V (Eg,V (−X|Ft)|Fs) ≤ −X + +� u +s +g(s, v, ¯Zt(s, v) dv − +� u +s +¯Zt(s, v) dBv. +(51) +Sub time-consistency (48) then follows by comparing (50) and (51) and by the unique- +ness of the solution of a BSVIE. +b) The case of time-consistency can be obtained by replacing inequalities with equalities +in the proof above. +Notice that a BSVIE of the form (50), that is with a final condition X that does +not depend on t and with a driver g independent of y, reduces to a BSDE when the +driver g(t, v, z) is constant in t. +An alternative proof of item a) could be given in terms of a Converse Comparison +Theorem for BSVIEs (similarly to Theorem 4.1 of Briand et al.[8] for BSDEs) if that +result was available. However, while for BSDEs the Converse Comparison Theorem +has been proved by Briand et al. [8], for BSVIEs we are not aware of any similar result. +We prove below that a Converse Comparison Theorem holds also for BSVIEs. +We recall that a BSVIE (41) is related to the following family of BSDEs parame- +terized by t: +η(r; t, ξt) = ξt + +� T +r +¯g(v, ζ(v; t); t)dv − +� T +r +ζ(v; t)dBv, +r ∈ [t, T] +(52) +where +ζ(v; t) = Z(t, v); +¯g(v, ζ(v; t); t) = g(t, v, Z(t, v)) +and +Yt = η(t; t, ξt). +We remark again that the drivers above do not depend on y. In this way, translation +invariance of the associated risk measure is not guaranteed. We are then able to prove +a Converse Comparison Theorem for BSVIEs. +27 + +Theorem 21 (Converse Comparison Theorem for BSVIEs) Let Y 1 +t +and Y 2 +t +be two +BSVIEs as in (41) with drivers g1 and g2 and terminal condition ξt and let η1 and η2 +be the corresponding families defined in (52). Assume that gi(·, v, ·) is continuous in v +for i = 1, 2. +a) If η1(r; t, ξt) ≤ η2(r; t, ξt) for any t ∈ [0, T], r ∈ [t, T] and ξt ∈ L2(FT ), then g1 ≤ g2, +that is g1(t, v, z) ≤ g2(t, v, z) for any t ∈ [0, T], v ∈ [t, T], z ∈ Rd. +b) If Y 1,X +t +≤ Y 2,X +t +for any t ∈ [0, T] and X ∈ L2(FT ) and if gi(t, v, 0) = 0 for i = 1, 2, +then g1 ≤ g2. +We recall that, for BSDEs, it is sufficient to require that Eg1(X) ≤ Eg2(X) holds +for any X ∈ L2(FT ) to guarantee that Eg1 +t (X) ≤ Eg2 +t (X) for any X ∈ L2(FT ). This is +mainly due to time-consistency of g-expectations. See Theorem 4.4 of Briand et al. [8] +for details. The proof of item b) is based on the argument above. +Proof. a) By (52), η1(·; t) and η2(·; t) are two BSDEs parameterized by t. By applying +the Converse Comparison Theorem for BSDEs (see Theorem 4.1 of Briand et al. [8]) +to η1(·; t), η2(·; t), it follows that ¯g1(v, ζ; t) ≤ ¯g2(v, ζ; t) for any v ≥ r ≥ t and ζ ∈ Rd. +Hence g1(t, v, z) ≤ g2(t, v, z) for any t ∈ [0, T], v ∈ [t, T], z ∈ Rd. +b) First of all, we prove that, given two BSDEs with drivers g1 and g2 satisfying +continuity in time and gi(v, 0) = 0 and a fixed t, +Eg1(X|Ft) ≤ Eg2(X|Ft) for any X ∈ L2(FT ) +⇒ Eg1(X|Fr) ≤ Eg2(X|Fr) for any X ∈ L2(FT ) and r ∈ [t, T]. +(53) +This implication extends Theorem 4.4 of Briand et al. [8] when t = 0. Assume now +that, for a given fixed t, Eg1(X|Ft) ≤ Eg2(X|Ft) for any X ∈ L2(FT ). By using a +similar approach to those in Theorem 4.4 of Briand et al. [8] and Lemma 4.5 of Coquet +et al. [9], +0 = Eg1 � +X − Eg1(X|Fr) +��� Ft +� +≤ +Eg2 � +X − Eg1(X|Fr) +��� Ft +� += +Eg2 � +Eg2 � +X − Eg1(X|Fr) +��� Fr +���� Ft +� += +Eg2 � +Eg2(X|Fr) − Eg1(X|Fr) +��� Ft +� +for any X ∈ L2(FT ) and r ∈ [t, T], where the first and the last equality are due +to translation invariance. Fix now r ∈ [t, T] arbitrarily and consider ξ = X1A with +A = {Eg2(X|Fr) < Eg1(X|Fr)} ∈ Fr. On the one hand, by the arguments above, +Eg2 � +Eg2(ξ|Fr) − Eg1(ξ|Fr) +��� Ft +� +≥ 0. +On the other hand, +Eg2 � +Eg2(ξ|Fr) − Eg1(ξ|Fr) +��� Ft +� += Eg2 � +1A +� +Eg2(X|Fr) − Eg1(X|Fr) +���� Ft +� +≤ Eg2(0|Ft) = 0, +28 + +because Egi(X1A|Fr) = 1AEgi(X|Fr) for any A ∈ Fr holds because of normalization +of gi (see Peng [20]) and the last equality is due to g2(t, v, 0) = 0. Hence, +Eg2 � +1A +� +Eg2(X|Fr) − Eg1(X|Fr) +���� Ft +� += 0. +Since 1A +� +Eg2(X|Fr) − Eg1(X|Fr) +� +≤ 0 and strictly negative with probability equal to +P(A), strict monotonicity of conditional g-expectation implies that +1A +� +Eg2(X|Fr) − Eg1(X|Fr) +� += 0, +P − a.s., +then Eg2(X|Fr) ≤ Eg1(X|Fr), P-a.s., for r ∈ [t, T]. This concludes the proof of (53). +Going back to BSVIEs, denote by ηt,X +i,r += ηi(r; t, X) and by ¯gt +i = ¯gi(·, ·; t) for +i = 1, 2, where η and ¯g are defined above. +Fix now t arbitrarily. +Assuming that +Y 1,X +t +≤ Y 2,X +t +for any X ∈ L2(FT ) is equivalent to assuming that ηt,X +1,t ≤ ηt,X +2,t or, also, +to E ¯gt +1(X|Ft) ≤ E ¯gt +2(X|Ft) for any X ∈ L2(FT ). By (53), it follows that Y 1,X +t +≤ Y 2,X +t +for any X ∈ L2(FT ) implies that E ¯gt +1(X|Fr) ≤ E ¯gt +2(X|Fr) for any r ∈ [t, T], hence +ηt,X +1,r +≤ ηt,X +2,r for any r ∈ [t, T]. +The thesis then follows by item a) where it is not +necessary to have ξt but it is enough to consider X ∈ L2(FT ). +4.1.2 +H-longevity +The next results investigate under which conditions on the driver g h-longevity of +(ρtu)t,u is fulfilled. +Corollary 22 If g(t, ·, ·) is decreasing in t and g(t, v, 0) ≥ 0 for any t ≤ v, then +h-longevity holds. +Proof. Since g(t, ·, ·) is decreasing in t, sub time-consistency follows by Theorem 20. +We then show here ρtu(0) ≥ 0 for any t ≤ u, so that we can conclude by Remark 4. +We prove now that g(t, v, 0) ≥ 0 for any t ≤ v implies that ρtu(0) ≥ 0 for any t ≤ u. +Indeed, +ρtu(0) = Yt = +� u +t +g(t, v, Z(t, v))dv − +� u +t +Z(t, v)dv. +Since the solution is (Yt = +� u +t g(t, v, 0)dv; Z(t, v) = 0), then ρtu(0) ≥ 0 by the assump- +tion g(t, v, 0) ≥ 0 for any t ≤ v. +In reality, we can do something more. +Proposition 23 If g(s, v, 0) ≥ 0 for any s ≤ v, then h-longevity holds. Furthermore, +γ(s, t, u, X) = E � +Qs,X +�� u +t g(s, v, 0)dv|Fs +� +for any s ≤ t ≤ u, where �Qs,X is a suitable +probability measure depending on X, with density +d �Qs,X +dP +≜ exp +� +−1 +2 +� u +s +|∆zg(s, v)|2dv + +� u +s +∆zg(s, v)dBv +� +. +29 + +Here above ∆zg(v) = (∆i +zg(v))i=1,...,d and +∆i +zg(v) ≜ g(s, v, Zu(s, v)) − g(s, v, ¯Zt(s, v)) +d +� +Zu,i(s, v) − ¯Zt,i(s, v) +� +1{Zu,i(s,v)̸= ¯Zt,i(s,v)}. +Proof. Let s ≤ t ≤ u and let X ∈ L2(Ft) be fixed arbitrarily. The risk measures ρst +and ρsu satisfy, respectively, the following BSVIEs: +ρst(X) += +−X + +� t +s +g(s, v, Zt(s, v))dv − +� t +s +Zt(s, v)dBv +ρsu(X) += +−X + +� u +s +g(s, v, Zu(s, v))dv − +� u +s +Zu(s, v)dBv. +Set now +¯Zt(s, v) = +� Zt(s, v); +s ≤ v ≤ t +0; +t < v ≤ u ; +�Z(s, v) = Zu(s, v) − ¯Zt(s, v). +Then +ρsu(X) − ρst(X) += +� u +s +[g(s, v, Zu(s, v)) − g(s, v, ¯Zt(s, v))]dv + +� u +t +g(s, v, ¯Zt(s, v))dv +− +� u +s +[Zu(s, v) − ¯Zt(s, v)]dBv − +� u +t +¯Zt(s, v)dBv += +� u +s +[g(s, v, Zu(s, v)) − g(s, v, ¯Zt(s, v))]dv + +� u +t +g(s, v, 0)dv +− +� u +s +�Z(s, v)dBv += +� u +s +∆zg(s, v) · �Z(s, v)dv − +� u +s +�Z(s, v)dBv + +� u +t +g(s, v, 0)dv. +(54) +Furthermore, (54) can be rewritten as a linear BSVIE +δρs = Γt,u +s ++ +� u +s +∆zg(s, v) · �Z(s, v)dv − +� u +s +�Z(s, v)dBv, +(55) +where δρs ≜ ρsu(X) − ρst(X) and Γt,u +s +≜ +� u +t g(s, v, 0)dv represents the final condition +at time u (which depends on t and also on s). +Since Γt,u ≥ 0 for any t by hypothesis and ∆zg(s, v) ∈ H2 +[0,T](Rd) by the assumption +of g Lipschitz in z, by the Comparison Theorem on BSVIEs (see Corollary 3.3 of Wang +and Yong [29]) it follows the longevity, i.e. +δρs ≥ 0 for any s ≤ t. +Moreover, by +applying Girsanov Theorem, (55) becomes +δρs += +� u +t +g(s, v, 0)dv + +� u +s +∆zg(s, v) · �Z(s, v)dv − +� u +s +�Z(s, v)dBv += +� u +t +g(s, v, 0)dv − +� u +s +�Z(s, v)dB +� +Qs,X +v +, +30 + +where B +� +Qs,X +v +≜ Bv − Bs − +� v +s ∆zg(s, v)dv, v ≥ s, is a Brownian motion with respect to +�Qs,X with initial value B +�Qs,X +s += 0. Hence, by taking the conditional expectation with +respect to �Qs,X, +γ(s, t, u, X) = δρs = E � +Qs,X +�� u +t +g(s, v, 0)dv +���� Fs +� +. +It then follows that ρsu(X) − ρst(X) = E � +Qs,X +�� u +t g(s, v, 0)dv|Fs +� +. +As discussed in Sections 2.2 and 3.1, γ may depend on the length of the time interval +[u, t], that is, γ(s, t, u, X) = γs,t(h, X) with h = u−t or, even, γs,t(h) independent from +X. The following example provides some cases covering the situation above. +Example 24 Let g(s, v, 0) ≥ 0 for any v ∈ [s, T]. +Hence, by the result above, h- +longevity holds and γ(s, t, u, X) = E � +Qs,X +�� u +t g(s, v, 0)dv|Fs +� +for any X ∈ L2(Ft). +a) If g(s, v, 0) = cs for any v ∈ [s, T], with cs ≥ 0, then cs is necessarily Fs- +measurable (since it should be measurable for any v ≥ s) and, consequently, +γ(s, t, u, X) = E � +Qs,X +�� u +t +g(s, v, 0)dv +���� Fs +� += (u − t)cs. +In other words, γ only depends on the evaluation time s and on h = u − t, that is, +roughly speaking, on the length of the time interval over which there is an uncorrect use +of the risk measure (ρsu versus ρst). +b) If g(s, v, 0) = exp(−rs v) for any v ∈ [0, T], with rs ≥ 0, then rs is necessarily +Fs-measurable (for the same arguments as above) and, consequently, +γ(s, t, u, X) = e−rst � +1 − e−rs(u−t)� +rs +. +Hence γ depends on the evaluation time s, on the “right” time horizon t (referring to +the measurability of X) and on the length of the time interval [t, u]. Compared to the +BSDE case (see Example 10), here γ depends also on the evaluation time s. This is +not surprising for BSVIEs. +Finally, we provide two examples of BSVIEs: the former with a linear driver, the +latter going beyond the Lipschitz case and similarly to Section 3.2.3. +Example 25 Consider the driver g(t, s, z) = a(t, s) · z + b(t, s) for any 0 ≤ t ≤ s ≤ T +and z ∈ Rd, where the Rd-valued process a(t, s) and 1-dimensional process b(t, s) are +given. By applying Girsanov Theorem, in the same line of Hu and Øksendal [16], the +BSVIE associated to the linear driver above becomes +Yt += +−X + +� T +t +[a(t, s) · Z(t, s) + b(t, s)]ds − +� T +t +Z(t, s)dBs += +−X + +� T +t +b(t, s)ds − +� T +t +Z(t, s)dB +� +Qt +s +31 + +where d � +Qt +dP ≜ exp +� +− 1 +2 +� T +t |a(t, s)|2ds + +� T +t a(t, s)dBs +� +and B +� +Qt +u +≜ Bu−Bt− +� u +t a(t, s)ds, +for u ≥ t, is a Brownian motion with respect to �Qt with initial value B +�Qt +t += 0. Hence, +by taking the conditional expectation with respect to �Qt, it holds that +Yt = E � +Qt +� +−X + +� T +t +b(t, s)ds +���� Ft +� +. +Chosing b(t, s) ≥ 0, the h-longevity holds. +Example 26 Consider the driver g(t, s, z) = b(t)|z|2 +2 + a(t, s) for any 0 ≤ t ≤ s ≤ T +and z ∈ Rd, where the deterministic function b is positive and the process a is given. +Hence +Yt += +−X + +� T +t +� +b(t)|Z(t, s)|2 +2 ++ a(t, s) +� +ds − +� T +t +Z(t, s)dBs += +−X + +� T +t +a(t, s)ds + +� T +t +b(t)|Z(t, s)|2 +2 +ds − +� T +t +Z(t, s)dBs +and, following the same arguments of [28], Example 3.1, it follows that +Yt = +1 +b(t) ln EP +� +exp +� +−b(t) +� +X − +� T +t +a(t, s)ds +�� ���Ft +� +. +Whenever a(t, s) is deterministic, Yt becomes +Yt = +1 +b(t) ln EP +� +e−b(t)X���Ft +� ++ +� T +t +a(t, s)ds, +that is a translation of the usual entropic risk measure. Choosing a(t, s) > 0, h-longevity +holds. +4.2 +Risk measures generated by a family of BSVIEs +Suppose that, for any t ≤ u, the risk measure ρtu comes from a BSVIE with a driver +gu depending on the maturity u. This means that +ρG +tu(X) = Egu,V (−X|Ft) , +X ∈ L2(Fu), +(56) +where Egu,V (ξ|Ft) denotes the Y -component of the solution (Yt, Z(t, s))t,s∈[0,T],s≥t of +the following BSVIE with driver gu: +Yt = ξ + +� u +t +gu(t, s, Z(t, s))ds − +� u +t +Z(t, s)dBs. +(57) +Assume now that G = (gu)u∈[0,T] is a family of drivers depending on the maturity u, +independent of y, Lipschitz, and convex in z. Each risk measure ρG +tu is of type (42). +32 + +By applying Theorem 19 with a driver gu parameterized by u, in a Brownian setting +ρG +tu can be represented as +ρtu(X) = ρG +tu(X) = ess.sup +Qt∈Qtu +� +EQt [−X|Ft] − αG +tu(Qt) +� +, +X ∈ L2(Fu), +where Qtu is defined in (43), g∗ +u(t, s, ·) denotes the convex conjugate of gu(t, s, ·) and +the minimal penalty functional is given by +αG +tu(Qt) = EQt +�� u +t +g∗ +u(t, s, q(t, s))ds +���� Ft +� +. +(58) +Furthermore, if gu(t, s, 0) = 0 for any t ≤ s ≤ u then ρtu(0) = 0 for any u. Differently +from the risk measures generated by a single BSVIE but similarly to those generated +by a family of BSDEs, in general gu(t, s, 0) = 0 for any t ≤ s ≤ u does not imply the +restriction property. +The following result shows that, for gu(t, s, 0) = 0 for any t, s, u, the restriction +property is satisfied only for risk measures induced by a single BSVIE. This result is +not surprising in view of Proposition 11 for the case of BSDEs with a family of drivers. +Proposition 27 Let gu(t, s, 0) = 0 for any t, s, u and let gu(·, s, ·) be continuous in s. +The restriction property (2) holds if and only if gu is constant in u. +Proof. +Assume that the restriction property holds, i.e. +ρtu(X) = ρtv(X) for any +t ≤ u ≤ v and X ∈ L2(Fu). Similarly to (52), denote by +ηt,X +ru = −X + +� u +r +¯gt +u(s, ζt +v)dv − +� u +r +ζt +vdBv, +r ∈ [t; u] +where +ζt +v = Z(t, v); +¯gt +u(v, ζt +v) = g(t, v, Z(t, v)) +and +ρtu(X) = ηt,X +tu . +Assumptions on gu guarantee that ¯gt +u(s, 0) = 0 for any s and that ¯gt +u(s, ·) is continuous +in s. Proceeding as in the proof of the Converse Comparison Theorem of Briand et +al. [8], Thm. 4.1, +¯gt +u(s, z) = lim +ε→0 +ηt,z·(Bs+ε−Bε) +su +ε +¯gt +v(s, z) = lim +ε→0 +ηt,z·(Bs+ε−Bε) +sv +ε +, +with convergence in L2, for any z ∈ Rd, u ≤ v and s ∈ [0, u]. +By extracting a +subsequence to obtain convergence P-a.s. and passing to the limit as ε → 0, it holds +that +ηt,z·(Bs+ε−Bε) +sv +ε += ηt,z·(Bs+ε−Bε) +su +ε +−→ ¯gt +u(s, z), +ǫ → 0, P-a.s. +where the equality is due to restriction. The thesis then follows because +ηt,z·(Bs+ε−Bε) +sv +ε +−→ ¯gt +v(s, z), +ǫ → 0, P-a.s. +The converse follows immediately by Proposition 18. +33 + +4.2.1 +Time-consistency +The following result provides a necessary and sufficient condition for a fully-dynamic +risk measure induced by a family of BSVIEs to satisfy sub time-consistency. Note that +the condition on the monotonicity of g·(t, ·, ·) is the same as for a BSVIE with a single +driver, while the condition on the monotonicity of the family gu is new. +Proposition 28 Let (ρtu)t,u be induced by a BSVIE with a family of drivers (gu)u∈[0,T] +as in (56). +a) (ρtu)t,u satisfies sub time-consistency if and only if both the family G is increasing +and g·(t, ·, ·) is decreasing in t. +b) (ρtu)t,u satisfies time-consistency if and only if G={g} and g·(t, ·, ·) is constant in t. +Proof. a) Assume sub time-consistency holds. By (58), the penalty term in the dual +representation of ρtu is given by +αG +tu(Qt) = EQt +�� u +t +g∗ +u(t, v, q(t, v))dv +���� Ft +� +for any Qt ∈ Qtu. Let s, t, u ∈ [0, T] with s ≤ t ≤ u and let Qs ∈ Qst, Qt ∈ Qtu be fixed +arbitrarily. Set now ¯Q the pasting of Qs on [s, t] and of Qt on [t, u], hence ¯Q ∈ Qsu. +Denote by q(s, v), q(t, v) and ¯q(s, v) the corresponding processes as in (43). From the +characterization of the penalty term for sub time-consistency in Proposition 5(iii) it +follows that +E ¯Q +�� u +s +g∗ +u(s, v, ¯q(s, v))dv +���� Fs +� +≤ E ¯Q +�� t +s +g∗ +t (s, v, ¯q(s, v))dv +���� Fs +� ++ E ¯Q +� +E ¯Q +�� u +t +g∗ +u(t, v, ¯q(t, v))dv +���� Ft +����� Fs +� +, +hence +0 ≤ EQs +�� t +s +[g∗ +t (s, v, ¯q(s, v)) − g∗ +u(s, v, ¯q(s, v))] dv +���� Fs +� ++EQs +� +EQt +�� u +t +[g∗ +u(t, v, ¯q(t, v)) − g∗ +u(s, v, ¯q(s, v))] dv +���� Ft +����� Fs +� +. +(59) +Since (59) should hold for any s ≤ t ≤ u and any Qs ∈ Qst, Qt ∈ Qtu, it follows that +� g∗ +t (s, v, ¯q) ≥ g∗ +u(s, v, ¯q), +for any s ≤ v ≤ t ≤ u and ¯q ∈ Rd +g∗ +u(t, v, ¯q) ≥ g∗ +u(s, v, ¯q), +for any s ≤ t ≤ v ≤ u and ¯q ∈ Rd +Hence, gu is increasing in u and g·(t, ·, ·) is decreasing in t. +Conversely, sub time-consistency of (ρtu)t,u induced by a family of BSVIEs can be +written in the following notation +Egt,V (Egu,V (−X|Ft)|Fs) ≤ Egu,V (−X|Fs) +(60) +34 + +for any s ≤ t ≤ u and X ∈ L2(Fu). By (57), the right-hand and left-hand sides of the +previous equation can be rewritten, respectively, as follows: +Egu,V (−X|Fs) = −X + +� u +s +gu(s, v, ˆZ(s, v)) dv − +� u +s +ˆZ(s, v) dBv +(61) +and +Egt,V (Egu,V (−X|Ft)|Fs) += +−X + +� u +t +gu(t, v, ˜Z(t, v)) dv − +� u +t +˜Z(t, v) dBv ++ +� t +s +gt(s, v, Z(s, v)) dv − +� t +s +Z(s, v) dBv += +−X + +� u +s +� +gt(s, v, Z(s, v))1[s,t](v) + gu(t, v, ˜Z(t, v))1(t,u](v) +� +dv +− +� u +s +� +Z(t, v)1[s,t](v) + ˜Z(t, v)1(t,u](v) +� +dBv. +(62) +Furthermore, (62) becomes +Egt,V (Egu,V (−X|Ft)|Fs) +≤ +−X + +� u +s +� +gt(s, v, Z(s, v))1[s,t](v) + gu(s, v, ˜Z(t, v))1(t,u](v) +� +dv +− +� u +s +� +Z(t, v)1[s,t](v) + ˜Z(t, v)1(t,u](v) +� +dBv +≤ +−X + +� u +s +� +gu(s, v, Z(s, v))1[s,t](v) + gu(s, v, ˜Z(t, v))1(t,u](v) +� +dv +− +� u +s +� +Z(t, v)1[s,t](v) + ˜Z(t, v)1(t,u](v) +� +dBv, +(63) +where the former inequality is due to decreasing monotonicity of g·(t, ·, ·), the latter +from increasing monotonicity of the family of drivers. By setting +¯Z(s, v) = Z(s, v)1[s,t](v) + ˜Z(t, v)1(t,u](v), +(63) becomes +Egt(Egu(−X|Ft)|Fs) ≤ −X + +� u +s +gu(s, v, ¯Z(s, v)) dv − +� u +s +¯Z(s, v) dBv. +(64) +Sub time-consistency then follows by comparing (64) and (61) and by the uniqueness +of the solution of a BSVIE. +b) The case of time-consistency can be obtained by replacing inequalities with equalities +in the proof above. +35 + +4.2.2 +H-longevity +The following result provides sufficient conditions for h-longevity, similarly to Propo- +sition 16 for BSDEs. +Proposition 29 a) If G is an increasing family of drivers, g·(t, ·, ·) is decreasing in t, +and ρtu(0) ≥ 0 for any t ≤ u, then (ρtu)t,u satisfies h-longevity. +b) If G is an increasing family of drivers and gu ≥ 0 for any u, then h-longevity holds. +Proof. a) follows by Remark 4 and Proposition 28. b) Let s ≤ t ≤ u and X ∈ L2(Ft) +be fixed arbitrarily. The risk measures ρst and ρsu satisfy, respectively, the following +BSVIEs: +ρst(X) += +−X + +� t +s +gt(s, v, Zt(s, v))dv − +� t +s +Zt(s, v)dBv +ρsu(X) += +−X + +� u +s +gu(s, v, Zu(s, v))dv − +� u +s +Zu(s, v)dBv. +Set now +¯Zt,u(s, v) = +� Zt(s, v); +s ≤ v ≤ t +0; +t < v ≤ u ; +�Z(s, v) = Zu(s, v) − ¯Zt,u(s, v). +Then +δρs += +ρsu(X) − ρst(X) += +� u +s +[gu(s, v, Zu(s, v)) − gt(s, v, ¯Zt,u(s, v))]dv + +� u +t +gt(s, v, 0)dv +− +� u +s +[Zu(s, v) − ¯Zt,u(s, v)]dBv +≥ +� u +s +[gu(s, v, Zu(s, v)) − gu(s, v, ¯Zt,u(s, v))]dv − +� u +s +�Z(s, v)dBv += +� u +s +∆zgu(s, v) · �Z(s, v)dv − +� u +s +�Z(s, v)dBv, +(65) +where the inequality is due to the hypothesis on the drivers and where +∆i +zgu(s, v) ≜ gu(s, v, Zu(s, v)) − gu(s, v, ¯Zt,u(s, v)) +d +� +Zu,i(s, v) − ¯Zt,u,i(s, v) +� +1{Zu,i(s,v)̸= ¯Zt,u,i(s,v)} +for i = 1, ..., d. By Girsanov Theorem, (65) becomes +δρs +≥ +− +� u +s +�Z(s, v)dB +� +Qs +v , +where d � +Qs +dP +≜ exp +� +− 1 +2 +� u +s |∆zgu(s, v)|2dv + +� u +s ∆zgu(s, v)dBv +� +and B +� +Qs +v +≜ Bv − Bs − +� v +s ∆zgu(s, v)dv, for v ≥ s, is a Brownian motion with respect to �Qs with initial value +B +� +Qs +s += 0. Hence, by taking the conditional expectation with respect to �Qs, +δρs ≥ E � +Qs +� +− +� u +s +�Z(s, v)dB +� +Qs +v +���� Fs +� += 0. +36 + +This completes the proof. +In the spirit of Example 17, we can easily extend Example 25 and Example 26 to +cover the case of families of BSVIEs. +Acknowledgements. +We thank Tomasz Bielecki, Matteo Burzoni, Igor Cialenco, +Alessandro Doldi, Nicole El Karoui, Mario Ghossoub, Michael Kupper, Felix Liebrich, +Max Nendel, and Frank Riedel for their interest and comments. The research leading +to these results has received funding from the Research Council of Norway (RCN) +within the project STORM - Stochastics for time-space risk models (nr. 274410). In +particular, the second author thanks this research group for the warm hospitality during +her visits. +References +[1] Acciaio, B., Penner, I. (2011). Dynamic risk measures. In Advanced mathematical +methods for finance (pp. 1–34). Springer, Berlin, Heidelberg. +[2] Agram, N. (2019). Dynamic risk measure for BSVIE with jumps and semimartin- +gale issues. Stochastic Analysis and Applications, 37(3), 361-376. +[3] Barrieu, P., El Karoui, N. (2009). Pricing, hedging and optimally designing deriva- +tives via minimization of risk measures. In: Volume on Indifference Pricing (ed: +Rene Carmona), Princeton University Press. +[4] Bielecki, T. R., Cialenco, I., Pitera, M. (2017). A survey of time consistency of +dynamic risk measures and dynamic performance measures in discrete time: LM- +measure perspective. Probability, Uncertainty and Quantitative Risk 2(1), 3. +[5] Bion-Nadal, J. (2008). Dynamic risk-measures: Time consistency and risk mea- +sures from BMO martingales. Finance and Stochastics, 12, 219–244. +[6] Bion-Nadal, J. (2009). Time consistent dynamic risk processes. Stochastic Pro- +cesses and their Applications, 119(2), 633–654. +[7] Bion-Nadal, J., Di Nunno, G. (2020). Fully-dynamic risk-indifference pricing and +no-good-deal bounds. SIAM Journal on Financial Mathematics, 11(2), 620–658. +[8] Briand, P., Coquet, F., Hu, Y., M´emin, J., Peng, S. (2000). A converse comparison +theorem for BSDEs and related properties of g-expectation. Electronic Communi- +cations in Probability, 5, 101–117. +[9] Coquet, F., Hu, Y., M´emin, J., Peng, S. (2002). Filtration-consistent nonlinear +expectations and related g-expectations. Probability Theory and Related Fields, +123(1), 1-27. +[10] Delbaen, F., Peng, S., Rosazza Gianin, E. (2010). Representation of the penalty +term of dynamic concave utilities. Finance and Stochastics, 14(3), 449–472. +37 + +[11] Detlefsen, K., Scandolo, G. (2005). Conditional and dynamic convex risk measures. +Finance and Stochastics, 9(4), 539–561. +[12] Di Bartolomeo, +D. (2017). +Risk, +Uncertainty and Time Horizon: +What +Most Risk Models Get Wrong! +Newport Seminar, June 2017. Available at: +https://www.northinfo.com/documents/770.pdf +[13] El Karoui, N., Peng, S., Quenez, M. C. (1997). Backward stochastic differential +equations in finance. Mathematical Finance 7(1), 1–71. +[14] F¨ollmer, H., Schied, A. (2011). Stochastic Finance, third edition, De Gruyter, +Berlin. +[15] Fuhrmann, S., Kupper, M., Nendel, M. (2022). Wasserstein perturbations of +Markovian transition semigroups. arXiv preprint, arXiv:2105.05655v2. +[16] Hu, Y., Øksendal, B. (2019). Linear Volterra backward stochastic integral equa- +tions. Stochastic Processes and their Applications, 129(2), 626-633. +[17] Jiang, L. (2008). Convexity, translation invariance and subadditivity for g- +expectations and related risk measures. The Annals of Applied Probability, 18(1), +245-258. +[18] Kobylanski, M. (2000). Backward stochastic differential equations and partial dif- +ferential equations with quadratic growth. The Annals of Probability, 28(2), 558- +602. +[19] Laeven, R. J., Stadje, M. (2014). Robust portfolio choice and indifference valua- +tion. Mathematics of Operations Research, 39(4), 1109-1141. +[20] Peng, S. (1997). BSDE and related g-expectations. El Karoui N., Mazliak L. Pit- +man Research Notes in Mathematics Series, (364), 141–159. +[21] Peng, S. (2005). Dynamically consistent nonlinear evaluations and expectations. +arXiv preprint math/0501415. +[22] Quenez, M. C., Sulem, A. (2013). BSDEs with jumps, optimization and appli- +cations to dynamic risk measures. Stochastic Processes and their Applications, +123(8), 3328-3357. +[23] Revuz, D., Yor, M. (2013). Continuous martingales and Brownian motion (Vol. +293). Springer Science & Business Media. +[24] Rosazza Gianin, E. (2006). Risk measures via g-expectations. Insurance: Mathe- +matics and Economics, 39(1), 19-34. +[25] Royer, M. (2006). Backward stochastic differential equations with jumps and re- +lated non-linear expectations. Stochastic processes and their applications, 116(10), +1358-1376. +38 + +[26] Yong, J. (2007). Continuous-time dynamic risk measures by backward stochastic +Volterra integral equations. Applicable Analysis, 86(11), 1429-1442. +[27] Yong, J. M. (2013). Backward stochastic Volterra integral equations—a brief sur- +vey. Applied Mathematics-A Journal of Chinese Universities, 28(4), 383-394. +[28] Wang, H., Sun, J., Yong, J. (2021). Recursive utility processes, dynamic risk +measures and quadratic backward stochastic Volterra integral equations. Applied +Mathematics & Optimization, 84(1), 145-190. +[29] Wang, T., Yong, J. (2015). Comparison theorems for some backward stochastic +Volterra integral equations. Stochastic Processes and their Applications, 125(5), +1756-1798. +39 + diff --git a/bNE4T4oBgHgl3EQfPQwx/content/tmp_files/load_file.txt b/bNE4T4oBgHgl3EQfPQwx/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a704221852a3a6b6a5fa84377f9fae32727ee93a --- /dev/null +++ b/bNE4T4oBgHgl3EQfPQwx/content/tmp_files/load_file.txt @@ -0,0 +1,1085 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf,len=1084 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='04971v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='PR] 12 Jan 2023 Fully-dynamic risk measures: horizon risk, time-consistency, and relations with BSDEs and BSVIEs Giulia Di Nunno∗† Emanuela Rosazza Gianin‡ January 12, 2023 Abstract In a dynamic framework, we identify a new concept associated with the risk of assessing the financial exposure by a measure that is not adequate to the ac- tual time horizon of the position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' This will be called horizon risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' We clarify that dynamic risk measures are subject to horizon risk, so we propose to use the fully-dynamic version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' To quantify horizon risk, we introduce h-longevity as an indicator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' We investigate these notions together with other properties of risk measures as normalization, restriction property, and different formulations of time- consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' We also consider these concepts for fully-dynamic risk measures gen- erated by backward stochastic differential equations (BSDEs), backward stochastic Volterra integral equations (BSVIEs), and families of these.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Within this study, we provide new results for BSVIEs such as a converse comparison theorem and the dual representation of the associated risk measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Keywords: Fully-dynamic risk measures, time-consistency, BSDEs, BSVIEs, con- verse comparison theorem for BSVIEs, dual representation, horizon risk, h-longevity MSC2020: 60H10, 60H20, 91B70, 91G70 1 Introduction Monetary risk assessment spans across time horizons with different length, from the very short ones for trading operations to the decades-long ones typical of sovereign wealth or pension funds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Hence the use of adequate risk measures across time as well as a time-consistent evaluation of risk are important factors in risk quantification and management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' We address the problem of using an appropriate risk evaluation for the given time horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' To explain, it is reasonable to think that it is not correct to use a risk measure ∗Department of Mathematics, University of Oslo, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Box 1053 Blindern, N-0316 Oslo Norway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Email: giulian@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='uio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='no †NHH - Norwegian School of Economics, Helleveien 30, N-5045 Bergen, Norway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' ‡Department of Statistics and Quantitative Methods, University of Milano Bicocca, via Bicocca degli Arcimboldi 8, 20126 Milano Italy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Email: emanuela.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='rosazza1@unimib.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='it 1 designed for long term positions to evaluate risks that occur in the short term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' This is considered a “common error” in risk quantification by some quants, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Indeed, bearing in mind that the purpose of risk assessment and management is not avoiding risk, but giving the rightful possibility to invest in risky assets with a reasonable control on what is acceptable to the investor, then it is clear that this form of “horizon risk” should be tackled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Indeed, in this work, we investigate horizon risk and we propose a way to quantify this via the here called horizon longevity, or h-longevity in short.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' For this we work with the general framework of fully-dynamic risk measures that embeds the classical dy- namic risk measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Indeed, a fully-dynamic risk measure is a family of risk measures (ρsu)0≤s≤u≤T (with T < ∞) indexed by two time parameters, the first represents the evaluation time, the second the horizon to which the risk pertains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Instead, a dynamic risk measure is a family of risk measures (ρs)0≤s≤T indexed by only one time parameter representing the time of risk evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' So we have that ρs = ρsT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' (1) Whenever we consider a risk X occurring at time t ∈ [0, T] (hence Ft-measurable for some given information flow), we can consider different risk evaluations at s, either ρst(X) or ρsu(X), for any other horizon u > t (even much later than t), since Ft ⊆ Fu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' According to the above, it is natural to penalize the use of the “wrong” risk measure, that is the one that does not pertain to the right time horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' We propose h-longevity to quantify this penalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' The impact of the length of the time horizon on risk evaluation has already been detected in the work [15] on epistemic uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' There, it comes out that ambiguity on the model choice is growing with the time horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' While [15] is not related with our considerations, it shows anyhow that the time horizon effects the precision on risk quantification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Another concept under investigation in this paper is time-consistency, which has different formulations and it plays an important role in dynamic risk evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' See e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' [1, 4, 6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' In our study we highlight that both h-longevity and time-consistency are linked to the restriction property (see [7]), that is ρst(X) = ρsu(X), for Ft-measurable X, u ≥ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' (2) Indeed, the presence of restriction induces that the different formulations of time- consistency are equivalent, while the absence of restriction provides the very possibility to introduce h-longevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' The restriction property is naturally satisfied by dynamic risk measures (ρs)s (see (1)) and this may be a reason for which the horizon risk has not been earlier identified in its own being and hence quantified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Similarly, also normalization, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' ρsu(0) = 0, (3) is discovered to play a crucial role in the relationships among the concepts above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Note that normalization is often standardly assumed in many financial risk evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' 2 In the first part of this paper, we work in full generality with an axiomatic setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' In the second part we concentrate on risk measures induced by backward dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' It is well-known the relationship between dynamic risk measures and backward stochastic differential equations (BSDEs), see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' [3, 17, 21, 22, 24, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Here we focus on the Brownian framework and use a BSDE with driver g to generate a fully-dynamic risk measure (ρsu)s,u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Then we study how the properties of the driver are connected to the concepts of time-consistency, h-longevity, and restriction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Since a fully-dynamic risk measure (ρsu)s,u depends on the horizon u, then we can also generate it from a family of BSDEs with drivers G = (gu)u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' In this way it is emphasized that each period [s, u] is associated to the BSDE with driver gu providing the risk evaluation ρsu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' In this framework we investigate again the relationships among the concepts of interest with the family of drivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Recently, backward stochastic Volterra integral equations (BSVIEs) have been sug- gested to generate dynamic risk measures, see [26, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' However, at that stage, some important results on BSVIEs were still lacking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Here we provide a converse comparison theorem for BSVIEs and a dual representation of risk measures induced by BSVIEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' With these two results we have a full picture of the relationship between the properties of the driver and those of the corresponding risk measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' We recall that a BSVIE is induced by a family of BSDEs, see [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' This has been a further motivation to study fully-dynamic risk measures induced by a BSVIE with Volterra driver g = g(t, s, ·), s ≥ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' As a final stage we have also considered families of BSVIEs with Volterra drivers G = (gu)u again to emphasize the role of the time horizon in the risk evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Working with families of backward equations has made evident the surprisingly crucial role of the restriction property in risk measurement across time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' For example, it turns out that the only way to have a strong (recursive) time-consistency is to work with fully-dynamic risk measures induced by a single standard BSDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Summarizing, in this work we have first identified horizon risk in its own being and proposed one way to quantify it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' We have hence considered fully-dynamic risk measures and studied the different forms of time-consistency and h-longevity, showing how the restriction property is actually playing a crucial role in these matters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Our work has covered both the general axiomatic approach and the risk measures generated by backward dynamics, both single and in families, both of standard and of Volterra type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' In this we have seen how the properties of the drivers reflect the properties of the fully-dynamic risk measures in respect of the risk evaluations across time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' In between we have also provided results on BSVIEs of independent interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' 2 Fully-dynamic risk measures, time-consistency, and hori- zon risk In the sequel, we will focus on fully-dynamic risk measures that have been recently introduced by Bion-Nadal and Di Nunno [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' See also [5], where this concept was actually simply called dynamic risk measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' 3 Definition 1 A fully-dynamic risk measure is a family (ρst)0≤s≤t≤T of risk measures indexed by two time parameters ρst : Lp(Ft) → Lp(Fs), with p ∈ [1, +∞], that are monotone, convex, Fs-translation invariant, and, for p = ∞, continuous from below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' In the definition above, continuity from below is assumed only in the case where p = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' In fact, for any p ∈ [1, ∞), it is implied by the other assumptions, see [7, Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Furthermore, fully-dynamic risk measures satisfy weak Fs-homogeneity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' 1Aρst(X) = 1Aρst(1AX), for any X ∈ Lp(Ft), A ∈ Fs (see [7, Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='6]) and have the following dual representations ρst(X) = ess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='max Q∈Qst {EQ[−X|Fs] − αst(Q)} (4) = ess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='max Q≪P : EP [αst(Q)]<∞ {EQ[−X|Fs] − αst(Q)} (5) where Qst ≜ {Q on Ft : Q ≪ P and Q|Fs ≡ P|Fs} (6) and αst(Q) ≜ ess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='sup X∈Lp(Ft) {EQ[−X|Fs] − ρst(X)} (7) is the minimal penalty functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' See [7, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='8] for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' We stress that, in general, the risk measures in the family of the fully dynamic risk measure are not normalized, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' ρst(0) ̸= 0, and the family does not satisfy the restriction property, that is ρrt(Y ) ̸= ρrs(Y ) for Y ∈ Lp(Fs), for s ≤ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' In view of the dynamic nature of the risk evaluation, we consider the relationships among the inter-temporal evaluations given by the fully-dynamic risk measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' For this, we recall here below the notions of time-consistency mostly used in the literature and their connections expressed in the present setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' See, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=', [1, 4, 6], among others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Strong time-consistency (or recursivity): for any s, t, u ∈ [0, T] with s ≤ t ≤ u, ρst(−ρtu(X)) = ρsu(X) for any X ∈ Lp(Fu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' (8) Observe that this notion is equivalent to the cocycle condition on the minimal penalties: αsu(Q) = αst(Q|Ft) + EQ[αtu(Q)|Fs], ∀s, t, u, Q ∈ Qsu, (9) together with the m-stability of the Radon-Nykodym derivatives associated to the cor- responding sets of measures (Qst)s,t, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' the probability measure S defined by dS dP = dQ dP dR dP , for all Q ∈ Qst, R ∈ Qtu (10) 4 belongs to Qsu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' See [5, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' The operation in (10) is called pasting of probability measures, also shortly denoted Q · R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Naturally, any measure S ∈ Qsu admits a representation in the form (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' In fact, we have dS dP = dS|Ft dP dR dP with dR dP = dS dP EP � dS dP |Ft �1A + 1Ac where A ≜ � ω ∈ Ω : EP � dS dP |Ft � (ω) > 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Order time-consistency: for any s, t, u ∈ [0, T] with s ≤ t ≤ u, ρtu(X) = ρtu(Y ), X, Y ∈ Lp(Fu) =⇒ ρsu(X) = ρsu(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' (11) We recall from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='3 and Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='4 of Bion-Nadal and Di Nunno [7] that, for fully-dynamic risk measures, strong time-consistency implies order time- consistency;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' furthermore, for normalized fully-dynamic risk measures strong time- consistency is equivalent to the restriction property plus order time-consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Motivated by the discussion on time-consistency done by Bion-Nadal and Di Nunno [7], where it was shown that risk indifference pricing does not satisfy strong time-consistency, but only a weaker version, we have decided to investigate this new axiom introduced in the work above mentioned, and that for the (non-normalized) fully-dynamic risk measures appears in the form: Weak time-consistency: for any s, t, u ∈ [0, T] with s ≤ t ≤ u, ρsu(ρtu(0) − ρtu(X)) = ρsu(X) for any X ∈ Lp(Fu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' (12) It is worth emphasizing that, differently from strong time-consistency, the formulation of weak time-consistency is in terms of risk measures with the same time horizon u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Notice that under the additional assumption of normalization (3), weak time- consistency (12) becomes ρsu(−ρtu(X)) = ρsu(X) for any X ∈ Lp(Fu), (13) while, under both normalization and the restriction property, weak time-consistency (12) reduces to the classical strong time-consistency (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Again from [7] and from the arguments above, it emerges that the restriction prop- erty has an extremely important role on time-consistency and on risk measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Once fully-dynamic risk measures are considered, hence the restriction property is dropped in general, it could be reasonable to impose the following axiom in order to take into account the increasing cost of riskiness for longer time horizons: Horizon longevity or h-longevity, for short: for any fixed s ∈ [0, T], ρst(X) ≤ ρsu(X) for any 0 ≤ t ≤ u, X ∈ Lp(Ft).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' (14) 5 Differently from weak time-consistency (12) that has an impact on the first time pa- rameter, horizon longevity (14) focuses on the behavior of the second time parameter, that is the time horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Our aim is to investigate weak time-consistency and h-longevity on fully-dynamic risk measures (ρst)0≤s≤t≤T from an axiomatic point of view as well as their impact on the relation between fully-dynamic risk measures and BSDEs or BSVIEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='1 Time-consistency The next results focus on weak time-consistency and on its characterizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Also we introduce the new concept of sub time-consistency, see Proposition 3, which will turn out to interplay with h-longevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Proposition 2 A fully-dynamic risk measure (ρst)0≤s≤t≤T is weakly time-consistent if and only if it satisfies order time-consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Assume order time-consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Let X ∈ Lp(Fu) and t ≤ u be arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' By Ft-translation invariance of ρtu it follows ρtu(X) = ρtu(0 − ρtu(X)) − ρtu(0) = ρtu(ρtu(0) − ρtu(X)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' (15) Since Y ≜ ρtu(0) − ρtu(X) ∈ Lp(Ft), (15) and order time-consistency imply that ρsu(X) = ρsu(Y ) for any s ≤ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' The thesis is therefore proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Conversely, suppose that ρtu(X) = ρtu(Y ) holds for some X, Y ∈ Lp(Fu) and t ≤ u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' By weak time-consistency it follows that ρsu(X) = ρsu(ρtu(0) − ρtu(X)) = ρsu(ρtu(0) − ρtu(Y )) = ρsu(Y ) for any s ≤ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Naturally, since strong time-consistency implies order time-consistency, the result above gives that strong time-consistency implies also weak time-consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Proposition 3 Let (αst)0≤s≤t≤T be the minimal penalty terms of (ρst)0≤s≤t≤T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' a) If (αru)0≤r≤u≤T satisfies, for all s ≤ t ≤ u, αsu(S) = αsu(Q) + EQ[αtu(R) − ess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='inf ¯R∈Qtu αtu( ¯R)|Fs], (16) for all Q ∈ Qsu, R ∈ Qtu, where S = Q|Ft · R ∈ Qsu is obtained by pasting Q on [s, t] and R on [t, u], then (ρst)0≤s≤t≤T is weakly time-consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' b) Weak time-consistency of (ρst)0≤s≤t≤T implies that, for all s ≤ t ≤ u, αsu(S) ≤ αsu(Q) + EQ[αtu(R) − ess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='inf ¯R∈Qtu αtu( ¯R)|Fs], (17) for all Q ∈ Qsu, R ∈ Qtu, where S = Q|Ft · R ∈ Qsu is obtained by pasting Q on [s, t] and R on [t, u].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' In (17) equality holds at least for the optimal scenarios in the dual representation (4)-(5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' 6 It is worth emphasizing that (16) is different from the usual cocycle condition (9) for two reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' First, it depends on the additional term ess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='inf ¯R∈Qtu αtu( ¯R), which is due to the non-normalization of the risk measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Second, both αsu(S) and αsu(Q) refer to the time horizon u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Note that ess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='inf ¯R∈Qtu αtu( ¯R) ∈ Lp(Ft) because of ρtu(0) = − ess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='inf ¯R∈Qtu αtu( ¯R) and, by assumption, ρtu(0) ∈ Lp(Ft).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' In terms of the normalized risk measure �ρst(X) ≜ ρst(X) − ρst(0) and of its minimal penalty function �αst(Q) ≜ αst(Q) − ess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='inf ¯R∈Qst αst( ¯R), condition (16) becomes �αsu(S) = �αsu(Q) + EQ[�αtu(R)|Fs], ∀s, t, u, Q ∈ Qsu, R ∈ Qtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' (18) Notice also that while weak time-consistency of a fully-dynamic risk measure (ρst)0≤s≤t≤T is equivalent to that of the corresponding normalized (�ρst)0≤s≤t≤T , the same is no more true for h-longevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' The reason is that the normalization terms ρst(0) and ρsu(0) are different in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' As pointed out by Bion-Nadal and Di Nunno [7], Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='12, also strong time-consistency of a fully-dynamic risk measure is not transferred to the normalized version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' In view of the above comments, for simplicity, we prove the result in terms of the normalized �ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' a) Assume that (18) holds true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' By the dual representation (4) of �ρ, �ρsu(−�ρtu(X)) = ess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='max Q∈Qsu {EQ[�ρtu(X)|Fs] − �αsu(Q)} = ess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='max Q∈Qsu {EQ[ess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='max R∈Qtu {ER[−X|Ft] − �αtu(R)}|Fs] − �αsu(Q)} = ess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='max Q∈Qsu,R∈Qtu {EQ[ER[−X|Ft]|Fs] − EQ[�αtu(R)|Fs] − �αsu(Q)} (19) = ess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='max S=Q|Ft·R: Q∈Qsu,R∈Qtu {ES[−X|Fs] − �αsu(S)} = �ρsu(X), (20) where S is obtained by pasting Q on [s, t] and R on [t, u].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Here above, (19) follows from the same arguments as in [6] and [11], while the first equality in (20) is due to (18) and to m-stability (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' b) Assume that weak time-consistency (13) holds for (�ρst)0≤s≤t≤T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' On the one hand, again by the dual representation (4), we have �ρsu(−�ρtu(X)) = ess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='max Q∈Qsu,R∈Qtu {EQ[ER[−X|Ft]|Fs] − EQ[�αtu(R)|Fs] − �αsu(Q)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' (21) 7 On the other hand, by weak time-consistency, �ρsu(−�ρtu(X)) = �ρsu(X) = ess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='max S∈Qsu {ES[−X|Fs] − �αsu(S)} = ess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='max S=Q|Ft·R: Q∈Qsu,R∈Qtu {EQ[ER[−X|Ft]|Fs] − �αsu(S)}, (22) which is due to the fact that, for any S ∈ Qsu, there exist Q ∈ Qsu, R ∈ Qtu such that S = Q|Ft · R and, vice versa, given Q ∈ Qsu, R ∈ Qtu, the pasting S = Q|Ft · R ∈ Qsu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Since �αsu(S) is the minimal penalty function on [s, u], (21) and (22) imply that �αsu(S) ≤ �αsu(Q) + EQ[�αtu(R)|Fs] (23) for any Q ∈ Qsu, R ∈ Qtu and the pasting S = Q|Ft · R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' It remains to prove that (17) holds with an equality at least for the optimal scenarios in the dual representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' By the arguments above, it is then enough to prove the reverse inequality in (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Since in the dual representation (4) the ess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='max is attained, it follows that �ρsu(−�ρtu(X)) = �ρsu(X) becomes EQ[ER[−X|Ft] − �αtu(R)|Fs] − �αsu(Q) = �ρsu(X) (24) for some R ∈ Qtu, Q ∈ Qsu (depending on X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' By defining S as the pasting Q · R of Q on [s, t] and R on [t, u], (24) reduces to ES[−X|Fs] − EQ[�αtu(R)|Fs] − �αsu(Q) = �ρsu(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' (25) From �ρsu(X) ≥ ES[−X|Fs] − �αsu(S) and (25), it follows that �αsu(S) ≥ �αsu(Q) + EQ[�αtu(R)|Fs].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Remark 4 If (ρst)0≤s≤t≤T satisfies weak time-consistency, h-longevity, and ρtu(0) ≤ 0 for any t, u, then ρst(−ρtu(X)) ≤ ρsu(X) for any X ∈ Lp(Fu), (26) called sub (strong) time-consistency in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Notice that ρtu(0) ≤ 0 is equiva- lent to ess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='infQ∈Qtu αtu(Q) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' On the other hand, sub (strong) time-consistency (26) together with ρtu(0) ≥ 0, for any t, u, imply h-longevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' For any ¯X ∈ Lp(Ft), indeed, ρst( ¯X) ≤ ρst( ¯X − ρtu(0)) = ρst(−ρtu( ¯X)) ≤ ρsu( ¯X), where the first inequality is due to ρtu(0) ≥ 0 and monotonicity, the last to sub time- consistency, while the equality follows by Ft-translation invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Proceeding similarly to Acciaio and Penner [1], it is easy to prove the relation between sub time-consistency and acceptance sets for fully-dynamic risk measures, here formulated as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' 8 Proposition 5 For any fully-dynamic risk measure (ρst)0≤s≤t≤T the following are equivalent: (i) Sub (strong) time-consistency, that is ρst(−ρtu(X)) ≤ ρsu(X) for any X ∈ Lp(Fu), s ≤ t ≤ u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' (ii) Asu ⊆ Ast + Atu where Asu ≜ {Z ∈ Lp(Fu) : ρsu(Z) ≤ 0, P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='} is the acceptance set associated to ρsu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' (iii) αsu(Q) ≤ αst(Q|Ft) + EQ[αtu(Q)|Fs] for any Q ∈ Qsu, s ≤ t ≤ u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' To summarize, the following implications among the different notions of time- consistency investigated above and h-longevity hold: strong TC ⇕ + normalization and restriction (see [7]) order TC ⇐⇒ weakly TC (see Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' 2) ⇓ + h-longevity and ρtu(0) ≤ 0 (see Remark 4) sub TC ⇓ + ρtu(0) ≥ 0 (see Remark 4) h-longevity 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='2 Horizon risk and h-longevity We recall that h-longevity (14) has been formulated as ρst(X) ≤ ρsu(X) for any 0 ≤ s ≤ t ≤ u, X ∈ Lp(Ft).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' From the formulation of h-longevity it is clear that the risk measure with longer horizon ρsu is relevant only restricted on Lp(Ft).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' In other words, longevity aims to penalize the risk measurement of a position done with the wrong risk measure, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' with the risk measure suitable for a longer horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' For this reason, in the sequel, we denote the restriction of ρsu on Lp(Ft) by ¯ρt su, while ¯αt su is the corresponding minimal penalty function, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' ¯αt su(Q) = ess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='sup X∈Lp(Ft) {EQ[−X|Fs] − ¯ρt su(X)}, Q ∈ Qst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' It then follows that ¯αt su(Q|Ft) ≤ αsu(Q), Q ∈ Qsu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' 9 In fact, αsu(Q) = ess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='sup X∈Lp(Fu) {EQ[−|XFs] − ρsu(X)} ≥ ess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='sup X∈Lp(Ft) {EQ[−X|Fs] − ¯ρt su(X)} = ¯αt su(Q|Ft).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Using the notation above, h-longevity naturally leads to ¯ρt su(X) = ρsu(X) = ρst(X) + γ(s, t, u, X) (27) for any 0 ≤ s ≤ t ≤ u, X ∈ Lp(Ft), and for a suitable Fs-measurable γ(s, t, u, X) ≥ 0 that may depend on the position X, on the time of evaluation s, on the time parameter t referring to the measurability of X, and on the “wrong” time horizon u used for evaluating X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' The term γ can be then seen as an indicator quantifying the horizon risk or, roughly speaking, as an additive term of adjustment/calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' It follows that, for any s ≤ t ≤ u and X ∈ Lp(Ft), γ(s, t, t, X) = 0 and, by translation invariance of the fully-dynamic risk measure, that γ(s, t, u, X + cs) = γ(s, t, u, X) for any cs ∈ Lp(Fs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' The restriction property (2) is then equivalent to γ(s, t, u, X) = 0 for all 0 ≤ s ≤ t ≤ u, X ∈ Lp(Ft).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' The following result characterizes longevity in terms of acceptance sets and of penalty functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Proposition 6 For a fully dynamic risk measure (ρst)s,t: (a) longevity is equivalent to Asu ∩ Lp(Ft) ⊆ Ast for any s ≤ t ≤ u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' (b) restriction is equivalent to Asu ∩ Lp(Ft) = Ast for any s ≤ t ≤ u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' (c) longevity implies that αst(Q) ≥ ¯αt su(Q) and also ¯αt su(Q) ≥ αst(Q) − ess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='supX∈Lp(Ft) γ(s, t, u, X), for any s ≤ t ≤ u and Q ∈ Qst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' (d) restriction implies that αst(Q) = ¯αt su(Q) for any s ≤ t ≤ u and Q ∈ Qst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' (a) For any X ∈ Asu ∩ Lp(Ft) it follows that ρsu(X) ≤ 0 and, by longevity, also that ρst(X) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Hence Asu ∩ Lp(Ft) ⊆ Ast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Conversely, from the representation of risk measures in terms of acceptance sets and from the condition on acceptance sets, it follows that, for any X ∈ Lp(Ft), ρsu(X) = ess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='inf{ms ∈ Lp(Fs)| ms + X ∈ Asu} ≥ ess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='inf{ms ∈ Lp(Fs)| ms + X ∈ Ast} = ρst(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' (b) Assume now restriction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' From (27), we see that restriction can be interpreted as h-longevity with γ(s, t, u, X) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Hence, from (a), it remains to prove the reverse inclusion for acceptance sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' For any X ∈ Ast (hence X ∈ Lp(Ft)) we have that 10 ρst(X) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' By restriction, ρsu(X) = ρst(X) ≤ 0, hence Asu ∩ Lp(Ft) ⊇ Ast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Con- versely, assume Asu ∩ Lp(Ft) = Ast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' From the representation of risk measures in terms of acceptance sets, we obtain that, for any X ∈ Lp(Ft), ρsu(X) = ess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='inf{ms ∈ Lp(Fs)| ms + X ∈ Asu} = ess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='inf{ms ∈ Lp(Fs)| ms + X ∈ Ast} = ρst(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' (c) Proceeding as in F¨ollmer and Schied [14], Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='16, by translation invariance of ρsu, we have αst(Q) = ess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='sup X∈Ast EQ[−X|Fs] for any s ≤ t and Q ∈ Qst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' By (a), h-longevity and translation invariance imply that, for any Q ∈ Qst, αst(Q) = ess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='sup X∈Ast EQ[−X|Fs] ≥ ess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='sup X∈Asu∩Lp(Ft) EQ[−X|Fs] = ess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='sup X∈Lp(Ft): ρsu(X)≤0 EQ[−X|Fs] = ¯αt su(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Moreover, in terms of the indicator γ in (27), we have that ¯αt su(Q) = ess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='sup X∈Lp(Ft) {EQ[−X|Fs] − ¯ρt su(X)} = ess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='sup X∈Lp(Ft) {EQ[−X|Fs] − ρst(X) − γ(s, t, u, X)} ≥ ess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='sup X∈Lp(Ft) {EQ[−X|Fs] − ρst(X)} − ess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='sup X∈Lp(Ft) γ(s, t, u, X) = αst(Q) − ess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='sup X∈Lp(Ft) γ(s, t, u, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' (d) By the equivalence between restriction and the condition on acceptance sets in (b), it follows that for any Q ∈ Qst αst(Q) = ess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='sup X∈Ast EQ[−X|Fs] = ess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='sup X∈Asu∩Lp(Ft) EQ[−X|Fs] = ¯αt su(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' The proof is then complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' With (c), we see that the greater is the term ess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='supX∈Lp(Ft) γ(s, t, u, X) for positions that are Ft-measurable, the lower is the lower bound αst(Q|Ft)−ess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='supX∈Lp(Ft) γ(s, t, u, X) of αsu(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' The term ess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='supX∈Lp(Ft) γ(s, t, u, X) can be then interpreted as the maxi- mal error (at the level of the penalty function) for a wrong use of ρsu for positions X belonging to Lp(Ft), that is, ρs· with a wrong time horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' 11 From a motivational point of view, one can imagine γ dependent on the time hori- zon, that is, γ(s, t, u, X) = γs,t(h, X) with h = u − t or, even, γs,t(h) independent from X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' In particular, h = u − t can be interpreted as the length of the time interval over which there is an uncorrect use of the risk measure (ρsu versus ρst).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' See Example 10 and Example 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' In what follows, we investigate whether BSDEs or BSVIEs can provide families of fully-dynamic risk measures satisfying sub or weak time-consistency and h-longevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' In particular, we study the cases of a single driver g and of a family of drivers (gt)t∈[0,T] depending on the time horizon t considered in ρst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' The former case will be shortened by BSDE (g) (or BSVIE (g)), the latter by BSDE (gt) (or BSVIE (gt)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' 3 Relation with BSDEs Hereafter, we consider a fully-dynamic risk measure induced by a single BSDE with driver g and also measures generated by a family of BSDEs associated to the drivers G = (gt)t∈[0,T] where the index t refers to the time horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' We will restrict our attention to L2 spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' On the probability space (Ω, F, P) we consider a d-dimensional Brownian motion (Bt)t∈[0,T] and the P-augmented natural filtration (Ft)t∈[0,T] of (Bt)t∈[0,T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' According to Peng [20], the solution (Yt, Zt)t∈[0,T] of the BSDE Yt = X + � T t g(s, Ys, Zs) ds − � T t Zs dBs (28) can be seen as an operator depending on the driver g and evaluated at the final condition X ∈ L2(FT ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' In our work, we consider a driver g : Ω × [0, T] × R × Rd → R satisfying the standard assumptions: adapted, uniformly Lipschitz, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' there exists a constant C > 0 such that, dP × dt-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=', |g(ω, t, y1, z1) − g(ω, t, y2, z2)| ≤ C(|y1 − y2| + |z1 − z2|), for any y1, y2 ∈ R, z1, z2 ∈ Rd, where | · | denotes the Euclidean norm in Rk for whatever k is relevant;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' E �� T 0 |g(s, 0, 0)|2 ds � < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Under these conditions, equation (28) admits a unique solution (Yt, Zt)t∈[0,T], with (Yt)t∈[0,T] ∈ H2 [0,T](R) and (Zt)t∈[0,T] ∈ H2 [0,T](Rd), where we have set H2 [a,b](Rk)≜ � adapted Rk-valued processes (ηs)s∈[a,b] :E �� b a |ηs|2 ds � <∞ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' 12 For further details see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=', El Karoui et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' In Peng [20, 21] and Rosazza Gianin [24] the relationship among BSDEs, nonlinear expectations and dynamic risk measures is detailed clarifying that the properties of g reflect the properties of the risk measures associated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' To summarize, in the financial context of monetary risk measures, if g in (28) does not depend on y, then the risk measure is translation invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Also g convex produces a convex risk measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' See Barrieu and El Karoui [3], Jiang [17], and Rosazza Gianin [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Beyond the Lipschitz condition on the driver, the connection between BSDEs and risk measures has been studied in terms of maximal solutions of BSDEs (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=', Bar- rieu and El Karoui [3] and Kobylanski [18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' When it comes to generalizations beyond the Brownian framework we can refer, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=', to Royer [25], Quenez and Sulem [22], and Laeven and Stadje [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='1 Risk measures generated by a single BSDE Consider the following BSDE with driver g, not depending on y, and terminal condition X ∈ L2(Fu): Yt = X + � u t g(s, Zs)ds − � u t ZsdBs, 0 ≤ t ≤ u, (29) where u ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' To simplify the notation, we will occasionally adopt the Peng’s notation where Eg (X|Ft) denotes the conditional g-expectation of X at time t, that is the Y -component of the solution (Y, Z) at time t of the BSDE above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='Then, we focus on risk measures of the following form: ρtu(X) = Eg (−X|Ft) , X ∈ L2(Fu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' (30) The condition g(t, 0) = 0, guarantees the restriction property as well as the nor- malization (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Hence, in our case we cannot assume g(t, 0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Proposition 7 The following statements are equivalent: (a) g(t, 0) = 0 for any t ∈ [0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' (b) ρ is normalized, see (3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' (c) ρ has the restriction property (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' (a) ⇒ (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' This implication follows immediately by remarking that (Yt, Zt) with Yt = Zt = 0 for any t is the unique solution when the terminal condition is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' See also Peng [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' (b) ⇒ (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Assume that ρtu(0) = 0 for any 0 ≤ t ≤ u ≤ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' It follows that 0 = Yt = � u t g(s, Zs)ds − � u t ZsdBs holds for any 0 ≤ t ≤ u ≤ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Hence � u t g(s, Zs)ds = � u t ZsdBs, for any 0 ≤ t ≤ u ≤ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' 13 Since a continuous martingale and a process of finite variation can be equal only if the martingale is constant (see Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='2 in Chapter 4 of Revuz and Yor [23]), then for any t the martingale Mt(u) = � u t ZsdBs, u ≥ t, starting from 0 is identically equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Hence Zs ≡ 0 for any s ≥ t and � u t g(s, Zs)ds ≡ 0 for any u ≥ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' By replacing Zs and deriving this last equation with respect to u, it follows that g(u, 0) = 0 for any u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' (a) ⇒ (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' This implication was proved in Peng [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Assume that X ∈ L2(Fu) and consider ρtu(X) and ρtv(X) for any v ≥ u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Let us denote by (Y X,u r , ZX,u r ) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' (Y X,v r , ZX,v r )) the solution corresponding to ρtu(X) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' ρtv(X)) at time r ≤ u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Since (Y X,v r , ZX,v r ) with Y X,v r = � Y X,u r ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' r ≤ u −X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' u < r ≤ v ZX,v(r, s) = � ZX,u r ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' s ≤ u 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' u < s ≤ v is a solution of ρtv(X) when g(t, 0) = 0 for any t ∈ [0, T], the restriction property follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' (c) ⇒ (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' By translation invariance and restriction, it holds that for any X ∈ L2(Ft) ⊆ L2(Fu) −X + ρtT (0) = ρtT (X) = ρtu(X) = −X + ρtu(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Hence ρtT (0) = ρtu(0) for any t ≤ u ≤ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Since ρtu is induced by the BSDE (29) via (30), it follows that ρtt(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Consequently, ρtu(0) = 0 for any t ≤ u ≤ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' By this the proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Note that while normalization refers to a single risk measure ρtu, restriction involves the whole family (ρtu)t,u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' It is already known that any BSDE satisfies both weak and strong time-consistency (see Barrieu and El Karoui [3], El Karoui et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' [13], Bion Nadal and Di Nunno [7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' From the arguments above, any BSDE with a driver g such that g(t, 0) = 0 for any t ∈ [0, T] satisfies normalization, restriction and strong time-consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Here below, we provide some examples where the driver is not normalized and (ρtu)t,u does not satisfies restriction, but only h-longevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Example 8 a) Consider the driver g(t, z) = a for any t ∈ [0, T], z ∈ Rd, with a ∈ R \\ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' It can be checked easily that, for any t ∈ [0, T] and X ∈ L2(FT ), ρtT (X) = EP � −X + � T t a ds ���� Ft � = EP [−X| Ft] + (T − t)a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' This means that, for a ̸= 0, (ρtu)t,u is not normalized and does not satisfy the restriction property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Instead, it satisfies h-longevity whenever a > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' b) Consider now the following driver: g(t, z) = bz + a for any t ∈ [0, T], z ∈ Rd, with a, b ∈ R \\ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' By Girsanov Theorem, −dYt = (bZt + a)dt − Zt dBt = a dt − Zt dBQ t 14 where E � dQ dP ���Ft � = exp � − 1 2b2t + b · Bt � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' It then follows easily that ρtT (X) = EQ [−X| Ft] + (T − t) a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' As before, for a ̸= 0, (ρtu)t,u is not normalized and does not satisfy the restriction property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Instead, it satisfies h-longevity whenever a = g(t, 0) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' The next result provides a sufficient condition on the driver g for h-longevity of (ρtu)t,u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Proposition 9 If g(v, 0) ≥ 0 for any v ∈ [0, T], then h-longevity holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Furthermore, for all s, u ∈ [0, T], s ≤ u, γ(s, t, u, X) = E � QX �� u t g(v, 0)dv|Fs � , s ≤ t ≤ u, X ∈ Lp(Ft), where �QX is a probability measure on Qsu depending on X equivalent to P, with density d �QX dP = exp � −1 2 � u s |∆zg(v)|2dv + � u s ∆zg(v)dBv � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Here above ∆zg(v) = (∆i zg(v))i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=',d and ∆i zg(v) ≜ g(v, Zu v ) − g(v, ¯Zt v) d(Zu,i v − ¯Zt,i v ) 1{Zu,i v ̸= ¯Zt,i v }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' The probability measure �QX here above can be interpreted as an h-longevity premium measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Let s ≤ t ≤ u and X ∈ Lp(Ft).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' The risk measures ρst(X) and ρsu(X) satisfy the following BSDEs: ρst(X) = −X + � t s g(v, Zt v)dv − � t s Zt vdBv ρsu(X) = −X + � u s g(v, Zu v )dv − � u s Zu v dBv, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Set now the Rd-valued process ¯Zt v = � Zt v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' v ≤ t 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' t < v ≤ u ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' �Zv = Zu v − ¯Zt v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Then ρsu(X) − ρst(X) = � u s [g(v, Zu v ) − g(v, ¯Zt v)]dv + � u t g(v, ¯Zt v)dv − � u s [Zu v − ¯Zt v]dBv − � u t ¯Zt vdBv = � u s [g(v, Zu v ) − g(v, ¯Zt v)]dv − � u s �ZvdBv + � u t g(v, 0)dv = � u s ∆zg(v) · �Zvdv − � u s �ZvdBv + � u t g(v, 0)dv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' (31) 15 Furthermore, (31) can be rewritten also as δρs = Γt,u + � u s ∆zg(v) · �Zvdv − � u s �ZvdBv, (32) where δρs ≜ ρsu(X) − ρst(X) and Γt,u ≜ � u t g(v, 0)dv represents the final condition at time u (which depends on t but not on s) of the linear BSDE (32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Since Γt,u ≥ 0 for any t by hypothesis and ∆zg(v) ∈ H2 [s,u](Rd) by the assumption of g Lipschitz in z, by Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='2 of El Karoui, Peng and Quenez [13] it follows that δρs ≥ 0 for any s ≤ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' By applying Girsanov Theorem, (31) becomes ρsu(X) − ρst(X) = � u s ∆zg(v) · �Zvdv − � u s �ZvdBv + � u t g(v, 0)dv = − � u s �ZvdB � QX v + � u t g(v, 0)dv, where B �QX v ≜ Bv − Bs − � v s ∆zg(r) dr, v ∈ [s, u], is a �QX-Brownian motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Hence, by taking the conditional expectation with respect to �QX, ρsu(X) − ρst(X) = E � QX � − � u s �ZvdB � QX v + � u t g(v, 0)dv ���Fs � = E � QX �� u t g(v, 0)dv ���Fs � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' By assumption on g(·, 0), it follows that ρsu(X) − ρst(X) ≥ 0 and that γ(s, t, u, X) = ρsu(X) − ρst(X) = E � QX �� u t g(v, 0)dv|Fs � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' As discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='2, γ may depend on the time horizon, that is, γ(s, t, u, X) = γs,t(h, X) with h = u − t or, even, γs,t(h) independent from X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' The following example provides some cases covering the situation above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Example 10 Let g(v, 0) ≥ 0 for any v ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Hence, by the result above, h-longevity holds and γ(s, t, u, X) = E � QX �� u t g(v, 0)dv|Fs � for any X ∈ L2(Ft).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' a) If g(v, 0) = c for any v ∈ [0, T], with c ≥ 0, then c is necessarily deterministic (since it should be measurable for any v ≥ 0) and, consequently, γ(s, t, u, X) = E � QX �� u t g(v, 0)dv ���Fs � = (u − t)c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Hence γ only depends on h = u − t, that is, roughly speaking, on the length of the time interval over which there is an uncorrect use of the risk measure (ρsu versus ρst).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' b) If g(v, 0) = exp(−r v) for any v ∈ [0, T], with r ≥ 0, then r is necessarily determin- istic (for the same arguments as above) and, consequently, γ(s, t, u, X) = e−rt � 1 − e−r(u−t)� r In other words, γ only depends on the “right” time horizon t (referring to the measur- ability of X) and on the length of the time interval [t, u].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' 16 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='2 Risk measures generated by a family of BSDEs Now we consider general risk measures induced by a family of BSDEs of type (28) with drivers G = (gu)u∈[0,T] depending on the time horizon u of ρtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' For later use, by increasing family G = (gu)u∈[0,T] it is meant, for any t ≤ u, gt(v, y, z) ≤ gu(v, y, z) for any v ∈ [0, t], y ∈ R, z ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Suppose that, for any t ≤ u, the risk measure ρtu comes from a gu-expectation with a driver depending on the maturity u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' This means that ρtu(X) = ρG tu(X) = Egu(−X|Ft), for any X ∈ L2(Fu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' (33) Assume now that (gu)u∈[0,T] is a family of drivers depending on the maturity u, independent of y, Lipschitz, and convex in z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Then the risk measure (ρG tu)t,u defined by (29), (33) satisfies monotonicity, convexity, continuity from above/below, and trans- lation invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Furthermore, for u ∈ [0, T], if gu(v, 0) = 0 for any v ≤ u, then ρG tu(0) = 0 for any t ≤ u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' In general, however, gu(v, 0) = 0 for any v ≤ u does not imply the restriction property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' In a Brownian setting, ρG tu can be represented as ρtu(X) = ρG tu(X) = ess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='sup Q∈Qtu � EQ [−X|Ft] − αG tu(Q) � , X ∈ L2(Fu), where Qtu is defined in (6) and the penalty term αG tu associated to ρG tu is given by αG tu(Q) = EQ �� u t g∗ u(v, qv)dv ���� Ft � , where g∗ u denotes the convex conjugate of the function gu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' See Delbaen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' The following result shows that, for gu(r, 0) = 0 for any r, u, the restriction property is satisfied only for risk measures induced by a BSDE with a single driver g (that is, constant with respect to the maturity time).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Proposition 11 For any u ∈ [0, T], let gu(v, 0) = 0 for any v ≤ u and let gu(v, ·) be continuous in v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' The restriction property (2) holds if and only if gu is constant in u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' If gu is constant in u, the restriction property follows directly by Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Conversely, assume that the restriction property holds, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' ρtu(X) = ρtv(X) for any t ≤ u ≤ v and X ∈ L2(Fu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Proceeding as in the proof of the Converse Comparison Theorem of Briand et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' [8], Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='1, gu(s, z) = lim ε→0 ρsu(−z · (Bs+ε − Bε)) ε gv(s, z) = lim ε→0 ρsv(−z · (Bs+ε − Bε)) ε 17 with convergence in L2, for any z ∈ Rd, u ≤ v and s ∈ [0, u].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' By extracting a subsequence to obtain convergence P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' and passing to the limit as ε → 0, it holds that ρsv(−z(Bs+ε − Bε)) ε = ρsu(−z(Bs+ε − Bε)) ε −→ gu(s, z), ǫ → 0, P − a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' where the equality is due to restriction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' The thesis then follows because ρsv(−z(Bs+ε − Bε)) ε −→ gv(s, z), ǫ → 0, P − a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' By this we end the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' As done in Example 8 for a single driver, we provide here below some examples where restriction fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Example 12 a) Consider the driver gu(t, z) = au for any t ∈ [0, u], z ∈ Rd, with au ∈ R \\ {0} depending on the maturity u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' It can be checked easily that, for any t ∈ [0, T] and X ∈ L2(FT ), ρtu(X) = EP [−X| Ft] + (u − t)au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' This means that, for au ̸= 0, (ρtu)t,u is not normalized and does not satisfy the re- striction property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Instead, it satisfies h-longevity whenever au > 0 is increasing in u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' b) Consider now the following driver: gu(t, z) = bz + au for any t ∈ [0, u], z ∈ Rd, with au, b ∈ R \\ {0} and au depending on the maturity u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Similarly to Example 8, it follows that ρtu(X) = EQ [−X| Ft] + (u − t)au, where E � dQ dP ��� Ft � = exp � − 1 2b2t + b · Bt � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' As before, therefore, for au ̸= 0, (ρtu)t,u is not normalized and does not satisfy the restriction property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Instead, it satisfies h-longevity whenever au > 0 is increasing in u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='1 Time-consistency Thanks to the Comparison Theorem (see El Karoui et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' [13]) and to the Converse Comparison Theorem (see Briand et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' [8]), the following result establishes that mono- tonicity of the family G = (gt)t∈[0,T] is equivalent to sub time-consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Theorem 13 Let (ρtu)t,u be induced by a BSDE with a family of drivers G = (gt)t∈[0,T] as in (33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' a) The family G is increasing if and only if (ρtu)t,u satisfies sub time-consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' b) G = {g} if and only if (ρtu)t,u satisfies strong time-consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' 18 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' a) By definition (33), sub time-consistency of (ρtu)t,u can be written as Egt(Egu(−X|Ft)|Fs) ≤ Egu(−X|Fs) (34) for any s ≤ t ≤ u and X ∈ L2(Fu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' By definition of Egt, the left-hand and right-hand sides of the previous equation can be rewritten as follows: Egt(Egu(−X|Ft)|Fs) = Egu(−X|Ft) + � t s gt(v, Zv) dv − � t s Zv dBv = −X + � u t gu(v, ˜Zv) dv − � u t ˜Zv dBv + � t s gt(v, Zv) dv − � t s Zv dBv and Egu(−X|Fs) = −X + � u s gu(v, ˆZv) dv − � u s ˆZv dBv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' (35) Set now ¯g(v, z) = � gt(v, z);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' v ∈ [0, t] gu(v, z);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' v ∈ (t, u] ¯Zv = � Zv;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' v ∈ [0, t] ˜Zv;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' v ∈ (t, u] By the arguments above, the left-hand side of (34) becomes Egt(Egu(−X|Ft)|Fs) = −X + � u s ¯g(v, ¯Zv) dv − � u s ¯Zv dBv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' (36) On the one hand, if ¯g ≤ gu on [0, u] (or, equivalently, gt ≤ gu on [0, t]), then by the Comparison theorem of BSDE (see El Karoui et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' [13]) sub time-consistency (34) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' On the other hand, by (35) and (36) and Converse Comparison Theorem of Briand et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' [8], sub time-consistency (34) implies that ¯g ≤ gu on [0, u], hence gt ≤ gu on [0, t].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' b) The case of time-consistency can be obtained by replacing inequalities with equalities in the proof above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' In other words, the result above guarantees that strong time-consistency is fulfilled if and only if the family of drivers reduces to a singleton, that is, the drivers gt do not depend on the maturity t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Remark 14 In the previous result, the proof of item a) can be done also by means of the penalty term characterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' In this case, one could use αG su(Q) = EQ �� u s g∗ u(v, qv)dv ���� Fs � and the fact that, by the Fenchel-Moreau biconjugate Theorem, the condition g∗ t ≥ g∗ u is equivalent to gt ≤ gu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Indeed, if g∗ t ≥ g∗ u then gt(s, z) = sup q∈Rd{q · z − g∗ t (s, q)} ≤ sup q∈Rd{q · z − g∗ u(s, q)} = gu(s, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' The converse implication can be checked similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' 19 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='2 H-longevity In order to investigate h-longevity, we need to compare BSDEs with different horizons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Hereafter, we suggest a comparison theorem for this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Note that in this result we consider more general BSDEs (28) with driver depending on y, as the result is of interest well beyond the use of BSDEs in risk measures modelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Let T1, T2 > 0 be two different time horizon with T2 > T1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Let (Y Ti,ξi t , ZTi,ξi t )t∈[0,T1] (for i = 1, 2) be the solution of a BSDE with final condition ξi ∈ L2(FTi) and driver gTi satisfying the usual assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' More precisely, Y Ti t = ξi + � Ti t gTi(s, Y Ti s , ZTi s )ds − � Ti t ZTi s dBs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' (37) In the terminology of Peng [20], Y Ti t = EgTi(ξi ��Ft).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Theorem 15 (Horizon Comparison Theorem) If gT2(s, y, z) ≥ gT1(s, y, z) for any s ∈ [0, T1], y ∈ R, z ∈ Rd and gT2(s, y, z) ≥ 0 for any s ∈ [T1, T2], y ∈ R, z ∈ Rd and ξ2 ≥ ξ1, then Y T2 t ≥ Y T1 t for any t ∈ [0, T1] and Y T2 t ≥ ξ1 for any t ∈ [T1, T2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' By (37), it follows that Y T2 t − Y T1 t = ξ2 − ξ1 + � T2 t gT2(s, Y T2 s , ZT2 s )ds − � T1 t gT1(s, Y T1 s , ZT1 s )ds − � T2 t ZT2 s dBs + � T1 t ZT1 s dBs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' (38) Let us now extend gT1(s, y, z), Y T1 s and ZT1 s also to [T1, T2] as follows: g(T1)(s, y, z) = � gT1(s, y, z);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' s ∈ [0, T1] 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' s ∈ (T1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' T2] Y T1 s = � Y T1 s ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' s ∈ [0, T1] ξ1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' s ∈ (T1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' T2] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Z T1 s = � ZT1 s ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' s ∈ [0, T1] 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' s ∈ (T1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' T2] For t ∈ [0, T2], equation (38) becomes then Y T2 t − Y T1 t = ξ2 − ξ1 + � T2 t � gT2(s, Y T2 s , ZT2 s ) − gT1(s, Y T1 s , Z T1 s ) � ds − � T2 t (ZT2 s − Z T1 s )dBs (39) Set now �Ys ≜ Y T2 s − Y T1 s , �ξ ≜ ξ2 − ξ1, �Zi s ≜ ZT2,i s − Z T1,i s for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=', d, and ∆sg = gT2(s, Y T1 s , Z T1 s ) − gT1(s, Y T1 s , Z T1 s ) ∆sy = gT2(s, Y T2 s , ZT2 s ) − gT2(s, Y T1 s , ZT2 s ) Y T2 s − Y T1 s 1{Y T2 s ̸=Y T1 s } (∆sz)i = gT2(s, Y T1 s , ZT2 s ) − gT2(s, Y T1 s , Z T1 s ) d(ZT2,i s − Z T1,i s ) 1{ZT2,i s ̸=Z T1,i s } 20 for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=', d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Hence (39) can be rewritten as � −d�Yt = � ∆ty �Yt + ∆tz · �Zt + ∆t g � dt − �ZtdBt �YT2 = �ξ (40) Since ∆ty ∈ H2 [0,T](R), ∆tz ∈ H2 [0,T](Rd) and ∆t g is in H2 [0,T](R) by the assumption of gTi Lipschitz, the solution of (40) exists uniquely by Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='2 of El Karoui, Peng and Quenez [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Moreover, since gT2(s, y, z) ≥ gT1(s, y, z) for any s ∈ [0, T2], y, z and �ξ ≥ 0, then �Yt ≥ 0 for any t ∈ [0, T2] which concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' The previous result guarantees that for fully-dynamic risk measures induced by g-expectations as in (33), h-longevity holds when gu is increasing in u and gu ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Proposition 16 a) If G is an increasing family of drivers and ρtu(0) ≥ 0 for any t ≤ u, then (ρtu)t,u satisfies h-longevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' b) If G is an increasing family of drivers and gt ≥ 0 for any t ∈ [0, T], then (ρtu)t,u satisfies h-longevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' a) follows by Remark 4 and Proposition 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' b) By Theorem 15, increasing monotonicity of the family G and gt ≥ 0 for any t ∈ [0, T] imply h-longevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Notice that requiring gu ≥ 0 implies that ρtu(X) ≥ EP[−X|Ft] for any X ∈ L2(Fu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Indeed, by definition of ρtu(X) as in (33) and from gu ≥ 0, it follows that ρtu(X) = EP � −X + � u t gu(v, Zv) dv − � u t Zv dBv ���� Ft � = EP [−X|Ft] + EP �� u t gu(v, Zv) dv ���� Ft � ≥ EP [−X|Ft] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='3 Beyond the Lipschitz driver: the entropic case Next, we consider the entropic risk measure, which is generated by a quadratic BSDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Even though the Lipschitz condition is clearly not satisfied in this case, the solution of the BSDE is still unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' This discussion on the uniqueness of solutions is developed in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' The entropic risk measure is particularly interesting because of its explicit connection with utility functions as investment preference criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' In the following example, we propose a variation to the classical entropic risk measure able to capture h-longevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Example 17 In the one-dimensional case, for every u ∈ [0, T], consider the driver gu(t, z) = bu 2 z2 + au(t) for any t ∈ [0, T], z ∈ R, with bu > 0 and au positive function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' We recall that bu is the reciprocal of the risk aversion parameter in the corresponding 21 utility function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' By using the explicit solution of the BSDE with driver ¯gu(t, z) = bu 2 z2, it can be checked easily that, for any t ∈ [0, T] and X ∈ L2(Fu), ρsu(X) = 1 bu ln (EP [exp(−buX)| Fs]) + � u s au(t)dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Hence, (ρsu)s,u is not normalized and does not satisfy the restriction property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Instead, h-longevity holds whenever (bu)u and (au)u are increasing in u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' See Proposition 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' 4 Relation with BSVIEs We have come now to consider a fully-dynamic risk measure induced by a single BSVIE with a driver g and also measures generated by a family of BSVIEs associated to the drivers G = (gt)t∈[0,T] where the index t refers to the time horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='1 Risk measures generated by a single BSVIE In the same spirit of risk measures generated by BSDEs, we can consider risk measures induced by BSVIEs as Agram [2] and Yong [26] suggested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Indeed, this can also be seen in connection with families of BSDEs as Yong [27] has pinpointed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Let us consider a BSVIE of type Yt = ξt + � T t g(t, s, Z(t, s)) ds − � T t Z(t, s) dBs, (41) where ξt ∈ L2(FT ), for all t ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' The driver g : Ω × ∆ × Rd → R, with ∆[0,T] ≜ {(t, s) ∈ [0, T] × [0, T] : s ≥ t}, is independent of y (so to guarantee translation invariance in the application to monetary risk measures) and it is such that for any t ∈ [0, T], z ∈ Rd, g(t, ·, z) is adapted;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' is uniformly Lipschitz, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' there exists a constant C > 0 such that |g(t, s, z1) − g(t, s, z2)| ≤ C|z1 − z2|, for any (t, s) ∈ ∆, z1, z2 ∈ Rd, P − a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' E �� T 0 �� T t g(t, s, 0) ds �2 dt � < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Under there assumptions, we have a unique solution (Y·, Z(·, ·)) ∈ H2 [0,T](R)×Z2 [0,T](Rd), where we have set Z2 [a,b](Rd) ≜ � Z : Ω × ∆[a,b] → Rd : Z(t, ·) is adapted for all t and E �� b a � b a |Z(t, s)|2 ds dt � < +∞ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' See Yong [27], Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='2, and [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Also Yong [27] introduced the following family of BSDEs parameterized by t and related to the BSVIE (41): η(r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' t, ξt) = ξt + � T r ¯g(v, ζ(v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' t)dv − � T r ζ(v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' t)dBv, r ∈ [t, T] 22 where ζ(v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' t) = Z(t, v);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' ¯g(v, ζ(v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' t) = g(t, v, Z(t, v)) and Yt = η(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' t, ξt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Although η(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' t, X) and Yt are closely related, Yt, solution of the BSVIE (41) may satisfy some specific properties, since it corresponds not to the whole family η(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' t, X), but only to η(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' t, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' As in Section 3, we assume convex drivers that guarantee convex risk measures satisfying monotonicity and translation invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' See Yong [26] and Agram [2], who also makes an extension to jump dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Then we focus on monetary risk measures of the type ρtu(X) = Eg,V (−X|Ft) , X ∈ L2(Fu), (42) where Eg,V (−X|Ft) denotes the Y -component of the solution of the BSVIE of type (41) with terminal condition ξt = −X, for all t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' As shown in the following result, for a driver independent on y the condition g(t, u, 0) = 0, for any t ≤ u, guarantees the restriction property as well as the nor- malization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Hence, in our case we cannot assume g(t, u, 0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Proposition 18 The following statements are equivalent: (a) g(t, u, 0) = 0 for any 0 ≤ t ≤ u ≤ T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' (b) ρ is normalized, see (3): (c) ρ satisfies the restriction property (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' (a) ⇒ (b) The implication is true because (Y·, , Z(·, ·)) with Yt = Z(t, s) = 0, s ≥ t is the unique solution corresponding to X = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' (b) ⇒ (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Assume that ρtu(0) = 0 for any 0 ≤ t ≤ u ≤ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' we have 0 = Yt = � u t g(t, s, Z(t, s))ds − � u t Z(t, s)dBs holds for any 0 ≤ t ≤ u ≤ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Hence � u t g(t, s, Z(t, s))ds = � u t Z(t, s)dBs, for any 0 ≤ t ≤ u ≤ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' As in the proof of Proposition 7, it follows that for any fixed t the martingale Mt(u) = � u t Z(t, s)dBs, with u ≥ t, starting from 0 should be identically to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Hence Z(t, s) ≡ 0 for any s ≥ t and � u t g(t, s, 0)ds ≡ 0 for any u ≥ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' By deriving this last equation with respect to u, it follows that g(t, u, 0) = 0 for any u ≥ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' (a) ⇒ (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Assume that X ∈ L2(Fu) and consider ρtu(X) and ρtv(X) for any v ≥ u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Let us denote by (Y X,u r , ZX,u(r, s)) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' (Y X,v r , ZX,v(r, s))) the solution corresponding to ρtu(X) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' ρtv(X)) at time r ≤ u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Since (Y X,v r , ZX,v(r, s)) with Y X,v r = � Y X,u r ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' r ≤ u −X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' u < r ≤ v ZX,v(r, s) = � ZX,u(r, s);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' s ≤ u 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' u < s ≤ v 23 is a solution of ρtv(X) whether g(t, u, 0) = 0 for any 0 ≤ t ≤ u ≤ T, the restriction property follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' (c) ⇒ (a) The argument is similar to the proof of the corresponding implication in Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Note that, for all t, the set of probability measures Qt on Fu such that dQt dP ≜ exp �1 2 � u t |q(t, s)|2ds − � u t q(t, s)dBs � , (43) for some stochastic process (q(t, s))t≤s≤u ∈ H2 [t,u](Rd) coincides with Qtu in a Brownian setting (see Revuz and Yor [23], Chapter VIII, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Theorem 19 (Dual representation) For any t ≤ u fixed, the risk measure ρtu induced by a BSVIE as in (42) has the following dual representation ρtu(X) = ess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='sup Qt∈Qtu {EQt [−X|Ft] − αtu(Qt)} , (44) where Qt corresponds to (q(t, s))t≤s≤u via (43), g∗(t, s, ·) denotes the convex conjugate of g(t, s, ·) and the penalty functional is given by αtu(Qt) = EQt �� u t g∗(t, s, q(t, s))ds ���� Ft � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' The present proof extends a similar one of Barrieu and El Karoui [3] to the case of BSVIEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Let t, u ∈ [0, T] with t ≤ u be fixed arbitrarily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' The evaluation of risk (42) can be written as Yt = −X + � u t [g(t, s, Z(t, s)) − q(t, s) · Z(t, s)]ds − � u t Z(t, s)(dBs − q(t, s)ds) = −X + � u t [g(t, s, Z(t, s)) − q(t, s) · Z(t, s)]ds − � u t Z(t, s)dBQt s and, by Girsanov Theorem, BQt u ≜ Bu − Bt − � u t q(t, s)ds, u ≥ t, is a Brownian motion with respect to Qt (see (43)) with initial value BQt t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Set now g∗(t, s, q) ≜ ess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='sup z∈Rd {q · z − g(t, s, z)}, q ∈ Rd, for s, t such that s ≥ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' We obtain that, for any fixed t, g∗(t, ·, q(t, ·)) is adapted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Furthermore, Yt = −X − � u t [q(t, s) · Z(t, s) − g(t, s, Z(t, s))]ds − � u t Z(t, s)dBQt s ≥ −X − � u t g∗(t, s, q(t, s))ds − � u t Z(t, s)dBQt s , 24 where the last inequality follows by the definition of g∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' By taking the conditional expectation with respect to Qt, it follows Yt ≥ EQt [−X|Ft] − EQt �� u t g∗(t, s, q(t, s))ds ���� Ft � , since EQt[ � u t Z(t, s)dBQt s |Ft] = 0 because (BQt s )s≥t is a Brownian motion with respect to Qt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Consequently, Yt ≥ ess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='sup Qt∈Qtu � EQt � − X|Ft � − EQt � � u t g∗(t, s, q(t, s))ds ��Ft �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' (45) It remains to prove the converse inequality in (45).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' In a similar way as in Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='4 (i) and Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='5 of Barrieu and El Karoui [3], for t fixed and any Z(t, s), s ≥ t, there exist some progressively measurable qZ(t, s) (associated to a Qt,Z via (43)) such that g(t, s, Z(t, s)) = qZ(t, s) · Z(t, s) − g∗(t, s, qZ(t, s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Hence, Yt = −X + � u t g(t, s, Z(t, s))ds − � u t Z(t, s)dBs = −X + � u t [qZ(t, s) · Z(t, s) − g∗(t, s, qZ(t, s))]ds − � u t Z(t, s)dBs = −X − � u t g∗(t, s, qZ(t, s))ds − � u t Z(t, s)dBQt,Z s , where BQt,Z s = Bs−Bt− � s t qZ(t, r)dr, t ≤ s ≤ u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' By taking the conditional expectation with respect to Qt,Z, it holds that Yt = EQt,Z � −X − � u t g∗(t, s, qZ(t, s))ds ���� Ft � ≤ ess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='sup Qt∈Qtu � EQt [−X|Ft] − EQt �� u t g∗(t, s, q(t, s))ds ���� Ft �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' (46) The thesis follows then by (45) and (46).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='1 Time-consistency The result below provides a necessary and sufficient condition on g for sub time- consistency of (ρtu)t,u induced by a BSVIE (g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Theorem 20 Let (ρtu)t,u be induced by a BSVIE with driver g as in (42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' a) The driver g(t, v, z) is decreasing in t for any v ∈ [t, u], z ∈ Rd (meaning that, for any s ≤ t, g(t, v, z) ≤ g(s, v, z)) if and only if (ρtu)t,u satisfies sub time-consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' b) If the driver g(t, ·, ·) is constant in t, then (ρtu)t,u satisfies time-consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' 25 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' a) Assume that sub time-consistency holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' By the dual representation of (ρtu)t,u in Proposition 19, the penalty term of ρtu is given by αtu(Qt) = EQt �� u t g∗(t, v, q(t, v))dv ���� Ft � for any Qt ∈ Qtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Let s, t, u ∈ [0, T] with s ≤ t ≤ u and let Qs ∈ Qst, Qt ∈ Qtu be fixed arbitrarily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Set now ¯Q = Qs · Qt the pasting of Qs on [s, t] and of Qt on [t, u], hence ¯Q ∈ Qsu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Denote by q(s, v), q(t, v) and ¯q(s, v) the corresponding processes as in (43).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' By applying the characterization of penalty term of sub time-consistent risk measures (see Proposition 5(iii)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' we obtain E ¯Q � � u s g∗(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' ¯q(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' v))dv ���Fs � ≤ E ¯Q � � t s g∗(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' ¯q(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' v))dv ���Fs � + E ¯Q � E ¯Q � � u t g∗(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' ¯q(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' v))dv ���Ft ����Fs � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' hence E ¯Q � � u t g∗(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' ¯q(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' v))dv ���Fs � ≤ E ¯Q � E ¯Q � � u t g∗(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' ¯q(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' v))dv ���Ft ����Fs � EQs � EQt � � u t g∗(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' ¯q(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' v))dv ���Ft ����Fs � ≤ EQs � EQt � � u t g∗(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' ¯q(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' v))dv ���Ft ����Fs � EQs � EQt � � u t � g∗(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' ¯q(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' v)) − g∗(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' ¯q(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' v)) � dv ���Ft ����Fs � ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' (47) Since (47) should hold for any s ≤ t ≤ u and any Qs ∈ Qst, Qt ∈ Qtu, it follows that g∗(t, v, ¯q(t, v)) ≥ g∗(s, v, ¯q(s, v)) for any s ≤ t, v ∈ [t, u], ¯q ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Hence, g(t, ·, ·) ≤ g(s, ·, ·) for any s ≤ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Conversely, assume that g(t, ·, ·) is decreasing in t, in the meaning specified above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' We are going to prove that ρst(−ρtu(X)) = Eg,V (Eg,V (−X|Ft)|Fs) ≤ Eg,V (−X|Fs) = ρsu(X) (48) for any s ≤ t ≤ u and X ∈ L2(Fu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' By definition of Eg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='V ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' the left-hand and right-hand sides of the previous equation can be rewritten as follows: Eg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='V (Eg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='V (−X|Ft)|Fs) = Eg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='V (−X|Ft) + � t s g(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Z(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' v)) dv − � t s Z(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' v) dBv = −X + � u t g(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' ˜Z(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' v) dv − � u t ˜Z(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' v) dBv + � t s g(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Z(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' v)) dv − � t s Z(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' v) dBv = −X + � u s � g(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Z(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' v))1[s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='t](v) + g(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' ˜Z(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' v))1(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='u](v) � dv − � u s � Z(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' v)1[s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='t](v) + ˜Z(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' v)1(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='u](v) � dBv (49) 26 and Eg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='V (−X|Fs) = −X + � u s g(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' ˆZ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' v) dv − � u s ˆZ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' v) dBv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' (50) Furthermore, by decreasing monotonicity of g(t, ·, ·) in t and by (49) it follows that Eg,V (Eg,V (−X|Ft)|Fs) ≤ −X + � u s � g(s, v, Z(s, v))1[s,t](v) + g(s, v, ˜Z(t, v))1(t,u](v) � dv − � u s � Z(s, v)1[s,t](v) + ˜Z(t, v)1(t,u](v) � dBv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Set now ¯Zt(s, v) = Z(s, v)1[s,t](v) + ˜Z(t, v)1(t,u](v), v ≥ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' By the arguments above, Eg,V (Eg,V (−X|Ft)|Fs) ≤ −X + � u s g(s, v, ¯Zt(s, v) dv − � u s ¯Zt(s, v) dBv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' (51) Sub time-consistency (48) then follows by comparing (50) and (51) and by the unique- ness of the solution of a BSVIE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' b) The case of time-consistency can be obtained by replacing inequalities with equalities in the proof above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Notice that a BSVIE of the form (50), that is with a final condition X that does not depend on t and with a driver g independent of y, reduces to a BSDE when the driver g(t, v, z) is constant in t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' An alternative proof of item a) could be given in terms of a Converse Comparison Theorem for BSVIEs (similarly to Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='1 of Briand et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' [8] for BSDEs) if that result was available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' However, while for BSDEs the Converse Comparison Theorem has been proved by Briand et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' [8], for BSVIEs we are not aware of any similar result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' We prove below that a Converse Comparison Theorem holds also for BSVIEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' We recall that a BSVIE (41) is related to the following family of BSDEs parame- terized by t: η(r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' t, ξt) = ξt + � T r ¯g(v, ζ(v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' t)dv − � T r ζ(v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' t)dBv, r ∈ [t, T] (52) where ζ(v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' t) = Z(t, v);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' ¯g(v, ζ(v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' t) = g(t, v, Z(t, v)) and Yt = η(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' t, ξt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' We remark again that the drivers above do not depend on y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' In this way, translation invariance of the associated risk measure is not guaranteed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' We are then able to prove a Converse Comparison Theorem for BSVIEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' 27 Theorem 21 (Converse Comparison Theorem for BSVIEs) Let Y 1 t and Y 2 t be two BSVIEs as in (41) with drivers g1 and g2 and terminal condition ξt and let η1 and η2 be the corresponding families defined in (52).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Assume that gi(·, v, ·) is continuous in v for i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' a) If η1(r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' t, ξt) ≤ η2(r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' t, ξt) for any t ∈ [0, T], r ∈ [t, T] and ξt ∈ L2(FT ), then g1 ≤ g2, that is g1(t, v, z) ≤ g2(t, v, z) for any t ∈ [0, T], v ∈ [t, T], z ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' b) If Y 1,X t ≤ Y 2,X t for any t ∈ [0, T] and X ∈ L2(FT ) and if gi(t, v, 0) = 0 for i = 1, 2, then g1 ≤ g2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' We recall that, for BSDEs, it is sufficient to require that Eg1(X) ≤ Eg2(X) holds for any X ∈ L2(FT ) to guarantee that Eg1 t (X) ≤ Eg2 t (X) for any X ∈ L2(FT ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' This is mainly due to time-consistency of g-expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' See Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='4 of Briand et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' [8] for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' The proof of item b) is based on the argument above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' a) By (52), η1(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' t) and η2(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' t) are two BSDEs parameterized by t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' By applying the Converse Comparison Theorem for BSDEs (see Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='1 of Briand et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' [8]) to η1(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' t), η2(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' t), it follows that ¯g1(v, ζ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' t) ≤ ¯g2(v, ζ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' t) for any v ≥ r ≥ t and ζ ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Hence g1(t, v, z) ≤ g2(t, v, z) for any t ∈ [0, T], v ∈ [t, T], z ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' b) First of all, we prove that, given two BSDEs with drivers g1 and g2 satisfying continuity in time and gi(v, 0) = 0 and a fixed t, Eg1(X|Ft) ≤ Eg2(X|Ft) for any X ∈ L2(FT ) ⇒ Eg1(X|Fr) ≤ Eg2(X|Fr) for any X ∈ L2(FT ) and r ∈ [t, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' (53) This implication extends Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='4 of Briand et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' [8] when t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Assume now that, for a given fixed t, Eg1(X|Ft) ≤ Eg2(X|Ft) for any X ∈ L2(FT ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' By using a similar approach to those in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='4 of Briand et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' [8] and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='5 of Coquet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' [9], 0 = Eg1 � X − Eg1(X|Fr) ��� Ft � ≤ Eg2 � X − Eg1(X|Fr) ��� Ft � = Eg2 � Eg2 � X − Eg1(X|Fr) ��� Fr ���� Ft � = Eg2 � Eg2(X|Fr) − Eg1(X|Fr) ��� Ft � for any X ∈ L2(FT ) and r ∈ [t, T], where the first and the last equality are due to translation invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Fix now r ∈ [t, T] arbitrarily and consider ξ = X1A with A = {Eg2(X|Fr) < Eg1(X|Fr)} ∈ Fr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' On the one hand, by the arguments above, Eg2 � Eg2(ξ|Fr) − Eg1(ξ|Fr) ��� Ft � ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' On the other hand, Eg2 � Eg2(ξ|Fr) − Eg1(ξ|Fr) ��� Ft � = Eg2 � 1A � Eg2(X|Fr) − Eg1(X|Fr) ���� Ft � ≤ Eg2(0|Ft) = 0, 28 because Egi(X1A|Fr) = 1AEgi(X|Fr) for any A ∈ Fr holds because of normalization of gi (see Peng [20]) and the last equality is due to g2(t, v, 0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Hence, Eg2 � 1A � Eg2(X|Fr) − Eg1(X|Fr) ���� Ft � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Since 1A � Eg2(X|Fr) − Eg1(X|Fr) � ≤ 0 and strictly negative with probability equal to P(A), strict monotonicity of conditional g-expectation implies that 1A � Eg2(X|Fr) − Eg1(X|Fr) � = 0, P − a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=', then Eg2(X|Fr) ≤ Eg1(X|Fr), P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=', for r ∈ [t, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' This concludes the proof of (53).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Going back to BSVIEs, denote by ηt,X i,r = ηi(r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' t, X) and by ¯gt i = ¯gi(·, ·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' t) for i = 1, 2, where η and ¯g are defined above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Fix now t arbitrarily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Assuming that Y 1,X t ≤ Y 2,X t for any X ∈ L2(FT ) is equivalent to assuming that ηt,X 1,t ≤ ηt,X 2,t or, also, to E ¯gt 1(X|Ft) ≤ E ¯gt 2(X|Ft) for any X ∈ L2(FT ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' By (53), it follows that Y 1,X t ≤ Y 2,X t for any X ∈ L2(FT ) implies that E ¯gt 1(X|Fr) ≤ E ¯gt 2(X|Fr) for any r ∈ [t, T], hence ηt,X 1,r ≤ ηt,X 2,r for any r ∈ [t, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' The thesis then follows by item a) where it is not necessary to have ξt but it is enough to consider X ∈ L2(FT ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='2 H-longevity The next results investigate under which conditions on the driver g h-longevity of (ρtu)t,u is fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Corollary 22 If g(t, ·, ·) is decreasing in t and g(t, v, 0) ≥ 0 for any t ≤ v, then h-longevity holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Since g(t, ·, ·) is decreasing in t, sub time-consistency follows by Theorem 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' We then show here ρtu(0) ≥ 0 for any t ≤ u, so that we can conclude by Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' We prove now that g(t, v, 0) ≥ 0 for any t ≤ v implies that ρtu(0) ≥ 0 for any t ≤ u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Indeed, ρtu(0) = Yt = � u t g(t, v, Z(t, v))dv − � u t Z(t, v)dv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Since the solution is (Yt = � u t g(t, v, 0)dv;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Z(t, v) = 0), then ρtu(0) ≥ 0 by the assump- tion g(t, v, 0) ≥ 0 for any t ≤ v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' In reality, we can do something more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Proposition 23 If g(s, v, 0) ≥ 0 for any s ≤ v, then h-longevity holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Furthermore, γ(s, t, u, X) = E � Qs,X �� u t g(s, v, 0)dv|Fs � for any s ≤ t ≤ u, where �Qs,X is a suitable probability measure depending on X, with density d �Qs,X dP ≜ exp � −1 2 � u s |∆zg(s, v)|2dv + � u s ∆zg(s, v)dBv � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' 29 Here above ∆zg(v) = (∆i zg(v))i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=',d and ∆i zg(v) ≜ g(s, v, Zu(s, v)) − g(s, v, ¯Zt(s, v)) d � Zu,i(s, v) − ¯Zt,i(s, v) � 1{Zu,i(s,v)̸= ¯Zt,i(s,v)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Let s ≤ t ≤ u and let X ∈ L2(Ft) be fixed arbitrarily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' The risk measures ρst and ρsu satisfy, respectively, the following BSVIEs: ρst(X) = −X + � t s g(s, v, Zt(s, v))dv − � t s Zt(s, v)dBv ρsu(X) = −X + � u s g(s, v, Zu(s, v))dv − � u s Zu(s, v)dBv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Set now ¯Zt(s, v) = � Zt(s, v);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' s ≤ v ≤ t 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' t < v ≤ u ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' �Z(s, v) = Zu(s, v) − ¯Zt(s, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Then ρsu(X) − ρst(X) = � u s [g(s, v, Zu(s, v)) − g(s, v, ¯Zt(s, v))]dv + � u t g(s, v, ¯Zt(s, v))dv − � u s [Zu(s, v) − ¯Zt(s, v)]dBv − � u t ¯Zt(s, v)dBv = � u s [g(s, v, Zu(s, v)) − g(s, v, ¯Zt(s, v))]dv + � u t g(s, v, 0)dv − � u s �Z(s, v)dBv = � u s ∆zg(s, v) · �Z(s, v)dv − � u s �Z(s, v)dBv + � u t g(s, v, 0)dv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' (54) Furthermore, (54) can be rewritten as a linear BSVIE δρs = Γt,u s + � u s ∆zg(s, v) · �Z(s, v)dv − � u s �Z(s, v)dBv, (55) where δρs ≜ ρsu(X) − ρst(X) and Γt,u s ≜ � u t g(s, v, 0)dv represents the final condition at time u (which depends on t and also on s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Since Γt,u ≥ 0 for any t by hypothesis and ∆zg(s, v) ∈ H2 [0,T](Rd) by the assumption of g Lipschitz in z, by the Comparison Theorem on BSVIEs (see Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='3 of Wang and Yong [29]) it follows the longevity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' δρs ≥ 0 for any s ≤ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Moreover, by applying Girsanov Theorem, (55) becomes δρs = � u t g(s, v, 0)dv + � u s ∆zg(s, v) · �Z(s, v)dv − � u s �Z(s, v)dBv = � u t g(s, v, 0)dv − � u s �Z(s, v)dB � Qs,X v , 30 where B � Qs,X v ≜ Bv − Bs − � v s ∆zg(s, v)dv, v ≥ s, is a Brownian motion with respect to �Qs,X with initial value B �Qs,X s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Hence, by taking the conditional expectation with respect to �Qs,X, γ(s, t, u, X) = δρs = E � Qs,X �� u t g(s, v, 0)dv ���� Fs � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' It then follows that ρsu(X) − ρst(X) = E � Qs,X �� u t g(s, v, 0)dv|Fs � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' As discussed in Sections 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='1, γ may depend on the length of the time interval [u, t], that is, γ(s, t, u, X) = γs,t(h, X) with h = u−t or, even, γs,t(h) independent from X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' The following example provides some cases covering the situation above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Example 24 Let g(s, v, 0) ≥ 0 for any v ∈ [s, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Hence, by the result above, h- longevity holds and γ(s, t, u, X) = E � Qs,X �� u t g(s, v, 0)dv|Fs � for any X ∈ L2(Ft).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' a) If g(s, v, 0) = cs for any v ∈ [s, T], with cs ≥ 0, then cs is necessarily Fs- measurable (since it should be measurable for any v ≥ s) and, consequently, γ(s, t, u, X) = E � Qs,X �� u t g(s, v, 0)dv ���� Fs � = (u − t)cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' In other words, γ only depends on the evaluation time s and on h = u − t, that is, roughly speaking, on the length of the time interval over which there is an uncorrect use of the risk measure (ρsu versus ρst).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' b) If g(s, v, 0) = exp(−rs v) for any v ∈ [0, T], with rs ≥ 0, then rs is necessarily Fs-measurable (for the same arguments as above) and, consequently, γ(s, t, u, X) = e−rst � 1 − e−rs(u−t)� rs .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Hence γ depends on the evaluation time s, on the “right” time horizon t (referring to the measurability of X) and on the length of the time interval [t, u].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Compared to the BSDE case (see Example 10), here γ depends also on the evaluation time s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' This is not surprising for BSVIEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Finally, we provide two examples of BSVIEs: the former with a linear driver, the latter going beyond the Lipschitz case and similarly to Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Example 25 Consider the driver g(t, s, z) = a(t, s) · z + b(t, s) for any 0 ≤ t ≤ s ≤ T and z ∈ Rd, where the Rd-valued process a(t, s) and 1-dimensional process b(t, s) are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' By applying Girsanov Theorem, in the same line of Hu and Øksendal [16], the BSVIE associated to the linear driver above becomes Yt = −X + � T t [a(t, s) · Z(t, s) + b(t, s)]ds − � T t Z(t, s)dBs = −X + � T t b(t, s)ds − � T t Z(t, s)dB � Qt s 31 where d � Qt dP ≜ exp � − 1 2 � T t |a(t, s)|2ds + � T t a(t, s)dBs � and B � Qt u ≜ Bu−Bt− � u t a(t, s)ds, for u ≥ t, is a Brownian motion with respect to �Qt with initial value B �Qt t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Hence, by taking the conditional expectation with respect to �Qt, it holds that Yt = E � Qt � −X + � T t b(t, s)ds ���� Ft � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Chosing b(t, s) ≥ 0, the h-longevity holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Example 26 Consider the driver g(t, s, z) = b(t)|z|2 2 + a(t, s) for any 0 ≤ t ≤ s ≤ T and z ∈ Rd, where the deterministic function b is positive and the process a is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Hence Yt = −X + � T t � b(t)|Z(t, s)|2 2 + a(t, s) � ds − � T t Z(t, s)dBs = −X + � T t a(t, s)ds + � T t b(t)|Z(t, s)|2 2 ds − � T t Z(t, s)dBs and, following the same arguments of [28], Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='1, it follows that Yt = 1 b(t) ln EP � exp � −b(t) � X − � T t a(t, s)ds �� ���Ft � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Whenever a(t, s) is deterministic, Yt becomes Yt = 1 b(t) ln EP � e−b(t)X���Ft � + � T t a(t, s)ds, that is a translation of the usual entropic risk measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Choosing a(t, s) > 0, h-longevity holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='2 Risk measures generated by a family of BSVIEs Suppose that, for any t ≤ u, the risk measure ρtu comes from a BSVIE with a driver gu depending on the maturity u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' This means that ρG tu(X) = Egu,V (−X|Ft) , X ∈ L2(Fu), (56) where Egu,V (ξ|Ft) denotes the Y -component of the solution (Yt, Z(t, s))t,s∈[0,T],s≥t of the following BSVIE with driver gu: Yt = ξ + � u t gu(t, s, Z(t, s))ds − � u t Z(t, s)dBs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' (57) Assume now that G = (gu)u∈[0,T] is a family of drivers depending on the maturity u, independent of y, Lipschitz, and convex in z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Each risk measure ρG tu is of type (42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' 32 By applying Theorem 19 with a driver gu parameterized by u, in a Brownian setting ρG tu can be represented as ρtu(X) = ρG tu(X) = ess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='sup Qt∈Qtu � EQt [−X|Ft] − αG tu(Qt) � , X ∈ L2(Fu), where Qtu is defined in (43), g∗ u(t, s, ·) denotes the convex conjugate of gu(t, s, ·) and the minimal penalty functional is given by αG tu(Qt) = EQt �� u t g∗ u(t, s, q(t, s))ds ���� Ft � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' (58) Furthermore, if gu(t, s, 0) = 0 for any t ≤ s ≤ u then ρtu(0) = 0 for any u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Differently from the risk measures generated by a single BSVIE but similarly to those generated by a family of BSDEs, in general gu(t, s, 0) = 0 for any t ≤ s ≤ u does not imply the restriction property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' The following result shows that, for gu(t, s, 0) = 0 for any t, s, u, the restriction property is satisfied only for risk measures induced by a single BSVIE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' This result is not surprising in view of Proposition 11 for the case of BSDEs with a family of drivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Proposition 27 Let gu(t, s, 0) = 0 for any t, s, u and let gu(·, s, ·) be continuous in s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' The restriction property (2) holds if and only if gu is constant in u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Assume that the restriction property holds, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' ρtu(X) = ρtv(X) for any t ≤ u ≤ v and X ∈ L2(Fu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Similarly to (52), denote by ηt,X ru = −X + � u r ¯gt u(s, ζt v)dv − � u r ζt vdBv, r ∈ [t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' u] where ζt v = Z(t, v);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' ¯gt u(v, ζt v) = g(t, v, Z(t, v)) and ρtu(X) = ηt,X tu .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Assumptions on gu guarantee that ¯gt u(s, 0) = 0 for any s and that ¯gt u(s, ·) is continuous in s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Proceeding as in the proof of the Converse Comparison Theorem of Briand et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' [8], Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='1, ¯gt u(s, z) = lim ε→0 ηt,z·(Bs+ε−Bε) su ε ¯gt v(s, z) = lim ε→0 ηt,z·(Bs+ε−Bε) sv ε , with convergence in L2, for any z ∈ Rd, u ≤ v and s ∈ [0, u].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' By extracting a subsequence to obtain convergence P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' and passing to the limit as ε → 0, it holds that ηt,z·(Bs+ε−Bε) sv ε = ηt,z·(Bs+ε−Bε) su ε −→ ¯gt u(s, z), ǫ → 0, P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' where the equality is due to restriction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' The thesis then follows because ηt,z·(Bs+ε−Bε) sv ε −→ ¯gt v(s, z), ǫ → 0, P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' The converse follows immediately by Proposition 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' 33 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='1 Time-consistency The following result provides a necessary and sufficient condition for a fully-dynamic risk measure induced by a family of BSVIEs to satisfy sub time-consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Note that the condition on the monotonicity of g·(t, ·, ·) is the same as for a BSVIE with a single driver, while the condition on the monotonicity of the family gu is new.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Proposition 28 Let (ρtu)t,u be induced by a BSVIE with a family of drivers (gu)u∈[0,T] as in (56).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' a) (ρtu)t,u satisfies sub time-consistency if and only if both the family G is increasing and g·(t, ·, ·) is decreasing in t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' b) (ρtu)t,u satisfies time-consistency if and only if G={g} and g·(t, ·, ·) is constant in t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' a) Assume sub time-consistency holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' By (58), the penalty term in the dual representation of ρtu is given by αG tu(Qt) = EQt �� u t g∗ u(t, v, q(t, v))dv ���� Ft � for any Qt ∈ Qtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Let s, t, u ∈ [0, T] with s ≤ t ≤ u and let Qs ∈ Qst, Qt ∈ Qtu be fixed arbitrarily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Set now ¯Q the pasting of Qs on [s, t] and of Qt on [t, u], hence ¯Q ∈ Qsu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Denote by q(s, v), q(t, v) and ¯q(s, v) the corresponding processes as in (43).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' From the characterization of the penalty term for sub time-consistency in Proposition 5(iii) it follows that E ¯Q �� u s g∗ u(s, v, ¯q(s, v))dv ���� Fs � ≤ E ¯Q �� t s g∗ t (s, v, ¯q(s, v))dv ���� Fs � + E ¯Q � E ¯Q �� u t g∗ u(t, v, ¯q(t, v))dv ���� Ft ����� Fs � , hence 0 ≤ EQs �� t s [g∗ t (s, v, ¯q(s, v)) − g∗ u(s, v, ¯q(s, v))] dv ���� Fs � +EQs � EQt �� u t [g∗ u(t, v, ¯q(t, v)) − g∗ u(s, v, ¯q(s, v))] dv ���� Ft ����� Fs � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' (59) Since (59) should hold for any s ≤ t ≤ u and any Qs ∈ Qst, Qt ∈ Qtu, it follows that � g∗ t (s, v, ¯q) ≥ g∗ u(s, v, ¯q), for any s ≤ v ≤ t ≤ u and ¯q ∈ Rd g∗ u(t, v, ¯q) ≥ g∗ u(s, v, ¯q), for any s ≤ t ≤ v ≤ u and ¯q ∈ Rd Hence, gu is increasing in u and g·(t, ·, ·) is decreasing in t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Conversely, sub time-consistency of (ρtu)t,u induced by a family of BSVIEs can be written in the following notation Egt,V (Egu,V (−X|Ft)|Fs) ≤ Egu,V (−X|Fs) (60) 34 for any s ≤ t ≤ u and X ∈ L2(Fu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' By (57), the right-hand and left-hand sides of the previous equation can be rewritten, respectively, as follows: Egu,V (−X|Fs) = −X + � u s gu(s, v, ˆZ(s, v)) dv − � u s ˆZ(s, v) dBv (61) and Egt,V (Egu,V (−X|Ft)|Fs) = −X + � u t gu(t, v, ˜Z(t, v)) dv − � u t ˜Z(t, v) dBv + � t s gt(s, v, Z(s, v)) dv − � t s Z(s, v) dBv = −X + � u s � gt(s, v, Z(s, v))1[s,t](v) + gu(t, v, ˜Z(t, v))1(t,u](v) � dv − � u s � Z(t, v)1[s,t](v) + ˜Z(t, v)1(t,u](v) � dBv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' (62) Furthermore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' (62) becomes Egt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='V (Egu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='V (−X|Ft)|Fs) ≤ −X + � u s � gt(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Z(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' v))1[s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='t](v) + gu(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' ˜Z(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' v))1(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='u](v) � dv − � u s � Z(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' v)1[s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='t](v) + ˜Z(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' v)1(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='u](v) � dBv ≤ −X + � u s � gu(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Z(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' v))1[s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='t](v) + gu(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' ˜Z(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' v))1(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='u](v) � dv − � u s � Z(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' v)1[s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='t](v) + ˜Z(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' v)1(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='u](v) � dBv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' (63) where the former inequality is due to decreasing monotonicity of g·(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' ·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' ·),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' the latter from increasing monotonicity of the family of drivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' By setting ¯Z(s, v) = Z(s, v)1[s,t](v) + ˜Z(t, v)1(t,u](v), (63) becomes Egt(Egu(−X|Ft)|Fs) ≤ −X + � u s gu(s, v, ¯Z(s, v)) dv − � u s ¯Z(s, v) dBv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' (64) Sub time-consistency then follows by comparing (64) and (61) and by the uniqueness of the solution of a BSVIE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' b) The case of time-consistency can be obtained by replacing inequalities with equalities in the proof above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' 35 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='2 H-longevity The following result provides sufficient conditions for h-longevity, similarly to Propo- sition 16 for BSDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Proposition 29 a) If G is an increasing family of drivers, g·(t, ·, ·) is decreasing in t, and ρtu(0) ≥ 0 for any t ≤ u, then (ρtu)t,u satisfies h-longevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' b) If G is an increasing family of drivers and gu ≥ 0 for any u, then h-longevity holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' a) follows by Remark 4 and Proposition 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' b) Let s ≤ t ≤ u and X ∈ L2(Ft) be fixed arbitrarily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' The risk measures ρst and ρsu satisfy, respectively, the following BSVIEs: ρst(X) = −X + � t s gt(s, v, Zt(s, v))dv − � t s Zt(s, v)dBv ρsu(X) = −X + � u s gu(s, v, Zu(s, v))dv − � u s Zu(s, v)dBv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Set now ¯Zt,u(s, v) = � Zt(s, v);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' s ≤ v ≤ t 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' t < v ≤ u ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' �Z(s, v) = Zu(s, v) − ¯Zt,u(s, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Then δρs = ρsu(X) − ρst(X) = � u s [gu(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Zu(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' v)) − gt(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' ¯Zt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='u(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' v))]dv + � u t gt(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' 0)dv − � u s [Zu(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' v) − ¯Zt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='u(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' v)]dBv ≥ � u s [gu(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Zu(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' v)) − gu(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' ¯Zt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='u(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' v))]dv − � u s �Z(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' v)dBv = � u s ∆zgu(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' v) · �Z(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' v)dv − � u s �Z(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' v)dBv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' (65) where the inequality is due to the hypothesis on the drivers and where ∆i zgu(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' v) ≜ gu(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Zu(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' v)) − gu(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' ¯Zt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='u(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' v)) d � Zu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='i(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' v) − ¯Zt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='i(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' v) � 1{Zu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='i(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='v)̸= ¯Zt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='i(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='v)} for i = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=', d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' By Girsanov Theorem, (65) becomes δρs ≥ − � u s �Z(s, v)dB � Qs v , where d � Qs dP ≜ exp � − 1 2 � u s |∆zgu(s, v)|2dv + � u s ∆zgu(s, v)dBv � and B � Qs v ≜ Bv − Bs − � v s ∆zgu(s, v)dv, for v ≥ s, is a Brownian motion with respect to �Qs with initial value B � Qs s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Hence, by taking the conditional expectation with respect to �Qs, δρs ≥ E � Qs � − � u s �Z(s, v)dB � Qs v ���� Fs � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' 36 This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' In the spirit of Example 17, we can easily extend Example 25 and Example 26 to cover the case of families of BSVIEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' We thank Tomasz Bielecki, Matteo Burzoni, Igor Cialenco, Alessandro Doldi, Nicole El Karoui, Mario Ghossoub, Michael Kupper, Felix Liebrich, Max Nendel, and Frank Riedel for their interest and comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' The research leading to these results has received funding from the Research Council of Norway (RCN) within the project STORM - Stochastics for time-space risk models (nr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' 274410).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' In particular, the second author thanks this research group for the warm hospitality during her visits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' References [1] Acciaio, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=', Penner, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Dynamic risk measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' In Advanced mathematical methods for finance (pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' 1–34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Springer, Berlin, Heidelberg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' [2] Agram, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Dynamic risk measure for BSVIE with jumps and semimartin- gale issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Stochastic Analysis and Applications, 37(3), 361-376.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' [3] Barrieu, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=', El Karoui, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Pricing, hedging and optimally designing deriva- tives via minimization of risk measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' In: Volume on Indifference Pricing (ed: Rene Carmona), Princeton University Press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' [4] Bielecki, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=', Cialenco, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=', Pitera, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' A survey of time consistency of dynamic risk measures and dynamic performance measures in discrete time: LM- measure perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Probability, Uncertainty and Quantitative Risk 2(1), 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' [5] Bion-Nadal, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Dynamic risk-measures: Time consistency and risk mea- sures from BMO martingales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Finance and Stochastics, 12, 219–244.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' [6] Bion-Nadal, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Time consistent dynamic risk processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Stochastic Pro- cesses and their Applications, 119(2), 633–654.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' [7] Bion-Nadal, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=', Di Nunno, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Fully-dynamic risk-indifference pricing and no-good-deal bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' SIAM Journal on Financial Mathematics, 11(2), 620–658.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' [8] Briand, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=', Coquet, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=', Hu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=', M´emin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=', Peng, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' A converse comparison theorem for BSDEs and related properties of g-expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Electronic Communi- cations in Probability, 5, 101–117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' [9] Coquet, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=', Hu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=', M´emin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=', Peng, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Filtration-consistent nonlinear expectations and related g-expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Probability Theory and Related Fields, 123(1), 1-27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' [10] Delbaen, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=', Peng, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=', Rosazza Gianin, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Representation of the penalty term of dynamic concave utilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Finance and Stochastics, 14(3), 449–472.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' 37 [11] Detlefsen, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=', Scandolo, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Conditional and dynamic convex risk measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Finance and Stochastics, 9(4), 539–561.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' [12] Di Bartolomeo, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Risk, Uncertainty and Time Horizon: What Most Risk Models Get Wrong!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Newport Seminar, June 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Available at: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='northinfo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='com/documents/770.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='pdf [13] El Karoui, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=', Peng, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=', Quenez, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' (1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Backward stochastic differential equations in finance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Mathematical Finance 7(1), 1–71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' [14] F¨ollmer, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=', Schied, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Stochastic Finance, third edition, De Gruyter, Berlin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' [15] Fuhrmann, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=', Kupper, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=', Nendel, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Wasserstein perturbations of Markovian transition semigroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' arXiv preprint, arXiv:2105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content='05655v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' [16] Hu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=', Øksendal, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Linear Volterra backward stochastic integral equa- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Stochastic Processes and their Applications, 129(2), 626-633.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' [17] Jiang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Convexity, translation invariance and subadditivity for g- expectations and related risk measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' The Annals of Applied Probability, 18(1), 245-258.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' [18] Kobylanski, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Backward stochastic differential equations and partial dif- ferential equations with quadratic growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' The Annals of Probability, 28(2), 558- 602.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' [19] Laeven, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=', Stadje, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Robust portfolio choice and indifference valua- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Mathematics of Operations Research, 39(4), 1109-1141.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' [20] Peng, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' (1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' BSDE and related g-expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' El Karoui N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=', Mazliak L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Pit- man Research Notes in Mathematics Series, (364), 141–159.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' [21] Peng, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Dynamically consistent nonlinear evaluations and expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' arXiv preprint math/0501415.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' [22] Quenez, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=', Sulem, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' BSDEs with jumps, optimization and appli- cations to dynamic risk measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Stochastic Processes and their Applications, 123(8), 3328-3357.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' [23] Revuz, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=', Yor, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Continuous martingales and Brownian motion (Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' 293).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Springer Science & Business Media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' [24] Rosazza Gianin, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Risk measures via g-expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Insurance: Mathe- matics and Economics, 39(1), 19-34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' [25] Royer, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Backward stochastic differential equations with jumps and re- lated non-linear expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Stochastic processes and their applications, 116(10), 1358-1376.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' 38 [26] Yong, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Continuous-time dynamic risk measures by backward stochastic Volterra integral equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Applicable Analysis, 86(11), 1429-1442.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' [27] Yong, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Backward stochastic Volterra integral equations—a brief sur- vey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Applied Mathematics-A Journal of Chinese Universities, 28(4), 383-394.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' [28] Wang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=', Sun, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=', Yong, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Recursive utility processes, dynamic risk measures and quadratic backward stochastic Volterra integral equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Applied Mathematics & Optimization, 84(1), 145-190.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' [29] Wang, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=', Yong, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Comparison theorems for some backward stochastic Volterra integral equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' Stochastic Processes and their Applications, 125(5), 1756-1798.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} +page_content=' 39' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE4T4oBgHgl3EQfPQwx/content/2301.04971v1.pdf'} diff --git a/dNE1T4oBgHgl3EQfLQNc/content/2301.02974v1.pdf b/dNE1T4oBgHgl3EQfLQNc/content/2301.02974v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..dc4ab180c1c3197645609ac179d79176e178c828 --- /dev/null +++ b/dNE1T4oBgHgl3EQfLQNc/content/2301.02974v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:b194f1b243478ca39d246636bf0441029a00c7ff1c9c193faddb805c63099ea4 +size 3048307 diff --git a/dNE1T4oBgHgl3EQfLQNc/vector_store/index.faiss b/dNE1T4oBgHgl3EQfLQNc/vector_store/index.faiss new file mode 100644 index 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+cRemote Sensing Technology, Technical University of Munich, Munich, 80333, Germany +A R T I C L E I N F O +Keywords: +Cloud removal +Data fusion +A B S T R A C T +In this paper, we introduce Planet-CR, a benchmark dataset for high-resolution cloud removal with +multi-modal and multi-resolution data fusion. Planet-CR is the first public dataset for cloud removal +to feature globally sampled high resolution optical observations, in combination with paired radar +measurements as well as pixel-level land cover annotations. It provides solid basis for exhaustive +evaluation in terms of generating visually pleasing textures and semantically meaningful structures. +With this dataset, we consider the problem of cloud removal in high resolution optical remote +sensing imagery by integrating multi-modal and multi-resolution information. Existing multi-modal +data fusion based methods, which assume the image pairs are aligned pixel-to-pixel, are hence not +appropriate for this problem. To this end, we design a new baseline named Align-CR to perform the +low-resolution SAR image guided high-resolution optical image cloud removal. It implicitly aligns the +multi-modal and multi-resolution data during the reconstruction process to promote the cloud removal +performance. The experimental results demonstrate that the proposed Align-CR method gives the best +performance in both visual recovery quality and semantic recovery quality. The project is available at +https://github.com/zhu-xlab/Planet-CR, and hope this will inspire future research. +1. Introduction +Remote sensing imagery has been receiving consider- +able attention as major promising prospects in various appli- +cations such as Earth observation and environmental mon- +itoring (Xia et al., 2018; Yuan et al., 2020; Requena-Mesa +et al., 2021; Xiong et al., 2022). However, haze and clouds +in the atmosphere affect the transmission of electromagnetic +signals and lead to a deficiency of surface information (Duan +and Li, 2020), severely hindering the potential of remote +sensing imagery. Cloud removal aims at reconstructing the +cloud-contaminated regions to counteract the degradations +caused by clouds, and thus becomes an indispensable pre- +processing step in the analysis of remote sensing imagery. As +a continuing concern within the remote sensing community, +many exciting progresses on this task have been extensively +reported, including many data-driven approaches using deep +learning (Ebel et al., 2020; Xu et al., 2022; Cresson et al., +2022). Benefiting from the open access policy of the Landsat +program and the Copernicus program, several datasets have +been proposed to promote the development of deep learning +in the field of cloud removal, like RICE-II (Lin et al., 2019) +and SEN12MS-CR (Ebel et al., 2020) datasets. However, +there are still two major issues that seriously limit the de- +velopment of cloud removal: +• Unavailability of high resolution imagery for existing +datasets. Recent advances in remote sensing have +xufang@whu.edu.cn (F. Xu); yilei.shi@tum.de (Y. Shi); +patrick.ebel@tum.de (P. Ebel); yangwen@whu.edu.cn (W. Yang); +xiaoxiang.zhu@tum.de (X.X. Zhu) +ORCID(s): +given rise to the next-generation satellites that pro- +vide optical imagery with superior spatial resolu- +tion (Kondmann et al., 2021; Cornebise et al., 2022). +Such data allows geometric analysis on a finer scale, +while posing new challenges for recovering cloud- +covered regions with corresponding level of details. +The existing datasets with relatively low spatial res- +olutions severely restrict the development of cloud +removal algorithms for recovering clear edge and +rich texture details of high resolution remote sensing +imagery. +• Lack of evaluation in generating semantically mean- +ingful structures. High-quality reconstructed cloud- +free images require both visually pleasing textures and +semantically meaningful structures. Existing work al- +most exclusively evaluates the quality of the recon- +structed images through image quality metrics like +PSNR and SSIM, i.e. evaluates the quality of the +reconstructed image in terms of generating visually +pleasing textures. While it is desirable to evaluate +the semantically meaningful structures to identify the +usability of the reconstructed images. +In this paper, we build a Planet-CR dataset to fulfill the +requirement of high resolution cloud removal datasets. High +resolution remote sensing imagery with extensive spatial- +temporal coverage is not easily available. Some satellites +offer very high resolution imagery only from specific lo- +cations, which makes it impractical to construct datasets +with a high diversity of globally distributed sites. By trad- +ing off the spatial-temporal coverage and the resolution +of satellite imagery, we collect data from Planet satellite +imagery (PBC, 2018–). Planet provides global daily data +Fang Xu, et al: Preprint submitted to Elsevier +Page 1 of 11 +arXiv:2301.03432v1 [cs.CV] 9 Jan 2023 + +aPlanet-CR +with a spatial resolution of 3푚. On the one hand, it allows +to acquire paired cloudy and cloud-free images with a very +short temporal offset. It could minimize surface changes that +may appear between the acquisitions of cloudy and cloud- +free images, thereby minimizing nuisances between cloudy +reference images and cloud-free target images. On the other +hand, it allows to acquire heterogeneous earth observation +data to encourage general-purpose cloud removal but not +along narrowly-defined and geo-spatially distinct regions of +interest. Meanwhile, the Planet-CR dataset collects corre- +sponding pixel-level land cover annotations from World- +Cover product (Zanaga et al., 2021). It offers the chance of +validating the effectiveness of the cloud removal methods +in generating semantically meaningful structures on a well- +established remote sensing task. +Cloud removal is a highly ill-posed problem due to +the loss of surface information. Previous studies have re- +ported that the ill-posedness can be reduced by resorting +to Synthetic Aperture Radar (SAR) data, which is cloud- +penetrable and inherently reflects the geometrical shapes of +ground objects (Ebel et al., 2020; Gao et al., 2020; Xu et al., +2022). However, SAR data is usually unaffordable (Atwood +and Garron, 2013), especially for global Earth data. Thanks +to the freely accessible global SAR data provided by the +European Space Agency, we further include the Sentinel- +1 SAR data in our Planet-CR dataset to promote cloud +removal of PlanetScope data. Compared with the existing +cloud removal datasets like SEN12MS-CR, which mostly +explore the fusion of Sentinel-1 SAR data and Sentinel-2 +optical data with the same resolution of 10푚, our Planet- +CR dataset helps to understand a more practical yet more +complex problem, i.e. multi-modal and multi-resolution data +fusion based cloud removal, in which the misalignment due +to resolution difference, field-of-view mismatch, disparity, +etc. has to be considered. +To address the problem of multi-modal and multi- +resolution data fusion based cloud removal, we propose a +novel method called Align-CR, where the low-resolution +SAR images guide the reconstruction of high-resolution +cloud-free images from cloudy images. Specifically, Align- +CR implicitly aligns the feature maps using the deformable +convolution (Dai et al., 2017) to compensate for the mis- +alignment between the multi-modal and multi-resolution +data. Based on the Planet-CR dataset, we benchmark rep- +resentative cloud removal methods, and analyze their per- +formance in generating visually pleasing textures by im- +age reconstruction metrics and in generating semantically +meaningful structures by a well-established semantic seg- +mentation task. Extensive evaluation demonstrates that the +Align-CR method achieves the best performance on the +vast majority of our benchmark tests. In summary, our +contributions are as follows: +• We construct a dataset Planet-CR collected specially +for cloud removal to advance the field, which is the +public dataset with the highest spatial resolution to +date. It comprises a large number of regions all around +the world with auxiliary SAR imagery and land cover +information. +• We propose a novel cloud removal algorithm, Align- +CR, which can better capture the correlations across +the multi-modal and multi-resolution data to recon- +struct the occluded regions. +• We benchmark state-of-the-art cloud removal algo- +rithms with the Planet-CR dataset, which can be used +as the baseline for future algorithm development, and +conduct extensive evaluations for recovered semantic +information on a well-established semantic segmenta- +tion task. +2. Related Work +2.1. Datasets for Cloud Removal +To promote the progress of deep learning-based cloud +removal, several datasets for cloud removal have been pro- +posed. We provide an overview of publicly available cloud +removal datasets, as shown in Tab. 1. RICE-I (Lin et al., +2019) contains 500 pairs of cloudy and cloud-free images +collected on Google Earth by setting whether to display the +cloud layer, which only contains filmy, partly-transparent +clouds. RICE-II (Lin et al., 2019) contains about 700 paired +images derived from the Landsat 8 OLI/TIRS dataset, where +the acquisition time of cloudy and cloud-free images at the +same location is less than 15 days. However, the size of both +datasets is relatively small, and the data in them is geograph- +ically and topographically homogeneous. STGAN (Sarukkai +et al., 2020) contains nearly 100, 000 paired Sentinel-2 im- +ages drawn from 17800 distinct tiles worldwide. The cloud +cover for each cloudy image is between 10% and 30%. It +excludes the images with insufficient visible ground upon +manual inspection. Then SEN12MS-CR (Ebel et al., 2020) +collects around 150, 000 samples, containing different types +of real-life clouds, from 169 non-overlapping AOIs sampled +across all inhabited continents during all meteorological +seasons. Each AOI is composed of a pair of orthorectified, +geo-referenced cloudy and cloud-free Sentinel-2 images. +And to develop cloud removal models that are robust to +extensive cloud coverage conditions, each AOI contains an +additional co-registered Sentinel-1 SAR image. The related +SEN12MS-CR-TS dataset (Ebel et al., 2022) is structured +likewise, while featuring multi-season repeated measure- +ments. However, a common limitation of the aforemen- +tioned datasets is that they are not stratified by land cover +types, making it impossible to assess the generalizability +of cloud removal methods over individual land cover type +that may be of interest to specific applications like vege- +tation monitoring (Rogan et al., 2002; Lima et al., 2019; +Ngadze et al., 2020) and water resources monitoring (Fisher +et al., 2016). To evaluate the effectiveness of cloud removal +methods over different land cover types, WHU2-CR (Li +et al., 2021) selects 36 locations from all over the world +according to three main land covers: urban, vegetation, and +bare land, and produces about 20, 000 pairs of cloudy and +cloud-free Sentinel-2 images. And Czerkawski et al. (2022) +construct a cloud removal dataset specially in the context +Fang Xu, et al: Preprint submitted to Elsevier +Page 2 of 11 + +Planet-CR +Table 1 +Comparison between Planet-CR and existing publicly available cloud removal datasets. LC is short for land cover map. +Dataset +Optical Image +w/ SAR w/ LC +source +resolution #AOIs width #images +RICE-I (Lin et al., 2019) +Google Earth +< 15푚 +/ +512 +500 + + +RICE-II (Lin et al., 2019) +Landsat-8 +30푚 +/ +512 +450 + + +STGAN (Sarukkai et al., 2020) +Sentinel-2 +10푚 +17,800 +256 +97,640 + + +SEN12MS-CR (Ebel et al., 2020) +Sentinel-2 +10푚 +169 +256 +122,218 + + +SEN12MS-CR-TS (Ebel et al., 2022) +Sentinel-2 +10푚 +53 +256 +15,578 + + +WHUS2-CR (Li et al., 2021) +Sentinel-2 +10푚 +36 +256 +17,182 + + +Scotland&India∗ (Czerkawski et al., 2022) +Sentinel-2 +10푚 +445 +256 +445 + + +Planet-CR +PlanetScope +3푚 +780 +300 +63,000 + + +* The cloudy observations in the dataset are simulated. +of crop monitoring. The dataset contains paired Sentinel-1 +and Sentinel-2 Images for 2 locations in Scotland and India. +Notably, the cloud regions in the dataset are simulated with +a cloud coverage between 10% and 50%. It will inevitably +lead to unrealistic representations that cannot generalize to +the scenario of real cloud-covered satellite imagery and their +spectral characteristics (Ebel et al., 2020). +In addition to the previously mentioned public datasets, +there are also several non-public datasets. For example, +Enomoto et al. (2017) create a dataset by synthesizing the +simulated cloud on the cloud-free images with eight com- +paratively cloudless WorldView-2 images for learning. Cres- +son et al. (2019) use Sentinel-2 images acquired over the +province of Tuy, Burkina Faso to construct experiments. +Gao et al. (2020) collect two simulated datasets with around +35% cloud cover based on Gaofen-2 optical imagery and +airborne optical imagery, respectively, and a real dataset +based on Sentinel-2 optical imagery. These datasets are +relatively small in size and contain a limited number of +scenarios. While featuring high resolution imagery, cloudy +observations are all simulated. +The majority of datasets containing real-life clouds are +based on medium-resolution Sentinel-2 data. It is hard to +develop and evaluate the removal of clouds in high resolu- +tion imagery with salient structures and abundant textured +features. Our Planet-CR dataset aims to advance the task of +cloud removal from high resolution imagery by releasing the +cloudy and cloud-free PlanetScope data with short time lags +in combination with Sentinel-1 SAR data and WorldCover +land cover maps. The inclusion of Sentinel-1 SAR data +enables the exploration of multi-modal fusion for cloud +removal, and the inclusion of WorldCover land cover product +enables to disentangle the performance over different land +cover types and evaluate the quality of recovered semantic +information. While the short revisit time may facilitate to +temporal mosaicing, it is not applicable to time-critical ap- +plications (Voigt et al., 2007). And when encountering con- +tinual cloudy days, cloud-free reference data from an adja- +cent period is largely unavailable (Xu et al., 2022). It is worth +noticing that Sarukkai et al. (2020) train a baseline land +classification model and evaluate its accuracy from cloudy, +cloud-free, and predicted cloud-free images, respectively, to +demonstrate the power of the predicted cloud-free images +for downstream use. However, cloud removal is a pixel-level +reconstruction task, an image-level classification task cannot +adequately reflect the quality of predicted cloud-free images. +Because the correct class can be inferred even if the image +is partially occluded (Gu et al., 2022). With our Planet- +CR dataset, a pixel-level classification task, i.e. semantic +segmentation, can be designed, which will be more suitable +for evaluating the power of the cloud removal methods on +semantic information recovery. +2.2. Algorithms for Cloud Removal +Cloud removal in optical remote sensing imagery is +a long-term research problem. Most early developments +address this problem by interpolation (Zhang et al., 2007; +Yu et al., 2011), filtering (Mitchell et al., 1977; Liu and +Hunt, 1984), or inpainting (Li et al., 2002; Helmer and +Ruefenacht, 2005). Currently, deep learning-based methods +are gaining considerable attention. They have the poten- +tial to solve many of the problems that arise in traditional +cloud removal methods and achieve impressive results. For +example, Multispectral conditional Generative Adversarial +Networks (McGANs) (Enomoto et al., 2017), leveraging the +remarkable generative capabilities of conditional Generative +Adversarial Networks (cGANs), remove simulated clouds +from Worldview-2 imagery by extending the input channels +of cGANs to be compatible with multispectral input. Cloud- +GAN (Singh and Komodakis, 2018) learns the mapping +between cloudy and cloud-free Sentinel-2 imagery using +a cyclic consistent generative adversarial network. RSC- +Net (Li et al., 2019) estimates the cloud-free output with +the contaminated Landsat-8 imagery based on an encoding- +decoding framework consisting of multiple residual convo- +lutional layers and residual deconvolutional layers. +As the clouds thicken and the cloud-covered region is +dominant, the task of removing the clouds becomes more +challenging. A series of work explore the potential of SAR +data as auxiliary data for cloud removal in optical imagery. +For example, DSen2-CR (Meraner et al., 2020) uses a deep +residual neural network to predict the target cloud-free opti- +cal image from the concatenation of Sentinel-1 SAR image +and Sentinel-2 optical image. Simulation-Fusion GAN (Gao +et al., 2020) fuse SAR and corrupted optical imagery with +two generative adversarial networks to acquire the cloud-free +Fang Xu, et al: Preprint submitted to Elsevier +Page 3 of 11 + +Planet-CR +Fig. 1: Visualization of the Planet-CR dataset. (a) Spatial distribution of the 780 AOIs of the Planet-CR dataset. (b) Cloud-free +optical observations from PlanetScope. (c) Cloudy optical observations from PlanetScope. (d) SAR observations from Sentinel-1. +(e) land cover maps from WorldCover. They are resized to the same size for better view. +results of simulated cloudy Gaofen-2 data and real cloudy +Sentinel-2 data. GLF-CR (Xu et al., 2022) incorporates the +contribution of Sentinel-1 SAR image in restoring reliable +texture details and maintaining global consistency to recon- +struct the occluded region of Sentinel-2 optical image. How- +ever, these methods are developed on data with relatively low +spatial resolutions. The problem of cloud removal in high- +resolution imagery remains largely underexplored. +3. Data +3.1. Curation of Planet-CR +To develop the cloud removal methods that are equally +applicable to heterogeneous Earth observation data, the +Planet-CR dataset curates 780 non-overlapping, highly rep- +resentative locations of AOIs that are distributed over all +continents and meteorological seasons of the globe, as +shown in Fig. 1. Each AOI is composed of a quartet of +orthorectified, geo-referenced cloudy and cloud-free optical +images, as well as the corresponding SAR image and land +cover map. +Optical data. The optical data of our dataset originates +from the PlanetScope satellite constellation operated by +Planet Labs. We gather the cloud-free and cloudy observa- +tions from the Level-3B top-of-atmosphere reflectance prod- +uct, which are radiometrically-, sensor-, and geometrically- +corrected. It provides four-band (Blue, Red, Green and Near- +Infrared) imagery with a spatial resolution of 3푚. In our +dataset, the average time interval between paired cloudy +and cloud-free data is 2.8 days. This is shorter than the +intra-mission revisiting period of Landsat-8 and Sentinel-2 +satellites, and minimizes the amount of land cover change +occurring between our paired data. And the cloudy data +features a diversity of clouds, ranging from semi-transparent +to dense, and from light to heavy cloud covering. In addition +to the cloudy and reference cloud-free data, we also collect +the cloud masks associated with cloudy PlanetScope images +according to the Unusable Data Mask (UDM) assets pro- +vided by Planet Labs. UDM masks give information about +which pixels in the images are clear or cloudy, permitting a +statistical evaluation of cloud coverage and enabling cloud +mask-guided methods to work. +SAR data. The SAR data of our dataset originates from the +Sentinel-1 mission operated by the European Space Agency. +We gather the Sentinel-1 data with a spatial resolution +of 10푚 from the Level-1 Ground Range Detected (GRD) +product archived by Google Earth Engine (Gorelick et al., +2017). The measurements are acquired in interferometric +wide swath (IW) mode with two polarization channels VV +(vertical transmit/vertical receive) and VH (vertical trans- +mit/horizontal receive), which are pre-processed with ther- +mal noise removal, radiometric calibration, terrain correc- +tion, and are converted to backscatter coefficients (휎◦) in +units of decibels (dB). +Land cover map. The land cover maps of our dataset origi- +nates from WorldCover, which is an open-access global land +cover product at 10푚 resolution released by the European +Space Agency. In this study, we reclassified the data into 6 +basic land cover types according to DeepGlobe 2018 (Demir +et al., 2018): forest land, rangeland, agriculture land, urban +land, barren land and water. +We partition the dataset into train and test splits to +allow for direct comparison with future works. A total of +780 AOIs are split into 660 scenes for training and 120 +for testing following a random global distribution. To train +the deep learning-based methods, we crop each AOI into +small patches using slide windows of sensor-specific sizes. +Since the spatial resolution of PlanetScope imagery is 3푚, +and the spatial resolution of Sentinel-1 imagery as well as +WorldCover land cover maps is 10푚, we set the correspond- +ing slide window sizes to 300 × 300 and 90 × 90 pixels. +For the purpose of uniform distribution over different cloud +coverage levels as well as land cover types, we finally select +60, 000 quartets of training samples and 3,000 quartets of +testing samples. The statistics about cloud coverage and land +cover types can be seen in Fig. 2. +3.2. Properties of Planet-CR +High Spatial Resolution. The Planet-CR dataset releases +the cloudy and cloud-free PlanetScope data with a spatial +Fang Xu, et al: Preprint submitted to Elsevier +Page 4 of 11 + + train +testPlanet-CR +(a) +(b) +Fig. 2: Statistics of the Planet-CR dataset. (a) Distribution of +cloud coverage. (b) Distribution of land cover types. +resolution of 3m. Compared to existing publicly available +cloud removal datasets which are mostly built on medium- +resolution Landsat-8 data or Sentinel-2 data, our dataset +redeems the current lack of cloud removal datasets with high +resolution, and advances the task of high-resolution cloud +removal that is still under-explored. +Multi-Modal and Multi-Resolution. The inclusion of Sen- +tinel-1 SAR data can provide auxiliary information to pro- +mote cloud removal of PlanetScope data. Compared with +the existing cloud removal datasets like SEN12MS-CR, +which mostly explore the fusion of Sentinel-1 SAR data and +Sentinel-2 optical data with the same resolution of 10m, +our Planet-CR dataset helps to understand a more practical +yet more complex problem, i.e., multi-modal and multi- +resolution data fusion based cloud removal. +Information on Land Cover. The inclusion of land cover +can disentangle the performance of cloud removal methods +over different land cover types on the one hand, and encour- +ages to design a pixel-level classification task to evaluate the +power of the cloud removal method in generating semanti- +cally meaningful structures on the other. +4. Methodology +4.1. Problem Statement +Given a cloudy image , the task of cloud removal +aims to reconstruct a clear image  revealing the complete +information content of the ground scene so that subsequent +analysis can be reliably performed. The basic strategy is to +deal with the cloud contamination in a single image without +additional information, i.e., the problem of single image +Cloud Removal(CR), which can be formulated as: + = CR() +(1) +It is a highly ill-posed problem, usually solved under the +assumption that the cloud-contaminated regions have similar +spectral/geometrical characteristics to the remaining parts +of the image. However, when it comes to the areas with +high-frequency texture or different land cover types, the +reconstruction performance cannot be guaranteed. Many +studies resort to SAR images that are cloud-penetrable and +inherently reflect the geometrical shapes of ground objects +as an a priori assumption to reduce the ill-posedness. Thus, +the problem of Multi-Modal data Fusion based Cloud +Removal (MMF-CR) is introduced. It restores the clear +image from a cloudy image and the corresponding SAR +image , formulated as: + = MMF-CR(Ω, Ω) +(2) +where Ω indicates the same spatial domain shared by the +cloudy image and the corresponding SAR image. The SAR +image guides most MMF-CR methods to restore the cloud- +free image from pixel-to-pixel aligned cloudy and SAR im- +ages. However, the geo-referenced cloudy images captured +by Planet satellites and SAR images captured by Sentinel-1 +satellites in the Planet-CR dataset do not meet the assump- +tion of pixel-to-pixel alignment, mainly caused by resolution +difference, field-of-view mismatch and disparity. The prob- +lem of Multi-Modal and multi-Resolution data Fusion +based Cloud Removal(MMRF-CR) is thus introduced: + = MMRF-CR(Ω푦, Ω푠|) +(3) +where Ω푦 and Ω푠 respectively denote the spatial domain of +the cloudy image and the SAR image, and  ∶ Ω푠 → +Ω푦 denotes the pixel-level correspondence mapping oper- +ator. To solve the problem of MMRF-CR, it is required to +precisely align the cloudy and SAR images for the cloud +removal process. +4.2. Align-CR Network +The proposed network, called Align-CR, adopts a two- +stream architecture to compensate for the missing informa- +tion in cloudy regions using ancillary SAR image, as shown +in Fig. 3. Since the SAR image has a lower resolution, an +upsampling operator is firstly employed to map it to the same +resolution as the optical image. Then, the cloudy optical +image and the upsampled SAR image are passed through +their respective feature extraction blocks to extract modality- +specific features 퐹 0 +표푝푡 and 퐹 0 +푠푎푟. After that, 퐹 0 +표푝푡 and 퐹 0 +푠푎푟 +are fed into 퐷 AlignFuse blocks to obtain knowledgeable +features with comprehensive information. The AlignFuse +block performs alignment and fusion sequentially in the +feature space. +̂퐹 푖 +푠푎푟 = 퐻Align(퐹 푖 +표푝푡, 퐹 푖 +푠푎푟), +(4) +퐹 푖+1 +표푝푡 , 퐹 푖+1 +푠푎푟 = 퐻fuse(퐹 푖 +표푝푡, ̂퐹 푖 +푠푎푟), +(5) +where 퐻Align(⋅) and 퐻fuse(⋅) denote the functions of the +alignment block and the fusion block, respectively, and +the details are shown in Sec. 4.2.1 and 4.2.2. Finally, all +intermediate features {퐹 푖 +표푝푡}퐷 +푖=1 are aggregated to reconstruct +the high-quality cloud-free image. +4.2.1. Alignment Block +Ideally, the task of MMRF-CR can be considered as +a two-step task that aligns the multi-modal and multi- +resolution data first and reconstruct the clear image follow- +ing Eq. 2 later. However, explicit pixel-to-pixel alignment +to ensure that the same pixels in the SAR image and the +cloudy image reflect the same ground target is very hard +Fang Xu, et al: Preprint submitted to Elsevier +Page 5 of 11 + +12 +train +test +10 +%) +Occurrrence ( +8 +4 +2 +0 +10 +CloudCoverageof Image Patches(%）35 +train +test +30 +25 +Occurrrence +20 +15 +10 +5 +Urban +Barrer +Wate +Land Cover TypesPlanet-CR +Fig. 3: Overview of the proposed Align-CR method. An upsampling operator is firstly employed to map the SAR image to the +same resolution as the optical image. Then, the cloudy optical image and the upsampled SAR image are passed through their +respective feature extraction (FE) blocks to extract modality-specific features. After that, the features are fed into 퐷 AlignFuse +blocks to obtain knowledgeable features with comprehensive information. The AlignFuse block performs alignment and fusion +sequentially in the feature space. Finally, the output of all AlignFuse blocks are concatenated and fed to the image reconstruction +(IR) block to restore the high-quality cloud-free image. +to achieve in our case. On the one hand, sharp edges in +the high-resolution optical image cannot be exactly aligned +with blurry edges in the low-resolution SAR image. On +the other hand, for non-co-orbital satellite data, it is very +difficult to resolve the unalignment caused by field-of-view +mismatch and disparity. Moreover, the occlusions in the +cloudy images complicates the pixel-to-pixel alignment. +Therefore, we implicitly learn the alignment process for +cloud-free image reconstruction. In this paper, we resort to +the deformable convolution to align the SAR features to the +optical features. Given two features to be aligned as input, +i.e., 퐹 푖 +표푝푡 and 퐹 푖 +푠푎푟, 푖 = 0, ⋯ , 퐷 −1, The core to solve feature +misalignment is to predict the offset with an offset prediction +module, +▵푝푖 = 퐻offset(퐹 푖 +표푝푡, 퐹 푖 +푠푎푟), +(6) +where 퐻offset denotes the function of the offset prediction +module, which can be implemented by general convolutional +layers. With the predicted offset, the SAR feature can be +warped to the optical feature using the deformable convo- +lution (DConv), +̂퐹 푖 +푠푎푟 = DConv(퐹 푖 +푠푎푟, ▵푝푖), +(7) +Specifically, we adopt the Pyramid, Cascading and De- +formable convolutions (PCD) (Wang et al., 2019) for the +alignment process. It performs the alignment in a pyramid +structure, i.e., aligning the features in lower scales with +coarse estimations first and propagating the aligned features +and learned offsets to higher scales to refine the estimations +later. Embedding it into the network, the network’s ability to +model transformations can be enhanced. +4.2.2. Fusion Block +The aligned features, i.e., 퐹 푖 +표푝푡 and ̂퐹 푖 +푠푎푟, are then fed +into the fusion block for the transfer of complementary +information. Similar to our prior work (Xu et al., 2022), we +exploit the power of SAR information from two aspects: +global fusion, to guide the global interactions among all +local optical windows; local fusion, to transfer the SAR +feature corresponding to cloudy areas to compensate for +the missing information. Specifically, each fusion block +contains an adapted SAR-guided global context interaction +(SGCI) block followed by a SAR-based local feature com- +pensation (SLFC) block. To reduce the complexity of the +SGCI block, instead of adding a Swin Transformer layer +(STL) (Liu et al., 2021) after each convolutional layer in +the dense connected layers, we only add a STL after the +convolutional layer in the local feature fusion of the residual +dense block (RDB) (Zhang et al., 2020) for cross-window +feature interaction. +4.2.3. Loss Function +We train the Align-CR network by minimizing the dif- +ference between the output of network 퐲 and the cloud-free +image ̂y temporally close to the input cloudy image. Though +the Planet-CR has avoided the surface changes that may +appear between the acquisitions of cloudy and cloud-free +images as much as possible through short time lag for data +collection, there are some inevitable nuisances determined +by the sunlight condition, acquisition geometry, humidity, +pollution, change of landscape, etc (Xu et al., 2022). In this +paper, we use the Charbonnier loss (Lai et al., 2017) which +can better handles outlier for training, and extra constraints +on cloudy regions are applied, + = (1 + 푤퐌) ⊙ ((퐲 − ̂y)2 + 휖2)훼 +(8) +where ⊙ denotes Hadamard product operator, 푤 denotes the +extra weight for cloudy regions, 퐌 denotes the cloudy mask, +휖 and 훼 are constants. +5. Evaluations +5.1. Experimental Settings +Preprocessing. Before the PlanetScope and Sentinel-1 data +are fed into the neural networks, we apply value clipping +to eliminate a small number of anomalous pixels and data +scaling to improve the stability of the neural networks. We +Fang Xu, et al: Preprint submitted to Elsevier +Page 6 of 11 + +AlignFuse block 1 +block D +Prediction +阳 +F1 +2 +Fo +block +FD +block +opt +opt +opt +lign +use +R +Cloudy +F +F0 +FO +F1 +FD +sar +sar +sar +sar + Upsample +Add +SARPlanet-CR +Table 2 +Quantitative results of visual recovery quality over different land cover types. +(a) results on objective assessment metrics, MAE and PSNR, to quantify the prediction error +per class MAE ↓ +MAE↓ +PSNR↑ +Forest +Rangeland Agriculture +Urban +Barren +Water +McGAN +0.0321 +0.0417 +0.0497 +0.0492 +0.0507 +0.0442 +0.0425 +26.4273 +SpA GAN +0.0358 +0.0376 +0.0393 +0.0444 +0.0423 +0.0425 +0.0381 +26.1448 +SAR-Opt-cGAN +0.0300 +0.0296 +0.0331 +0.0352 +0.0344 +0.0414 +0.0323 +28.2106 +DSen2-CR +0.0289 +0.0293 +0.0311 +0.0352 +0.0340 +0.0353 +0.0298 +28.7832 +GLF-CR +0.0231 +0.0257 +0.0293 +0.0329 +0.0322 +0.0274 +0.0256 +29.9848 +Ours (wo/ SAR) +0.0261 +0.0259 +0.0296 +0.0321 +0.0315 +0.0369 +0.0272 +29.6859 +Ours (wo/ Align) +0.0238 +0.0251 +0.0282 +0.0318 +0.0311 +0.0311 +0.0255 +30.0107 +Ours (Align-CR) +0.0219 +0.0243 +0.0277 +0.0324 +0.0311 +0.0257 +0.0238 +30.4785 +(b) results on subjective assessment metrics, SAM and SSIM, to quantify spectral and structural similarity +per class SAM ↓ +SAM↓ +SSIM↑ +Forest +Rangeland Agriculture +Urban +Barren +Water +McGAN +9.5005 +12.1237 +13.4662 +14.6771 +14.4427 +15.6362 +12.2471 +0.7975 +SpA GAN +8.9910 +8.7126 +8.4814 +10.1953 +9.6112 +18.1517 +9.2522 +0.8181 +SAR-Opt-cGAN +7.4595 +6.4506 +6.7722 +8.1655 +7.1440 +11.0905 +7.2188 +0.8602 +DSen2-CR +7.1437 +6.5581 +6.3116 +7.7517 +7.2008 +9.7014 +6.6527 +0.8806 +GLF-CR +5.7830 +5.7131 +5.7403 +7.8324 +7.0525 +9.7931 +5.6189 +0.9063 +Ours (wo/ SAR) +7.0319 +6.2599 +6.3504 +7.6640 +7.0522 +10.1132 +6.5402 +0.8930 +Ours (wo/ Align) +6.0453 +5.7382 +5.7558 +7.5235 +6.8470 +9.6844 +5.7816 +0.9021 +Ours (Align-CR) +5.5766 +5.5408 +5.6306 +7.8660 +7.0693 +9.5205 +5.4183 +0.9124 +clip the values of all bands of the PlanetScope data to [0, +10, 000] and divide by 10, 000 for all bands. We clip the VV +and VH polarizations of the Sentinel-1 data to values [-25, +0] and [-32.5, 0], respectively, and rescale them to the range +[0, 1]. All experiments in this paper use these preprocessing +steps, following previous best practices (Meraner et al., +2020; Ebel et al., 2020). +Implementation Details. The proposed Align-CR network +is implemented using Pytorch and trained on 2 NVIDIA +Geforce RTX 3090 GPUs with a batch size of 12. During +training, we randomly crop the samples into 160 × 160 +patches. The Adam optimizer is used and the maximum +epoch of training iterations is set to 30. The learning rate is +set to 10−4 for the whole network except for the Alignment +blocks where the learning rate is set to a smaller value of +10−5. And the learning rates decays by 50% every 5 epochs +after the first 10 epochs. For the network architecture, the +upsampling operator adopts the nearest neighbor interpola- +tion and the number of the AlignFuse blocks 퐷 is set to 6. +For the loss function, 푤, 휖 and 훼 is set to 5, 10−3 and 0.45, +respectively. +Baselines In this paper, we compare the proposed method +with 5 baseline methods on the Planet-CR dataset with +the proposed data splits, including the single image cloud +removal methods, McGAN (Enomoto et al., 2017) and SpA +GAN (Pan, 2020), and the multi-modal data fusion based +cloud removal methods, SAR-Opt-cGAN (Grohnfeldt et al., +2018), DSen2-CR (Meraner et al., 2020) and GLF-CR (Xu +et al., 2022). Since existing multi-modal data fusion based +cloud removal methods require the input SAR images to +be of the same spatial resolution as the input optical im- +ages, upsampling the SAR images in the Planet-CR dataset +is necessary for these algorithms to work properly. Here, +all multi-modal data fusion based cloud removal methods +utilize the SAR images upsampled by the nearest neighbor +interpolation as input. Moreover, to determine the benefits of +including auxiliary low-resolution SAR images, we train the +Align-CR network without use of the SAR images, denoted +as w/o SAR. And to validate the superiority of Align-CR +in integrating multi-modal and multi-resolution information, +we train the Align-CR network by removing the Alignment +blocks in the AlignFuse blocks, denoted as w/o Align. +5.2. Evaluation of Visual Recovery Quality +To evaluate the quality of the reconstructed images in +terms of generating visually pleasing textures, we report +mean absolute error (MAE) and peak signal-to-noise ratio +(PSNR) to objectively quantify the prediction error, and +spectral angle mapper (SAM) and the structural similarity +index measure (SSIM) to subjectively measure the quality of +reconstructed images from spectral similarity and structural +similarity, respectively. Notably, with the benefit of the land +cover annotations, we disentangle the performance of cloud +removal methods over different land cover types by pixel- +wise MAE and SAM metrics, as shown in Tab. 2. And we +choose two scenes to qualitatively evaluate visual recovery +quality, as shown in Fig. 4. We can see that the methods +McGAN, SpA GAN and SAR-Opt-cGAN, which were de- +veloped on relatively small datasets with geographically and +Fang Xu, et al: Preprint submitted to Elsevier +Page 7 of 11 + +Planet-CR +Fig. 4: Qualitative results of visual recovery quality for two different samples. For each sample, from top-left to bottom-right +are respectively the cloudy image, the SAR image, the land cover map, the results from McGAN, SpA GAN, SAR-Opt-cGAN, +DSen2-CR, GLF-CR, w/o SAR, w/o Align, Align-CR, and the cloud-free image. +Fig. 5: Quantitative results of visual recovery quality over different cloud cover levels in terms of the MAE, PSNR, SAM, and +SSIM quality metrics. +topographically homogeneous data, perform poorly in terms +of generalizability on our Planet-CR dataset. Their results +have relatively limited spectral fidelity. It indicates the need +to take the point of global distribution into consideration +when creating a practical dataset. Our method achieves the +best performance overall, which can restore images with +more details and fewer artifacts. It indicates the effective- +ness and superiority of our method on restoring cloud-free +images. Moreover, we compare the results over different land +cover types. We can find that removing clouds over urban +land that has highly complex geometrical structures is more +challenging than others. And we can find that the results over +water do not perform well on SAM but relatively well on +Fang Xu, et al: Preprint submitted to Elsevier +Page 8 of 11 + +Cloudy +SAR +Land Cover +McGAN +SpA GAN +SAR-Opt-cGAN +DSen2-CR +GLF-CR +Ours (w/o SAR) +Ours(w/oAlign +Ours (Align-CR) +Cloud-FreeCloudy +SAR +Cover +McGAN +SpA GAN +SAR-Opt-cGAN +DSen2-CR +GLF-CR +Ours (w/o SAR) +Ours (w/o Align) +Ours (Align-CR) +Cloud-Free(a)MAE(↓) +(b) PSNR （↑) +(c) SAM (↓ ) +(d)SSIM （个) +0.92 +4.2 +12 +30 +MAE (10-2) +3.6 +(dB) +10 +0.88 +SAM ( +SSIM +PSNR +3.0 +28 +8 +0.84 +2.4 +6 +26 +0.80 +2040 +60 +80100 +2040 +60 +80100 +2040 +60 +80100 +20 +40 +60 +80100 +CloudCoverage(%) +CloudCoverage(%) +CloudCoverage(%) +CloudCoverage(%) +McGAN +SAR-Opt-cGAN +GLF-CR +Ours (w/o Align) +SpA GAN +DSen2-CR +Ours(w/oSAR) +Ours (Align-CR)Planet-CR +Table 3 +The performance of semantic recovery quality over different cloud cover levels. +0∼20% +20∼40% +40∼60% +60∼80% +80∼100% +Overall +mIoU +PA +mIoU +PA +mIoU +PA +mIoU +PA +mIoU +PA +mIoU +PA +Cloud-Free +44.63 62.68 +43.66 65.44 +41.85 64.40 +40.58 64.16 +42.98 66.70 +43.24 64.66 +Cloudy +18.89 34.08 +13.26 24.84 +10.24 19.27 +8.62 +16.21 +7.89 +13.00 +11.72 21.46 +McGAN +17.63 35.13 +17.84 34.08 +15.51 31.99 +15.30 32.06 +12.52 39.88 +16.62 34.64 +SpA GAN +24.07 41.37 +21.93 41.94 +17.31 37.66 +14.03 35.18 +10.44 28.99 +16.15 36.97 +SAR-Opt-cGAN +12.78 31.43 +11.25 28.84 +9.85 +28.59 +9.10 +28.99 +9.15 +28.09 +10.44 29.20 +DSen2-CR +33.11 43.96 +30.17 39.03 +26.05 37.73 +22.94 35.91 +23.10 41.92 +28.11 39.72 +GLF-CR +33.99 47.66 +31.54 44.96 +27.02 40.90 +22.44 39.49 +22.95 45.19 +28.75 43.63 +Ours (w/o SAR) +31.85 45.72 +25.14 39.83 +20.91 37.34 +16.95 37.02 +11.58 41.02 +20.97 40.20 +Ours (w/o Align) 33.80 46.94 +30.85 41.61 +26.76 38.97 +22.39 38.69 +23.94 44.67 +28.47 42.19 +Ours (Align-CR) +37.32 51.16 +32.90 45.40 +28.93 43.22 +24.17 42.37 +23.43 46.69 +30.51 45.78 +MAE. It indicates the challenge in maintaining the spectral +fidelity over water. +To determine what contributes to the superior perfor- +mance of the proposed method, we analyze the effectiveness +of each component by comparing the proposed method with +its variants, i.e., w/o SAR and w/o Align. We can find that +w/o SAR tends to generate undesirable artifacts for cloud- +covered regions due to the lack of ground information. While +Align-CR exploits the geometrical information embedded in +the SAR images, which can reconstruct the ground object. +We further compare the cloud removal performance of w/o +Align to Align-CR. When the Alignment blocks are re- +moved, the gain of integrating low-resolution SAR informa- +tion is reduced. In regards to the misalignment, such as the +river border in the first sample and the junction of different +land cover types in the second sample in Fig. 4, w/o Align +tends to generate blurring artifacts, while the reconstructed +results of Align-CR have sharper edges. It demonstrates the +superiority of our proposed Align-CR method in integrating +multi-modal and multi-resolution information. +We further assess the reconstruction performance on +different cloud cover levels, as shown in Fig. 5. The perfor- +mance of all methods, except McGAN, decreases roughly +as the percentage of cloud cover increases. Among them, +the performance of the methods with the benefit of SAR +images degrades more slowly than the one without, since +the utilization of SAR images can alleviate the decline to +some extent. Align-CR performs favorably when compared +with all baseline methods. When the Alignment blocks are +removed, w/o Align behaves very similarly to GLF-CR, +while GLF-CR contains more Transformer layers for global +fusion and thus performs slightly better in terms of SSIM. +Our method Align-CR, which aligns the multi-modal and +multi-resolution data during the reconstruction process, can +better exploit the power of SAR information. It steadily +outperforms w/o Align on all cloud cover levels. +5.3. Evaluation of Semantic Recovery Quality +In addition to evaluating in generating visually pleas- +ing textures, we further evaluate in generating semantically +meaningful structures, where the semantic information is +important for future analytical applications. In this paper, we +Fig. 6: The discrepancy between the results with the cloud-free +images as input and the results with the predicted cloud-free +images as well as cloudy images as input. +evaluate the quality of the recovered semantic information +through a well-established land cover semantic segmenta- +tion model. Using the cloud-free PlanetScope images and +associated land cover annotations, we train a land cover +semantic segmentation model with the same data splits +adopted for the experiments on cloud removal. The model is +based on DeepLabv3plus (Chen et al., 2018) with ResNet- +50 (He et al., 2016) as the backbone network. With the +cloud-free images, cloudy images and predicted cloud-free +images from benchmarked models as inputs respectively, we +evaluate the performance of the trained land cover semantic +segmentation model in predicting the correct class of each +pixel. Ideally, the results with the predicted images as input +should be as consistent as possible with the results with +the cloud-free images as input, i.e., the closer they are to +the results with the cloud-free images as input, the better +the corresponding cloud removal method performs in terms +of semantic recovery. We report the mean intersection over +union (mIoU) and pixel accuracy (PA) over varying levels +of cloud cover in Tab. 3, where mIoU can better deal with +the class imbalance issue. Additionally, we show the discrep- +ancy between the results with the cloud-free images as input +Fang Xu, et al: Preprint submitted to Elsevier +Page 9 of 11 + +(a)mlou difference（→) +(b)Acc difference（→) +35 +50 +30 +25 +40 +lou +Acc +20 +30 +15 +20 +10 +10 +20 +40 +60 +80 +100 +20 +40 +60 +80 +100 +Cloud Coverage(%) +Cloud Coverage (%) +McGAN +DSen2-CR +Ours (w/o Align) +SpA GAN +GLF-CR +Ours (Align-CR) +SAR-Opt-cGAN +Ours(w/oSAR) +CloudyPlanet-CR +and the results with the predicted cloud-free images as well +as cloudy images as input in Fig. 6. +Clearly, the existence of clouds deteriorates the seman- +tic analysis, and the degradation is more severe when the +cloud cover level is higher. The benchmarked cloud removal +models can to some extent counteract the degradation, in +which Align-CR generally performs the best. Notably, we +can observe that the trained land cover semantic segmen- +tation model performs better with the predicted images of +McGAN and SpA GAN as input than with the predicted +images of SAR-Opt-cGAN as input. While SAR-Opt-cGAN +performs better than McGAN and SpA GAN in terms of +all visual quality metrics, as shown in Fig. 5. It indicates +that the metrics for measuring the visual recovery quality +can not adequately reflect the performance of cloud removal +methods in terms of semantic recovery. We can also find +this by comparing the performance of w/o Align and Align- +CR. Align-CR steadily outperforms w/o Align on all cloud +cover levels in terms of all visual quality metrics. However, +it does not perform as well as w/o Align on the images with +cloud cover 80% to 100% in terms of mIoU. What’s more, +we can find that its performance in terms of mIoU is more +superior to that of w/o Align when more prior information +from cloud-free regions is available, since there is more +information available for alignment. +6. Discussion +Due to the shortcomings in semantic annotations, the +vast majority of current studies quantify the goodness of +cloud removal methods exclusively by metrics that assess the +visual similarity between two images. Towards the direction +of boosting the performance on the visual metrics, most +methods use the the loss functions that are constructed based +on these metrics to guide the training of cloud removal +models, e.g., L1 loss function computes the mean absolute +error (Xu et al., 2022). It will motivate the predicted cloud- +free image to move towards over-smoothness and lead to +the loss of semantic information. As we have validated in +Sec. 5.3, the current visual metrics cannot adequately eval- +uate the quality of recovered semantic information. Though +our method performs the best in terms of semantic recovery, +there is still a large gap between its predicted cloud-free +images and real cloud-free images. Therefore, it is very +important for the cloud removal field to design a loss func- +tion that can guide the recovery of semantic information. +Future work could consider the correlation between cloud +removal and downstream task, and design the loss function +by balancing the semantic context and image details to guide +the reconstruction process. +7. Conclusion +We introduce Planet-CR, a multi-modal and multi- +resolution dataset for cloud removal in high-resolution op- +tical remote sensing imagery. We offer the dataset as open- +source with the purpose of developing new cloud removal +approaches towards high resolution by integrating multi- +modal and multi-resolution information. We address the +problem of multi-modal and multi-resolution data fusion +based cloud removal by proposing a novel method called +Align-CR, which implicitly aligns the feature maps during +the reconstruction process to compensate for the misalign- +ment between the multi-modal and multi-resolution data. +The experimental results in both visual recovery quality and +semantic recovery quality demonstrate the effectiveness and +superiority of the proposed method than the existing rep- +resentative cloud removal methods. We believe the Planet- +CR dataset will prove beneficial for the community and our +evaluations provide valuable directions for future research. +Acknowledgements +The work of W. Yang is supported by the National +Natural Science Foundation of China (NSFC) Regional In- +novation and Development Joint Fund (No. U22A2010). +The work of X. Zhu is jointly supported by the European +Research Council (ERC) under the European Union’s Hori- +zon 2020 research and innovation programme (grant agree- +ment No. [ERC-2016-StG-714087], Acronym: So2Sat), by +the Helmholtz Association through the Framework of the +Helmholtz Excellent Professorship “Data Science in Earth +Observation - Big Data Fusion for Urban Research”(grant +number: W2-W3-100), by the German Federal Ministry of +Education and Research (BMBF) in the framework of the +international future AI lab "AI4EO – Artificial Intelligence +for Earth Observation: Reasoning, Uncertainties, Ethics and +Beyond" (grant number: 01DD20001) and by German Fed- +eral Ministry for Economic Affairs and Climate Action in the +framework of the "national center of excellence ML4Earth" +(grant number: 50EE2201C). +References +Atwood, D., Garron, J., 2013. Addressing three fallacies about synthetic +aperture radar. Eos, Transactions American Geophysical Union 94, 471– +472. +Chen, L.C., Zhu, Y., Papandreou, G., Schroff, F., Adam, H., 2018. Encoder- +decoder with atrous separable convolution for semantic image segmen- +tation, in: European Conference on Computer Vision, pp. 801–818. +Cornebise, J., Oršolić, I., Kalaitzis, F., 2022. Open high-resolution satellite +imagery: The worldstrat dataset–with application to super-resolution. +arXiv:2207.06418 . +Cresson, R., Ienco, D., Gaetano, R., Ose, K., Minh, D.H.T., 2019. Optical +image gap filling using deep convolutional autoencoder from optical and +radar images, in: IEEE International Geoscience and Remote Sensing +Symposium, pp. 218–221. +Cresson, R., Narçon, N., Gaetano, R., Dupuis, A., Tanguy, Y., May, S., +Commandre, B., 2022. Comparison of convolutional neural networks +for cloudy optical images reconstruction from single or multitemporal +joint sar and optical images. arXiv:2204.00424 . +Czerkawski, M., Upadhyay, P., Davison, C., Werkmeister, A., Cardona, J., +Atkinson, R., Michie, C., Andonovic, I., Macdonald, M., Tachtatzis, C., +2022. Deep internal learning for inpainting of cloud-affected regions in +satellite imagery. Remote Sensing 14. +Dai, J., Qi, H., Xiong, Y., Li, Y., Zhang, G., Hu, H., Wei, Y., 2017. +Deformable convolutional networks, in: IEEE International Conference +on Computer Vision, pp. 764–773. +Fang Xu, et al: Preprint submitted to Elsevier +Page 10 of 11 + +Planet-CR +Demir, I., Koperski, K., Lindenbaum, D., Pang, G., Huang, J., Basu, S., +Hughes, F., Tuia, D., Raskar, R., 2018. Deepglobe 2018: A challenge +to parse the earth through satellite images, in: IEEE Conference on +Computer Vision and Pattern Recognition Workshops, pp. 172–181. +Duan, C., Li, R., 2020. Multi-head linear attention generative adversarial +network for thin cloud removal. arXiv:2012.10898 . +Ebel, P., Meraner, A., Schmitt, M., Zhu, X.X., 2020. Multisensor data fusion +for cloud removal in global and all-season sentinel-2 imagery. IEEE +Transactions on Geoscience and Remote Sensing 59, 5866–5878. +Ebel, P., Xu, Y., Schmitt, M., Zhu, X.X., 2022. Sen12ms-cr-ts: A remote- +sensing data set for multimodal multitemporal cloud removal. +IEEE +Transactions on Geoscience and Remote Sensing 60, 1–14. doi:10.1109/ +TGRS.2022.3146246. +Enomoto, K., Sakurada, K., Wang, W., Fukui, H., Matsuoka, M., Nakamura, +R., Kawaguchi, N., 2017. +Filmy cloud removal on satellite imagery +with multispectral conditional generative adversarial nets, in: IEEE +Conference on Computer Vision and Pattern Recognition Workshops, +pp. 48–56. +Fisher, A., Flood, N., Danaher, T., 2016. Comparing landsat water index +methods for automated water classification in eastern australia. Remote +Sensing of Environment 175, 167–182. +Gao, J., Yuan, Q., Li, J., Zhang, H., Su, X., 2020. Cloud removal with fusion +of high resolution optical and sar images using generative adversarial +networks. Remote Sensing 12, 191. +Gorelick, N., Hancher, M., Dixon, M., Ilyushchenko, S., Thau, D., Moore, +R., 2017. Google Earth Engine: Planetary-scale geospatial analysis for +everyone. Remote Sensing of Environment 202, 18–27. +Grohnfeldt, C., Schmitt, M., Zhu, X., 2018. +A conditional generative +adversarial network to fuse sar and multispectral optical data for cloud +removal from sentinel-2 images, in: IEEE International Geoscience and +Remote Sensing Symposium, pp. 1726–1729. +Gu, Z., Ebel, P., Yuan, Q., Schmitt, M., Zhu, X.X., 2022. Explicit haze & +cloud removal for global land cover classification, in: IEEE Conference +on Computer Vision and Pattern Recognition Workshops. +He, K., Zhang, X., Ren, S., Sun, J., 2016. +Deep residual learning for +image recognition, in: IEEE Conference on Computer Vision and Pattern +Recognition, pp. 770–778. +Helmer, E.H., Ruefenacht, B., 2005. Cloud-free satellite image mosaics +with regression trees and histogram matching. Photogrammetric Engi- +neering & Remote Sensing 71, 1079–1089. +Kondmann, L., Toker, A., Ruß wurm, M., Camero, A., Peressuti, D., +Milcinski, G., Mathieu, P.P., Longepe, N., Davis, T., Marchisio, G., Leal- +Taixé, L., Zhu, X., 2021. Denethor: The dynamicearthnet dataset for +harmonized, inter-operable, analysis-ready, daily crop monitoring from +space, in: Neural Information Processing Systems Track on Datasets and +Benchmarks. +Lai, W.S., Huang, J.B., Ahuja, N., Yang, M.H., 2017. Deep laplacian pyra- +mid networks for fast and accurate super-resolution, in: IEEE Conference +on Computer Vision and Pattern Recognition, pp. 5835–5843. +Li, J., Wu, Z., Hu, Z., Li, Z., Wang, Y., Molinier, M., 2021. Deep learning +based thin cloud removal fusing vegetation red edge and short wave +infrared spectral information for sentinel-2a imagery. Remote Sensing +13, 157. +Li, M., Liew, S.C., Kwoh, L.K., 2002. +Generating "cloud free" and +"cloud-shadow free" mosaic for spot panchromatic images, in: IEEE +International Geoscience and Remote Sensing Symposium, pp. 2480– +2482. +Li, W., Li, Y., Chen, D., Chan, J.C.W., 2019. Thin cloud removal with +residual symmetrical concatenation network. +ISPRS Journal of Pho- +togrammetry and Remote Sensing 153, 137–150. +Lima, T.A., Beuchle, R., Langner, A., Grecchi, R.C., Griess, V.C., Achard, +F., 2019. +Comparing sentinel-2 msi and landsat 8 oli imagery for +monitoring selective logging in the brazilian amazon. Remote Sensing +11, 961. +Lin, D., Xu, G., Wang, X., Wang, Y., Sun, X., Fu, K., 2019. A remote +sensing image dataset for cloud removal. arXiv:1901.00600 . +Liu, Z., Hunt, B.R., 1984. A new approach to removing cloud cover from +satellite imagery. Computer Vision, Graphics, and Image Processing 25, +252–256. +Liu, Z., Lin, Y., Cao, Y., Hu, H., Wei, Y., Zhang, Z., Lin, S., Guo, B., +2021. Swin transformer: Hierarchical vision transformer using shifted +windows, in: IEEE International Conference on Computer Vision, pp. +10012–10022. +Meraner, A., Ebel, P., Zhu, X.X., Schmitt, M., 2020. Cloud removal in +sentinel-2 imagery using a deep residual neural network and sar-optical +data fusion. ISPRS Journal of Photogrammetry and Remote Sensing +166, 333–346. +Mitchell, O., Delp, E.J., Chen, P.L., 1977. Filtering to remove cloud cover +in satellite imagery. IEEE Transactions on Geoscience Electronics 15, +137–141. +Ngadze, F., Mpakairi, K.S., Kavhu, B., Ndaimani, H., Maremba, M.S., +2020. Exploring the utility of sentinel-2 msi and landsat 8 oli in burned +area mapping for a heterogenous savannah landscape. PLOS ONE 15, +1–13. +Pan, H., 2020. +Cloud removal for remote sensing imagery via spatial +attention generative adversarial network. arXiv:2009.13015 . +PBC, P.L., 2018–. Planet application program interface: In space for life on +earth. URL: https://api.planet.com. +Requena-Mesa, C., Benson, V., Reichstein, M., Runge, J., Denzler, J., 2021. +Earthnet2021: A large-scale dataset and challenge for earth surface +forecasting as a guided video prediction task., in: Proceedings of the +IEEE Conference on Computer Vision and Pattern Recognition, pp. +1132–1142. +Rogan, J., Franklin, J., Roberts, D.A., 2002. +A comparison of methods +for monitoring multitemporal vegetation change using thematic mapper +imagery. Remote Sensing of Environment 80, 143–156. +Sarukkai, V., Jain, A., Uzkent, B., Ermon, S., 2020. +Cloud removal in +satellite images using spatiotemporal generative networks, in: IEEE +Winter Conference on Applications of Computer Vision, pp. 1785–1794. +Singh, P., Komodakis, N., 2018. Cloud-gan: Cloud removal for sentinel-2 +imagery using a cyclic consistent generative adversarial networks, in: +IEEE International Geoscience and Remote Sensing Symposium, pp. +1772–1775. +Voigt, S., Kemper, T., Riedlinger, T., Kiefl, R., Scholte, K., Mehl, H., 2007. +Satellite image analysis for disaster and crisis-management support. +IEEE Transactions on Geoscience and Remote Sensing 45, 1520–1528. +Wang, X., Chan, K.C., Yu, K., Dong, C., Loy, C.C., 2019. Edvr: Video +restoration with enhanced deformable convolutional networks, in: IEEE +Conference on Computer Vision and Pattern Recognition. +Xia, G.S., Bai, X., Ding, J., Zhu, Z., Belongie, S., Luo, J., Datcu, M., Pelillo, +M., Zhang, L., 2018. Dota: A large-scale dataset for object detection +in aerial images, in: IEEE Conference on Computer Vision and Pattern +Recognition, pp. 3974–3983. +Xiong, Z., Zhang, F., Wang, Y., Shi, Y., Zhu, X.X., 2022. +Earthnets: +Empowering ai in earth observation. arXiv:2210.04936 . +Xu, F., Shi, Y., Ebel, P., Yu, L., Xia, G.S., Yang, W., Zhu, X.X., 2022. Glf-cr: +Sar-enhanced cloud removal with global–local fusion. ISPRS Journal of +Photogrammetry and Remote Sensing 192, 268–278. +Yu, C., Chen, L., Su, L., Fan, M., Li, S., 2011. +Kriging interpolation +method and its application in retrieval of modis aerosol optical depth, +in: International Conference on Geoinformatics, pp. 1–6. +Yuan, Q., Shen, H., Li, T., Li, Z., Li, S., Jiang, Y., Xu, H., Tan, W., Yang, Q., +Wang, J., et al., 2020. Deep learning in environmental remote sensing: +Achievements and challenges. Remote Sensing of Environment 241, +111716. +Zanaga, D., Van De Kerchove, R., De Keersmaecker, W., Souverijns, N., +Brockmann, C., Quast, R., Wevers, J., Grosu, A., Paccini, A., Vergnaud, +S., et al., 2021. Esa worldcover 10 m 2020 v100 . +Zhang, C., Li, W., Travis, D., 2007. Gaps-fill of slc-off landsat etm+ satellite +image using a geostatistical approach. International Journal of Remote +Sensing 28, 5103–5122. +Zhang, Y., Tian, Y., Kong, Y., Zhong, B., Fu, Y., 2020. Residual dense +network for image restoration. IEEE Transactions on Pattern Analysis +and Machine Intelligence 43, 2480–2495. +Fang Xu, et al: Preprint submitted to Elsevier +Page 11 of 11 + diff --git a/etE1T4oBgHgl3EQfyQXE/content/tmp_files/load_file.txt b/etE1T4oBgHgl3EQfyQXE/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..92c11e37b0b4460a69d23b379a19fae1327b6e23 --- /dev/null +++ b/etE1T4oBgHgl3EQfyQXE/content/tmp_files/load_file.txt @@ -0,0 +1,1159 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf,len=1158 +page_content='High-Resolution Cloud Removal with Multi-Modal and Multi-Resolution Data Fusion: A New Baseline and Benchmark Fang Xua,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Yilei Shic,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Patrick Ebelb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Wen Yanga and Xiao Xiang Zhub aSchool of Electronic Information,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Wuhan University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Wuhan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' 430072,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' China bData Science in Earth Observation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Technical University of Munich,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Munich,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' 80333,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Germany cRemote Sensing Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Technical University of Munich,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Munich,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' 80333,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Germany A R T I C L E I N F O Keywords: Cloud removal Data fusion A B S T R A C T In this paper,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' we introduce Planet-CR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' a benchmark dataset for high-resolution cloud removal with multi-modal and multi-resolution data fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Planet-CR is the first public dataset for cloud removal to feature globally sampled high resolution optical observations, in combination with paired radar measurements as well as pixel-level land cover annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' It provides solid basis for exhaustive evaluation in terms of generating visually pleasing textures and semantically meaningful structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' With this dataset, we consider the problem of cloud removal in high resolution optical remote sensing imagery by integrating multi-modal and multi-resolution information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Existing multi-modal data fusion based methods, which assume the image pairs are aligned pixel-to-pixel, are hence not appropriate for this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' To this end, we design a new baseline named Align-CR to perform the low-resolution SAR image guided high-resolution optical image cloud removal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' It implicitly aligns the multi-modal and multi-resolution data during the reconstruction process to promote the cloud removal performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' The experimental results demonstrate that the proposed Align-CR method gives the best performance in both visual recovery quality and semantic recovery quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' The project is available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='com/zhu-xlab/Planet-CR, and hope this will inspire future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Introduction Remote sensing imagery has been receiving consider- able attention as major promising prospects in various appli- cations such as Earth observation and environmental mon- itoring (Xia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Yuan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Requena-Mesa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Xiong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' However, haze and clouds in the atmosphere affect the transmission of electromagnetic signals and lead to a deficiency of surface information (Duan and Li, 2020), severely hindering the potential of remote sensing imagery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Cloud removal aims at reconstructing the cloud-contaminated regions to counteract the degradations caused by clouds, and thus becomes an indispensable pre- processing step in the analysis of remote sensing imagery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' As a continuing concern within the remote sensing community, many exciting progresses on this task have been extensively reported, including many data-driven approaches using deep learning (Ebel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Cresson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Benefiting from the open access policy of the Landsat program and the Copernicus program, several datasets have been proposed to promote the development of deep learning in the field of cloud removal, like RICE-II (Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2019) and SEN12MS-CR (Ebel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2020) datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' However, there are still two major issues that seriously limit the de- velopment of cloud removal: Unavailability of high resolution imagery for existing datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Recent advances in remote sensing have xufang@whu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='cn (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Xu);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' yilei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='shi@tum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='de (Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Shi);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' patrick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='ebel@tum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='de (P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Ebel);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' yangwen@whu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='cn (W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Yang);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' xiaoxiang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='zhu@tum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='de (X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Zhu) ORCID(s): given rise to the next-generation satellites that pro- vide optical imagery with superior spatial resolu- tion (Kondmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Cornebise et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Such data allows geometric analysis on a finer scale, while posing new challenges for recovering cloud- covered regions with corresponding level of details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' The existing datasets with relatively low spatial res- olutions severely restrict the development of cloud removal algorithms for recovering clear edge and rich texture details of high resolution remote sensing imagery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Lack of evaluation in generating semantically mean- ingful structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' High-quality reconstructed cloud- free images require both visually pleasing textures and semantically meaningful structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Existing work al- most exclusively evaluates the quality of the recon- structed images through image quality metrics like PSNR and SSIM, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' evaluates the quality of the reconstructed image in terms of generating visually pleasing textures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' While it is desirable to evaluate the semantically meaningful structures to identify the usability of the reconstructed images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' In this paper, we build a Planet-CR dataset to fulfill the requirement of high resolution cloud removal datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' High resolution remote sensing imagery with extensive spatial- temporal coverage is not easily available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Some satellites offer very high resolution imagery only from specific lo- cations, which makes it impractical to construct datasets with a high diversity of globally distributed sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' By trad- ing off the spatial-temporal coverage and the resolution of satellite imagery, we collect data from Planet satellite imagery (PBC, 2018–).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Planet provides global daily data Fang Xu, et al: Preprint submitted to Elsevier Page 1 of 11 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='03432v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='CV] 9 Jan 2023 aPlanet-CR with a spatial resolution of 3푚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' On the one hand, it allows to acquire paired cloudy and cloud-free images with a very short temporal offset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' It could minimize surface changes that may appear between the acquisitions of cloudy and cloud- free images, thereby minimizing nuisances between cloudy reference images and cloud-free target images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' On the other hand, it allows to acquire heterogeneous earth observation data to encourage general-purpose cloud removal but not along narrowly-defined and geo-spatially distinct regions of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Meanwhile, the Planet-CR dataset collects corre- sponding pixel-level land cover annotations from World- Cover product (Zanaga et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' It offers the chance of validating the effectiveness of the cloud removal methods in generating semantically meaningful structures on a well- established remote sensing task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Cloud removal is a highly ill-posed problem due to the loss of surface information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Previous studies have re- ported that the ill-posedness can be reduced by resorting to Synthetic Aperture Radar (SAR) data, which is cloud- penetrable and inherently reflects the geometrical shapes of ground objects (Ebel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' However, SAR data is usually unaffordable (Atwood and Garron, 2013), especially for global Earth data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Thanks to the freely accessible global SAR data provided by the European Space Agency, we further include the Sentinel- 1 SAR data in our Planet-CR dataset to promote cloud removal of PlanetScope data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Compared with the existing cloud removal datasets like SEN12MS-CR, which mostly explore the fusion of Sentinel-1 SAR data and Sentinel-2 optical data with the same resolution of 10푚, our Planet- CR dataset helps to understand a more practical yet more complex problem, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' multi-modal and multi-resolution data fusion based cloud removal, in which the misalignment due to resolution difference, field-of-view mismatch, disparity, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' has to be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' To address the problem of multi-modal and multi- resolution data fusion based cloud removal, we propose a novel method called Align-CR, where the low-resolution SAR images guide the reconstruction of high-resolution cloud-free images from cloudy images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Specifically, Align- CR implicitly aligns the feature maps using the deformable convolution (Dai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2017) to compensate for the mis- alignment between the multi-modal and multi-resolution data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Based on the Planet-CR dataset, we benchmark rep- resentative cloud removal methods, and analyze their per- formance in generating visually pleasing textures by im- age reconstruction metrics and in generating semantically meaningful structures by a well-established semantic seg- mentation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Extensive evaluation demonstrates that the Align-CR method achieves the best performance on the vast majority of our benchmark tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' In summary, our contributions are as follows: We construct a dataset Planet-CR collected specially for cloud removal to advance the field, which is the public dataset with the highest spatial resolution to date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' It comprises a large number of regions all around the world with auxiliary SAR imagery and land cover information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' We propose a novel cloud removal algorithm, Align- CR, which can better capture the correlations across the multi-modal and multi-resolution data to recon- struct the occluded regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' We benchmark state-of-the-art cloud removal algo- rithms with the Planet-CR dataset, which can be used as the baseline for future algorithm development, and conduct extensive evaluations for recovered semantic information on a well-established semantic segmenta- tion task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Related Work 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Datasets for Cloud Removal To promote the progress of deep learning-based cloud removal, several datasets for cloud removal have been pro- posed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' We provide an overview of publicly available cloud removal datasets, as shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' RICE-I (Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2019) contains 500 pairs of cloudy and cloud-free images collected on Google Earth by setting whether to display the cloud layer, which only contains filmy, partly-transparent clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' RICE-II (Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2019) contains about 700 paired images derived from the Landsat 8 OLI/TIRS dataset, where the acquisition time of cloudy and cloud-free images at the same location is less than 15 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' However, the size of both datasets is relatively small, and the data in them is geograph- ically and topographically homogeneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' STGAN (Sarukkai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2020) contains nearly 100, 000 paired Sentinel-2 im- ages drawn from 17800 distinct tiles worldwide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' The cloud cover for each cloudy image is between 10% and 30%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' It excludes the images with insufficient visible ground upon manual inspection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Then SEN12MS-CR (Ebel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2020) collects around 150, 000 samples, containing different types of real-life clouds, from 169 non-overlapping AOIs sampled across all inhabited continents during all meteorological seasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Each AOI is composed of a pair of orthorectified, geo-referenced cloudy and cloud-free Sentinel-2 images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' And to develop cloud removal models that are robust to extensive cloud coverage conditions, each AOI contains an additional co-registered Sentinel-1 SAR image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' The related SEN12MS-CR-TS dataset (Ebel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2022) is structured likewise, while featuring multi-season repeated measure- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' However, a common limitation of the aforemen- tioned datasets is that they are not stratified by land cover types, making it impossible to assess the generalizability of cloud removal methods over individual land cover type that may be of interest to specific applications like vege- tation monitoring (Rogan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Lima et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Ngadze et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2020) and water resources monitoring (Fisher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' To evaluate the effectiveness of cloud removal methods over different land cover types, WHU2-CR (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2021) selects 36 locations from all over the world according to three main land covers: urban, vegetation, and bare land, and produces about 20, 000 pairs of cloudy and cloud-free Sentinel-2 images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' And Czerkawski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' (2022) construct a cloud removal dataset specially in the context Fang Xu, et al: Preprint submitted to Elsevier Page 2 of 11 Planet-CR Table 1 Comparison between Planet-CR and existing publicly available cloud removal datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' LC is short for land cover map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Dataset Optical Image w/ SAR w/ LC source resolution #AOIs width #images RICE-I (Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2019) Google Earth < 15푚 / 512 500 \x17 \x17 RICE-II (Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2019) Landsat-8 30푚 / 512 450 \x17 \x17 STGAN (Sarukkai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2020) Sentinel-2 10푚 17,800 256 97,640 \x17 \x17 SEN12MS-CR (Ebel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2020) Sentinel-2 10푚 169 256 122,218 \x13 \x17 SEN12MS-CR-TS (Ebel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2022) Sentinel-2 10푚 53 256 15,578 \x13 \x17 WHUS2-CR (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2021) Sentinel-2 10푚 36 256 17,182 \x17 \x17 Scotland&India∗ (Czerkawski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2022) Sentinel-2 10푚 445 256 445 \x13 \x17 Planet-CR PlanetScope 3푚 780 300 63,000 \x13 \x13 The cloudy observations in the dataset are simulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' of crop monitoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' The dataset contains paired Sentinel-1 and Sentinel-2 Images for 2 locations in Scotland and India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Notably, the cloud regions in the dataset are simulated with a cloud coverage between 10% and 50%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' It will inevitably lead to unrealistic representations that cannot generalize to the scenario of real cloud-covered satellite imagery and their spectral characteristics (Ebel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' In addition to the previously mentioned public datasets, there are also several non-public datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' For example, Enomoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' (2017) create a dataset by synthesizing the simulated cloud on the cloud-free images with eight com- paratively cloudless WorldView-2 images for learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Cres- son et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' (2019) use Sentinel-2 images acquired over the province of Tuy, Burkina Faso to construct experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' (2020) collect two simulated datasets with around 35% cloud cover based on Gaofen-2 optical imagery and airborne optical imagery, respectively, and a real dataset based on Sentinel-2 optical imagery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' These datasets are relatively small in size and contain a limited number of scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' While featuring high resolution imagery, cloudy observations are all simulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' The majority of datasets containing real-life clouds are based on medium-resolution Sentinel-2 data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' It is hard to develop and evaluate the removal of clouds in high resolu- tion imagery with salient structures and abundant textured features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Our Planet-CR dataset aims to advance the task of cloud removal from high resolution imagery by releasing the cloudy and cloud-free PlanetScope data with short time lags in combination with Sentinel-1 SAR data and WorldCover land cover maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' The inclusion of Sentinel-1 SAR data enables the exploration of multi-modal fusion for cloud removal, and the inclusion of WorldCover land cover product enables to disentangle the performance over different land cover types and evaluate the quality of recovered semantic information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' While the short revisit time may facilitate to temporal mosaicing, it is not applicable to time-critical ap- plications (Voigt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' And when encountering con- tinual cloudy days, cloud-free reference data from an adja- cent period is largely unavailable (Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' It is worth noticing that Sarukkai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' (2020) train a baseline land classification model and evaluate its accuracy from cloudy, cloud-free, and predicted cloud-free images, respectively, to demonstrate the power of the predicted cloud-free images for downstream use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' However, cloud removal is a pixel-level reconstruction task, an image-level classification task cannot adequately reflect the quality of predicted cloud-free images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Because the correct class can be inferred even if the image is partially occluded (Gu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' With our Planet- CR dataset, a pixel-level classification task, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' semantic segmentation, can be designed, which will be more suitable for evaluating the power of the cloud removal methods on semantic information recovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Algorithms for Cloud Removal Cloud removal in optical remote sensing imagery is a long-term research problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Most early developments address this problem by interpolation (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2011), filtering (Mitchell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 1977;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Liu and Hunt, 1984), or inpainting (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Helmer and Ruefenacht, 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Currently, deep learning-based methods are gaining considerable attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' They have the poten- tial to solve many of the problems that arise in traditional cloud removal methods and achieve impressive results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' For example, Multispectral conditional Generative Adversarial Networks (McGANs) (Enomoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2017), leveraging the remarkable generative capabilities of conditional Generative Adversarial Networks (cGANs), remove simulated clouds from Worldview-2 imagery by extending the input channels of cGANs to be compatible with multispectral input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Cloud- GAN (Singh and Komodakis, 2018) learns the mapping between cloudy and cloud-free Sentinel-2 imagery using a cyclic consistent generative adversarial network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' RSC- Net (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2019) estimates the cloud-free output with the contaminated Landsat-8 imagery based on an encoding- decoding framework consisting of multiple residual convo- lutional layers and residual deconvolutional layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' As the clouds thicken and the cloud-covered region is dominant, the task of removing the clouds becomes more challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' A series of work explore the potential of SAR data as auxiliary data for cloud removal in optical imagery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' For example, DSen2-CR (Meraner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2020) uses a deep residual neural network to predict the target cloud-free opti- cal image from the concatenation of Sentinel-1 SAR image and Sentinel-2 optical image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Simulation-Fusion GAN (Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2020) fuse SAR and corrupted optical imagery with two generative adversarial networks to acquire the cloud-free Fang Xu, et al: Preprint submitted to Elsevier Page 3 of 11 Planet-CR Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' 1: Visualization of the Planet-CR dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' (a) Spatial distribution of the 780 AOIs of the Planet-CR dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' (b) Cloud-free optical observations from PlanetScope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' (c) Cloudy optical observations from PlanetScope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' (d) SAR observations from Sentinel-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' (e) land cover maps from WorldCover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' They are resized to the same size for better view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' results of simulated cloudy Gaofen-2 data and real cloudy Sentinel-2 data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' GLF-CR (Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2022) incorporates the contribution of Sentinel-1 SAR image in restoring reliable texture details and maintaining global consistency to recon- struct the occluded region of Sentinel-2 optical image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' How- ever, these methods are developed on data with relatively low spatial resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' The problem of cloud removal in high- resolution imagery remains largely underexplored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Data 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Curation of Planet-CR To develop the cloud removal methods that are equally applicable to heterogeneous Earth observation data, the Planet-CR dataset curates 780 non-overlapping, highly rep- resentative locations of AOIs that are distributed over all continents and meteorological seasons of the globe, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Each AOI is composed of a quartet of orthorectified, geo-referenced cloudy and cloud-free optical images, as well as the corresponding SAR image and land cover map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Optical data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' The optical data of our dataset originates from the PlanetScope satellite constellation operated by Planet Labs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' We gather the cloud-free and cloudy observa- tions from the Level-3B top-of-atmosphere reflectance prod- uct, which are radiometrically-, sensor-, and geometrically- corrected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' It provides four-band (Blue, Red, Green and Near- Infrared) imagery with a spatial resolution of 3푚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' In our dataset, the average time interval between paired cloudy and cloud-free data is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='8 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' This is shorter than the intra-mission revisiting period of Landsat-8 and Sentinel-2 satellites, and minimizes the amount of land cover change occurring between our paired data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' And the cloudy data features a diversity of clouds, ranging from semi-transparent to dense, and from light to heavy cloud covering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' In addition to the cloudy and reference cloud-free data, we also collect the cloud masks associated with cloudy PlanetScope images according to the Unusable Data Mask (UDM) assets pro- vided by Planet Labs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' UDM masks give information about which pixels in the images are clear or cloudy, permitting a statistical evaluation of cloud coverage and enabling cloud mask-guided methods to work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' SAR data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' The SAR data of our dataset originates from the Sentinel-1 mission operated by the European Space Agency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' We gather the Sentinel-1 data with a spatial resolution of 10푚 from the Level-1 Ground Range Detected (GRD) product archived by Google Earth Engine (Gorelick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' The measurements are acquired in interferometric wide swath (IW) mode with two polarization channels VV (vertical transmit/vertical receive) and VH (vertical trans- mit/horizontal receive), which are pre-processed with ther- mal noise removal, radiometric calibration, terrain correc- tion, and are converted to backscatter coefficients (휎◦) in units of decibels (dB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Land cover map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' The land cover maps of our dataset origi- nates from WorldCover, which is an open-access global land cover product at 10푚 resolution released by the European Space Agency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' In this study, we reclassified the data into 6 basic land cover types according to DeepGlobe 2018 (Demir et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2018): forest land, rangeland, agriculture land, urban land, barren land and water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' We partition the dataset into train and test splits to allow for direct comparison with future works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' A total of 780 AOIs are split into 660 scenes for training and 120 for testing following a random global distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' To train the deep learning-based methods, we crop each AOI into small patches using slide windows of sensor-specific sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Since the spatial resolution of PlanetScope imagery is 3푚, and the spatial resolution of Sentinel-1 imagery as well as WorldCover land cover maps is 10푚, we set the correspond- ing slide window sizes to 300 × 300 and 90 × 90 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' For the purpose of uniform distribution over different cloud coverage levels as well as land cover types, we finally select 60, 000 quartets of training samples and 3,000 quartets of testing samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' The statistics about cloud coverage and land cover types can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Properties of Planet-CR High Spatial Resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' The Planet-CR dataset releases the cloudy and cloud-free PlanetScope data with a spatial Fang Xu, et al: Preprint submitted to Elsevier Page 4 of 11 train testPlanet-CR (a) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' 2: Statistics of the Planet-CR dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' (a) Distribution of cloud coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' (b) Distribution of land cover types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' resolution of 3m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Compared to existing publicly available cloud removal datasets which are mostly built on medium- resolution Landsat-8 data or Sentinel-2 data, our dataset redeems the current lack of cloud removal datasets with high resolution, and advances the task of high-resolution cloud removal that is still under-explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Multi-Modal and Multi-Resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' The inclusion of Sen- tinel-1 SAR data can provide auxiliary information to pro- mote cloud removal of PlanetScope data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Compared with the existing cloud removal datasets like SEN12MS-CR, which mostly explore the fusion of Sentinel-1 SAR data and Sentinel-2 optical data with the same resolution of 10m, our Planet-CR dataset helps to understand a more practical yet more complex problem, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', multi-modal and multi- resolution data fusion based cloud removal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Information on Land Cover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' The inclusion of land cover can disentangle the performance of cloud removal methods over different land cover types on the one hand, and encour- ages to design a pixel-level classification task to evaluate the power of the cloud removal method in generating semanti- cally meaningful structures on the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Methodology 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Problem Statement Given a cloudy image \ue245, the task of cloud removal aims to reconstruct a clear image \ue232 revealing the complete information content of the ground scene so that subsequent analysis can be reliably performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' The basic strategy is to deal with the cloud contamination in a single image without additional information, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', the problem of single image Cloud Removal(CR), which can be formulated as: \ue232 = CR(\ue245) (1) It is a highly ill-posed problem, usually solved under the assumption that the cloud-contaminated regions have similar spectral/geometrical characteristics to the remaining parts of the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' However, when it comes to the areas with high-frequency texture or different land cover types, the reconstruction performance cannot be guaranteed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Many studies resort to SAR images that are cloud-penetrable and inherently reflect the geometrical shapes of ground objects as an a priori assumption to reduce the ill-posedness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Thus, the problem of Multi-Modal data Fusion based Cloud Removal (MMF-CR) is introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' It restores the clear image from a cloudy image and the corresponding SAR image \ue23f, formulated as: \ue232 = MMF-CR(\ue245Ω, \ue23fΩ) (2) where Ω indicates the same spatial domain shared by the cloudy image and the corresponding SAR image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' The SAR image guides most MMF-CR methods to restore the cloud- free image from pixel-to-pixel aligned cloudy and SAR im- ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' However, the geo-referenced cloudy images captured by Planet satellites and SAR images captured by Sentinel-1 satellites in the Planet-CR dataset do not meet the assump- tion of pixel-to-pixel alignment, mainly caused by resolution difference, field-of-view mismatch and disparity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' The prob- lem of Multi-Modal and multi-Resolution data Fusion based Cloud Removal(MMRF-CR) is thus introduced: \ue232 = MMRF-CR(\ue245Ω푦, \ue23fΩ푠|\ue239) (3) where Ω푦 and Ω푠 respectively denote the spatial domain of the cloudy image and the SAR image, and \ue239 ∶ Ω푠 → Ω푦 denotes the pixel-level correspondence mapping oper- ator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' To solve the problem of MMRF-CR, it is required to precisely align the cloudy and SAR images for the cloud removal process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Align-CR Network The proposed network, called Align-CR, adopts a two- stream architecture to compensate for the missing informa- tion in cloudy regions using ancillary SAR image, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Since the SAR image has a lower resolution, an upsampling operator is firstly employed to map it to the same resolution as the optical image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Then, the cloudy optical image and the upsampled SAR image are passed through their respective feature extraction blocks to extract modality- specific features 퐹 0 표푝푡 and 퐹 0 푠푎푟.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' After that, 퐹 0 표푝푡 and 퐹 0 푠푎푟 are fed into 퐷 AlignFuse blocks to obtain knowledgeable features with comprehensive information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' The AlignFuse block performs alignment and fusion sequentially in the feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' ̂퐹 푖 푠푎푟 = 퐻Align(퐹 푖 표푝푡, 퐹 푖 푠푎푟), (4) 퐹 푖+1 표푝푡 , 퐹 푖+1 푠푎푟 = 퐻fuse(퐹 푖 표푝푡, ̂퐹 푖 푠푎푟), (5) where 퐻Align(⋅) and 퐻fuse(⋅) denote the functions of the alignment block and the fusion block, respectively, and the details are shown in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Finally, all intermediate features {퐹 푖 표푝푡}퐷 푖=1 are aggregated to reconstruct the high-quality cloud-free image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Alignment Block Ideally, the task of MMRF-CR can be considered as a two-step task that aligns the multi-modal and multi- resolution data first and reconstruct the clear image follow- ing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' 2 later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' However, explicit pixel-to-pixel alignment to ensure that the same pixels in the SAR image and the cloudy image reflect the same ground target is very hard Fang Xu, et al: Preprint submitted to Elsevier Page 5 of 11 12 train test 10 %) Occurrrence ( 8 4 2 0 10 CloudCoverageof Image Patches(%）35 train test 30 25 Occurrrence 20 15 10 5 Urban Barrer Wate Land Cover TypesPlanet-CR Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' 3: Overview of the proposed Align-CR method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' An upsampling operator is firstly employed to map the SAR image to the same resolution as the optical image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Then, the cloudy optical image and the upsampled SAR image are passed through their respective feature extraction (FE) blocks to extract modality-specific features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' After that, the features are fed into 퐷 AlignFuse blocks to obtain knowledgeable features with comprehensive information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' The AlignFuse block performs alignment and fusion sequentially in the feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Finally, the output of all AlignFuse blocks are concatenated and fed to the image reconstruction (IR) block to restore the high-quality cloud-free image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' to achieve in our case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' On the one hand, sharp edges in the high-resolution optical image cannot be exactly aligned with blurry edges in the low-resolution SAR image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' On the other hand, for non-co-orbital satellite data, it is very difficult to resolve the unalignment caused by field-of-view mismatch and disparity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Moreover, the occlusions in the cloudy images complicates the pixel-to-pixel alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Therefore, we implicitly learn the alignment process for cloud-free image reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' In this paper, we resort to the deformable convolution to align the SAR features to the optical features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Given two features to be aligned as input, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 퐹 푖 표푝푡 and 퐹 푖 푠푎푟, 푖 = 0, ⋯ , 퐷 −1, The core to solve feature misalignment is to predict the offset with an offset prediction module, ▵푝푖 = 퐻offset(퐹 푖 표푝푡, 퐹 푖 푠푎푟), (6) where 퐻offset denotes the function of the offset prediction module, which can be implemented by general convolutional layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' With the predicted offset, the SAR feature can be warped to the optical feature using the deformable convo- lution (DConv), ̂퐹 푖 푠푎푟 = DConv(퐹 푖 푠푎푟, ▵푝푖), (7) Specifically, we adopt the Pyramid, Cascading and De- formable convolutions (PCD) (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2019) for the alignment process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' It performs the alignment in a pyramid structure, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', aligning the features in lower scales with coarse estimations first and propagating the aligned features and learned offsets to higher scales to refine the estimations later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Embedding it into the network, the network’s ability to model transformations can be enhanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Fusion Block The aligned features, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 퐹 푖 표푝푡 and ̂퐹 푖 푠푎푟, are then fed into the fusion block for the transfer of complementary information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Similar to our prior work (Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2022), we exploit the power of SAR information from two aspects: global fusion, to guide the global interactions among all local optical windows;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' local fusion, to transfer the SAR feature corresponding to cloudy areas to compensate for the missing information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Specifically, each fusion block contains an adapted SAR-guided global context interaction (SGCI) block followed by a SAR-based local feature com- pensation (SLFC) block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' To reduce the complexity of the SGCI block, instead of adding a Swin Transformer layer (STL) (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2021) after each convolutional layer in the dense connected layers, we only add a STL after the convolutional layer in the local feature fusion of the residual dense block (RDB) (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2020) for cross-window feature interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Loss Function We train the Align-CR network by minimizing the dif- ference between the output of network 퐲 and the cloud-free image ̂y temporally close to the input cloudy image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Though the Planet-CR has avoided the surface changes that may appear between the acquisitions of cloudy and cloud-free images as much as possible through short time lag for data collection, there are some inevitable nuisances determined by the sunlight condition, acquisition geometry, humidity, pollution, change of landscape, etc (Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' In this paper, we use the Charbonnier loss (Lai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2017) which can better handles outlier for training, and extra constraints on cloudy regions are applied, \ue238 = (1 + 푤퐌) ⊙ ((퐲 − ̂y)2 + 휖2)훼 (8) where ⊙ denotes Hadamard product operator, 푤 denotes the extra weight for cloudy regions, 퐌 denotes the cloudy mask, 휖 and 훼 are constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Evaluations 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Experimental Settings Preprocessing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Before the PlanetScope and Sentinel-1 data are fed into the neural networks, we apply value clipping to eliminate a small number of anomalous pixels and data scaling to improve the stability of the neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' We Fang Xu, et al: Preprint submitted to Elsevier Page 6 of 11 AlignFuse block 1 block D Prediction 阳 F1 2 Fo block FD block opt opt opt lign use R Cloudy F F0 FO F1 FD sar sar sar sar Upsample Add SARPlanet-CR Table 2 Quantitative results of visual recovery quality over different land cover types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' (a) results on objective assessment metrics, MAE and PSNR, to quantify the prediction error per class MAE ↓ MAE↓ PSNR↑ Forest Rangeland Agriculture Urban Barren Water McGAN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='0321 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} 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on subjective assessment metrics, SAM and SSIM, to quantify spectral and structural similarity per class SAM ↓ SAM↓ SSIM↑ Forest Rangeland Agriculture Urban Barren Water McGAN 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='5005 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='1237 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='4662 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='6771 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='4427 15.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='1655 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='1440 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='0905 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='2188 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='8602 DSen2-CR 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='1437 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='5581 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='3116 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='7517 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='2008 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='7014 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='6527 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='8806 GLF-CR 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='7830 5.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='8930 Ours (wo/ Align) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='0453 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='7382 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='7558 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='5235 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='8470 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='6844 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='7816 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='9021 Ours (Align-CR) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='5766 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='5408 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='6306 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='8660 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='0693 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='5205 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='4183 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='9124 clip the values of all bands of the PlanetScope data to [0, 10, 000] and divide by 10, 000 for all bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' We clip the VV and VH polarizations of the Sentinel-1 data to values [-25, 0] and [-32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='5, 0], respectively, and rescale them to the range [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' All experiments in this paper use these preprocessing steps, following previous best practices (Meraner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Ebel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Implementation Details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' The proposed Align-CR network is implemented using Pytorch and trained on 2 NVIDIA Geforce RTX 3090 GPUs with a batch size of 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' During training, we randomly crop the samples into 160 × 160 patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' The Adam optimizer is used and the maximum epoch of training iterations is set to 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' The learning rate is set to 10−4 for the whole network except for the Alignment blocks where the learning rate is set to a smaller value of 10−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' And the learning rates decays by 50% every 5 epochs after the first 10 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' For the network architecture, the upsampling operator adopts the nearest neighbor interpola- tion and the number of the AlignFuse blocks 퐷 is set to 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' For the loss function, 푤, 휖 and 훼 is set to 5, 10−3 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='45, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Baselines In this paper, we compare the proposed method with 5 baseline methods on the Planet-CR dataset with the proposed data splits, including the single image cloud removal methods, McGAN (Enomoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2017) and SpA GAN (Pan, 2020), and the multi-modal data fusion based cloud removal methods, SAR-Opt-cGAN (Grohnfeldt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2018), DSen2-CR (Meraner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2020) and GLF-CR (Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Since existing multi-modal data fusion based cloud removal methods require the input SAR images to be of the same spatial resolution as the input optical im- ages, upsampling the SAR images in the Planet-CR dataset is necessary for these algorithms to work properly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Here, all multi-modal data fusion based cloud removal methods utilize the SAR images upsampled by the nearest neighbor interpolation as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Moreover, to determine the benefits of including auxiliary low-resolution SAR images, we train the Align-CR network without use of the SAR images, denoted as w/o SAR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' And to validate the superiority of Align-CR in integrating multi-modal and multi-resolution information, we train the Align-CR network by removing the Alignment blocks in the AlignFuse blocks, denoted as w/o Align.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Evaluation of Visual Recovery Quality To evaluate the quality of the reconstructed images in terms of generating visually pleasing textures, we report mean absolute error (MAE) and peak signal-to-noise ratio (PSNR) to objectively quantify the prediction error, and spectral angle mapper (SAM) and the structural similarity index measure (SSIM) to subjectively measure the quality of reconstructed images from spectral similarity and structural similarity, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Notably, with the benefit of the land cover annotations, we disentangle the performance of cloud removal methods over different land cover types by pixel- wise MAE and SAM metrics, as shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' And we choose two scenes to qualitatively evaluate visual recovery quality, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' We can see that the methods McGAN, SpA GAN and SAR-Opt-cGAN, which were de- veloped on relatively small datasets with geographically and Fang Xu, et al: Preprint submitted to Elsevier Page 7 of 11 Planet-CR Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' 4: Qualitative results of visual recovery quality for two different samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' For each sample, from top-left to bottom-right are respectively the cloudy image, the SAR image, the land cover map, the results from McGAN, SpA GAN, SAR-Opt-cGAN, DSen2-CR, GLF-CR, w/o SAR, w/o Align, Align-CR, and the cloud-free image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' 5: Quantitative results of visual recovery quality over different cloud cover levels in terms of the MAE, PSNR, SAM, and SSIM quality metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' topographically homogeneous data, perform poorly in terms of generalizability on our Planet-CR dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Their results have relatively limited spectral fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' It indicates the need to take the point of global distribution into consideration when creating a practical dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Our method achieves the best performance overall, which can restore images with more details and fewer artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' It indicates the effective- ness and superiority of our method on restoring cloud-free images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Moreover, we compare the results over different land cover types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' We can find that removing clouds over urban land that has highly complex geometrical structures is more challenging than others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' And we can find that the results over water do not perform well on SAM but relatively well on Fang Xu, et al: Preprint submitted to Elsevier Page 8 of 11 Cloudy SAR Land Cover McGAN SpA GAN SAR-Opt-cGAN DSen2-CR GLF-CR Ours (w/o SAR) Ours(w/oAlign Ours (Align-CR) Cloud-FreeCloudy SAR Cover McGAN SpA GAN SAR-Opt-cGAN DSen2-CR GLF-CR Ours (w/o SAR) Ours (w/o Align) Ours (Align-CR) Cloud-Free(a)MAE(↓) (b) PSNR （↑) (c) SAM (↓ ) (d)SSIM （个) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='92 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='2 12 30 MAE (10-2) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='6 (dB) 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='88 SAM ( SSIM PSNR 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='0 28 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='84 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='4 6 26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='80 2040 60 80100 2040 60 80100 2040 60 80100 20 40 60 80100 CloudCoverage(%) CloudCoverage(%) CloudCoverage(%) CloudCoverage(%) McGAN SAR-Opt-cGAN GLF-CR Ours (w/o Align) SpA GAN DSen2-CR Ours(w/oSAR) Ours (Align-CR)Planet-CR Table 3 The performance of semantic recovery quality over different cloud cover levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' 0∼20% 20∼40% 40∼60% 60∼80% 80∼100% Overall mIoU PA mIoU PA mIoU PA mIoU PA mIoU PA mIoU PA Cloud-Free 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='63 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='68 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='66 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='44 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='85 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='40 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='58 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='16 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='98 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='70 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='24 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='66 Cloudy 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='89 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='08 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='26 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='84 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='24 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='27 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='62 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='21 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='89 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='00 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='72 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='46 McGAN 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='63 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='13 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='84 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='08 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='51 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='99 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='30 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='06 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='52 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='88 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='62 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='64 SpA GAN 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='07 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='37 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='93 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='94 17.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='97 SAR-Opt-cGAN 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='78 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='43 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='25 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='84 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='85 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='59 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='10 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='99 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='15 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='09 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='44 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='20 DSen2-CR 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='11 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='96 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='17 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='03 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='05 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='73 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='94 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='91 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='10 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='92 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='11 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='72 GLF-CR 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='99 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='66 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='54 44.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='75 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='63 Ours (w/o SAR) 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='85 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='72 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='14 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='83 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='91 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='34 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='95 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='02 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='58 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='02 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='97 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='20 Ours (w/o Align) 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='80 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='94 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='85 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='61 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='76 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='97 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='39 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='69 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='94 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='67 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='47 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='19 Ours (Align-CR) 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='32 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='16 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='90 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='40 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='93 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='22 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='17 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='37 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='43 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='69 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='51 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='78 MAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' It indicates the challenge in maintaining the spectral fidelity over water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' To determine what contributes to the superior perfor- mance of the proposed method, we analyze the effectiveness of each component by comparing the proposed method with its variants, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', w/o SAR and w/o Align.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' We can find that w/o SAR tends to generate undesirable artifacts for cloud- covered regions due to the lack of ground information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' While Align-CR exploits the geometrical information embedded in the SAR images, which can reconstruct the ground object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' We further compare the cloud removal performance of w/o Align to Align-CR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' When the Alignment blocks are re- moved, the gain of integrating low-resolution SAR informa- tion is reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' In regards to the misalignment, such as the river border in the first sample and the junction of different land cover types in the second sample in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' 4, w/o Align tends to generate blurring artifacts, while the reconstructed results of Align-CR have sharper edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' It demonstrates the superiority of our proposed Align-CR method in integrating multi-modal and multi-resolution information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' We further assess the reconstruction performance on different cloud cover levels, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' The perfor- mance of all methods, except McGAN, decreases roughly as the percentage of cloud cover increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Among them, the performance of the methods with the benefit of SAR images degrades more slowly than the one without, since the utilization of SAR images can alleviate the decline to some extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Align-CR performs favorably when compared with all baseline methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' When the Alignment blocks are removed, w/o Align behaves very similarly to GLF-CR, while GLF-CR contains more Transformer layers for global fusion and thus performs slightly better in terms of SSIM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Our method Align-CR, which aligns the multi-modal and multi-resolution data during the reconstruction process, can better exploit the power of SAR information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' It steadily outperforms w/o Align on all cloud cover levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Evaluation of Semantic Recovery Quality In addition to evaluating in generating visually pleas- ing textures, we further evaluate in generating semantically meaningful structures, where the semantic information is important for future analytical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' In this paper, we Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' 6: The discrepancy between the results with the cloud-free images as input and the results with the predicted cloud-free images as well as cloudy images as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' evaluate the quality of the recovered semantic information through a well-established land cover semantic segmenta- tion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Using the cloud-free PlanetScope images and associated land cover annotations, we train a land cover semantic segmentation model with the same data splits adopted for the experiments on cloud removal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' The model is based on DeepLabv3plus (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2018) with ResNet- 50 (He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2016) as the backbone network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' With the cloud-free images, cloudy images and predicted cloud-free images from benchmarked models as inputs respectively, we evaluate the performance of the trained land cover semantic segmentation model in predicting the correct class of each pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Ideally, the results with the predicted images as input should be as consistent as possible with the results with the cloud-free images as input, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', the closer they are to the results with the cloud-free images as input, the better the corresponding cloud removal method performs in terms of semantic recovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' We report the mean intersection over union (mIoU) and pixel accuracy (PA) over varying levels of cloud cover in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' 3, where mIoU can better deal with the class imbalance issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Additionally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' we show the discrep- ancy between the results with the cloud-free images as input Fang Xu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' et al: Preprint submitted to Elsevier Page 9 of 11 (a)mlou difference（→) (b)Acc difference（→) 35 50 30 25 40 lou Acc 20 30 15 20 10 10 20 40 60 80 100 20 40 60 80 100 Cloud Coverage(%) Cloud Coverage (%) McGAN DSen2-CR Ours (w/o Align) SpA GAN GLF-CR Ours (Align-CR) SAR-Opt-cGAN Ours(w/oSAR) CloudyPlanet-CR and the results with the predicted cloud-free images as well as cloudy images as input in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Clearly, the existence of clouds deteriorates the seman- tic analysis, and the degradation is more severe when the cloud cover level is higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' The benchmarked cloud removal models can to some extent counteract the degradation, in which Align-CR generally performs the best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Notably, we can observe that the trained land cover semantic segmen- tation model performs better with the predicted images of McGAN and SpA GAN as input than with the predicted images of SAR-Opt-cGAN as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' While SAR-Opt-cGAN performs better than McGAN and SpA GAN in terms of all visual quality metrics, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' It indicates that the metrics for measuring the visual recovery quality can not adequately reflect the performance of cloud removal methods in terms of semantic recovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' We can also find this by comparing the performance of w/o Align and Align- CR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Align-CR steadily outperforms w/o Align on all cloud cover levels in terms of all visual quality metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' However, it does not perform as well as w/o Align on the images with cloud cover 80% to 100% in terms of mIoU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' What’s more, we can find that its performance in terms of mIoU is more superior to that of w/o Align when more prior information from cloud-free regions is available, since there is more information available for alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Discussion Due to the shortcomings in semantic annotations, the vast majority of current studies quantify the goodness of cloud removal methods exclusively by metrics that assess the visual similarity between two images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Towards the direction of boosting the performance on the visual metrics, most methods use the the loss functions that are constructed based on these metrics to guide the training of cloud removal models, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', L1 loss function computes the mean absolute error (Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' It will motivate the predicted cloud- free image to move towards over-smoothness and lead to the loss of semantic information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' As we have validated in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='3, the current visual metrics cannot adequately eval- uate the quality of recovered semantic information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Though our method performs the best in terms of semantic recovery, there is still a large gap between its predicted cloud-free images and real cloud-free images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Therefore, it is very important for the cloud removal field to design a loss func- tion that can guide the recovery of semantic information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Future work could consider the correlation between cloud removal and downstream task, and design the loss function by balancing the semantic context and image details to guide the reconstruction process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Conclusion We introduce Planet-CR, a multi-modal and multi- resolution dataset for cloud removal in high-resolution op- tical remote sensing imagery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' We offer the dataset as open- source with the purpose of developing new cloud removal approaches towards high resolution by integrating multi- modal and multi-resolution information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' We address the problem of multi-modal and multi-resolution data fusion based cloud removal by proposing a novel method called Align-CR, which implicitly aligns the feature maps during the reconstruction process to compensate for the misalign- ment between the multi-modal and multi-resolution data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' The experimental results in both visual recovery quality and semantic recovery quality demonstrate the effectiveness and superiority of the proposed method than the existing rep- resentative cloud removal methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' We believe the Planet- CR dataset will prove beneficial for the community and our evaluations provide valuable directions for future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Acknowledgements The work of W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Yang is supported by the National Natural Science Foundation of China (NSFC) Regional In- novation and Development Joint Fund (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' U22A2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' The work of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Zhu is jointly supported by the European Research Council (ERC) under the European Union’s Hori- zon 2020 research and innovation programme (grant agree- ment No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' [ERC-2016-StG-714087],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Acronym: So2Sat),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' by the Helmholtz Association through the Framework of the Helmholtz Excellent Professorship “Data Science in Earth Observation - Big Data Fusion for Urban Research”(grant number: W2-W3-100),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' by the German Federal Ministry of Education and Research (BMBF) in the framework of the international future AI lab "AI4EO – Artificial Intelligence for Earth Observation: Reasoning,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Uncertainties,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Ethics and Beyond" (grant number: 01DD20001) and by German Fed- eral Ministry for Economic Affairs and Climate Action in the framework of the "national center of excellence ML4Earth" (grant number: 50EE2201C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' References Atwood, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Garron, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Addressing three fallacies about synthetic aperture radar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Eos, Transactions American Geophysical Union 94, 471– 472.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Chen, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Zhu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Papandreou, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Schroff, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Adam, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Encoder- decoder with atrous separable convolution for semantic image segmen- tation, in: European Conference on Computer Vision, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' 801–818.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Cornebise, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Oršolić, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Kalaitzis, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Open high-resolution satellite imagery: The worldstrat dataset–with application to super-resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' arXiv:2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='06418 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Cresson, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Ienco, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Gaetano, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Ose, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Minh, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Optical image gap filling using deep convolutional autoencoder from optical and radar images, in: IEEE International Geoscience and Remote Sensing Symposium, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' 218–221.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Cresson, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Narçon, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Gaetano, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Dupuis, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Tanguy, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', May, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Commandre, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Comparison of convolutional neural networks for cloudy optical images reconstruction from single or multitemporal joint sar and optical images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' arXiv:2204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='00424 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Czerkawski, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Upadhyay, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Davison, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Werkmeister, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Cardona, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Atkinson, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Michie, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Andonovic, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Macdonald, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Tachtatzis, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Deep internal learning for inpainting of cloud-affected regions in satellite imagery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Remote Sensing 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Dai, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Qi, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Xiong, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Li, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Zhang, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Hu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Wei, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Deformable convolutional networks, in: IEEE International Conference on Computer Vision, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' 764–773.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Fang Xu, et al: Preprint submitted to Elsevier Page 10 of 11 Planet-CR Demir, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Koperski, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Lindenbaum, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Pang, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Huang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Basu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Hughes, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Tuia, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Raskar, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Deepglobe 2018: A challenge to parse the earth through satellite images, in: IEEE Conference on Computer Vision and Pattern Recognition Workshops, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' 172–181.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Duan, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Li, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Multi-head linear attention generative adversarial network for thin cloud removal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' arXiv:2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='10898 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Ebel, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Meraner, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Schmitt, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Zhu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Multisensor data fusion for cloud removal in global and all-season sentinel-2 imagery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' IEEE Transactions on Geoscience and Remote Sensing 59, 5866–5878.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Ebel, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Xu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Schmitt, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Zhu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Sen12ms-cr-ts: A remote- sensing data set for multimodal multitemporal cloud removal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' IEEE Transactions on Geoscience and Remote Sensing 60, 1–14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='1109/ TGRS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='3146246.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Enomoto, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Sakurada, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Wang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Fukui, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Matsuoka, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Nakamura, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Kawaguchi, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Filmy cloud removal on satellite imagery with multispectral conditional generative adversarial nets, in: IEEE Conference on Computer Vision and Pattern Recognition Workshops, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' 48–56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Fisher, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Flood, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Danaher, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Comparing landsat water index methods for automated water classification in eastern australia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Remote Sensing of Environment 175, 167–182.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Gao, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Yuan, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Li, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Zhang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Su, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Cloud removal with fusion of high resolution optical and sar images using generative adversarial networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Remote Sensing 12, 191.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Gorelick, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Hancher, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Dixon, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Ilyushchenko, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Thau, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Moore, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Google Earth Engine: Planetary-scale geospatial analysis for everyone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Remote Sensing of Environment 202, 18–27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Grohnfeldt, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Schmitt, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Zhu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' A conditional generative adversarial network to fuse sar and multispectral optical data for cloud removal from sentinel-2 images, in: IEEE International Geoscience and Remote Sensing Symposium, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' 1726–1729.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Gu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Ebel, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Yuan, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Schmitt, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Zhu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Explicit haze & cloud removal for global land cover classification, in: IEEE Conference on Computer Vision and Pattern Recognition Workshops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' He, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Zhang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Ren, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Sun, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Deep residual learning for image recognition, in: IEEE Conference on Computer Vision and Pattern Recognition, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' 770–778.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Helmer, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Ruefenacht, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Cloud-free satellite image mosaics with regression trees and histogram matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Photogrammetric Engi- neering & Remote Sensing 71, 1079–1089.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Kondmann, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Toker, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Ruß wurm, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Camero, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Peressuti, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Milcinski, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Mathieu, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Longepe, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Davis, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Marchisio, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Leal- Taixé, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Zhu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Denethor: The dynamicearthnet dataset for harmonized, inter-operable, analysis-ready, daily crop monitoring from space, in: Neural Information Processing Systems Track on Datasets and Benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Lai, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Huang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Ahuja, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Yang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Deep laplacian pyra- mid networks for fast and accurate super-resolution, in: IEEE Conference on Computer Vision and Pattern Recognition, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' 5835–5843.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Li, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Wu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Hu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Li, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Wang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Molinier, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Deep learning based thin cloud removal fusing vegetation red edge and short wave infrared spectral information for sentinel-2a imagery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Remote Sensing 13, 157.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Li, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Liew, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Kwoh, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Generating "cloud free" and "cloud-shadow free" mosaic for spot panchromatic images, in: IEEE International Geoscience and Remote Sensing Symposium, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' 2480– 2482.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Li, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Li, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Chen, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Chan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Thin cloud removal with residual symmetrical concatenation network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' ISPRS Journal of Pho- togrammetry and Remote Sensing 153, 137–150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Lima, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Beuchle, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Langner, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Grecchi, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Griess, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Achard, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Comparing sentinel-2 msi and landsat 8 oli imagery for monitoring selective logging in the brazilian amazon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Remote Sensing 11, 961.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Lin, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Xu, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Wang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Wang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Sun, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Fu, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' A remote sensing image dataset for cloud removal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' arXiv:1901.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='00600 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Liu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Hunt, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 1984.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' A new approach to removing cloud cover from satellite imagery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Computer Vision, Graphics, and Image Processing 25, 252–256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Liu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Lin, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Cao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Hu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Wei, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Zhang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Lin, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Guo, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Swin transformer: Hierarchical vision transformer using shifted windows, in: IEEE International Conference on Computer Vision, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' 10012–10022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Meraner, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Ebel, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Zhu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Schmitt, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Cloud removal in sentinel-2 imagery using a deep residual neural network and sar-optical data fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' ISPRS Journal of Photogrammetry and Remote Sensing 166, 333–346.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Mitchell, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Delp, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Chen, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 1977.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Filtering to remove cloud cover in satellite imagery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' IEEE Transactions on Geoscience Electronics 15, 137–141.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Ngadze, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Mpakairi, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Kavhu, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Ndaimani, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Maremba, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Exploring the utility of sentinel-2 msi and landsat 8 oli in burned area mapping for a heterogenous savannah landscape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' PLOS ONE 15, 1–13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Pan, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Cloud removal for remote sensing imagery via spatial attention generative adversarial network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' arXiv:2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='13015 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' PBC, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2018–.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Planet application program interface: In space for life on earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' URL: https://api.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Requena-Mesa, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Benson, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Reichstein, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Runge, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Denzler, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Earthnet2021: A large-scale dataset and challenge for earth surface forecasting as a guided video prediction task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', in: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' 1132–1142.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Rogan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Franklin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Roberts, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' A comparison of methods for monitoring multitemporal vegetation change using thematic mapper imagery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Remote Sensing of Environment 80, 143–156.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Sarukkai, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Jain, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Uzkent, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Ermon, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Cloud removal in satellite images using spatiotemporal generative networks, in: IEEE Winter Conference on Applications of Computer Vision, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' 1785–1794.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Singh, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Komodakis, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Cloud-gan: Cloud removal for sentinel-2 imagery using a cyclic consistent generative adversarial networks, in: IEEE International Geoscience and Remote Sensing Symposium, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' 1772–1775.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Voigt, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Kemper, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Riedlinger, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Kiefl, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Scholte, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Mehl, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Satellite image analysis for disaster and crisis-management support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' IEEE Transactions on Geoscience and Remote Sensing 45, 1520–1528.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Wang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Chan, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Yu, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Dong, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Loy, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Edvr: Video restoration with enhanced deformable convolutional networks, in: IEEE Conference on Computer Vision and Pattern Recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Xia, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Bai, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Ding, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Zhu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Belongie, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Luo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Datcu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Pelillo, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Zhang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Dota: A large-scale dataset for object detection in aerial images, in: IEEE Conference on Computer Vision and Pattern Recognition, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' 3974–3983.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Xiong, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Zhang, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Wang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Shi, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Zhu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Earthnets: Empowering ai in earth observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' arXiv:2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='04936 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Xu, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Shi, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Ebel, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Yu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Xia, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Yang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Zhu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content='X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Glf-cr: Sar-enhanced cloud removal with global–local fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' ISPRS Journal of Photogrammetry and Remote Sensing 192, 268–278.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Yu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Chen, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Su, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Fan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Li, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Kriging interpolation method and its application in retrieval of modis aerosol optical depth, in: International Conference on Geoinformatics, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' 1–6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Yuan, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Shen, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Li, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Li, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Li, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Jiang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Xu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Tan, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Yang, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Deep learning in environmental remote sensing: Achievements and challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Remote Sensing of Environment 241, 111716.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Zanaga, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Van De Kerchove, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', De Keersmaecker, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Souverijns, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Brockmann, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Quast, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Wevers, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Grosu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Paccini, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Vergnaud, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Esa worldcover 10 m 2020 v100 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Zhang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Li, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Travis, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Gaps-fill of slc-off landsat etm+ satellite image using a geostatistical approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' International Journal of Remote Sensing 28, 5103–5122.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Zhang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Tian, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Kong, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Zhong, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', Fu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=', 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Residual dense network for image restoration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' IEEE Transactions on Pattern Analysis and Machine Intelligence 43, 2480–2495.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} +page_content=' Fang Xu, et al: Preprint submitted to Elsevier Page 11 of 11' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE1T4oBgHgl3EQfyQXE/content/2301.03432v1.pdf'} diff --git a/fNFAT4oBgHgl3EQf7x4q/content/tmp_files/2301.08746v1.pdf.txt b/fNFAT4oBgHgl3EQf7x4q/content/tmp_files/2301.08746v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..76f4b5a73a94ae5861088072a67bbb2eac06dfed --- /dev/null +++ b/fNFAT4oBgHgl3EQf7x4q/content/tmp_files/2301.08746v1.pdf.txt @@ -0,0 +1,983 @@ +Characterization of a deep-depletion 4K x 4K CCD Detector System +designed for ADFOSC +Dimple* 1, 2, T. S. Kumar1, A. Omar1, K. Misra1 +1Aryabhatta Research Institute of observational sciences, Manora Peak, Nainital 263001, India. +2Department of Physics, Deen Dayal Upadhyaya Gorakhpur University, Gorakhpur-273009, India. +Abstract. +We present the characterization of the CCD system developed for the ADFOSC instrument on the 3.6m +Devasthal Optical Telescope (DOT). We describe various experiments performed to tune the CCD controller param- +eters to obtain optimum performance in single and four-port readout modes. Different methodologies employed for +characterizing the performance parameters of the CCD, including bias stability, noise, defects, linearity, and gain, are +described here. The CCD has grade-0 characteristics at temperatures close to its nominal operating temperature of +−120◦C. The overall system is linear with a regression coefficient of 0.9999, readout noise of 6 electrons, and a gain +value close to unity. We demonstrate a method to calculate the dark signal using the gradient in the bias frames at +lower temperatures. Using the optimized setting, we verify the performance of the CCD detector system on-sky using +the ADFOSC instrument mounted on the 3.6m DOT. Some science targets were observed to evaluate the detector’s +performance in both imaging and spectroscopic modes. +Keywords: characterization, ADFOSC, CCD . +*Dimple, dimplepanchal96@gmail.com +1 Introduction +The 3.6m DOT was commissioned at the Devasthal observatory of Aryabhatta Research Institute +of observational sciencES (ARIES), Nainital (India).1 The Devasthal Observatory is situated in the +Himalayan regions of Uttarakhand at ∼ 2450 meter above the mean sea level with geographical co- +ordinates of 29◦.360 N, 79◦.690 E. This location lies in the middle of the 180◦-wide longitude-gap +between the Canary Islands (∼ 20◦ W) and Eastern Australia (∼ 160◦ E), making it suitable for +observations of time-critical astronomical events due to the availability of several moderate aper- +ture telescopes. The DOT uses a f/9 Ritchey-Chr +′etien (RC) system supported on an alt-azimuth +mount.2,3 The aperture of this telescope is appropriate for medium-resolution spectroscopy and +observations of faint sources. A low-dispersion spectrograph-cum-imager, ARIES Devasthal Faint +Object Spectrograph (ADFOSC), has been developed in ARIES for spectroscopy and imaging of +the celestial objects.4 The spectrograph uses a fixed focal reducer, which converts the incoming +1 +arXiv:2301.08746v1 [astro-ph.IM] 20 Jan 2023 + +f/9 optical beam from the telescope into a faster ∼ f/4.2 beam. The spectrograph can be used +in both spectroscopic and imaging modes by selecting the instrument’s corresponding optical el- +ements (e.g. filters, grism, slit, etc.) with the help of a GUI-based instrument control software.5 +In either mode, a charge-coupled device (CCD) is required to detect and record the data. A CCD +detector system/imager has been designed and assembled in ARIES in technical collaboration with +the Herzberg Institute of Astrophysics (HIA), Canada. +We performed a detailed characterization of the CCD system before commissioning it for sci- +entific observations, both in the laboratory and on the sky. This included estimating parameters +like bias level, readout noise, and thermal noise in the dark room. We then performed iterative +experiments in the laboratory to optimize the overall system performance and verified the CCD for +cosmetic defects. We demonstrate a method to calculate the dark signal of the CCD at different +temperatures using the bias frames. As the CCD is developed for the ADFOSC instrument, we +also estimated the spectral dispersion on the detector using the lamp spectra. After optimization +in the laboratory environment, we performed similar experiments over the night sky on the 3.6m +DOT to verify the on-sky performance of the detector system. +The paper discusses the different methodologies employed for characterizing the performance +of the CCD system. The test setup used for performing different tests to optimize the system +parameters is detailed in section 3. We also discuss various experiments performed to determine +and optimize the CCD characteristics. The integration of the CCD system with the ADFOSC +instrument and results of the on-sky tests are discussed in section 4. To evaluate the performance +of the CCD system on science targets, we observed transient sources during the observing cycle +2020C2 of the 3.6m DOT. The results of the scientific observations are presented in section 5. +2 + +Fig 1 The ADFOSC CCD camera setup in the laboratory. The Camera consists of a CCD detector, a controller, a +pressure gauge, a dewar which is cooled using a heat-exchange cryogenic system. +3 + +Table 1 CCD Characteristics +CCD Characteristic +Value +CCD +e2v 231 − 84 / Grade-0 +Pixel Size +15 µm +Readout Frequency +160 kHz +Operating Temperature +−120◦C +Bias level +1134 ± 2.62 ADU +Gain +1.00 ± 0.04 e−/ADU +Readout Noise +6 e−/pixel +Full well capacity +408 ke−/pixel +Saturation level +65535 ADU +2 The CCD detector system +The CCD is a 4096×4096 format back-illuminated e2v 231-84 CCD sensor having square pixels of +15-micron size. It is a deep-depleted sensor capable of enhancing the sensitivity toward the longer +wavelengths (∼ 700 − 1000 nm ) of the optical spectrum. The CCD has an imaging area of 61.4 +× 61.4 mm2, providing a Field of View (FoV) of ∼ 13.6 × 13.6 arcmin2 and a spectral dispersion +in the range 0.1 − 0.2 nm/pixel. The CCD has four readout ports (0, 1, 2, 3). The ∼ 16 million +pixels can be read from any of the four amplifiers or through four amplifiers simultaneously. The +four-port readout decreases the readout time by a factor of four. However, it requires additional +processing to match the bias levels of the four quadrants. Since the readout noise would differ for +the four different amplifiers, each quadrant’s respective signal-to-noise ratio (SNR) would also be +different. In the case of the ADFOSC instrument, we have implemented a single port readout via +port-0, which provides the lowest readout noise. The readout frequency is fixed at ∼ 160 kHz, +providing a readout time of ∼ 104 sec. A Bonn shutter is mounted at the camera entrance with an +aperture size of 100 mm ×100 mm. The shutter employs servo motors for fast operation and offers +4 + +uniform exposure at the detector plane. Using this shutter, a minimum exposure time of ∼ 1 ms is +possible with an uncertainty of 300 µs. The detector system consists of the CCD detector, a clock- +shaping fan-out circuit board, and a generic Astronomical Research Cameras (ARC) controller1 +to generate the suitable clock and bias voltages for the detector. The controller hardware has two +video cards for reading four ports with 16-bit Analog to digital converter (ADC) units to interface +and digitize the four channels of the CCD. Additionally, different bin settings are implemented to +read the image in different binning patterns for photometry and spectroscopy. +The CCD sensor is cooled to −120◦C to minimize the dark signal. The CCD dewar is evacu- +ated for several hours using an oil-free turbo molecular vacuum pump before deep cooling. The +dewar is equipped with a Pirani vacuum gauge for monitoring its vacuum pressure. Once the vac- +uum reaches below ∼ 1 x 10−3 Torr, the closed-cycle (Joule-Thomson) cryogenic heat-exchange +system supplied by Brooks Automation, USA, is switched on. The overall process of cryo-cooling +the CCD from an ambient temperature of 25◦C to −120◦C takes between 4 to 5 hours. This tem- +perature is stabilized and held constant within 0.01◦C using a Lakeshore 335 Proportional Integral +Derivative (PID) temperature controller2. For this purpose, a small heater and a temperature sen- +sor are mounted on the cold plate below the sensor. A charcoal-filled getter is used to absorb +outgassing inside the dewar. The charcoal gets activated to absorb gases at cryogenic temperatures +and helps attain a high vacuum. An ultimate vacuum of ∼ 3 x 10−7 Torr is usually attained with +this system at −120◦C. +1http://astro-cam.com +2http://irtfweb.ifa.hawaii.edu/˜s2/software/gpib-eth-ls335/335_Manual.pdf +5 + +3 Characterization of the CCD system +Detailed characterization of the CCD includes the estimation of bias level, bias stability, readout +noise (RN), gain, defects, linearity, and saturation level of the CCD. This section describes the +laboratory-based test setup and the experiments performed to determine these parameters. We +tested the CCD performance using all four ports, numbered zero to three. However, the read noise +is found to be the lowest for port-0; hence single port readout mode using port-0 is currently being +used for acquiring the scientific data. The paper focuses on characterizing parameters for port-0 of +the CCD system. +3.1 Test setup and Data Acquisition +We set up the CCD system on an electrostatic discharge (ESD) safe, dark room of the ARIES +optics laboratory for performing the experiments. A light-emitting diode (LED) operated at a +constant regulated voltage was used as an illumination source for the experiment. We covered +the CCD window with an Aluminium plate with a pinhole for the light to enter. We fixed the +source on the pinhole to avoid any fluctuations in light intensity due to any change in the source’s +position. The sub-systems, namely, the temperature controller, cryogenic pump, pressure gauge, +etc., were carefully grounded to a common point to avoid noise entering from ground loops. The +entire system was again reconfigured when the ADFOSC was mounted on the 3.6m DOT for sky +tests. We acquired the data using the Owl3 software provided along with the ARC controller. The +software offers different dialog boxes to control the controller parameters, including gain, readout, +binning, etc., and saves the acquired images in standard Flexible Image Transport System (FITS) +format. Different modules of Python6 like Astropy7–9 and ccdproc10 were used to process +3http://www.astro-cam.com/Gen3Software.php +6 + +Fig 2 Master bias of the CCD created using median combining 50 bias frames. +the FITS image files. +3.2 Bias level and readout noise +A positive offset is generally provided to the CCD electronics to avoid negative counts in the output +of the CCD. The mean offset value, or the bias value, is optimized in a way that it is large enough +to avoid the non-linear regime of the CCD amplifiers but not too large to reduce the dynamic range +7 + +1125 +1130 +1135 +1140 +1145 +Counts (ADU) +0.00 +0.02 +0.04 +0.06 +0.08 +0.10 +0.12 +0.14 +0.16 +Pixel Density +Fig 3 Histogram of the master bias frame with a mean value of 1134.01 ± 2.62 counts. +of the CCD. We set the bias level slightly above thousand counts to balance the above factors. +We estimated the bias level of the CCD using the bias frames, which are the CCD images +with a zero-second exposure. Several bias frames were acquired, and we used fifty of them to +generate a master bias frame by taking their median value. We did this using Mediancombine +task of ccdproc software module. Fig. 2 shows the median bias frame, and the corresponding +histogram is shown in Fig. 3. The width of the distribution represents the RN of the CCD, which +is the number of electrons introduced by the readout electronics while reading out each pixel. We +estimated the RN using the Janesick method (equation 1).11 We created a difference image using +8 + +two bias images and estimated its standard deviation (σBias1−Bias2). The explanation of the Gain +value used in equation 1 is provided in section 3.5. We found the RN value to be 6.20 analog to +digital units (ADU) or 6.20 e−1 for the gain value of ∼ 1. However, the achieved noise is more +than twice the value in the e2v datasheet at 160kHz. We observed some interference patterns +in the image. These patterns are likely to be responsible for this increased noise. The probable +cause could be ground loops outside and inside the dewar, length of cables and imperfect shielding +scheme. These have been controlled to a large extent by iteratively evaluating various schemes like +shortening the cables and star connecting the ground points of the auxiliary devices like chiller, +pressure gauge, temperature controller etc. Also, grounding permutations were tried with the four- +port video cables. Finally, the shielding of the video cables was grounded only at the controller end +and left open at the CCD connector end, which resulted in a lower noise floor. There is still scope +for improvement as the ground loops inside the dewar have not been evaluated. This evaluation +will be attempted later since the CCD has already been commissioned for observations. +RN = Gain × σBias1−Bias2 +√ +2 +, +(1) +We calculated the standard deviation of 50 bias frames to verify this RN value, as shown in Fig. +4. The mean of these standard deviation values is 6.38 ADU, which is consistent with the value +calculated using equation 1. +9 + +0 +10 +20 +30 +40 +50 +Frame Number +5.0 +5.5 +6.0 +6.5 +7.0 +7.5 +8.0 +Standard Deviation (ADU) +Mean = 6.3 ADU +Standard devation +Fig 4 A plot showing the variation of the standard deviation of 50 bias frames +10 + +3.3 Linearity +The CCD system should have a linear response to the incident light for scientific observations. +However, several factors can introduce non-linearity in CCD performance. The controller clock +periods and overlaps should be timed for complete charge transfer during the readout process. +Moreover, there should be a delay between this transfer and the correlated double sampling instant +to allow the transients to settle to avoid any induced noise or glitch and introduce non-linearity. +We verified the signal waveform using a 1 GHz digital oscilloscope to ensure the above before +connecting the interface board to the detector. Other factors critical for linearity are bias voltages: +voltage output drain (VOD) and voltage reset drain (VRD). The CCD manufacturer provided a +range of values for the voltages, VOD from 25 to 31 volts and VRD from 16 to 19 volts. To check +the behaviour of the CCD at different voltages, we initially set these voltages near minimum values +and iteratively increased these voltages within this range. We rejected some of the voltage combi- +nations that provided very low-bias levels. For other combinations, experiments were performed +to check the linearity of the CCD. +We used an LED source (section 3.1) to illuminate the CCD and acquired images with an +incremental increase in exposure time. We obtained a pair of images for each exposure time to +detect any variation in the source intensity. We noticed that the counts are identical for the pair of +images. +We estimated the non-linearity (the relative difference between our measurements and the best- +fit linear curves) by fitting a linear function to the variation of mean counts with exposure time for +each voltage combination. We used the linregress function from the stats library under +Python for this purpose. Fig. 5 shows the non-linearity curves for various combinations of VOD +11 + +2000 +4000 +6000 +8000 +10000 +12000 +Counts (ADU) +−10 +0 +10 +20 +30 +Non − linearity % +. +10000 +20000 +30000 +40000 +50000 +60000 +−2.0 +−1.5 +−1.0 +−0.5 +0.0 +0.5 +1.0 +1.5 +VOD = 28, VRD = 17 +VOD = 28.5, VRD = 16.5 +VOD = 28.5, VRD = 17 +VOD = 28.5, VRD = 17.5 +VOD = 29, VRD = 16.5 +VOD = 29, VRD = 17 +VOD = 29, VRD = 17.5 +VOD = 29, VRD = 18 +Fig 5 Non-linearity curves at different operating voltages in the lower count regime (left panel) and in the higher count +region (right panel). Non-linearity is the minimum for a combination of VOD=29 volts and VRD=16.5 volts. +and VRD in different count regions. For most voltage combinations, the non-linearity is negligible +in the higher count regime. The non-linearity, however, shows up in the lower count regime and +is significant for certain voltages. For a combination of VOD = 29 volts and VRD = 16.5 volts, +the non-linearity is the lowest. For this voltage combination, the value of the regression coefficient +(R2) is 0.9999, which is almost equal to unity (see Fig. 6). We considered this voltage combination +as the optimum value for the CCD system. +3.4 Saturation level +The maximum capacity of a CCD pixel to store the photo-electrons is its full well capacity, beyond +which the pixels saturate. Since the available 16-bit ADC of the controller saturates at a value +of 65535, the controller’s gain setting helps to select the dynamic range. As we are interested in +accurate photometry of faint objects, a gain of unity is selected, constraining the saturation point +to 65535. The selection of system gain is discussed in section 3.5. Illumination of the CCD +until its saturation point limited the counts to 65535, the saturation point of the 16-bit ADC. It is +12 + +0 +2 +4 +6 +8 +10 +12 +Exposure Time (sec) +0 +10000 +20000 +30000 +40000 +50000 +60000 +70000 +Mean Counts (ADU) +R2 = 0.9999 +Best fit +Saturation level +Lab data +Fig 6 Linearity curve at VOD = 29 volts and VRD = 16.5 volts. A linear fit to the data gives the regression coefficient +as 0.9999. The horizontal dashed line indicates the saturation level. +13 + +demonstrated and shown in Fig. 6 where a bright source illuminates the detector, and the ADC +saturates at 65535 counts. If the science cases demand the utilization of full well capacity, the user +can select a gain setting close to 3 or higher electrons per ADU. +3.5 The system gain +The gain of a CCD system is defined in terms of ADU, which corresponds to the number of +electrons assigned to one digital unit in the output image. The available gain values are 1, 2, 5, +and 10 e−/ADU in the controller. The gain values can be selected using the software at runtime. +The saturation level of the CCD should correspond to the saturation level of the ADC to utilize the +full well capacity. Since the full well capacity of the CCD is 408 ke− (as mentioned in the result +sheet of the supplied detector), a gain of 10 is suitable to match the saturation levels. However, +for the detection of photon-limited faint objects, a gain of 1 is implemented using the controller +parameters. +The electronic gain of the CCD system is the product of gain values introduced by each stage +of the readout electronics. The inherent gain of the CCD, defined by the output capacitor, is 7 +µ V/e−. A series of Op-amp stages within the controller further amplify this gain. Initially, it is +preamplified with a gain of 4 and passed through a gain selection stage, offering a range of gain +values: 1, 2, 4.75, and 9.5. A bias adjustment stage after the integrator provides a gain of 2. Hence, +an amplification of 56 µ V/e− is obtained with these four stages. Since the 16-bit ADC operating +at a reference voltage of 10 V provides a bin size of 152.588 µ V/ADU, the integration of the +Op-amp integrator is adjusted to provide an additional gain factor of 2.725 to achieve the desired +system gain of 1 e−/ADU. Since the integrator time can only be adjusted in increments of 40 ns, +the closest possible value of 0.998 e−/ADU is set. +14 + +0 +10000 +20000 +30000 +Mean Counts (ADU) +102 +103 +104 +Variance (ADU2) +Gain = 1.00 ± 0.04 +Lab data +Fig 7 Photon transfer curve (PTC) of the CCD obtained in the laboratory environment. The measured value of the +gain is 1.00 ± 0.04 e−/ADU. +We experimentally verified this gain setting using the Janesick method11 given by equation 2 +where (S) is the mean of the signal acquired by the CCD, and σ2 +S is the variance. +σ2 +S = S +G + σ2 +R, +(2) +We acquired a pair of images at each exposure and estimated the mean signal after bias subtrac- +tion and cosmic-ray removal from the image. Further, these images were normalized by subtracting +15 + +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Pixel Number +×106 +0 +200 +400 +600 +800 +1000 +1200 +1400 +Deviation From Zeroth Pixel Counts (ADU) +−50◦C +Fig 8 Masterbias at −50◦C and deviation of the counts of all pixels from the zeroth pixel counts. A clear gradient can +be seen in the image as the last pixels get more time to generate the dark signal. +one image from the other for compensating the flat-field effect. We used the resulting image to es- +timate the variance of the signal (σ2 +S). Fig. 7 shows the photon transfer curve (PTC) for the CCD. +To estimate the gain, the PTC was fitted with a linear function using linregress. The estimated +gain is 1.00 ± 0.04 e−/ADU, which matches the electronic gain value of the system within the +errorbar. +3.6 Thermal noise +In the CCD detectors, electron charge-density increases exponentially with an increase in temper- +ature due to the thermal generation of electrons. A CCD must be cooled optimally to minimize the +dark signal. To determine this optimum temperature, we calculated the dark signal at various tem- +peratures ranging from −35◦C to −120◦C using the bias frames. We acquired several bias frames +at different temperatures and generated master bias frames at each temperature. The left panel of +Fig. 8 shows the master bias frame at −50◦C. A gradient in counts is visible in the master bias +16 + +due to the finite readout time of the CCD. The zeroth pixel gets the lowest time to generate dark +electrons. The last pixel accumulates dark counts over full readout time, hence has the maximum +number of thermally generated electrons. If the readout time and the gain are known, then by com- +paring the counts of the first and the last pixel, we can measure the number of electrons generated +per pixel per second. Using this method, we estimated the dark signal at different temperatures. +Temp. +(◦C) +Dark Current +(e-/pix/sec) +Temp. +(◦C) +Dark Current +(e-/pix/sec) +-35.0 +39.89 ± 0.017 +-60.0 +1.03 ± 0.002 +-40.0 +21.30 ± 0.009 +-65.0 +0.42 ± 0.002 +-42.5 +15.36 ± 0.007 +-70.0 +0.18 ± 0.002 +-50.0 +5.37 ± 0.003 +-80.0 +0.10 ± 0.001 +-55.0 +2.89 ± 0.002 +-90.0 +0.05 ± 0.001 +-57.5 +1.87 ± 0.002 +-95.0 +0.01 ± 0.002 +Table 2 Dark current at different temperatures as estimated using the gradient of bias frames. +The right panel of Fig. 8 shows the deviation of counts in each pixel from the zeroth counts. +The farther the pixel number is from the readout port, the larger the dark count and the larger the +deviation from the zeroth counts. To determine the slope of this gradient, we fitted a polynomial +in counts vs. pixel number data using the polyfit function of Python. It is seen that a linear +function provides the best fit, as shown in the left panel of Fig. 8. We used the slope to calculate +the difference in counts between the first and the last pixel. We divided this difference by the total +readout time to obtain dark counts generated per second. Since the bias frames were acquired in +4 × 4 binning, it was further scaled by a factor of 16 after subtracting the RN. Below −100◦C +temperature, the thermal noise becomes less than RN; hence, we could estimate the dark signal +values up to −95◦C. The dark signal values at different temperatures are listed in table 2. +As shown in Fig. 9, the dark signal varies exponentially with temperature. Below -80°C, the +17 + +−90 +−80 +−70 +−60 +−50 +−40 +Temperature (◦C) +0 +5 +10 +15 +20 +25 +30 +35 +40 +Dark Signal (e−/pix/sec) +Fig 9 Variation of dark signal with temperature. The dark signal is negligible below −80◦C. +dark signal is negligible, suggesting that the CCD can be used below this temperature with minimal +thermal noise. +3.7 CCD defects +The CCD may have some pixels that might not respond to light optimally due to defects in the +CCD structure. These can be point defects, hot defects, or dark/dead pixels. We are employing +a grade-0 CCD detector (the CCD detector with minimum possible defects) as mentioned by the +manufacturer. To check the CCD for point and column defects, we examined the response of all +the pixels at different temperatures. Fig. 10 shows the deviation of counts from the mean bias +counts for each pixel of the CCD operated at different temperatures (the method to obtain these +plots is described in section 3.6). Some pixels are seen to behave differently at higher temperatures +18 + +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +×106 +0 +5000 +10000 +15000 +20000 +−35◦C +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +×106 +0 +500 +1000 +1500 +−50◦C +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +×106 +0 +100 +200 +−60◦C +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +×106 +−50 +0 +50 +100 +150 +−70◦C +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +×106 +0 +50 +100 +150 +−80◦C +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +×106 +0 +50 +100 +150 +−90◦C +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +×106 +0 +50 +100 +−100◦C +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +×106 +−40 +−20 +0 +20 +40 +−110◦C +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +×106 +−10 +0 +10 +20 +−120◦C +Pixel Number +Deviation From Zeroth Pixel Counts (ADU) +Fig 10 The response of CCD pixels at different temperatures is shown in the figure. The deviation of the counts from +the zeroth counts is fitted with a polynomial (black line). The dark signal is calculated from the best fit. There are a +few pixels which are behaving differently at higher temperatures. At lower temperatures, the CCD behaves as grade-0 +CCD. +19 + +exhibiting high counts. Though they appear to be hot pixels, the counts are found to decrease +with decreasing temperature. Eventually, below −110◦C, the CCD acts as a nominal grade-0 CCD +without any point and column defects. +4 On-sky verification +After optimizing the performance of the CCD system in the laboratory, we verified the on-sky +performance. The CCD was integrated with the ADFOSC instrument and mounted on the axial +port of 3.6m DOT. This section describes the estimation of gain, linearity, bias level, and bias +stability using on sky observation with the instrument. +4.1 Bias Stability +We calculated the bias level using the methodology described in section 3. The mean bias level +equal to 1133.85±2.48 matches the laboratory estimated value, i.e. 1134.01±2.62. Since fluctua- +tion in bias level can introduce errors in photometric estimates, we acquired and examined several +bias frames to check the stability of the bias. Fig. 4 shows the variation of the mean bias level +for 30 different nights spread across an observing cycle of three months. The mean bias level +fluctuates within a fraction of a count, ensuring bias stability. +4.2 Linearity and gain +We chose the standard field available at the zenith at the time of observations to validate the lin- +earity and gain of the CCD. Multiple images of Landolt standard field SA11012 were acquired in +r-band with an exposure time ranging from 5 sec to 100 sec. Before using the images for character- +ization purposes, we pre-processed the images with basic steps of bias subtraction, flat correction +and cosmic-ray removal using the ccdproc. Fig. 12 shows the pre-processed CCD image of the +20 + +0 +5 +10 +15 +20 +25 +30 +Frame Number +1133.00 +1133.25 +1133.50 +1133.75 +1134.00 +1134.25 +1134.50 +1134.75 +1135.00 +Mean Bias Counts (ADU) +Mean Bias Level (Sky) +Mean Bias Level (Lab) +Fig 11 Variation within the mean counts of bias frames during different nights. The bias level is stable within one +count. The red and black dotted lines show the mean bias level estimated in the lab and on-sky. +21 + +Fig 12 CCD image of the Landolt standard field SA110. The standard stars are between 10 to 16 mag in V-band. +22 + +0 +20 +40 +60 +80 +100 +Exposure Time (sec) +0 +5000 +10000 +15000 +20000 +25000 +30000 +35000 +Mean Counts (ADU) +R2 = 0.9997 +On Sky +Fig 13 Variation of mean counts with exposure time from on-sky experiments. The black line represents the best linear +fit with a regression coefficient of 0.9997. +field SA110, which contains both bright and faint stars (with magnitudes ranging from 10 to 16 +mag in the V-band). +We used the faint stars to check the linearity in the lower count region and the bright stars to +estimate the saturation level of the CCD. Fig. 13 shows CCD linearity with R2 = 0.9997 and a +non-linearity percentage of 0.30. The CCD system is seen to saturate at 65535 counts for a gain of +1 e−/ADU. +The estimated gain value using the method described in section 3.5 is 1.04 ± 0.13 e−/ADU, +which is close to the value estimated in the laboratory. On-sky gain estimation is also affected by +23 + +0 +5000 +10000 +15000 +Mean Counts (ADU) +103 +104 +Variance (ADU2) +Gain = 1.04 ± 0.13 +On Sky +Fig 14 Photon transfer curve (PTC) of the CCD as obtained from the sky experiments. The gain value is estimated as +1.04 ± 0.13 e−/ADU. +24 + +Fig 15 Field of GRB 210217A afterglow imaged with the ADFOSC in r-band. +the sky variation, which results in a slightly higher error bar. The mean-variance plot is shown in +Fig. 14. +5 Performance of the CCD +We used the CCD for imaging and spectroscopic observations of various science targets after opti- +mizing it in the lab and successfully verifying it on-sky. This section demonstrates the performance +of the CCD system in both imaging and spectroscopic modes with observations of GRB and Su- +pernovae sources. The on-sky performance of ADFOSC on different science targets is discussed +in more detail in Omar et al. 2019.4 +25 + +GRB210217A(DOT)0 +500 +1000 +1500 +2000 +2500 +3000 +Column Number +1000 +2000 +3000 +4000 +5000 +6000 +7000 +8000 +Counts +5000 +6000 +7000 +8000 +9000 +10000 +Wavelength (˚A) +1000 +2000 +3000 +4000 +5000 +6000 +7000 +8000 +Counts +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Residual (˚A) +Fig 16 The lamp spectrum of Mercury Argon lamp using 770R grating. The left panel shows the spectrum in pixel +scale. The vertical lines indicate the column numbers for the first and last emission lines identified in the spectrum in +the left figure. The right panel shows the spectrum in wavelength scale. +5.1 Imaging +We observed the optical afterglow of GRB 210217A using the ADFOSC in imaging mode. These +observations were performed on 18th February 2021 in the r-band at 23:59:18 UT, at ∼ 1.7 days +after the burst. Owing to the faintness and rapid decay rate of GRB afterglows, a series of eight +images, each with an exposure time of 300 seconds, were acquired. The images were stacked +after pre-processing (as described in the previous section) to improve the signal-to-noise ratio. The +optical afterglow is visible in the stacked image as shown in Fig. 15. The photometric estimate of +the afterglow is 22.32 ± 0.16 mag (AB). +5.2 Spectroscopy +The spectrograph provides three sets of gratings: 300 gr/mm, 420 gr/mm, and 600 gr/mm. We ac- +quired the lamp spectra using the Mercury Argon (HgAr) calibration lamp to estimate the spectral +dispersion. We used the find peaks module of scipy13 to extract the spectral peaks from the +obtained spectra. We identified the column number corresponding to each peak and compared it +with the wavelength-calibrated lamp spectrum atlas. We defined initial polynomial solutions using +26 + +4000 +5000 +6000 +7000 +8000 +9000 +10000 +11000 +Rest Wavelength (Å) +Normalised flux+const +365.2 days +253.9 days +80.15 days +CaII +OI +FeII +Na I +H +CoII +CaII +CaII +OI +ScII +FeII +FeII +Shane, D=3 m +DOT, D=3.6 m +LCO, D=2 m +Fig 17 Spectrum of SN 2020jfo obtained using ADFOSC at ∼ 254 days after the discovery of supernova. We identified +different absorption lines in the spectrum and compared these with the spectra taken from other instruments/telescopes. +these matched wavelength pairs and calculated the best-fit polynomial coefficients to transform +between the column number and wavelength. +The left panel of Fig. 16 shows the lamp spectrum obtained using a 300gr/nm grating element +in pixel scale. The right panel shows the calibrated spectrum in the wavelength scale. A spec- +tral dispersion of 0.20 nm/pixel is estimated for this grating. For gratings elements, 420 gr/mm +and 600 gr/mm, the estimated values of spectral dispersion are 0.14 nm/pixel and 0.10 nm/pixel, +respectively. +We acquired the spectrum of the supernova SN 2020jfo using 1.5 +′′ slit and 420 gr/mm grating +with an exposure time of 900 sec on 13th January 2021 at 23:12:53 UT. Ailawadhi et al.14 describe +27 + +the spectral reduction technique. The absorption features in the spectrum obtained by ADFOSC +were identified and matched with the spectra obtained from other telescopes, as shown in Fig. 17. +It is noticed that spectral features at longer wavelengths are visible, indicating the high sensitivity +of the CCD detector (deep-depleted) in the long-wavelength regime. +6 Conclusions +We present the methodology employed to characterize the performance of a CCD system devel- +oped for integrating with a low dispersion spectrograph instrument, ADFOSC, on the 3.6m DOT. +Various experiments were initially performed in the laboratory to characterize and optimize differ- +ent critical parameters of the CCD system. We also verified the estimated parameters on the sky +by mounting the instrument on the 3.6m DOT. We evaluated the bias level during the on-sky tests +and examined its stability over several observing nights. We experimentally identified an optimum +combination of bias voltages: VOD and VRD, to operate the CCD with minimum non-linearity. +The readout performance of the CCD is satisfactory. However, some interference patterns in the +image contribute to readout noise. Through different experiments, we tuned and verified the gain +parameter corresponding to 1 e−/ADU for detecting faint objects. We calculated the dark cur- +rent at different temperatures using bias frames at lower temperatures and established an optimum +operating temperature of the CCD. The CCD acts as a grade-0 detector with no hot pixels at op- +timum temperature. The regression coefficient values and the gain parameter obtained on-sky are +consistent with the values obtained in the laboratory. After verifying the satisfactory performance, +we observed the science targets both in imaging and spectroscopic modes. We carried out the +imaging of GRB 210217A15 field and the spectroscopy of supernova SN 2020jfo using ADFOSC +successfully.14 +28 + +Acknowledgments +We thank Greg Burley and Tim Hardy from NRC Herzberg Astronomy and Astrophysics Research +Centre, Canada, for their help in developing the CCD system. We thank the ARIES 3.6 m DOT +engineering team and staff members for providing the necessary support during development, ver- +ification, and commissioning work. We would also like to thank Dr. Raya Dastidar for helping us +with spectroscopic data reduction. +References +1 B. Kumar, A. Omar, G. Maheswar, et al., “3.6-m Devasthal Optical Telescope Project: Com- +pletion and first results,” Bulletin de la Societe Royale des Sciences de Liege 87, 29–41 +(2018). +2 R. Sagar, B. Kumar, and A. Omar, “The 3.6 metre devasthal optical telescope:from inception +to realization,” Current Science 117, 365 (2019). +3 N. Ninane, C. Flebus, and B. Kumar, “The 3,6 m Indo-Belgian Devasthal Optical Telescope: +general description,” in Ground-based and Airborne Telescopes IV, L. M. Stepp, R. Gilmozzi, +and H. J. Hall, Eds., Society of Photo-Optical Instrumentation Engineers (SPIE) Conference +Series 8444, 84441V (2012). +4 A. Omar, T. S. Kumar, B. Krishna Reddy, et al., “First-light images from low disper- +sion spectrograph-cum-imager on 3.6-meter Devasthal Optical Telescope,” arXiv e-prints , +arXiv:1902.05857 (2019). +5 T. S. Kumar, “Design and development of control unit and software for the ADFOSC instru- +ment of the 3.6 m Devasthal optical telescope,” in Ground-based and Airborne Instrumen- +29 + +tation for Astronomy VI, C. J. Evans, L. Simard, and H. Takami, Eds., 9908, 1515 – 1520, +International Society for Optics and Photonics, SPIE (2016). +6 G. Van Rossum and F. L. Drake Jr, Python reference manual, Centrum voor Wiskunde en +Informatica Amsterdam (1995). +7 Astropy Collaboration, T. P. Robitaille, E. J. Tollerud, et al., “Astropy: A community Python +package for astronomy,” aap 558, A33 (2013). +8 Astropy Collaboration, A. M. Price-Whelan, B. M. Sip˝ocz, et al., “The Astropy Project: +Building an Open-science Project and Status of the v2.0 Core Package,” aj 156, 123 (2018). +9 Astropy Collaboration, A. M. Price-Whelan, P. L. Lim, et al., “The Astropy Project: Sustain- +ing and Growing a Community-oriented Open-source Project and the Latest Major Release +(v5.0) of the Core Package,” apj 935, 167 (2022). +10 M. Craig, S. Crawford, M. Seifert, et al., “astropy/ccdproc: v1.3.0.post1,” (2017). +11 J. R. Janesick, Scientific charge-coupled devices (2001). +12 A. U. Landolt, “UBVRI Photometric Standard Stars in the Magnitude Range 11.5 ¡ V ¡ 16.0 +Around the Celestial Equator,” aj 104, 340 (1992). +13 P. Virtanen, R. Gommers, T. E. Oliphant, et al., “SciPy 1.0: Fundamental Algorithms for +Scientific Computing in Python,” Nature Methods 17, 261–272 (2020). +14 B. Ailawadhi, R. Dastidar, K. Misra, et al., “Photometric and spectroscopic analysis of the +Type II SN 2020jfo with a short plateau,” arXiv e-prints , arXiv:2211.02823 (2022). +15 Dimple, K. Misra, A. Ghosh, et al., “GRB 210217a: a short or a long GRB?,” Journal of +Astrophysics and Astronomy 43 (2022). +30 + +7 Biography +• Dimple is a PhD student at the ARIES, Nainital. Her research interest mainly focuses on +Gamma-Ray Bursts (GRBs) and their associated counterparts: the gravitational waves and +the supernovae. She uses multiwavelength data from gamma-ray to optical for her research +work. +• Dr. Tripurari S. Kumar completed his Ph. D. in Systems and Control Engineering from +IIT Bombay and serves at ARIES Nainital as head of engineering division primarily working +on telescope system engineering aspects. After initially spending two years in Tata Motors +Ltd., an automobile industry, he joined ARIES and spent more than eighteen years working +on various developmental activities for the ground-based telescopes. His work focuses on +development of mechatronics, optoelectronics, motion control and software for telescopes +and backend instruments, CCD camera systems, adaptive optics etc. He was lead engineer +for development of faint object spectrograph system for the 360-cm Devasthal optical tele- +scope and for assembly integration and verification team of the 360-cm telescope. Currently, +he is leading the system engineer aspects of ground-based telescope projects at ARIES which +includes upgradation of 360-cm and 130-cm telescopes, development of a new 50 cm space +situational awareness telescope jointly with ISRO, adaptive optics for the 360-cm telescope +etc. He is one of the members in the India TMT project from ARIES and leading motion +control aspects of the WFOS subsystems. +• Dr. Amitesh Omar is a scientist working at ARIES, India. His research interests focus on +how galaxy evolve in different environments in the Universe. He uses both optical and radio +telescopes for his research works. He also takes interests in ’in-country’ development of +31 + +complex back-end instruments for the optical telescopes. +• Dr. Kuntal Misra is a scientist working at ARIES, India. Her research interests are focused +on studying highly energetic transient astrophysical sources and understanding their progen- +itors. She is also interested in transient search programs using survey data mainly from the +4.0m International Liquid Mirror Telescope (ILMT) located in ARIES. +List of Figures +1 +The ADFOSC CCD camera setup in the laboratory. The Camera consists of a +CCD detector, a controller, a pressure gauge, a dewar which is cooled using a heat- +exchange cryogenic system. +2 +Master bias of the CCD created using median combining 50 bias frames. +3 +Histogram of the master bias frame with a mean value of 1134.01 ± 2.62 counts. +4 +A plot showing the variation of the standard deviation of 50 bias frames +5 +Non-linearity curves at different operating voltages in the lower count regime (left +panel) and in the higher count region (right panel). Non-linearity is the minimum +for a combination of VOD=29 volts and VRD=16.5 volts. +6 +Linearity curve at VOD = 29 volts and VRD = 16.5 volts. A linear fit to the data +gives the regression coefficient as 0.9999. The horizontal dashed line indicates the +saturation level. +7 +Photon transfer curve (PTC) of the CCD obtained in the laboratory environment. +The measured value of the gain is 1.00 ± 0.04 e−/ADU. +32 + +8 +Masterbias at −50◦C and deviation of the counts of all pixels from the zeroth pixel +counts. A clear gradient can be seen in the image as the last pixels get more time +to generate the dark signal. +9 +Variation of dark signal with temperature. The dark signal is negligible below +−80◦C. +10 +The response of CCD pixels at different temperatures is shown in the figure. The +deviation of the counts from the zeroth counts is fitted with a polynomial (black +line). The dark signal is calculated from the best fit. There are a few pixels which +are behaving differently at higher temperatures. At lower temperatures, the CCD +behaves as grade-0 CCD. +11 +Variation within the mean counts of bias frames during different nights. The bias +level is stable within one count. The red and black dotted lines show the mean bias +level estimated in the lab and on-sky. +12 +CCD image of the Landolt standard field SA110. The standard stars are between +10 to 16 mag in V-band. +13 +Variation of mean counts with exposure time from on-sky experiments. The black +line represents the best linear fit with a regression coefficient of 0.9997. +14 +Photon transfer curve (PTC) of the CCD as obtained from the sky experiments. +The gain value is estimated as 1.04 ± 0.13 e−/ADU. +15 +Field of GRB 210217A afterglow imaged with the ADFOSC in r-band. +33 + +16 +The lamp spectrum of Mercury Argon lamp using 770R grating. The left panel +shows the spectrum in pixel scale. The vertical lines indicate the column numbers +for the first and last emission lines identified in the spectrum in the left figure. The +right panel shows the spectrum in wavelength scale. +17 +Spectrum of SN 2020jfo obtained using ADFOSC at ∼ 254 days after the dis- +covery of supernova. We identified different absorption lines in the spectrum and +compared these with the spectra taken from other instruments/telescopes. +List of Tables +1 +CCD Characteristics +2 +Dark current at different temperatures as estimated using the gradient of bias frames. +34 + diff --git a/fNFAT4oBgHgl3EQf7x4q/content/tmp_files/load_file.txt b/fNFAT4oBgHgl3EQf7x4q/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ad83394c3b8b3ae48c135f8a25a2338d5a05d9cd --- /dev/null +++ b/fNFAT4oBgHgl3EQf7x4q/content/tmp_files/load_file.txt @@ -0,0 +1,717 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf,len=716 +page_content='Characterization of a deep-depletion 4K x 4K CCD Detector System designed for ADFOSC Dimple* 1, 2, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Kumar1, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Omar1, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Misra1 1Aryabhatta Research Institute of observational sciences, Manora Peak, Nainital 263001, India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' 2Department of Physics, Deen Dayal Upadhyaya Gorakhpur University, Gorakhpur-273009, India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' We present the characterization of the CCD system developed for the ADFOSC instrument on the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='6m Devasthal Optical Telescope (DOT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' We describe various experiments performed to tune the CCD controller param- eters to obtain optimum performance in single and four-port readout modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Different methodologies employed for characterizing the performance parameters of the CCD, including bias stability, noise, defects, linearity, and gain, are described here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' The CCD has grade-0 characteristics at temperatures close to its nominal operating temperature of −120◦C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' The overall system is linear with a regression coefficient of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='9999, readout noise of 6 electrons, and a gain value close to unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' We demonstrate a method to calculate the dark signal using the gradient in the bias frames at lower temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Using the optimized setting, we verify the performance of the CCD detector system on-sky using the ADFOSC instrument mounted on the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='6m DOT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Some science targets were observed to evaluate the detector’s performance in both imaging and spectroscopic modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Keywords: characterization, ADFOSC, CCD .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Dimple, dimplepanchal96@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='com 1 Introduction The 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='6m DOT was commissioned at the Devasthal observatory of Aryabhatta Research Institute of observational sciencES (ARIES), Nainital (India).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='1 The Devasthal Observatory is situated in the Himalayan regions of Uttarakhand at ∼ 2450 meter above the mean sea level with geographical co- ordinates of 29◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='360 N, 79◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='690 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' This location lies in the middle of the 180◦-wide longitude-gap between the Canary Islands (∼ 20◦ W) and Eastern Australia (∼ 160◦ E), making it suitable for observations of time-critical astronomical events due to the availability of several moderate aper- ture telescopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' The DOT uses a f/9 Ritchey-Chr ′etien (RC) system supported on an alt-azimuth mount.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='2,3 The aperture of this telescope is appropriate for medium-resolution spectroscopy and observations of faint sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' A low-dispersion spectrograph-cum-imager, ARIES Devasthal Faint Object Spectrograph (ADFOSC), has been developed in ARIES for spectroscopy and imaging of the celestial objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='4 The spectrograph uses a fixed focal reducer, which converts the incoming 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='08746v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='IM] 20 Jan 2023 f/9 optical beam from the telescope into a faster ∼ f/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='2 beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' The spectrograph can be used in both spectroscopic and imaging modes by selecting the instrument’s corresponding optical el- ements (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' filters, grism, slit, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=') with the help of a GUI-based instrument control software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='5 In either mode, a charge-coupled device (CCD) is required to detect and record the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' A CCD detector system/imager has been designed and assembled in ARIES in technical collaboration with the Herzberg Institute of Astrophysics (HIA), Canada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' We performed a detailed characterization of the CCD system before commissioning it for sci- entific observations, both in the laboratory and on the sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' This included estimating parameters like bias level, readout noise, and thermal noise in the dark room.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' We then performed iterative experiments in the laboratory to optimize the overall system performance and verified the CCD for cosmetic defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' We demonstrate a method to calculate the dark signal of the CCD at different temperatures using the bias frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' As the CCD is developed for the ADFOSC instrument, we also estimated the spectral dispersion on the detector using the lamp spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' After optimization in the laboratory environment, we performed similar experiments over the night sky on the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='6m DOT to verify the on-sky performance of the detector system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' The paper discusses the different methodologies employed for characterizing the performance of the CCD system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' The test setup used for performing different tests to optimize the system parameters is detailed in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' We also discuss various experiments performed to determine and optimize the CCD characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' The integration of the CCD system with the ADFOSC instrument and results of the on-sky tests are discussed in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' To evaluate the performance of the CCD system on science targets, we observed transient sources during the observing cycle 2020C2 of the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='6m DOT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' The results of the scientific observations are presented in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' 2 Fig 1 The ADFOSC CCD camera setup in the laboratory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' The Camera consists of a CCD detector, a controller, a pressure gauge, a dewar which is cooled using a heat-exchange cryogenic system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' 3 Table 1 CCD Characteristics CCD Characteristic Value CCD e2v 231 − 84 / Grade-0 Pixel Size 15 µm Readout Frequency 160 kHz Operating Temperature −120◦C Bias level 1134 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='62 ADU Gain 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='00 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='04 e−/ADU Readout Noise 6 e−/pixel Full well capacity 408 ke−/pixel Saturation level 65535 ADU 2 The CCD detector system The CCD is a 4096×4096 format back-illuminated e2v 231-84 CCD sensor having square pixels of 15-micron size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' It is a deep-depleted sensor capable of enhancing the sensitivity toward the longer wavelengths (∼ 700 − 1000 nm ) of the optical spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' The CCD has an imaging area of 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='4 × 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='4 mm2, providing a Field of View (FoV) of ∼ 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='6 × 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='6 arcmin2 and a spectral dispersion in the range 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='2 nm/pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' The CCD has four readout ports (0, 1, 2, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' The ∼ 16 million pixels can be read from any of the four amplifiers or through four amplifiers simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' The four-port readout decreases the readout time by a factor of four.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' However, it requires additional processing to match the bias levels of the four quadrants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Since the readout noise would differ for the four different amplifiers, each quadrant’s respective signal-to-noise ratio (SNR) would also be different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' In the case of the ADFOSC instrument, we have implemented a single port readout via port-0, which provides the lowest readout noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' The readout frequency is fixed at ∼ 160 kHz, providing a readout time of ∼ 104 sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' A Bonn shutter is mounted at the camera entrance with an aperture size of 100 mm ×100 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' The shutter employs servo motors for fast operation and offers 4 uniform exposure at the detector plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Using this shutter, a minimum exposure time of ∼ 1 ms is possible with an uncertainty of 300 µs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' The detector system consists of the CCD detector, a clock- shaping fan-out circuit board, and a generic Astronomical Research Cameras (ARC) controller1 to generate the suitable clock and bias voltages for the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' The controller hardware has two video cards for reading four ports with 16-bit Analog to digital converter (ADC) units to interface and digitize the four channels of the CCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Additionally, different bin settings are implemented to read the image in different binning patterns for photometry and spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' The CCD sensor is cooled to −120◦C to minimize the dark signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' The CCD dewar is evacu- ated for several hours using an oil-free turbo molecular vacuum pump before deep cooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' The dewar is equipped with a Pirani vacuum gauge for monitoring its vacuum pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Once the vac- uum reaches below ∼ 1 x 10−3 Torr, the closed-cycle (Joule-Thomson) cryogenic heat-exchange system supplied by Brooks Automation, USA, is switched on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' The overall process of cryo-cooling the CCD from an ambient temperature of 25◦C to −120◦C takes between 4 to 5 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' This tem- perature is stabilized and held constant within 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='01◦C using a Lakeshore 335 Proportional Integral Derivative (PID) temperature controller2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' For this purpose, a small heater and a temperature sen- sor are mounted on the cold plate below the sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' A charcoal-filled getter is used to absorb outgassing inside the dewar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' The charcoal gets activated to absorb gases at cryogenic temperatures and helps attain a high vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' An ultimate vacuum of ∼ 3 x 10−7 Torr is usually attained with this system at −120◦C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' 1http://astro-cam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='com 2http://irtfweb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='ifa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='hawaii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='edu/˜s2/software/gpib-eth-ls335/335_Manual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='pdf 5 3 Characterization of the CCD system Detailed characterization of the CCD includes the estimation of bias level, bias stability, readout noise (RN), gain, defects, linearity, and saturation level of the CCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' This section describes the laboratory-based test setup and the experiments performed to determine these parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' We tested the CCD performance using all four ports, numbered zero to three.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' However, the read noise is found to be the lowest for port-0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' hence single port readout mode using port-0 is currently being used for acquiring the scientific data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' The paper focuses on characterizing parameters for port-0 of the CCD system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='1 Test setup and Data Acquisition We set up the CCD system on an electrostatic discharge (ESD) safe, dark room of the ARIES optics laboratory for performing the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' A light-emitting diode (LED) operated at a constant regulated voltage was used as an illumination source for the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' We covered the CCD window with an Aluminium plate with a pinhole for the light to enter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' We fixed the source on the pinhole to avoid any fluctuations in light intensity due to any change in the source’s position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' The sub-systems, namely, the temperature controller, cryogenic pump, pressure gauge, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=', were carefully grounded to a common point to avoid noise entering from ground loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' The entire system was again reconfigured when the ADFOSC was mounted on the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='6m DOT for sky tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' We acquired the data using the Owl3 software provided along with the ARC controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' The software offers different dialog boxes to control the controller parameters, including gain, readout, binning, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=', and saves the acquired images in standard Flexible Image Transport System (FITS) format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Different modules of Python6 like Astropy7–9 and ccdproc10 were used to process 3http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='astro-cam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='com/Gen3Software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='php 6 Fig 2 Master bias of the CCD created using median combining 50 bias frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' the FITS image files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='2 Bias level and readout noise A positive offset is generally provided to the CCD electronics to avoid negative counts in the output of the CCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' The mean offset value, or the bias value, is optimized in a way that it is large enough to avoid the non-linear regime of the CCD amplifiers but not too large to reduce the dynamic range 7 1125 1130 1135 1140 1145 Counts (ADU) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='16 Pixel Density Fig 3 Histogram of the master bias frame with a mean value of 1134.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='01 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='62 counts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' of the CCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' We set the bias level slightly above thousand counts to balance the above factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' We estimated the bias level of the CCD using the bias frames, which are the CCD images with a zero-second exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Several bias frames were acquired, and we used fifty of them to generate a master bias frame by taking their median value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' We did this using Mediancombine task of ccdproc software module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' 2 shows the median bias frame, and the corresponding histogram is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' The width of the distribution represents the RN of the CCD, which is the number of electrons introduced by the readout electronics while reading out each pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' We estimated the RN using the Janesick method (equation 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='11 We created a difference image using 8 two bias images and estimated its standard deviation (σBias1−Bias2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' The explanation of the Gain value used in equation 1 is provided in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' We found the RN value to be 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='20 analog to digital units (ADU) or 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='20 e−1 for the gain value of ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' However, the achieved noise is more than twice the value in the e2v datasheet at 160kHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' We observed some interference patterns in the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' These patterns are likely to be responsible for this increased noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' The probable cause could be ground loops outside and inside the dewar, length of cables and imperfect shielding scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' These have been controlled to a large extent by iteratively evaluating various schemes like shortening the cables and star connecting the ground points of the auxiliary devices like chiller, pressure gauge, temperature controller etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Also, grounding permutations were tried with the four- port video cables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Finally, the shielding of the video cables was grounded only at the controller end and left open at the CCD connector end, which resulted in a lower noise floor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' There is still scope for improvement as the ground loops inside the dewar have not been evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' This evaluation will be attempted later since the CCD has already been commissioned for observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' RN = Gain × σBias1−Bias2 √ 2 , (1) We calculated the standard deviation of 50 bias frames to verify this RN value, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' The mean of these standard deviation values is 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='38 ADU, which is consistent with the value calculated using equation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' 9 0 10 20 30 40 50 Frame Number 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='5 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='0 Standard Deviation (ADU) Mean = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='3 ADU Standard devation Fig 4 A plot showing the variation of the standard deviation of 50 bias frames 10 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='3 Linearity The CCD system should have a linear response to the incident light for scientific observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' However, several factors can introduce non-linearity in CCD performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' The controller clock periods and overlaps should be timed for complete charge transfer during the readout process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Moreover, there should be a delay between this transfer and the correlated double sampling instant to allow the transients to settle to avoid any induced noise or glitch and introduce non-linearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' We verified the signal waveform using a 1 GHz digital oscilloscope to ensure the above before connecting the interface board to the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Other factors critical for linearity are bias voltages: voltage output drain (VOD) and voltage reset drain (VRD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' The CCD manufacturer provided a range of values for the voltages, VOD from 25 to 31 volts and VRD from 16 to 19 volts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' To check the behaviour of the CCD at different voltages, we initially set these voltages near minimum values and iteratively increased these voltages within this range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' We rejected some of the voltage combi- nations that provided very low-bias levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' For other combinations, experiments were performed to check the linearity of the CCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' We used an LED source (section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='1) to illuminate the CCD and acquired images with an incremental increase in exposure time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' We obtained a pair of images for each exposure time to detect any variation in the source intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' We noticed that the counts are identical for the pair of images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' We estimated the non-linearity (the relative difference between our measurements and the best- fit linear curves) by fitting a linear function to the variation of mean counts with exposure time for each voltage combination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' We used the linregress function from the stats library under Python for this purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' 5 shows the non-linearity curves for various combinations of VOD 11 2000 4000 6000 8000 10000 12000 Counts (ADU) −10 0 10 20 30 Non − linearity % .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' 10000 20000 30000 40000 50000 60000 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='0 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='5 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='5 VOD = 28, VRD = 17 VOD = 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='5, VRD = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='5 VOD = 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='5, VRD = 17 VOD = 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='5, VRD = 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='5 VOD = 29, VRD = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='5 VOD = 29, VRD = 17 VOD = 29, VRD = 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='5 VOD = 29, VRD = 18 Fig 5 Non-linearity curves at different operating voltages in the lower count regime (left panel) and in the higher count region (right panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Non-linearity is the minimum for a combination of VOD=29 volts and VRD=16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='5 volts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' and VRD in different count regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' For most voltage combinations, the non-linearity is negligible in the higher count regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' The non-linearity, however, shows up in the lower count regime and is significant for certain voltages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' For a combination of VOD = 29 volts and VRD = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='5 volts, the non-linearity is the lowest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' For this voltage combination, the value of the regression coefficient (R2) is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='9999, which is almost equal to unity (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' We considered this voltage combination as the optimum value for the CCD system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='4 Saturation level The maximum capacity of a CCD pixel to store the photo-electrons is its full well capacity, beyond which the pixels saturate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Since the available 16-bit ADC of the controller saturates at a value of 65535, the controller’s gain setting helps to select the dynamic range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' As we are interested in accurate photometry of faint objects, a gain of unity is selected, constraining the saturation point to 65535.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' The selection of system gain is discussed in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Illumination of the CCD until its saturation point limited the counts to 65535, the saturation point of the 16-bit ADC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' It is 12 0 2 4 6 8 10 12 Exposure Time (sec) 0 10000 20000 30000 40000 50000 60000 70000 Mean Counts (ADU) R2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='9999 Best fit Saturation level Lab data Fig 6 Linearity curve at VOD = 29 volts and VRD = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='5 volts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' A linear fit to the data gives the regression coefficient as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='9999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' The horizontal dashed line indicates the saturation level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' 13 demonstrated and shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' 6 where a bright source illuminates the detector, and the ADC saturates at 65535 counts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' If the science cases demand the utilization of full well capacity, the user can select a gain setting close to 3 or higher electrons per ADU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='5 The system gain The gain of a CCD system is defined in terms of ADU, which corresponds to the number of electrons assigned to one digital unit in the output image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' The available gain values are 1, 2, 5, and 10 e−/ADU in the controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' The gain values can be selected using the software at runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' The saturation level of the CCD should correspond to the saturation level of the ADC to utilize the full well capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Since the full well capacity of the CCD is 408 ke− (as mentioned in the result sheet of the supplied detector), a gain of 10 is suitable to match the saturation levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' However, for the detection of photon-limited faint objects, a gain of 1 is implemented using the controller parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' The electronic gain of the CCD system is the product of gain values introduced by each stage of the readout electronics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' The inherent gain of the CCD, defined by the output capacitor, is 7 µ V/e−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' A series of Op-amp stages within the controller further amplify this gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Initially, it is preamplified with a gain of 4 and passed through a gain selection stage, offering a range of gain values: 1, 2, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='75, and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' A bias adjustment stage after the integrator provides a gain of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Hence, an amplification of 56 µ V/e− is obtained with these four stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Since the 16-bit ADC operating at a reference voltage of 10 V provides a bin size of 152.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='588 µ V/ADU, the integration of the Op-amp integrator is adjusted to provide an additional gain factor of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='725 to achieve the desired system gain of 1 e−/ADU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Since the integrator time can only be adjusted in increments of 40 ns, the closest possible value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='998 e−/ADU is set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' 14 0 10000 20000 30000 Mean Counts (ADU) 102 103 104 Variance (ADU2) Gain = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='00 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='04 Lab data Fig 7 Photon transfer curve (PTC) of the CCD obtained in the laboratory environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' The measured value of the gain is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='00 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='04 e−/ADU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' We experimentally verified this gain setting using the Janesick method11 given by equation 2 where (S) is the mean of the signal acquired by the CCD, and σ2 S is the variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' σ2 S = S G + σ2 R, (2) We acquired a pair of images at each exposure and estimated the mean signal after bias subtrac- tion and cosmic-ray removal from the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Further, these images were normalized by subtracting 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='0 Pixel Number ×106 0 200 400 600 800 1000 1200 1400 Deviation From Zeroth Pixel Counts (ADU) −50◦C Fig 8 Masterbias at −50◦C and deviation of the counts of all pixels from the zeroth pixel counts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' A clear gradient can be seen in the image as the last pixels get more time to generate the dark signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' one image from the other for compensating the flat-field effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' We used the resulting image to es- timate the variance of the signal (σ2 S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' 7 shows the photon transfer curve (PTC) for the CCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' To estimate the gain, the PTC was fitted with a linear function using linregress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' The estimated gain is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='00 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='04 e−/ADU, which matches the electronic gain value of the system within the errorbar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='6 Thermal noise In the CCD detectors, electron charge-density increases exponentially with an increase in temper- ature due to the thermal generation of electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' A CCD must be cooled optimally to minimize the dark signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' To determine this optimum temperature, we calculated the dark signal at various tem- peratures ranging from −35◦C to −120◦C using the bias frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' We acquired several bias frames at different temperatures and generated master bias frames at each temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' The left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' 8 shows the master bias frame at −50◦C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' A gradient in counts is visible in the master bias 16 due to the finite readout time of the CCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' The zeroth pixel gets the lowest time to generate dark electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' The last pixel accumulates dark counts over full readout time, hence has the maximum number of thermally generated electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' If the readout time and the gain are known, then by com- paring the counts of the first and the last pixel, we can measure the number of electrons generated per pixel per second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Using this method, we estimated the dark signal at different temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Temp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' (◦C) Dark Current (e-/pix/sec) Temp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' (◦C) Dark Current (e-/pix/sec) 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='0 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='89 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='017 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='03 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='002 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='0 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='30 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='009 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='42 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='002 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='36 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='007 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='18 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='002 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='37 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='003 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='10 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='001 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='89 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='002 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='05 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='001 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='87 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='002 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='01 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='002 Table 2 Dark current at different temperatures as estimated using the gradient of bias frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' The right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' 8 shows the deviation of counts in each pixel from the zeroth counts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' The farther the pixel number is from the readout port, the larger the dark count and the larger the deviation from the zeroth counts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' To determine the slope of this gradient, we fitted a polynomial in counts vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' pixel number data using the polyfit function of Python.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' It is seen that a linear function provides the best fit, as shown in the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' We used the slope to calculate the difference in counts between the first and the last pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' We divided this difference by the total readout time to obtain dark counts generated per second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Since the bias frames were acquired in 4 × 4 binning, it was further scaled by a factor of 16 after subtracting the RN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Below −100◦C temperature, the thermal noise becomes less than RN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' hence, we could estimate the dark signal values up to −95◦C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' The dark signal values at different temperatures are listed in table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' 9, the dark signal varies exponentially with temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Below -80°C, the 17 −90 −80 −70 −60 −50 −40 Temperature (◦C) 0 5 10 15 20 25 30 35 40 Dark Signal (e−/pix/sec) Fig 9 Variation of dark signal with temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' The dark signal is negligible below −80◦C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' dark signal is negligible, suggesting that the CCD can be used below this temperature with minimal thermal noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='7 CCD defects The CCD may have some pixels that might not respond to light optimally due to defects in the CCD structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' These can be point defects, hot defects, or dark/dead pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' We are employing a grade-0 CCD detector (the CCD detector with minimum possible defects) as mentioned by the manufacturer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' To check the CCD for point and column defects, we examined the response of all the pixels at different temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' 10 shows the deviation of counts from the mean bias counts for each pixel of the CCD operated at different temperatures (the method to obtain these plots is described in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Some pixels are seen to behave differently at higher temperatures 18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='0 ×106 0 5000 10000 15000 20000 −35◦C 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='0 ×106 0 500 1000 1500 −50◦C 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='0 ×106 0 100 200 −60◦C 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='0 ×106 −50 0 50 100 150 −70◦C 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='0 ×106 0 50 100 150 −80◦C 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='0 ×106 0 50 100 150 −90◦C 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='0 ×106 0 50 100 −100◦C 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='0 ×106 −40 −20 0 20 40 −110◦C 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='0 ×106 −10 0 10 20 −120◦C Pixel Number Deviation From Zeroth Pixel Counts (ADU) Fig 10 The response of CCD pixels at different temperatures is shown in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' The deviation of the counts from the zeroth counts is fitted with a polynomial (black line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' The dark signal is calculated from the best fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' There are a few pixels which are behaving differently at higher temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' At lower temperatures, the CCD behaves as grade-0 CCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' 19 exhibiting high counts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Though they appear to be hot pixels, the counts are found to decrease with decreasing temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Eventually, below −110◦C, the CCD acts as a nominal grade-0 CCD without any point and column defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' 4 On-sky verification After optimizing the performance of the CCD system in the laboratory, we verified the on-sky performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' The CCD was integrated with the ADFOSC instrument and mounted on the axial port of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='6m DOT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' This section describes the estimation of gain, linearity, bias level, and bias stability using on sky observation with the instrument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='1 Bias Stability We calculated the bias level using the methodology described in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' The mean bias level equal to 1133.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='85±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='48 matches the laboratory estimated value, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' 1134.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='01±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Since fluctua- tion in bias level can introduce errors in photometric estimates, we acquired and examined several bias frames to check the stability of the bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' 4 shows the variation of the mean bias level for 30 different nights spread across an observing cycle of three months.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' The mean bias level fluctuates within a fraction of a count, ensuring bias stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='2 Linearity and gain We chose the standard field available at the zenith at the time of observations to validate the lin- earity and gain of the CCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Multiple images of Landolt standard field SA11012 were acquired in r-band with an exposure time ranging from 5 sec to 100 sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Before using the images for character- ization purposes, we pre-processed the images with basic steps of bias subtraction, flat correction and cosmic-ray removal using the ccdproc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' 12 shows the pre-processed CCD image of the 20 0 5 10 15 20 25 30 Frame Number 1133.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='00 1133.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='25 1133.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='50 1133.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='75 1134.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='00 1134.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='25 1134.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='50 1134.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='75 1135.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='00 Mean Bias Counts (ADU) Mean Bias Level (Sky) Mean Bias Level (Lab) Fig 11 Variation within the mean counts of bias frames during different nights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' The bias level is stable within one count.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' The red and black dotted lines show the mean bias level estimated in the lab and on-sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' 21 Fig 12 CCD image of the Landolt standard field SA110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' The standard stars are between 10 to 16 mag in V-band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' 22 0 20 40 60 80 100 Exposure Time (sec) 0 5000 10000 15000 20000 25000 30000 35000 Mean Counts (ADU) R2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='9997 On Sky Fig 13 Variation of mean counts with exposure time from on-sky experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' The black line represents the best linear fit with a regression coefficient of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='9997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' field SA110, which contains both bright and faint stars (with magnitudes ranging from 10 to 16 mag in the V-band).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' We used the faint stars to check the linearity in the lower count region and the bright stars to estimate the saturation level of the CCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' 13 shows CCD linearity with R2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='9997 and a non-linearity percentage of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' The CCD system is seen to saturate at 65535 counts for a gain of 1 e−/ADU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' The estimated gain value using the method described in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='5 is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='04 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='13 e−/ADU, which is close to the value estimated in the laboratory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' On-sky gain estimation is also affected by 23 0 5000 10000 15000 Mean Counts (ADU) 103 104 Variance (ADU2) Gain = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='04 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='13 On Sky Fig 14 Photon transfer curve (PTC) of the CCD as obtained from the sky experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' The gain value is estimated as 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='04 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='13 e−/ADU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' 24 Fig 15 Field of GRB 210217A afterglow imaged with the ADFOSC in r-band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' the sky variation, which results in a slightly higher error bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' The mean-variance plot is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' 5 Performance of the CCD We used the CCD for imaging and spectroscopic observations of various science targets after opti- mizing it in the lab and successfully verifying it on-sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' This section demonstrates the performance of the CCD system in both imaging and spectroscopic modes with observations of GRB and Su- pernovae sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' The on-sky performance of ADFOSC on different science targets is discussed in more detail in Omar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='4 25 GRB210217A(DOT)0 500 1000 1500 2000 2500 3000 Column Number 1000 2000 3000 4000 5000 6000 7000 8000 Counts 5000 6000 7000 8000 9000 10000 Wavelength (˚A) 1000 2000 3000 4000 5000 6000 7000 8000 Counts 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='0 Residual (˚A) Fig 16 The lamp spectrum of Mercury Argon lamp using 770R grating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' The left panel shows the spectrum in pixel scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' The vertical lines indicate the column numbers for the first and last emission lines identified in the spectrum in the left figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' The right panel shows the spectrum in wavelength scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='1 Imaging We observed the optical afterglow of GRB 210217A using the ADFOSC in imaging mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' These observations were performed on 18th February 2021 in the r-band at 23:59:18 UT, at ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='7 days after the burst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Owing to the faintness and rapid decay rate of GRB afterglows, a series of eight images, each with an exposure time of 300 seconds, were acquired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' The images were stacked after pre-processing (as described in the previous section) to improve the signal-to-noise ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' The optical afterglow is visible in the stacked image as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' The photometric estimate of the afterglow is 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='32 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='16 mag (AB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='2 Spectroscopy The spectrograph provides three sets of gratings: 300 gr/mm, 420 gr/mm, and 600 gr/mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' We ac- quired the lamp spectra using the Mercury Argon (HgAr) calibration lamp to estimate the spectral dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' We used the find peaks module of scipy13 to extract the spectral peaks from the obtained spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' We identified the column number corresponding to each peak and compared it with the wavelength-calibrated lamp spectrum atlas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' We defined initial polynomial solutions using 26 4000 5000 6000 7000 8000 9000 10000 11000 Rest Wavelength (Å) Normalised flux+const 365.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='2 days 253.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='9 days 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='15 days CaII OI FeII Na I H CoII CaII CaII OI ScII FeII FeII Shane, D=3 m DOT, D=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='6 m LCO, D=2 m Fig 17 Spectrum of SN 2020jfo obtained using ADFOSC at ∼ 254 days after the discovery of supernova.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' We identified different absorption lines in the spectrum and compared these with the spectra taken from other instruments/telescopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' these matched wavelength pairs and calculated the best-fit polynomial coefficients to transform between the column number and wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' The left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' 16 shows the lamp spectrum obtained using a 300gr/nm grating element in pixel scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' The right panel shows the calibrated spectrum in the wavelength scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' A spec- tral dispersion of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='20 nm/pixel is estimated for this grating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' For gratings elements, 420 gr/mm and 600 gr/mm, the estimated values of spectral dispersion are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='14 nm/pixel and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='10 nm/pixel, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' We acquired the spectrum of the supernova SN 2020jfo using 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='5 ′′ slit and 420 gr/mm grating with an exposure time of 900 sec on 13th January 2021 at 23:12:53 UT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Ailawadhi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='14 describe 27 the spectral reduction technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' The absorption features in the spectrum obtained by ADFOSC were identified and matched with the spectra obtained from other telescopes, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' It is noticed that spectral features at longer wavelengths are visible, indicating the high sensitivity of the CCD detector (deep-depleted) in the long-wavelength regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' 6 Conclusions We present the methodology employed to characterize the performance of a CCD system devel- oped for integrating with a low dispersion spectrograph instrument, ADFOSC, on the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='6m DOT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Various experiments were initially performed in the laboratory to characterize and optimize differ- ent critical parameters of the CCD system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' We also verified the estimated parameters on the sky by mounting the instrument on the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='6m DOT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' We evaluated the bias level during the on-sky tests and examined its stability over several observing nights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' We experimentally identified an optimum combination of bias voltages: VOD and VRD, to operate the CCD with minimum non-linearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' The readout performance of the CCD is satisfactory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' However, some interference patterns in the image contribute to readout noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Through different experiments, we tuned and verified the gain parameter corresponding to 1 e−/ADU for detecting faint objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' We calculated the dark cur- rent at different temperatures using bias frames at lower temperatures and established an optimum operating temperature of the CCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' The CCD acts as a grade-0 detector with no hot pixels at op- timum temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' The regression coefficient values and the gain parameter obtained on-sky are consistent with the values obtained in the laboratory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' After verifying the satisfactory performance, we observed the science targets both in imaging and spectroscopic modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' We carried out the imaging of GRB 210217A15 field and the spectroscopy of supernova SN 2020jfo using ADFOSC successfully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='14 28 Acknowledgments We thank Greg Burley and Tim Hardy from NRC Herzberg Astronomy and Astrophysics Research Centre, Canada, for their help in developing the CCD system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' We thank the ARIES 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='6 m DOT engineering team and staff members for providing the necessary support during development, ver- ification, and commissioning work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' We would also like to thank Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Raya Dastidar for helping us with spectroscopic data reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' References 1 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Kumar, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Omar, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Maheswar, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=', “3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='6-m Devasthal Optical Telescope Project: Com- pletion and first results,” Bulletin de la Societe Royale des Sciences de Liege 87, 29–41 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' 2 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Sagar, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Kumar, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Omar, “The 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='6 metre devasthal optical telescope:from inception to realization,” Current Science 117, 365 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' 3 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Ninane, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Flebus, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Kumar, “The 3,6 m Indo-Belgian Devasthal Optical Telescope: general description,” in Ground-based and Airborne Telescopes IV, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Stepp, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Gilmozzi, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Hall, Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=', Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series 8444, 84441V (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' 4 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Omar, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Kumar, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Krishna Reddy, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=', “First-light images from low disper- sion spectrograph-cum-imager on 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='6-meter Devasthal Optical Telescope,” arXiv e-prints , arXiv:1902.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='05857 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' 5 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Kumar, “Design and development of control unit and software for the ADFOSC instru- ment of the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='6 m Devasthal optical telescope,” in Ground-based and Airborne Instrumen- 29 tation for Astronomy VI, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Evans, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Simard, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Takami, Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=', 9908, 1515 – 1520, International Society for Optics and Photonics, SPIE (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' 6 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Van Rossum and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Drake Jr, Python reference manual, Centrum voor Wiskunde en Informatica Amsterdam (1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' 7 Astropy Collaboration, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Robitaille, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Tollerud, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=', “Astropy: A community Python package for astronomy,” aap 558, A33 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' 8 Astropy Collaboration, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Price-Whelan, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Sip˝ocz, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=', “The Astropy Project: Building an Open-science Project and Status of the v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='0 Core Package,” aj 156, 123 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' 9 Astropy Collaboration, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Price-Whelan, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Lim, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=', “The Astropy Project: Sustain- ing and Growing a Community-oriented Open-source Project and the Latest Major Release (v5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='0) of the Core Package,” apj 935, 167 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' 10 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Craig, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Crawford, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Seifert, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=', “astropy/ccdproc: v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='post1,” (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' 11 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Janesick, Scientific charge-coupled devices (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' 12 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Landolt, “UBVRI Photometric Standard Stars in the Magnitude Range 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='5 ¡ V ¡ 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='0 Around the Celestial Equator,” aj 104, 340 (1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' 13 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Virtanen, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Gommers, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Oliphant, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=', “SciPy 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='0: Fundamental Algorithms for Scientific Computing in Python,” Nature Methods 17, 261–272 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' 14 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Ailawadhi, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Dastidar, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Misra, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=', “Photometric and spectroscopic analysis of the Type II SN 2020jfo with a short plateau,” arXiv e-prints , arXiv:2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='02823 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' 15 Dimple, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Misra, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Ghosh, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=', “GRB 210217a: a short or a long GRB?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=',” Journal of Astrophysics and Astronomy 43 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' 30 7 Biography Dimple is a PhD student at the ARIES, Nainital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Her research interest mainly focuses on Gamma-Ray Bursts (GRBs) and their associated counterparts: the gravitational waves and the supernovae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' She uses multiwavelength data from gamma-ray to optical for her research work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Tripurari S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Kumar completed his Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' in Systems and Control Engineering from IIT Bombay and serves at ARIES Nainital as head of engineering division primarily working on telescope system engineering aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' After initially spending two years in Tata Motors Ltd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=', an automobile industry, he joined ARIES and spent more than eighteen years working on various developmental activities for the ground-based telescopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' His work focuses on development of mechatronics, optoelectronics, motion control and software for telescopes and backend instruments, CCD camera systems, adaptive optics etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' He was lead engineer for development of faint object spectrograph system for the 360-cm Devasthal optical tele- scope and for assembly integration and verification team of the 360-cm telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Currently, he is leading the system engineer aspects of ground-based telescope projects at ARIES which includes upgradation of 360-cm and 130-cm telescopes, development of a new 50 cm space situational awareness telescope jointly with ISRO, adaptive optics for the 360-cm telescope etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' He is one of the members in the India TMT project from ARIES and leading motion control aspects of the WFOS subsystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Amitesh Omar is a scientist working at ARIES, India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' His research interests focus on how galaxy evolve in different environments in the Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' He uses both optical and radio telescopes for his research works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' He also takes interests in ’in-country’ development of 31 complex back-end instruments for the optical telescopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Kuntal Misra is a scientist working at ARIES, India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Her research interests are focused on studying highly energetic transient astrophysical sources and understanding their progen- itors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' She is also interested in transient search programs using survey data mainly from the 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='0m International Liquid Mirror Telescope (ILMT) located in ARIES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' List of Figures 1 The ADFOSC CCD camera setup in the laboratory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' The Camera consists of a CCD detector, a controller, a pressure gauge, a dewar which is cooled using a heat- exchange cryogenic system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' 2 Master bias of the CCD created using median combining 50 bias frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' 3 Histogram of the master bias frame with a mean value of 1134.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='01 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='62 counts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' 4 A plot showing the variation of the standard deviation of 50 bias frames 5 Non-linearity curves at different operating voltages in the lower count regime (left panel) and in the higher count region (right panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' Non-linearity is the minimum for a combination of VOD=29 volts and VRD=16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='5 volts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' 6 Linearity curve at VOD = 29 volts and VRD = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='5 volts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' A linear fit to the data gives the regression coefficient as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='9999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' The horizontal dashed line indicates the saturation level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' 7 Photon transfer curve (PTC) of the CCD obtained in the laboratory environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' The measured value of the gain is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='00 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='04 e−/ADU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' 32 8 Masterbias at −50◦C and deviation of the counts of all pixels from the zeroth pixel counts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' A clear gradient can be seen in the image as the last pixels get more time to generate the dark signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' 9 Variation of dark signal with temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' The dark signal is negligible below −80◦C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' 10 The response of CCD pixels at different temperatures is shown in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' The deviation of the counts from the zeroth counts is fitted with a polynomial (black line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' The dark signal is calculated from the best fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' There are a few pixels which are behaving differently at higher temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' At lower temperatures, the CCD behaves as grade-0 CCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' 11 Variation within the mean counts of bias frames during different nights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' The bias level is stable within one count.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' The red and black dotted lines show the mean bias level estimated in the lab and on-sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' 12 CCD image of the Landolt standard field SA110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' The standard stars are between 10 to 16 mag in V-band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' 13 Variation of mean counts with exposure time from on-sky experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' The black line represents the best linear fit with a regression coefficient of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='9997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' 14 Photon transfer curve (PTC) of the CCD as obtained from the sky experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' The gain value is estimated as 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='04 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content='13 e−/ADU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' 15 Field of GRB 210217A afterglow imaged with the ADFOSC in r-band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' 33 16 The lamp spectrum of Mercury Argon lamp using 770R grating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' The left panel shows the spectrum in pixel scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' The vertical lines indicate the column numbers for the first and last emission lines identified in the spectrum in the left figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' The right panel shows the spectrum in wavelength scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' 17 Spectrum of SN 2020jfo obtained using ADFOSC at ∼ 254 days after the dis- covery of supernova.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' We identified different absorption lines in the spectrum and compared these with the spectra taken from other instruments/telescopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNFAT4oBgHgl3EQf7x4q/content/2301.08746v1.pdf'} +page_content=' List of Tables 1 CCD Characteristics 2 Dark current at different temperatures as estimated using the gradient of bias frames.' metadata={'source': 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+DAMIANO MASSELLA1,*, MICHAEL WALLACE2, RONALD BROEKE2, AND FRANCISCO DIAZ1,3 +1University of Vigo, El Telecomunication - Campus Universitario As Lagoas, 36310 Vigo, Spain +2Bright Photonics BV, Eindhoven, Netherlands. +3AtlanTTic research center, El Telecomunication - Campus Universitario As Lagoas, 36310 Vigo, Spain +*Corresponding author: damiano.massella@uvigo.es +Compiled January 11, 2023 +Optical injection locking of integrated ring lasers +promises to be a key technology of the future, but it is +still lacking experimental verification. In this paper we +present two monolithically integrated photonic circuits +that experimentally realize Optically injection locking +of ring lasers. The presented circuits pave the way for +future application in Quantum Key Distribution and +coherent communication. +© 2023 Optica Publishing Group +http://dx.doi.org/10.1364/ao.XX.XXXXXX +1. INTRODUCTION +In recent years the discussion regarding communication security +has taken a central role as the importance of telecommunication +grows. Quantum Key Distribution (QKD) promises to solve the +problem of security in communication, an issue which is given +more significance with the advent of quantum computers[1]. +At the same time, due to ever increasing demands on data trans- +mission there has been growing interest in Optical Injection +Locking (OIL) of lasers[2]. Using this technique it is possible to +increase the modulation bandwidth of a laser up to 100GHz[3]. +The bandwidth of transmitters can be increased by multiplexing +a large number of channels however, this is not scalable using +bulk optics. Photonic Integrated Circuits (PICs), consist of a +large number of optical components integrated on a single chip, +are a key technology to overcome this limitation [4]. +QKD transmitters have been realized in both Indium Phosphide +and Silicon showing the advantage of integration in terms of +stability and miniaturization of single channels. The work of +Sibson et al [5] reports on the integration of a QKD transmitter +without the use of modulators and based on an OIL technique +first demonstrated in [6]. In this paper we report the first exper- +imental realization OIL of monolithically integrated ring laser +systems. An extensive study of this kind of system has been +reported before but no experimental realization has been show +yet [7–9]. +In this work we show two different circuits designed for two +different applications. The first circuit is consist of a Distributed +Feedback (DFB) Laser is OIL to a ring laser. This system is +intended for mainly QKD purposes. Coherent communication +is also possible however, the modulation scheme would be too +complex[10]. +Our second circuit is realized using 2 ring lasers that are feed +using a Distributed Bragg Reflector (DBR) laser. This circuit has +been designed to be a coherent communication transmitter and +thus be able to modulate both the ring lasers simultaneously. +With respect to the circuit presented by Paraiso et all [10], the +use of ring lasers instead of DFB lasers carries some significant +advantages. +Firstly, the ring lasers don’t need to be tuned like in the case of +single mode lasers. And secondly, when OIL a ring laser we +ensure the unidirectional emission from it. Consequently, there +are no reflections from the slave laser to the master laser, like in +the DFB case [3]. +2. CIRCUIT DESIGN +The first circuit is presented in figure 1. In this circuit we use a +simple DFB laser to inject a ring laser. The notation for the ring +lasers follows the one used for the mask design, consequently in +the ring laser of circuit one is labelled as ’ring 3’. The simplicity +of this circuit is its main advantage with respect to other QKD cir- +cuits. In the schematics of this section we have omitted the MMI +blocks and the photodiodes used to monitor the state of the laser. +Fig. 1. This schematic represent the first design of OIL ring +laser. The ring laser is injected using a DFB laser. +We used a standard DFB laser designed to emit at 1550nm, +arXiv:2301.03684v1 [physics.optics] 9 Jan 2023 + +SOALetter +Optics Letters +2 +the left output of it is terminated using a photodiode. This +photodiode will be used to characterize the DFB laser emitted +power. +On the ring we have used custom MMI blocks, they are +designed to split power unequally, 85% on the cross state and +15% on the bar. We have chosen this particular ratio to reduce +the losses inside the ring and have a shorter gain section. The +MMI used in the rings are all 2x2 and the additional port is +terminated using a photodiode. The photodiodes inside the ring +can be used to monitor both the clockwise directed power and +the counter-clockwise. The SOA section is 400µm long and it +has been connected using RF lines in order to be able to drive +the ring laser to higher frequencies and verify the modulation +bandwidth enhancement due to OIL. A spot size converter is +placed at the output of the circuit to minimize the losses when +coupling to fiber. +The second circuit is schematized in figure 2. +In this case we decided to exploit the two outputs of a DBR laser +to OIL two ring lasers. The output of the two rings are then +combined to a single output. This schematic can be used for +both QKD and Quadrature Phase Shift Keying (QPSK). +The main advantage with respect to the first circuit is the pos- +sibility to use the modulation of the rings for encoding the bits +of the QPSK scheme. We have added a phase modulator in the +output of one of the ring laser in order to have the π/2 phase +shift for QPSK. This additional phase modulation section is con- +stituted by an thermo-optical phase modulator of length 450µm. +This length is sufficient to ensure the π/2 phase difference be- +tween the two rings. The modulation bandwidth is enhanced by +the OIL since we are going to modulate the two slave lasers and +not the master laser. +The ring lasers used for the two circuits are identical in dimen- +sions and components used. The constituents components of +the DBR laser have lengths of: 30µm DBR mirrors, 400µm SOA +and 200µm phase modulation section. +Fig. 2. This schematic represents the second design of OIL ring +laser. The two ring lasers are injected using a DBR laser. +The circuits presented in this letter are fitted in an MPW cell +of dimensions 8mm x 2mm, highlighting the small footprint of +both of them respect to traditional modulator based circuits. +3. MEASUREMENTS AND DISCUSSION +In the following section we are going to present the experimental +results on the basic functionalities of the circuit. +A. DFB and single ring circuit +When analyzing the emission of the DFB laser it became +immediately evident that there had been some problem in +the fabrication. In fact, the threshold is above 60mA against +Fig. 3. Photo of the realized circuit under a microscope. We +can clearly see the metal tracks for electrical connection and +the waveguides below them. +the 25mA expected from the design manual. This has been +confirmed from the fabrication report and in general we can +observe a lot of variability in the behaviour of this component. +We have measured other DFB laser fabricated in the same run +and the threshold current ranges from 44mA to 60mA with +maximum emitted power above 1mW only in one case. +To estimate the MMI performances we drove the DFB at +80mA, resulting in a current at the photodiode terminating the +laser of −80µA whereas, the photodiode in the cross port of the +MMI registered −60µA. Both the photodiodes were biased at +−2V during the measurement. +Assuming a 1dB loss for this kind of component we reach the +result that the splitting ratio of the MMI is 83% for the cross port +and 17% for the through port. This ratio is close to the designed +ratio of 85 − 15. +Fig. 4. Emission spectra of the ring 3 as measured in the lab. +The single emission peak is due to cross-gain saturation and +to fabrication defects. We can notice the frequency comb in the +low power regime. +Before we discuss the OIL of the ring laser we have to +address it’s free running properties. +Figure 4 reports the +experimental spectrum recorded by coupling a fiber to the +chip and driving the ring3 SOA at 100mA. We can notice that +in this case the laser is single mode with lots of small peaks +representing the cavity resonances. The laser emission is not +stable and tends to jump and side modes appear and disappear +following non-predictable fluctuations. +This is due to the +intrinsic multi-mode nature of the laser. + +SOA +phase +SOA +SOAarmem +DeslEring3@100mA +-20 +Mean output power [dBm] +-30 +-40 +50 +-60 +-70 +-80 +-90 +1544 +1546 +1548 +1550 +1552 +1554 +Wavelength[nm]Letter +Optics Letters +3 +The next step of the experimental analysis is to verify the OIL +of the system. +Figure 5 reports the resulting spectra from optical injection of the +ring laser. The spectra is obtained applying a current of 100mA +to the ring3 SOA and 97mA to the DFB. The result proves the +OIL, since we do not see any other peak and the main peak is +shifted respect to figure 4. Comparing the two figures we can +also observe that the secondary peak seen in figure 5 at 1545nm +disappears. +Fig. 5. Spectrum of the ring3 laser at 100mA when it is injected +with the DFB at 97mA. We can notice the shift in wavelength +with respect to figure 4 that is due to the optical injection of +the DFB. +An additional proof of OIL is obtained by comparing the pho- +todiodes recorded power. In fact, the optical injection of a ring +laser ensures the unidirectionality of the light inside the cavity, +completely suppressing the CW mode. In this case, we have +measured the current at the photodiode in the CW direction and +turned on and off the DFB laser. Without DFB laser injection the +ring laser is bidirectional: the current on the CW photodiode +is −0.160mA. When the DFB laser is turned on at 97mA, the +current measured on the CW photodiode is −0.002mA, demon- +strating the high suppression of the CW light and the successful +OIL. +B. DBR and double ring circuit +We will start our analysis from ring2, this ring is identical +to ring1 and really similar to ring3 analyzed in the previous +section. The main difference with ring3 is the presence, in ring1 +and ring2, of the feedback of the DBR mirror of the master laser. +This changes the behaviour of the ring, giving it a preferred +lasing direction. +The spectrum of Ring2 is presented in figure 6. +Similarly, to the previous ring laser we notice a single peak. This +can be account by small reflections due to fabrication defects +can lead to different cavity length and ultimately to single mode +lasing. A second effect not taken into account is the cross-gain +saturation effect that tends to make only a single wavelength +lase. +It is typical of a multimode laser to change lasing frequency +based on small changes in temperature and conditions of the +cavity, leading to a highly unstable emission. +Fig. 6. Emission spectra measured at 60mA current injection +on the ring2 SOA. As in the previous case the ring exhibits +single mode operation due to small imperfections in the fabri- +cation and cross-gain saturation. +The design of the DBR laser was focused on having a high +power output sacrificing the single mode emission. In fact, we +will use the OIL not only for isolating a single frequency of the +slave laser but also to isolate a single frequency of the master +laser. The challenging part will then be to lock both ring1 and +ring2 laser to the same master frequency. +We can now discuss the OIL scheme. We first prove the +locking with only one of the rings. +Figure 7 shows the output of ring1 laser when injected with the +DBR laser. +The laser emission is single mode with a side-mode suppression +ratio of 45dB demonstrating the locking of the two lasers. In +fact, we do not see the multimode emission of the DBR laser nor +any sign of the multimode of the ring laser. This shows that this +system can be used for OIL effectively. +Like in the case of the DFB and ring3 the current applied to the +SOA is critical and influences the locking mechanism, but we +can find multiple combinations of currents that achieve OIL. +This system is already capable of QKD like the previous one, +but we also want to test if we are able to lock both ring1 and +ring2 to the same DBR laser mode at the same time. +When the ring2 laser is activated we notice that there is a +series of other peaks in addition to the peak from ring1 in the +spectrum. We have varied the current applied to the different +lasers in order to get to stable locking of both ring lasers, but +the best results so far has been not convincing. These results are +clearly not usable for QPSK since for this application we need a +single wavelength. The main question raising from this plot is +why the injection doesn’t work for both rings ? +As a final test we have decided to vary the phase modulator +in the DBR laser in order to check the influence of it in the lock- +ing of the lasers. +We have decided to drive the rings at the same current of 80mA +and only vary current applied to the DBR laser SOA section and +phase section. The best recorded spectra is visible in figure 8, + +ring3@100mA-DFB@97mA +20 +Mean output power [dBm] +-30 +-40 +50 +60 +-70 +-80 +90 +1544 +1546 +1548 +1550 +1552 +1554 +Wavelength [nm]ring2@60mA +-20 +-30 +Mean output power [dBm] +-40 +-50 +-70 +80 +06- +1544 +1546 +1548 +1550 +1552 +1554 +1556 +Wavelength[nm]Letter +Optics Letters +4 +Fig. 7. Emission spectra of ring1 at 55mA when injected with +the DBR laser at 40mA. We can notice the single mode emis- +sion resulting from the locking of the DBR and the ring. +obtained with 77mA on the SOA and 29mA on the phase mod- +ulator. The resulting spectrum shows a main peak and a few +side peaks 35dB lower. From this spectrum is still clear that the +system is still not necessary locked and the three laser influence +each other. In fact if we turn ring1 off the output spectra goes +back to be multimode. This gives the indication that the rings +themselves are influencing the emission of one another, even if +in an OIL system they should be lasing in a single direction as +proven for ring3. +Further investigation in this system is necessary and the photodi- +odes inside ring1 and ring2 should be monitored to understand +if there is light travelling in the counter propagating mode of +the laser. +Fig. 8. Spetrum, ring1 and ring2 are injected at 80mA, the SOA +of the DBR laser is driven at 77mA and the phase modulation +section of the DBR laser is driven at 29mA. This combination +of factors has been chosen to realize optical injection in both +the rings. +4. CONCLUSIONS +In conclusion, we have designed two photonic integrated cir- +cuits for QKD and QPSK both based on OIL mechanism. +The first circuit presented is composed by a DFB and a ring laser. +We have characterized the laser’s free running behaviour. Using +the integrated photodiodes and an optical spectrum analyzer +we have demonstrated OIL between the two lasers, resulting +in single mode, single direction emission. Using the DFB and +the two photodiodes it has been possible to estimate the power +splitting ratio of the custom MMI, in agreement with the de- +signed specifications. This is the first experimental verification +of monolithically integrated OIL system using ring lasers. +The second circuit is composed of two ring lasers and a central +DBR laser. Using OIL we want to achieve high speed modula- +tion of the ring lasers simultaneously. We have characterized the +identical ring lasers and DBR laser separately. We have demon- +strated the OIL of one ring laser with the DBR laser, but when +moving to locking both ring lasers to the master DBR laser we +achieved partial locking only using the phase tuning section. +In future work we want to measure the modulation bandwidth +of both circuits and further investigate the double ring locking +system. +5. ACKNOWLEDGEMENTS +Project developed in the framework of the European Doctor- +ate in Indium Phosphide PIC Fabrication Technology (EDIFY) +project. +6. DISCLOSURES +The authors declare no conflicts of interest. +REFERENCES +1. +J. Qiu, Nature 508, 441 (2014). +2. +H. Liu, C. F. Lam, and C. Johnson, “Scaling Optical Interconnects in +Datacenter Networks Opportunities and Challenges for WDM,” in 2010 +18th IEEE Symposium on High Performance Interconnects, (IEEE, +Mountain View, CA, USA, 2010), pp. 113–116. +3. +Z. Liu and R. Slavik, J. Light. Technol. 38, 43 (2020). +4. +A. Tanaka, M. Fujiwara, K.-i. Yoshino, S. Takahashi, Y. Nambu, +A. Tomita, S. Miki, T. Yamashita, Z. Wang, M. Sasaki, and A. Tajima, +IEEE J. Quantum Electron. 48, 542 (2012). +5. +P. Sibson, C. Erven, M. Godfrey, S. Miki, T. Yamashita, M. Fujiwara, +M. Sasaki, H. Terai, M. G. Tanner, C. M. Natarajan, R. H. Hadfield, J. L. +O’Brien, and M. G. Thompson, Nat Commun 8, 13984 (2017). +6. +Z. L. Yuan, B. Fröhlich, M. Lucamarini, G. L. Roberts, J. F. Dynes, and +A. J. Shields, Phys. Rev. X 6, 031044 (2016). +7. +B. Zhang, R. Zhang, S. Xu, C. Luo, and B. Qiu, IEEE Photonics J. 14, +1 (2022). Conference Name: IEEE Photonics Journal. +8. +O. Duzgol, G. Kyritsis, and N. Zakhleniuk, IET Optoelectronics 11, 58 +(2017). +9. +L. Chrostowski and W. Shi, J. Light. Technol. 26, 3355 (2008). Confer- +ence Name: Journal of Lightwave Technology. +10. +T. K. Paraïso, I. De Marco, T. Roger, D. G. Marangon, J. F. Dynes, +M. Lucamarini, Z. Yuan, and A. J. Shields, npj Quantum Inf 5, 1 (2019). + +ring1@55mADBR@40mA +-30 +40 +50 +-60 +-70 +-80 +06- +1544 +1546 +1548 +1550 +1552 +1554 +1556 +Wavelength[nm]ing1-2@80mADBR@77mAphmod@29mA +-10 +20 +Mean output power [dBm] +-30 +-40 +-50 +-60 +70 +-80 +-90 +1544 +1546 +1548 +1550 +1552 +1554 +1556 +Wavelength[nm] \ No newline at end of file diff --git a/lNE2T4oBgHgl3EQfIwaA/content/tmp_files/load_file.txt b/lNE2T4oBgHgl3EQfIwaA/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7b2fc1c510a17a85297e9c48cbc788b37e95d378 --- /dev/null +++ b/lNE2T4oBgHgl3EQfIwaA/content/tmp_files/load_file.txt @@ -0,0 +1,275 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf,len=274 +page_content='Letter Optics Letters 1 Monolithically integrated circuits for Optical injection locking of ring lasers with QKD and QPSK applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' DAMIANO MASSELLA1,*, MICHAEL WALLACE2, RONALD BROEKE2, AND FRANCISCO DIAZ1,3 1University of Vigo, El Telecomunication - Campus Universitario As Lagoas, 36310 Vigo, Spain 2Bright Photonics BV, Eindhoven, Netherlands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' 3AtlanTTic research center, El Telecomunication - Campus Universitario As Lagoas, 36310 Vigo, Spain Corresponding author: damiano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content='massella@uvigo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content='es Compiled January 11, 2023 Optical injection locking of integrated ring lasers promises to be a key technology of the future, but it is still lacking experimental verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' In this paper we present two monolithically integrated photonic circuits that experimentally realize Optically injection locking of ring lasers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' The presented circuits pave the way for future application in Quantum Key Distribution and coherent communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' © 2023 Optica Publishing Group http://dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content='doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content='1364/ao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content='XX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content='XXXXXX 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' INTRODUCTION In recent years the discussion regarding communication security has taken a central role as the importance of telecommunication grows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' Quantum Key Distribution (QKD) promises to solve the problem of security in communication, an issue which is given more significance with the advent of quantum computers[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' At the same time, due to ever increasing demands on data trans- mission there has been growing interest in Optical Injection Locking (OIL) of lasers[2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' Using this technique it is possible to increase the modulation bandwidth of a laser up to 100GHz[3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' The bandwidth of transmitters can be increased by multiplexing a large number of channels however, this is not scalable using bulk optics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' Photonic Integrated Circuits (PICs), consist of a large number of optical components integrated on a single chip, are a key technology to overcome this limitation [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' QKD transmitters have been realized in both Indium Phosphide and Silicon showing the advantage of integration in terms of stability and miniaturization of single channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' The work of Sibson et al [5] reports on the integration of a QKD transmitter without the use of modulators and based on an OIL technique first demonstrated in [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' In this paper we report the first exper- imental realization OIL of monolithically integrated ring laser systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' An extensive study of this kind of system has been reported before but no experimental realization has been show yet [7–9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' In this work we show two different circuits designed for two different applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' The first circuit is consist of a Distributed Feedback (DFB) Laser is OIL to a ring laser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' This system is intended for mainly QKD purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' Coherent communication is also possible however, the modulation scheme would be too complex[10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' Our second circuit is realized using 2 ring lasers that are feed using a Distributed Bragg Reflector (DBR) laser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' This circuit has been designed to be a coherent communication transmitter and thus be able to modulate both the ring lasers simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' With respect to the circuit presented by Paraiso et all [10], the use of ring lasers instead of DFB lasers carries some significant advantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' Firstly, the ring lasers don’t need to be tuned like in the case of single mode lasers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' And secondly, when OIL a ring laser we ensure the unidirectional emission from it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' Consequently, there are no reflections from the slave laser to the master laser, like in the DFB case [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' CIRCUIT DESIGN The first circuit is presented in figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' In this circuit we use a simple DFB laser to inject a ring laser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' The notation for the ring lasers follows the one used for the mask design, consequently in the ring laser of circuit one is labelled as ’ring 3’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' The simplicity of this circuit is its main advantage with respect to other QKD cir- cuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' In the schematics of this section we have omitted the MMI blocks and the photodiodes used to monitor the state of the laser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' This schematic represent the first design of OIL ring laser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' The ring laser is injected using a DFB laser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' We used a standard DFB laser designed to emit at 1550nm, arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content='03684v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content='optics] 9 Jan 2023 SOALetter Optics Letters 2 the left output of it is terminated using a photodiode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' This photodiode will be used to characterize the DFB laser emitted power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' On the ring we have used custom MMI blocks, they are designed to split power unequally, 85% on the cross state and 15% on the bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' We have chosen this particular ratio to reduce the losses inside the ring and have a shorter gain section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' The MMI used in the rings are all 2x2 and the additional port is terminated using a photodiode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' The photodiodes inside the ring can be used to monitor both the clockwise directed power and the counter-clockwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' The SOA section is 400µm long and it has been connected using RF lines in order to be able to drive the ring laser to higher frequencies and verify the modulation bandwidth enhancement due to OIL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' A spot size converter is placed at the output of the circuit to minimize the losses when coupling to fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' The second circuit is schematized in figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' In this case we decided to exploit the two outputs of a DBR laser to OIL two ring lasers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' The output of the two rings are then combined to a single output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' This schematic can be used for both QKD and Quadrature Phase Shift Keying (QPSK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' The main advantage with respect to the first circuit is the pos- sibility to use the modulation of the rings for encoding the bits of the QPSK scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' We have added a phase modulator in the output of one of the ring laser in order to have the π/2 phase shift for QPSK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' This additional phase modulation section is con- stituted by an thermo-optical phase modulator of length 450µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' This length is sufficient to ensure the π/2 phase difference be- tween the two rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' The modulation bandwidth is enhanced by the OIL since we are going to modulate the two slave lasers and not the master laser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' The ring lasers used for the two circuits are identical in dimen- sions and components used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' The constituents components of the DBR laser have lengths of: 30µm DBR mirrors, 400µm SOA and 200µm phase modulation section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' This schematic represents the second design of OIL ring laser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' The two ring lasers are injected using a DBR laser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' The circuits presented in this letter are fitted in an MPW cell of dimensions 8mm x 2mm, highlighting the small footprint of both of them respect to traditional modulator based circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' MEASUREMENTS AND DISCUSSION In the following section we are going to present the experimental results on the basic functionalities of the circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' DFB and single ring circuit When analyzing the emission of the DFB laser it became immediately evident that there had been some problem in the fabrication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' In fact, the threshold is above 60mA against Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' Photo of the realized circuit under a microscope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' We can clearly see the metal tracks for electrical connection and the waveguides below them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' the 25mA expected from the design manual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' This has been confirmed from the fabrication report and in general we can observe a lot of variability in the behaviour of this component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' We have measured other DFB laser fabricated in the same run and the threshold current ranges from 44mA to 60mA with maximum emitted power above 1mW only in one case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' To estimate the MMI performances we drove the DFB at 80mA, resulting in a current at the photodiode terminating the laser of −80µA whereas, the photodiode in the cross port of the MMI registered −60µA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' Both the photodiodes were biased at −2V during the measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' Assuming a 1dB loss for this kind of component we reach the result that the splitting ratio of the MMI is 83% for the cross port and 17% for the through port.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' This ratio is close to the designed ratio of 85 − 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' Emission spectra of the ring 3 as measured in the lab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' The single emission peak is due to cross-gain saturation and to fabrication defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' We can notice the frequency comb in the low power regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' Before we discuss the OIL of the ring laser we have to address it’s free running properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' Figure 4 reports the experimental spectrum recorded by coupling a fiber to the chip and driving the ring3 SOA at 100mA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' We can notice that in this case the laser is single mode with lots of small peaks representing the cavity resonances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' The laser emission is not stable and tends to jump and side modes appear and disappear following non-predictable fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' This is due to the intrinsic multi-mode nature of the laser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' SOA phase SOA SOAarmem DeslEring3@100mA 20 Mean output power [dBm] 30 40 50 60 70 80 90 1544 1546 1548 1550 1552 1554 Wavelength[nm]Letter Optics Letters 3 The next step of the experimental analysis is to verify the OIL of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' Figure 5 reports the resulting spectra from optical injection of the ring laser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' The spectra is obtained applying a current of 100mA to the ring3 SOA and 97mA to the DFB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' The result proves the OIL, since we do not see any other peak and the main peak is shifted respect to figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' Comparing the two figures we can also observe that the secondary peak seen in figure 5 at 1545nm disappears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' Spectrum of the ring3 laser at 100mA when it is injected with the DFB at 97mA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' We can notice the shift in wavelength with respect to figure 4 that is due to the optical injection of the DFB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' An additional proof of OIL is obtained by comparing the pho- todiodes recorded power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' In fact, the optical injection of a ring laser ensures the unidirectionality of the light inside the cavity, completely suppressing the CW mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' In this case, we have measured the current at the photodiode in the CW direction and turned on and off the DFB laser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' Without DFB laser injection the ring laser is bidirectional: the current on the CW photodiode is −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content='160mA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' When the DFB laser is turned on at 97mA, the current measured on the CW photodiode is −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content='002mA, demon- strating the high suppression of the CW light and the successful OIL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' DBR and double ring circuit We will start our analysis from ring2, this ring is identical to ring1 and really similar to ring3 analyzed in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' The main difference with ring3 is the presence, in ring1 and ring2, of the feedback of the DBR mirror of the master laser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' This changes the behaviour of the ring, giving it a preferred lasing direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' The spectrum of Ring2 is presented in figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' Similarly, to the previous ring laser we notice a single peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' This can be account by small reflections due to fabrication defects can lead to different cavity length and ultimately to single mode lasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' A second effect not taken into account is the cross-gain saturation effect that tends to make only a single wavelength lase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' It is typical of a multimode laser to change lasing frequency based on small changes in temperature and conditions of the cavity, leading to a highly unstable emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' Emission spectra measured at 60mA current injection on the ring2 SOA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' As in the previous case the ring exhibits single mode operation due to small imperfections in the fabri- cation and cross-gain saturation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' The design of the DBR laser was focused on having a high power output sacrificing the single mode emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' In fact, we will use the OIL not only for isolating a single frequency of the slave laser but also to isolate a single frequency of the master laser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' The challenging part will then be to lock both ring1 and ring2 laser to the same master frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' We can now discuss the OIL scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' We first prove the locking with only one of the rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' Figure 7 shows the output of ring1 laser when injected with the DBR laser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' The laser emission is single mode with a side-mode suppression ratio of 45dB demonstrating the locking of the two lasers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' In fact, we do not see the multimode emission of the DBR laser nor any sign of the multimode of the ring laser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' This shows that this system can be used for OIL effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' Like in the case of the DFB and ring3 the current applied to the SOA is critical and influences the locking mechanism, but we can find multiple combinations of currents that achieve OIL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' This system is already capable of QKD like the previous one, but we also want to test if we are able to lock both ring1 and ring2 to the same DBR laser mode at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' When the ring2 laser is activated we notice that there is a series of other peaks in addition to the peak from ring1 in the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' We have varied the current applied to the different lasers in order to get to stable locking of both ring lasers, but the best results so far has been not convincing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' These results are clearly not usable for QPSK since for this application we need a single wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' The main question raising from this plot is why the injection doesn’t work for both rings ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' As a final test we have decided to vary the phase modulator in the DBR laser in order to check the influence of it in the lock- ing of the lasers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' We have decided to drive the rings at the same current of 80mA and only vary current applied to the DBR laser SOA section and phase section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' The best recorded spectra is visible in figure 8, ring3@100mA-DFB@97mA 20 Mean output power [dBm] 30 40 50 60 70 80 90 1544 1546 1548 1550 1552 1554 Wavelength [nm]ring2@60mA 20 30 Mean output power [dBm] 40 50 70 80 06- 1544 1546 1548 1550 1552 1554 1556 Wavelength[nm]Letter Optics Letters 4 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' Emission spectra of ring1 at 55mA when injected with the DBR laser at 40mA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' We can notice the single mode emis- sion resulting from the locking of the DBR and the ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' obtained with 77mA on the SOA and 29mA on the phase mod- ulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' The resulting spectrum shows a main peak and a few side peaks 35dB lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' From this spectrum is still clear that the system is still not necessary locked and the three laser influence each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' In fact if we turn ring1 off the output spectra goes back to be multimode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' This gives the indication that the rings themselves are influencing the emission of one another, even if in an OIL system they should be lasing in a single direction as proven for ring3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' Further investigation in this system is necessary and the photodi- odes inside ring1 and ring2 should be monitored to understand if there is light travelling in the counter propagating mode of the laser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' Spetrum, ring1 and ring2 are injected at 80mA, the SOA of the DBR laser is driven at 77mA and the phase modulation section of the DBR laser is driven at 29mA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' This combination of factors has been chosen to realize optical injection in both the rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' CONCLUSIONS In conclusion, we have designed two photonic integrated cir- cuits for QKD and QPSK both based on OIL mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' The first circuit presented is composed by a DFB and a ring laser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' We have characterized the laser’s free running behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' Using the integrated photodiodes and an optical spectrum analyzer we have demonstrated OIL between the two lasers, resulting in single mode, single direction emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' Using the DFB and the two photodiodes it has been possible to estimate the power splitting ratio of the custom MMI, in agreement with the de- signed specifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' This is the first experimental verification of monolithically integrated OIL system using ring lasers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' The second circuit is composed of two ring lasers and a central DBR laser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' Using OIL we want to achieve high speed modula- tion of the ring lasers simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' We have characterized the identical ring lasers and DBR laser separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' We have demon- strated the OIL of one ring laser with the DBR laser, but when moving to locking both ring lasers to the master DBR laser we achieved partial locking only using the phase tuning section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' In future work we want to measure the modulation bandwidth of both circuits and further investigate the double ring locking system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' ACKNOWLEDGEMENTS Project developed in the framework of the European Doctor- ate in Indium Phosphide PIC Fabrication Technology (EDIFY) project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' DISCLOSURES The authors declare no conflicts of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' REFERENCES 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' Qiu, Nature 508, 441 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' Liu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' Lam, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' Johnson, “Scaling Optical Interconnects in Datacenter Networks Opportunities and Challenges for WDM,” in 2010 18th IEEE Symposium on High Performance Interconnects, (IEEE, Mountain View, CA, USA, 2010), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' 113–116.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' Liu and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' Slavik, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' Light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' Technol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' 38, 43 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' Tanaka, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' Fujiwara, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content='-i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' Yoshino, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' Takahashi, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' Nambu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' Tomita, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' Miki, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' Yamashita, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' Wang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' Sasaki, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' Tajima, IEEE J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' Quantum Electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' 48, 542 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' Sibson, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' Erven, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' Godfrey, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' Miki, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' Yamashita, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' Fujiwara, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' Sasaki, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' Terai, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' Tanner, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' Natarajan, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' Hadfield, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' O’Brien, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' Thompson, Nat Commun 8, 13984 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' Yuan, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' Fröhlich, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' Lucamarini, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' Roberts, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' Dynes, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' Shields, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' X 6, 031044 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' Zhang, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' Zhang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' Xu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' Luo, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' Qiu, IEEE Photonics J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' 14, 1 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' Conference Name: IEEE Photonics Journal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' Duzgol, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' Kyritsis, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' Zakhleniuk, IET Optoelectronics 11, 58 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' Chrostowski and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' Shi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' Light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' Technol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' 26, 3355 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' Confer- ence Name: Journal of Lightwave Technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' Paraïso, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' De Marco, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' Roger, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' Marangon, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' Dynes, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' Lucamarini, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' Yuan, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' Shields, npj Quantum Inf 5, 1 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} +page_content=' ring1@55mADBR@40mA 30 40 50 60 70 80 06- 1544 1546 1548 1550 1552 1554 1556 Wavelength[nm]ing1-2@80mADBR@77mAphmod@29mA 10 20 Mean output power [dBm] 30 40 50 60 70 80 90 1544 1546 1548 1550 1552 1554 1556 Wavelength[nm]' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNE2T4oBgHgl3EQfIwaA/content/2301.03684v1.pdf'} diff --git a/ltE3T4oBgHgl3EQf6Avi/vector_store/index.pkl b/ltE3T4oBgHgl3EQf6Avi/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..843b6e18b76c173567ee438652ca26bca95bdf21 --- /dev/null +++ b/ltE3T4oBgHgl3EQf6Avi/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:d00fcc0cc9a85816c9d9860dddb8d3ca821c159a6e70f80af85805ee30089730 +size 92715 diff --git a/mNE0T4oBgHgl3EQf8ALY/vector_store/index.faiss b/mNE0T4oBgHgl3EQf8ALY/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..0b6769d275e39c0f010d62ad40fd6baf2e5b60a7 --- /dev/null +++ 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sha256:232398ac99ce5fa9e1eb9344cf8689de9c25d62852990c07d8467455875eaeda +size 435504 diff --git a/mdE2T4oBgHgl3EQfzAi5/content/tmp_files/2301.04127v1.pdf.txt b/mdE2T4oBgHgl3EQfzAi5/content/tmp_files/2301.04127v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..359c630f8139d156d99c24c40a3b0cfc61295bc1 --- /dev/null +++ b/mdE2T4oBgHgl3EQfzAi5/content/tmp_files/2301.04127v1.pdf.txt @@ -0,0 +1,1429 @@ +arXiv:2301.04127v1 [math.AG] 10 Jan 2023 +LINES ON K3-QUARTICS VIA TRIANGULAR SETS +ALEX DEGTYAREV AND S�LAWOMIR RAMS +Abstract. We prove the sharp upper bound of at most 52 lines on a complex +K3-surface of degree 4 with a non-empty singular locus. We also classify the +configurations of more than 48 lines on smooth complex quartics. +1. Introduction +Our main goal is to present an approach to study large line configurations on +complex projective K3-quartics. In particular, we prove the following theorem. +Theorem 1.1 (see §7.2). Let X4 ⊂ P3(C) be a degree 4 K3-surface with non-empty +singular locus. Then X4 contains at most 52 lines. Moreover, each K3-quartic with +at least 49 lines contains four coplanar lines. +The above bound is sharp: the existence of a complex K3-quartic with 52 lines +and non-empty singular locus (two simple nodes) was shown by the first named +author in 2016 (via Torelli’s theorem, see [6, Theorem 1.10]) and the equation of +the surface in question was found by D. Veniani, see [25, Example 5.5]. +We conjecture that the quartic surface discovered in [6, Theorem 1.10] is the +only quartic that attains the bound of Theorem 1.1, but the proof of this fact is +beyond the scope of this paper. +It is well-known that the complexity of large line configurations on projective +K3-surfaces decreases as the degree d of the polarization grows. In particular, a +complete classification of close to maximal configurations is known for octics (see +[2, Theorem 1.1]) and sextics (to appear in [3]): the respective upper bounds are +32 and 36 in the presence of a singularity vs. 36 and 42 in the smooth case. In +contrast, even though quartic surfaces with singular points have been a subject of +intensive study ever since the 19-th century (see, e.g., the classical treatise [12]), +hardly anything is known about large line configurations on such surfaces. The +main reason is the existence of the so-called triangular configurations on quartics +(see §2.4 for the definition) — a property that drastically increases the complexity +of the problem. Here, we circumvent this difficulty with the help of the so-called +triangular sets introduced in §3. +One can easily check that the degree-d Fermat surface (over C) contains exactly +3d2 lines for d > 2. Moreover, for almost all integers d the Fermat surface is the +best known example of a smooth complex projective surface with many lines, and +the question whether smooth degree-d surfaces with more lines exist remains open. +To illustrate the power of our approach, we refine the results of [8] and classify +2000 Mathematics Subject Classification. Primary: 14J28; Secondary: 14J27, 14N25. +Key words and phrases. K3-surface, quartic, elliptic pencil, integral lattice, discriminant form. +A.D. was partially supported by the T¨UB˙ITAK grant 118F413. S.R. was partially supported +by the National Science Centre, Poland, Opus grant no. 2018/31/B/ST1/02857. +1 + +2 +ALEX DEGTYAREV AND S�LAWOMIR RAMS +Table 1. Smooth complex quartics with at least 49 lines +Γ +|Aut Γ| +|Sym X4| +(r, c) +NS(X4)⊥ +X64 +4608 +(192, 1493)6 +(1, 0) +[8, 4, 8] +X′ +60 +480 +(60, 5)2 +(1, 0) +[4, 2, 16] +X′′ +60 +240 +(60, 5)2 +(0, 1) +[4, 1, 14] +X56 +128 +(16, 11)4 +(0, 1) +[8, 0, 8] +Y56 +64 +(16, 8)2 +(1, 0) +∗[2, 0, 32] +Q56 +384 +(48, 50)2 +(1, 0) +[4, 2, 16] +X54 +384 +(24, 12)2 +(1, 0) +[4, 0, 24] +Q54 +48 +(8, 5) +(1, 0) +[4, 2, 20] +X′ +52 +24 +(3, 1) +(1, 0) +[8, 4, 12] +X′′ +52 +36 +(6, 1) +(1, 0) +[4, 2, 20] +X′′′ +52 +320 +(20, 3)4 +(1, 0) +[10, 0, 10] +Xv +52 +32 +(4, 1)2 +(1, 0) +[10, 4, 10] +Y′ +52 +8 +(4, 1)2 +(1, 0) +∗[2, 0, 38] +(0, 1) +[8, 2, 10] +Y′′ +52 +8 +(4, 2)2 +(1, 0) +∗[2, 1, 40] +(0, 1) +[4, 1, 20] +(0, 1) +[8, 1, 10] +Z52 +384 +(12, 3)2 +(1, 0) +∗U(2) ⊕ [24] +(52) +384 +(8, 5) +(1, 0) +[−8] ⊕ [4, 2, 4] +Q′ +52 +64 +(8, 5) +(1, 0) +[4, 0, 24] +Q′′ +52 +64 +(16, 11) +(0, 1) +[8, 4, 12] +Q′′′ +52 +96 +(4, 1)6 +(1, 0) +[10, 5, 10] +X51 +12 +(6, 1) +(0, 1) +[4, 1, 22] +(1, 0) +[6, 3, 16] +X′ +50 +18 +(3, 1) +(1, 0) +[4, 2, 28] +X′′ +50 +12 +(3, 1) +(2, 0) +[4, 0, 24] +X′′′ +50 +16 +(2, 1)2 +(0, 1) +[4, 0, 24] +Z50 +160 +(10, 1)2 +(1, 0) +U(5) ⊕ [4] +(50) +96 +(8, 5) +(1, 0) +∗U(2) ⊕ [28] +Z49 +36 +(3, 1) +(1, 0) +∗U(2) ⊕ [28] +all configurations of at least 49 lines on smooth quartics (i.e., the configurations +that are larger than the one on the Fermat quartic). Remarkably, compared to +[8], we found but three new configurations: one of rank 20 (Q′′′ +52 previously found +in [6]) and two of rank 19 (designated as (52) and (50) in Table 1). On the other +hand, there are at least 28 configurations of 48 lines on smooth quartics, giving +yet another reason why 48 is a reasonable threshold (cf. also Proposition 5.6 and +Remark 4.11 below). +Theorem 1.2 (see §7.3). Up to isomorphism, there are 26 configurations of at least +49 lines on smooth quartic surfaces, see Table 1. They are realized by 34 singular +(aka projectively rigid) surfaces (18 real and 8 pairs of complex conjugate) and five +connected 1-parameter families. +As a consequence, we answer a question left open in [8, Addendum 1.4]. + +LINES ON K3-QUARTICS VIA TRIANGULAR SETS +3 +Addendum 1.3 (see §7.4). The complete list of values taken by the number of real +lines on a real smooth quartic is +{0, 1, . . ., 49, 50, 52, 56}. +The configurations of more than 48 real lines on a real smooth quartic are those +marked with a ∗ in Table 1. +Listed in Table 1 are: +• the name of the configuration Γ (mostly following [8]); the subscript always +refers to the number of lines (vertices of Γ); +• the size of the group Aut Γ of abstract graph automorphisms of Γ; +• the group Sym X4 of simplectic automorphisms of a generic quartic X4 with +the given Fano graph, in the form (size, index), referring to the SmallGroup +library in GAP [9]; the superscript is the index of Sym X4 in the full group +Aut(X4, h) of projective automorphisms of X4 (if greater than 1); +• the numbers (r, c) of, respectively, real and pairs of complex conjugate +components of the equilinear moduli space; +• the (generic, if rk ⩾ 3) transcendental lattice T := NS(X4)⊥; it is marked +with a ∗ if the corresponding deformation family has a real quartic with all +lines real (see [8, Lemma 3.8]). +If T is not determined by Γ, each lattice is listed in a separate row (following the +main entry), and the numbers (r, c) of components are itemized accordingly. +As in [5], we use the following notation for common integral lattices: +• [a] := Zu is the lattice of rank 1 given by the condition u2 = a; +• [a, b, c] := Zu + Zv, u2 = a, u · v = b, v2 = c, is a lattice of rank 2; when it +is positive definite, we assume that 0 < a ⩽ c and 0 ⩽ 2b ⩽ a: then, u is a +shortest vector, v is a next shortest one, and the triple (a, b, c) is unique; +• U := [0, 1, 0] is the unimodular even lattice of rank 2; +• L(n) denotes the lattice obtained by the scaling of a given lattice L by a +fixed integer n ∈ Z. +In general, we maintain the standard notation for various objects associated to a +lattice L (the determinant, discriminant group, etc.) —see, e.g., [1, 16]. The inertia +indices of the quadratic form L ⊗ R are denoted by σ±,0(L). +1.1. Contents of the paper. Roughly, the paper consists of two parts: the dis- +crete one (§2–§6) and the geometric one (§7). +Our approach is a refinement of the technique developed in [8, 2], and we recall +the necessary facts and introduce certain technical terms (e.g., acceptable graphs) +in §2. +Then, in §3, we define the main technical tool, viz. the triangular set, +and discuss methods of extending a given graph by a collection of triangular sets. +Finally, after those preparations, we present the proof of Proposition 3.12, which is +the discrete counterpart of the most difficult case of Theorem 1.1. +§4 is a digression: we restrict our attention to the case of smooth lattices (i.e., +we assume that the lattice contains no exceptional divisors) and apply triangular +sets to classify geometric Fano graphs with at least 49 vertices. +In §5 and §6, we turn back to the general case (with exceptional divisors allowed) +and study the properties of triangular free Fano graphs. +Finally, in §7 we recall the definition of the Fano graph (resp. extended Fano +graph) of a surface and its relation to the geometricity of the Fano graph of a + +4 +ALEX DEGTYAREV AND S�LAWOMIR RAMS +lattice, see Theorem 7.4 (resp. Theorem 7.7) and prove the principal results of the +paper, viz. Theorem 1.1 and Theorem 1.2. +1.2. History of the problem. As mentioned, configurations of lines (or, more +generally, smooth rational curves) on quartic surfaces in P3 have been a subject of +intensive study ever since the 19-th century. Still, the methods of Italian school were +not efficient enough to deal with the classification of large line configurations on +such surfaces. It was not until the last decade that the theory of elliptic fibrations, +Mordell–Weil groups, Torelli’s theorem and progress in algorithmic methods in the +theory of lattices led to a substantial progress in the case of smooth quartics: sharp +bound for the number of lines over fields of characteristic p ̸= 2, 3 (see [20, 18, 8]), +p = 3 (see [17]), p = 2 (see [7]), the classification of large configurations (see [8]), +explicit equations of surfaces with many lines (see [24] and the bibliography therein). +Strangely enough, the Bogomolov–Miyaoka–Yau inequality yields no bounds in the +case of quartics (see [14]). +In contrast, in spite of long interest (see, e.g., the classical text [12]), far less is +known in the case of quartic surfaces with singular points — essentially, it was only +shown that, over a field of characteristic p ̸= 2, the number of lines on a quartic +with singularities cannot exceed the maximal number of lines on a smooth quartics +— see [23, 25, 11]. +For p = 2 the maximal number of lines on a quartic with +singularities is 68 (vs. 60 in the smooth case, see [22]) and we do know projective +models of surfaces that attain this maximum (see [17]). +A refinement of the method pioneered in [8] led to the complete picture of large +line configurations on smooth degree-d K3-surfaces for d > 2 in [5]. +Vinberg’s +algorithms combined with the above methods yield a means to classify the large +configurations of lines on degree-d K3-surfaces with at worst Du Val singularities +for d > 4 (see [2, 3]). The methods of [2] are not sufficient to deal with the case +of quartics (i.e., d = 4): the existence of triangles (i.e., ˜A2-configurations) of lines +and the fact that, on quartic surfaces, said triangles may interlace lead to numerous +configurations that are excluded on degree-d K3-surfaces for d > 4. In the present +paper, we discuss an approach to deal with such configurations. However, in order +to keep our exposition compact, we apply our method to find the maximal number +of lines on a complex K3-quartics with non-empty singular locus, but we do not +try to classify all configurations of 52 lines. +1.3. Acknowledgements. This paper was mostly written during our research stay +at the Max-Planck-Institut f¨ur Mathematik, Bonn. We are grateful to MPIM for +creating perfect working conditions. S.R. thanks IM PAN (Cracow, Poland) for the +support that enabled him to complete this project. +2. Preliminaries +In this section we recall the main technical tools that we use in our work. To +shorten the exposition, we focus on the case of 4-polarized 2-admissible lattices and +graphs. The details and more general statements can be found in [2]. To keep the +exposition continuous,we assume the reader familiar with the basics of the theory +of K3-surfaces, (−2)-curves, etc. and adopt a formal graph-theoretical language. +The relation of graphs considered in §2–§6 to the problem at hand, i.e., lines on +quartic surfaces, is briefly discussed in §7.1 below, right before the proofs of the +principal results of the paper. + +LINES ON K3-QUARTICS VIA TRIANGULAR SETS +5 +2.1. Polarized lattices. Recall that a nondegenerate lattice S is called hyperbolic +if σ+S = 1. A polarized lattice S ∋ h is a hyperbolic lattice S equipped with a +distinguished vector h of positive square; the square h2 is called the degree of the +polarization and S is said to be h2-polarized. Here we assume h2 = 4, so whenever +we speak of a polarized lattice we mean a 4-polarized lattice. +Furthermore, we +confine ourselves to lines and exceptional divisors (resp. only lines in §4), leaving +out smooth rational curves of higher degree. +Remark 2.1. We make frequent use of the following obvious observation: if S is a +hyperbolic lattice, then any sublattice N ⊂ S is either semidefinite (and then one +has rk ker N = 1) or nondegenerate. +For a polarized lattice and n = 0, 1, one defines the sets +rootn(S, h) := +� +r ∈ S +�� r2 = −2, r · h = n +� +. +As in [2, § 2.2], we put rt(S, h) ⊂ h⊥ ⊂ S (resp. C+(S, h)) to denote the sublattice +generated by root0(S, h) (resp. the positive cone). +Recall that every connected +component ∆♯ of +C+(S, h) ∖ +� +r2=−2 +r⊥ +is a fundamental polyhedron for the group generated by reflections of S. +By definition, rt(S, h) is a root lattice and each fixed Weyl chamber ∆ for (the +group generated by reflections of) rt(S, h) gives rise to a distinguished fundamental +polyhedron ∆♯. +We put {∆} to denote the ”outward” roots orthogonal to the +walls of ∆ and define the (plain) Fano graph of the polarized lattice (S, h) with a +distinguished Weyl chamber ∆ for rt(S, h) as the set of vertices +(2.2) +Fn∆(S, h) := {∆♯}1 := +� +l ∈ root1(S, h) +�� l · e ⩾ 0 for all e ∈ {∆} +� +, +with two vertices l1 ̸= l2 connected by an edge of multiplicity l1 · l2. The bi-colored +extended Fano graph is defined as +(2.3) +Fnex +∆ (S, h) := {∆♯}1 ∪ {∆}, +with the same convention about the multiplicities of the edges and vertices v colored +according to the value v · h ∈ {0, 1}. +Definition 2.4. Let S ∋ h be a polarized lattice and let Γ be a subset of root1(S, h). +(1) A Weyl chamber ∆ is called compatible with Γ if Γ ⊂ Fn∆(S, h). +(2) A root r ∈ root0(S, h) is called separating with respect to Γ if there is a +pair of vertices u, v ∈ Γ separated by r, so that r · u > 0 and r · v < 0. +Finally, in order to use general theory of K3-surfaces in the sequel we need the +following definition. +Definition 2.5. A polarized lattice S ∋ h is called: +(1) admissible, if there is no vector p ∈ S such that p2 = 0 and p · h = 2; +(2) geometric, if it is admissible and there exists a primitive isometry +S ֒→ L := 2E8 ⊕ 3U. + +6 +ALEX DEGTYAREV AND S�LAWOMIR RAMS +2.2. Subgeometric and geometric graphs. Let Γ be a (plain) graph. To Γ we +associate the polarized lattice +(2.6) +F(Γ) := (ZΓ + Zh)/ ker, +h2 = 4, +h · v = 1 for v ∈ Γ. +where ZΓ is the lattice freely generated by the vertices v ∈ Γ, so that u · v = n +when u ̸= v are connected by an n-fold edge, and v2 = −2 for each v ∈ Γ. +Convention 2.7. As in [2, § 4], we speak of polarized graphs Γ (omitting the degree +which is fixed to equal 4), and we apply to Γ the lattice theoretic terminology such +as the rank rk Γ := rk F(Γ) etc. Furthermore, we treat the vertices of Γ as vectors +in F(Γ): e.g., u·v ∈ Z stands for the multiplicity of the edge [u, v], and we say that +u, v ∈ Γ intersect if u · v = 1. The only exception from this rule is the classification +of graphs according to the inertia indices of ZΓ rather than F(Γ) (which is always +assumed hyperbolic): thus, we say that Γ is +• elliptic, if σ+(ZΓ) = σ0(ZΓ) = 0, +• parabolic, if σ+(ZΓ) = 0 and σ0(ZΓ) > 0, and +• hyperbolic, if σ+(ZΓ) = 1 (no assumption on σ0). +Recall that any connected elliptic (resp. parabolic) graph is a Dynkin diagram (resp. +affine Dynkin diagram); as in [5], we order the isomorphism classes of affine Dynkin +diagrams according to their Milnor number, followed by A < D < E. Recall also +that, for each connected parabolic subgraph Σ, there is a unique positive minimal +generator κΣ ∈ ker ZΣ; it has the form κΣ = � mcc, c ∈ Σ, with all mc > 0. +We define the perturbation order on the set of (isomorphism classes of) elliptic +and parabolic graphs: Γ′ ⊳ Γ′′ if Γ′ is isomorphic to an induced subgraph of Γ′′. +Given an isotropic subgroup K ⊂ discr F(Γ) (aka kernel), we consider the fi- +nite index extension F(Γ, K) of F(Γ) by K (cf. [16]). The pair (Γ, K) is said to +be extensible if it admits a compatible Weyl chamber ∆ for rt(F(Γ, K), h) (see +Definition 2.4): +Γ ⊂ Fn∆ F(Γ, K). +Recall that, by [2, Lemma 3.4], we have +(2.8) +(Γ, K) is extensible if and only if F(Γ, K) has no separating roots +(see Definition 2.4); moreover, if this is the case, the compatible Weyl chamber +∆ ⊂ rt F(Γ, K) is unique. Therefore, for an extensible pair (Γ, K) one can define +its saturation and extended saturation +sat(Γ, K) := Fn∆ F(Γ, K), +satex(Γ, K) := Fnex +∆ F(Γ, K). +A graph Γ (resp. pair (Γ, K)) is called admissible if it is extensible and the lattice +F(Γ) (resp. F(Γ, K)) is admissible. Then, an isotropic subgroup K ⊂ discr F(Γ) is +called a geometric kernel if the lattice F(Γ, K) is geometric. We follow [2] and put +G(Γ) := +� +K ⊂ discr F(Γ) +�� K is geometric +� +. +After those preparations we can recall the following definition. +Definition 2.9. Let Γ be a graph. +(1) We call Γ subgeometric if the set G(Γ) is non-empty. +(2) A subgeometric graph Γ is called geometric if Γ ∼= sat(Γ, K) for a certain +kernel K ∈ G(Γ′). +(3) A bi-colored graph Γ′ is geometric if Γ′ ∼= Fnex +∆ (S, h) for some geometric +polarized lattice S ∋ h. + +LINES ON K3-QUARTICS VIA TRIANGULAR SETS +7 +2.3. Algorithms. In many arguments we use computer-aided test to check that a +given graph satisfies certain conditions. All algorithms are found in [2, Appendix]. +Given a graph Γ, one can check whether: +(1) F(Γ) is hyperbolic, i.e., σ+F(Γ) = 1; straight as it is, this can often be +done fast in bulk, see [2, Lemmas A.3, A.4]; +(2) Γ is extensible and admissible, see the master test in [2, § A.1.1]; +(3) Γ is (sub-)geometric, see [2, § A.1.2]. +The principal tool of [2] was starting from a sufficiently large initial graph Γ0 +and extending it by adding one vertex at a time, see § A.4 in loc. cit. This is what is +done in this paper, too, see §5 and §6, where, without much explanation, we merely +state the updated results. The principal novelty of this paper is §3, where, due to +the more complicated geometric nature of the problem, extra vertices have to be +added in groups of up to three. The details are discussed in §3.3. +Remark 2.10. A technical, but crucial part of our (as well as any lattice-based) +approach is the fact that a geometric graph Γ has rk Γ ⩽ 20. It follows that, if +rk Γ′ = 20, any geometric overgraph Γ ⊃ Γ′ would be of the form sat(Γ′, K) for +some K ∈ G(Γ′), and all such finite index extensions of F(Γ′) can easily be found +using [16] (see the saturation lists in [2, § A.1.3]). Therefore, we introduce another +technical term: +Γ is acceptable if it is subgeometric and rk Γ ⩽ 19. +It is understood that, after each step of every algorithm, only acceptable graphs are +left for the further processing (and, thus, we do not need to add dozens of vertices +to reach line counts over 50), whereas each intermediate graph Γ of rank rk Γ = 20 +is excluded upon computing its saturation list and recording all “interesting” (cf. +Remark 3.24 below) geometric overgraphs to a global master list. +2.4. Classification of graphs in terms of girth. Following [5, 8], we subdivide +parabolic and hyperbolic graphs Γ according to the type of the minimal (in the sense +of Convention 2.7) affine Dynkin diagram Σ ⊂ Γ. The most important classes can +also be characterised in terms of the girth girth(Γ) (the length of a shortest cycle +in Γ, with the convention that the girth of a forest is ∞). Thus, Γ is called +• triangular, or ˜A2-, if girth(Γ) = 3, +• quadrangular, or ˜A3-, if girth(Γ) = 4, +• pentagonal, or ˜A4-, if girth(Γ) = 5, +• astral, or ˜D4-, if girth(Γ) ⩾ 6 and Γ has a vertex of valency ⩾ 4. +All other graphs are locally elliptic, i.e., one has val v ⩽ 3 for each vertex v ∈ Γ +(and we assume girth(Γ) ⩾ 6 to exclude a few trivial cases). +Given a graph Γ and a distinguished connected parabolic subgraph Σ ⊂ Γ, the +pencil Π induced by Σ is defined as +(2.11) +Π := Π(Γ ⊃ Σ) := Σ ∪ +� +l ∈ Γ +�� l · c = 0 for all vertices c ∈ Σ +� +. +This graph Π ⊃ Σ is parabolic as it is orthogonal to κΣ (see Convention 2.7 and +Remark 2.1). We have Γ = Π ∪ sec∗, where +(2.12) +sec∗ := sec∗(Γ ⊃ Σ) := +� +l ∈ Γ ∖ Σ +�� l · c = 1 for a vertex c ∈ Σ +� +. +The elements of sec∗ are called the (multi-)sections, or m-sections of Π, where the +integer m := l · κΣ is the multiplicity of a section l. If m = 1, the section is called + +8 +ALEX DEGTYAREV AND S�LAWOMIR RAMS +simple, otherwise, multiple. Fixing an order Σ = (c1, . . . , cn), we also consider +(2.13) +seci := sec(Γ ⊃ Σ ∋ ci) := +� +l ∈ Γ ∖ Σ +�� l · cj = δij for cj ∈ Σ +� +, +sec∗ +i := sec∗(Γ ⊃ Σ ∋ ci) := +� +l ∈ Γ ∖ Σ +�� l · ci = 1 +� +⊃ seci, +where 1 ⩽ i ⩽ n and δij is the Kronecker symbol. We have +(2.14) +each set sec∗ +i ⊃ seci, i = 1, . . . , n, is either elliptic or parabolic, +as it is orthogonal to the isotropic vector h − ci ̸= 0 (see Remark 2.1). +In the future, we almost never use the correct, but long notation referring to the +full flag, as Γ ⊃ Σ = (c1, . . . , cn) are always assumed fixed. +3. Triangular sets +A triangular set, or △-set, is an induced subgraph of an admissible graph whose +all connected components are of type ˜A2, A3, A2, or A1. Clearly, each triangular +pencil is a △-set, but not vice versa: a △-set may also be elliptic, i.e., have no +type ˜A2 components. Combinatorially, a △-set Θ is uniquely of the form +(3.1) +˜a2 ˜A2 ⊕ a3A3 ⊕ a2A2 ⊕ a1A1, +(˜a2, a3, a2, a1) ∈ N4, +and the coefficient quadruple (˜a2, a3, a2, a1) determines Θ up to isomorphism. The +isomorphism classes of △-sets (or coefficient quadruples) are called patterns. +We define the cardinality |θ| of a pattern θ as that of any of its representatives +and introduce the following order on the set of patterns: +(3.2) +θ′ ≺ θ′′ +iff +� +|θ′| < |θ′′| or +|θ′| = |θ′′| and (˜a′ +2, a′ +3, a′ +2, a′ +1) > (˜a′′ +2, a′′ +3, a′′ +2, a′′ +1), +where the coefficient quadruples are compared lexicographically. (Pay attention to +the reverse lexicographic order!) This order is not to be mixed with the perturbation +order defined in Convention 2.7. The latter, for △-sets, is easily described in terms +of the coefficient quadruples: θ′ ⊳ θ′′ if and only if (˜a′ +2, a′ +3, a′ +2, a′ +1) is obtained from +(˜a′′ +2, a′′ +3, a′′ +2, a′′ +1) by a finite sequence of elementary perturbations of the form +(˜a2, a3, a2, a1) �→ (˜a2, a3, a2, a1) + δ, +δ ∈ +� +(0, 0, −1, 1), (0, −1, 1, 0), (0, −1, 0, 2), (−1, 0, 1, 0) +� +provided that, at each step, the quadruple remains in N4. +3.1. Constructing triangular graphs. Let Γ be a triangular admissible graph, +cf. §2.4. Fix a type ˜A2 fiber Σ = (c1, c2, c3) ⊂ Γ and consider the pencil Π and +sets seci, i = 1, 2, 3, see (2.11) and (2.13), respectively. +Lemma 3.3. A triangular pencil Π has at most one multiple section; hence, +0 ⩽ |Γ| − |Π ∪ sec1 ∪ sec2 ∪ sec3| ⩽ 1. +Furthermore, if a multiple section s exists, it is disjoint from each seci, i = 1, 2, 3. +Proof. Consider a multiple section s and let e := h − c1 − c2 − c3 − s. +If s · c1 = s · c2 = s · c3 = 1, then e is orthogonal to h and each ci, i = 1, 2, 3. +Hence, e = 0 (see Remark 2.1) and any other line l intersects exactly one of c1, c2, +c3, s, implying both statements. + +LINES ON K3-QUARTICS VIA TRIANGULAR SETS +9 +Likewise, if s · c1 = s · c2 = 1 and s · c3 = 0, then e is an exceptional divisor and +e · s = e · c3 = 1 > 0. Hence, any other line l intersects at most one of c1, c2, c3, s, +as otherwise l · e < 0 and e would separate s and l, see (2.8). +□ +Lemma 3.4. Each set seci, i = 1, 2, 3, is a △-set. +Proof. In view of (2.14), it suffices to rule out the connected components of seci +containing ˜A3, A4, or D4. Each of the offending graphs has a chain +s1•−− +s2•−− +s3• +and another vertex l adjacent to at least one of s1, s2, s3. Then s := s1 + s2 + s3 +is a root, s · h = s · ci = 3, and l · s > 0; hence, e := −h + ci + s is an exceptional +divisor separating s1 and l, see (2.8). +□ +Now, our goal is to describe all geometric graphs Γ of size |Γ| ⩾ 53. In view of +Lemma 3.3, it suffices to consider trigonal pencils Σ ⊂ Π ⊂ Γ such that +|Π| + |sec1| + |sec2| + |sec3| ⩾ 52. +Clearly, we can also assume that Π ⊂ Γ is maximal with respect to (3.2) and the +edges c1, c2, c3 of Σ are ordered so that sec1 ≽ sec2 ≽ sec3. These assumptions give +rise to the following compatibility conditions on the patterns π ∋ Π and θi ∋ seci: +π ⊢ θ1 +if +3|θ1| + |π| ⩾ 52 and +� +θ1 ≼ π or θ1 is elliptic +� +; +(3.5) +(π ⊢ θ1) ⊢ θ2 +if +2|θ2| + |θ1| + |π| ⩾ 52 and θ2 ≼ θ1; +(3.6) +(π ⊢ θ1 ⊢ θ2) ⊢ θ3 +if +|θ3| + |θ2| + |θ1| + |π| ⩾ 52 and θ3 ≼ θ2. +(3.7) +In (3.5), we assume one of the terms, π or θ1, fixed and treat the condition as a +restriction on the other term. The other two conditions are restrictions on the last +term provided that the parenthesized part is fixed. +Since △-sets appear as sets of sections, for an ordered fiber ˜A2 ∼= Σ = (c1, c2, c3) +and △-set Θ we define Σ ⊔i Θ, where i = 1, 2, 3, as the graph obtained from the +disjoint union of Σ and Θ by connecting ci to each vertex v ∈ Θ by a simple edge. +This construction extends to patterns, producing an isomorphism class of graphs. +Checking the parameter quadruples (˜a2, a3, a2, a1) one-by-one, it is fairly easy to +compute the sets of patterns +P := {subgeometric triangular pencils}/∼=, and +T := {△-sets Θ such that ˜A2 ⊔i Θ is subgeometric}/∼=. +(To simplify the computation, for P one can start from Shimada’s list [21] of Jaco- +bian elliptic K3-surfaces, and for T one can take into account the bound val ci ⩽ 20 +found in [23].) Then, condition (3.5) becomes a binary relation from P to T . The +other conditions also descend to patterns, as do the rank functions: +(3.8) +rk(π) = |Π| − ˜a2(Π) + 2, +Π ∈ π ∈ P +rk( ˜A2 ⊔i θ) = |Θ| − ˜a2(Θ) + 4, +Θ ∈ θ ∈ T , +where ˜a2(·) stands for the number of parabolic components. +The next lemma is an immediate consequence of this computation. For the last +statement, we merely list all geometric extensions (e.g. using algorithms from [2, +Appendix A]) of the four graphs of rank 18 or 19, see (3.8); in fact, the sharp bound +in Lemma 3.9(3) is |Γ| ⩽ 29. +Lemma 3.9. In a geometric graph Γ, for any type ˜A2 fiber Σ one has: + +10 +ALEX DEGTYAREV AND S�LAWOMIR RAMS +(1) |seci| ⩽ 18 for each i = 1, 2, 3; +(2) if seci is elliptic, then |seci| ⩽ 15; +(3) if seci is elliptic and |seci| > 13, then |Γ| ⩽ 52. +⊳ +Remark 3.10. In view of Lemma 3.9, the compatibility condition (3.5) simplifies +to the form +π ⊢ θ1 +if +3|θ1| + |π| ⩾ 52 and θ1 ≼ π +more consistent with (3.6) and (3.7). Indeed, if θ1 is elliptic, we can assume that +|θ1| ⩽ 13; then necessarily |π| ⩾ 13, see (3.5), and θ1 ≼ π holds automatically. +Lemma 3.11. If Π ⊂ Γ is a maximal, with respect to (3.2), triangular pencil and +|Γ| > 52, then |Π| ⩾ 14. +Proof. As explained in Remark 3.10, in view of Lemma 3.9 we have |Π| ⩾ 13 and +|sec3| ⩽ |sec2| ⩽ |sec1| ⩽ |Π|. If |Π| = 13, then |sec3| = |sec2| = |sec1| = 13 by (3.5) +and, moreover, Π must have a multiple section s. Assuming that s · c1 = s · c2 = 1, +so that c1, c2, s constitute a triangle, the union (c1, c2, s) ⊔ sec3 ⊂ Γ is a triangular +pencil (see Lemma 3.3) with 16 vertices, contradicting the maximality of Π. +□ +3.2. Abundant collections. Let Γ0 ⊃ Σ = (c1, c2, c3) ∼= ˜A2 be a triangular +graph, i = ∅, 1, 2, 3 a parameter, and θ ∈ P (if i = ∅) or θ ∈ T (otherwise) a +pattern. An overgraph Γ ⊃ Γ0 is said to represent (Γ0, θ)i if +sec(Γ ⊃ Σ ∋ ci) ∈ θ and Γ ∖ seci = Γ0, +if i = 1, 2, 3 +Π(Γ ⊃ Σ) ∈ θ and Γ ∖ Π = Γ0 ∖ Σ, +if i = ∅. +A pair (Γ0, θ)i is called abundant if it cannot be represented by an acceptable graph, +see Remark 2.10. Both notions extend to a collection G of graphs: Γ represents +(G, θ)i if it represents (Γ0, θ)i for a graph Γ0 ∈ G, and (G, θ)i is abundant if so +is each (Γ0, θ)i, Γ0 ∈ G. Iterating, we extend both notions to a compatible (i.e., +satisfying the compatibility conditions from §3.1) collection of patterns +π ∈ P, +θi ∈ T , i = 1, . . . , n ⩽ 3. +Clearly, if π ⊢ . . . ⊢ θn is abundant, so is any π′ ⊢ . . . ⊢ θ′ +n with π ⊳ π′, . . . , +θn ⊳ θ′ +n. +Our proof of Theorem 1.1 essentially reduces to applying the algorithm in §3.3 +below to show that any compatible collection π ⊢ θ1 ⊢ θ2 ⊢ θ3 is abundant: indeed, +conditions (3.5)–(3.7) guarantee that on the way we will encounter all subgeometric +graphs Γ such that either |Γ| > 52 or rk Γ = 20, and in the latter case it would +suffice to analyze all geometric saturations of Γ. To this end, we introduce the +inductive notion of a ruled out collection: +• any abundant compatible collection is considered ruled out; +• in general, a compatible collection π ⊢ θ1 ⊢ . . . ⊢ θn, n ⩽ 2, is ruled out if +so is any compatible extension (π ⊢ θ1 ⊢ . . . ⊢ θn) ⊢ θn+1, θn+1 ∈ T . +By a machine aided computation, we establish the following statement; its proof is +given in §3.6, after we collect all the necessary facts in §3.3, §3.4 and §3.5. +Proposition 3.12 (see §3.6). Each compatible pair π ⊢ θ1, where +π ∈ P14 := +� +π ∈ P +�� |π| ⩾ 14 +� +and +θ ∈ T , +is ruled out. + +LINES ON K3-QUARTICS VIA TRIANGULAR SETS +11 +By the very definition, the assertion of Proposition 3.12 means that, for each +representative Γ of any compatible collection π ⊢ θ1 ⊢ θ2 ⊢ θ3, π ∈ P14, one has +either |Γ| = 52 or rk Γ = 20, and, moreover, all such representatives are encountered +in the course of the proof. The latter fact enables us to obtain the complete list +of representatives Γ of compatible collections π ⊢ θ1 ⊢ θ2 ⊢ θ3, π ∈ P14, such that +|Γ| > 52 and rk Γ = 20 (see Addendum 3.23). +3.3. Extending a graph by a triangular set. The heart of the computation is +an algorithm extending a given subgeometric graph Γ0 by a given pattern θ ∈ T , the +goal being listing all subgeometric overgraphs Γ ⊃ Γ0 representing (Γ0, θ)i (where +i = ∅, 1, 2, 3 is also fixed, see §3.2). The elements of Γ ∖ Γ0 are referred to as +sections, whereas the connected components of Γ ∖ Γ0 ∈ θ are polysections. The +algorithm is similar to that of [2], except that we can no longer guarantee that the +sections are pairwise disjoint. Therefore, instead of adding to Γ one section at time, +we fix θ in advance and add whole polysections, in the order ˜A2, A3, A2, A1. +Convention 3.13. From now on, following [2, § A.3], we identify a section v with +its support +supp v := +� +u ∈ Γ0 +�� v · u = 1 +� +and thus treat it as a subset of Γ0. We also keep the notation +Γ0 ⊔ v +and +Γ0 ⊔ v(m) +for a multiset v (which we no longer assume sorted) and (|v| × |v|)-matrix m. As +explained in §2.3, only acceptable graphs are retained after each step. +In practice, we start with computing the group G0 := Aut Γ0 and set +(3.14) +S(Γ0) := +� +s ⊂ Γ0 +�� s ∩ Σ = fixed +� +of sections of Γ0 satisfying extra conditions imposed by the problem at hand. (Here, +fixed ⊂ Σ is a certain subset fixed in advance. We can also take into account a +few obvious geometric restrictions, but this is not crucial: “wrong” sections are +immediately ruled out by the preliminary tests in §2.3(1). We omit many other +technical tweaks, referring to the code [4] as the ultimate source.) Then, running +the tests cited in §2.3(2) and (3), we compute the sets +(3.15) +A1(Γ0) := +� +v ∈ S(Γ0) +�� Γ0 ⊔ v is acceptable +� +, +m(Γ0) := +� +v ∈ A1(Γ0)dim m �� Γ0 ⊔ v(m) is acceptable +� +, +where m = A2, ˜A2, 2A1, A3 (in this order). Certainly, the tests are applied to a +single representative of each G0-orbit; in what follows (cf., e.g., Remark 3.19) this +convention is taken for granted. This computation is aborted if a “required” list is +empty (e.g., if A2(Γ0) = ∅ whereas θ contains ˜A2 or A3, cf. the next remark). +Remark 3.16 (a technical detail). The set A2(Γ0) is used in the computation of +˜A2(Γ0): we consider only those triples (v1, v2, v3) for which (vi, vj) ∈ A2(Γ0) for +all 1 ⩽ i < j ⩽ 3. Likewise, both A2(Γ0) and 2A1(Γ0) are used in the computation +of A3(Γ0). Furthermore, 2A1(Γ0) is used at all subsequent steps: when iterating +Γn−1 ⊔ v(mn) = +� +Γ0 ⊔ u(·) +� +⊔ v(mn), +in (3.17) below, we check first that (u, v) ∈ 2A1(Γ0) for all u ∈ u, v ∈ v. + +12 +ALEX DEGTYAREV AND S�LAWOMIR RAMS +Now, let θ = ˜a2 ˜A2 + a3A3 + a2A2 + a1A1, so that N := ˜a2 + a3 + a2 + a1 is +the number of components, and let m1 ⩾ . . . ⩾ mN be the types of the components +of θ ordered via ˜A2 > A3 > A2 > A1). Then, we start from +S0 := {Γ0} +and run the computation in up to N steps. +Step n ⩾ 1: for each graph Γn−1 ∈ Sn−1, we pick a single representative v of each +(Aut Γn−1)-orbit on mn(Γ0) and use the tests of §2.3(2) and (3) to compute +(3.17) +Sn(Γn−1) := +� +Γv := Γn−1 ⊔ v(mn) +�� Γv is acceptable +� +. +The step concludes by uniting all sets Sn(Γn−1), Γn−1 ∈ Sn−1, obtained followed +by retaining a single representative of each graph isomorphism class. +The algorithm terminates either upon the completion of Step N (resulting in +a list SN(Γ0) to be processed by other means) or when one of the previous steps +results in an empty list Sn(Γ0) = ∅, implying that (Γ0, θ) is abundant. +Remark 3.18. Both auto- and isomorphisms of graphs are computed using the +digraph package in GAP [9]. All morphisms are restricted: we assume the fiber Σ +fixed as a set and, typically, one or two edges ci ∈ Σ fixed pointwise, so that the +set S(Γ0) in (3.14) be invariant. +Remark 3.19 (a technical detail). Each time the matrix m changes, i.e., whenever +mn > mn+1, we recompute the (relevant) sets m(Γn) for each graph Γn ∈ Sn and +use these new lists in the subsequent steps. Instead of starting from scratch, as +in the case of Γ0, we merely run the tests on the ready lists m(Γm) for the last +subgraph Γm ⊂ Γn for which they have been computed. +3.4. Processing several patterns. The material of this section is of a purely +technical nature; however, it is the tweak described here that makes the computa- +tion much faster and eventually helps it to terminate reasonably fast. +Typically, we fix a subgeometric graph Γ0 and try to rule out a whole collection +of patterns T (Γ0). Since patterns tend to have similar initial sequences, processing +them all one-by-one would force us to repeat the same steps of the computation over +and over again. To avoid the repetition and remove a number of redundant steps, we +sort the patterns in the direct lexicographic order and process them simultaneously, +organizing the computation into four layers: the outermost ˜A2, A3, A2, and the +innermost A1. +Each inner layer starts from a certain intermediate graph Γ and processes a +collection of patterns T (Γ). If the algorithm terminates prematurely, at a certain +pattern θ, we conclude that (Γ, θ) is abundant, and hence so is (Γ, θ′) whenever +θ ⊳ θ′. This fact is reported to the previous layer, where the information is consol- +idated and often results in excluding the graph Γ and/or some patterns from the +further consideration. We refer to the code [4] for the precise details (we implement +each next layer as a hook within the previous one, where it is used to modify the +intermediate lists); here, we merely illustrate the paradigm by the following simple +example. +Example 3.20. Assume that the patterns to be considered are +T (Γ0) = +� +2A2 ⊕ 6A1, 3A2 ⊕ 4A1, 4A2 ⊕ 2A1 +� +, + +LINES ON K3-QUARTICS VIA TRIANGULAR SETS +13 +so that only two layers of computation are required. We run the first two steps of +the A2-layer, resulting, say, in a list S2 = {Γ′ +2, Γ′′ +2}, and switch to the A1-layer for +each of the two graphs. Assume that this inner layer terminates at +• step 4 for Γ′ +2 ⇒ (Γ′ +2, 4A1), (Γ′ +2, A2 ⊕ 3A1), (Γ′ +2, 2A2 ⊕ 2A1) are abundant, +• step 5 for Γ′′ +2 ⇒ (Γ′′ +2, 5A1), (Γ′′ +2, A2 ⊕ 4A1) are abundant. +(Obviously, 4A1 ⊳ A2 ⊕ 3A1 ⊳ 2A2 ⊕ 2A1 and 5A1 ⊳ A2 ⊕ 4A1.) We conclude +that Γ′ +2 can be excluded from S2 and that both 2A2 ⊕ 6A1 and 3A2 ⊕ 4A1 can be +excluded from T (Γ0). Therefore, we can run two more steps of the A2-layer on the +new reduced list {Γ′′ +2}, followed by the A1-layer on the result. (The A1-layer after +Step 3 can be skipped as θ = 3A2 ⊕ 4A1 has already been ruled out!) +If it were not for Γ′′ +2 (e.g., if the A1-layer terminated at a step ⩽ 4 for each of the +two graphs), we would have stopped immediately, as all elements of T (Γ0) would +have been ruled out by the A1-layer after Step 2. +3.5. The aggressive version. In certain cases, one can argue that, in order to +achieve the goal |Γ| ⩾ 53, the overgraph Γ ⊃ Γ0 must be spanned over Γ0 by a few +pairwise disjoint vertices independent over Γ0. (Precisely, this condition means that +F(Γ) ⊗ Q is generated over F(Γ0) ⊗ Q by (rk Γ − rk Γ0) pairwise disjoint vertices.) +In this case, we switch to the aggressive version of the algorithm, i.e., we +• add disjoint vertices only (the A1-layer), +• disregard the extra vertices that do not increase rank, and +• check the saturation lists of all intermediate graphs, including Γ0, +cf. the progressive mode in [2, § A.4.4]. +Precisely, this approach is used +• in the proof of Lemma 3.9(3): we add up to two disjoint vertices, and +• at the steps fixed = {c2} or {c3} in §3.6 below (cf. also Remark 4.11), when +extending a graph Γ0 of the submaximal rank rk Γ0 = 19. +3.6. Proof of Proposition 3.12. As stated, the proof is an explicit machine aided +computation using the algorithm described in §3.3 and §3.4. It runs in two steps: +first, for each pattern θ1 ∈ T , we rule out almost all compatible patterns π ∈ P14; +the set of these patterns is denoted by P− +14(θ1). Then, for each π ∈ P14, we rule +out the remaining patterns θ1 ∈ T such that π ⊢ θ1 and π /∈ P− +14(θ1). +Remark 3.21. The choice of the set P− +14(θ1) at the first step looks quite arbitrary, +and indeed so it is. As a rule, we let π ∈ P− +14(θ1) if π ⊢ θ1 and +rk π ⩽ rk(Σ ⊔1 θ1). +However, a few border cases are subject to further manual tweaking, which is based +on experiments. More precisely, depending on the values r := rk(Σ⊔1 θ1), n := |θ1|, +the following coefficient quadruples (˜a2, a3, a2, a1) are excluded from P− +14(θ1): +(r, n) = (12, 11): +(∗, ∗, ∗, ∗), +(12, 12): +(∗, ∗, ∗, ∗), +(13, 11): +(∗, ∗, ∗, ∗), +(13, 12): +(3, ∗, ∗, ∗), (2, ∗, ∗, ∗), (1, ∗, ∗, ∗), +(13, 13): +(2, ∗, ∗, ∗), (1, ∗, ∗, ∗), +(14, 11): +(∗, ∗, ∗, ∗), + +14 +ALEX DEGTYAREV AND S�LAWOMIR RAMS +(14, 12): +(3, ∗, ∗, ∗), (2, ∗, ∗, ∗), (1, ∗, ∗, ∗), +(14, 13): +(2, ∗, ∗, ∗), (1, ∗, ∗, ∗), +(14, 14): +(2, ∗, ∗, ∗), (1, ∗, ∗, ∗), +(15, 11): +(∗, ∗, ∗, ∗), +(15, 13): +(2, 0, 0, ∗), (1, 0, ∗, ∗), +(15, 14): +(2, 1, 0, ∗), (2, 0, ∗, ∗), (1, 0, ∗, ∗), +(15, 15): +(1, ∗, ∗, ∗) +(where, as usual, ∗ means any value). +At the first step, we start with the graph +Γ0 := Σ ⊔1 Θ1, +Θ1 ∈ θ1 ∈ T , +and use §3.4 to find all graphs Γ representing (Γ0, π)∅, π ∈ P− +14(θ1). Technically, +we extend Γ0 by a pattern θ such that ˜A2 ⊕ θ = π. Thus, we let fixed = ∅ in (3.14) +and use graph auto-/isomorphisms preserving Σ ∋ c1 (see Remark 3.18). For each +graph Γ on the resulting list SN, we consider the full set T (Γ) of patterns θ2 ∈ T +satisfying (3.6), and run the same algorithm with fixed = {c2} and graph morphisms +preserving c1 and c2. +Remark 3.22. Strictly speaking, we should have run the algorithm once more, +using fixed = {c3} and patterns θ3 satisfying (3.7). However, our thresholds are +chosen so that the list SN resulting from the first run consists of relatively few +graphs of rank 19 (for which the aggressive version is used, see §3.5) and very few +graphs of rank 18, for which the algorithm terminates fast and rules everything out. +At the second step, we start with a pencil +Γ0 := Π ∈ π ∈ P14 +and use §3.4 to find all graphs Γ representing (Γ0, θ1)1, π ⊢ θ1 and π /∈ P− +14(θ1). We +let fixed = {c1} in (3.14) and use graph morphisms preserving Σ ∋ c1. As above, +the relatively few graphs obtained, all of rank 19 or 18, are ruled out by the next +run, using fixed = {c2} and θ2 ∈ T compatible in the sense of (3.6). This completes +the proof of Proposition 3.12, as well as of Addendum 3.23 below. +□ +As explained right after Proposition 3.12, before discarding a graph Γ of rank 20, +we analyze its geometric finite index extensions and record those of size greater +than 52. The result of this analysis is stated below. +Addendum 3.23. Let Γ be a geometric representative of a compatible collection +π ⊢ θ1 ⊢ θ2 ⊢ θ3, π ∈ P14, such that rk Γ = 20 and |Γ| > 52. Then Γ is one of the +eight smooth configurations found in [8], see the first eight rows of Table 1. +⊳ +Remark 3.24. In order to produce a plethora of examples of large configurations +of lines, when discarding the graphs of rank 20 (see Remark 2.10) we collected all +extended graphs with at least 48 lines or at least six exceptional divisors; the results +are found in [4]. In particular, in addition to the surfaces listed in Table 1, we found +but one quartic with 52 lines and two nodes and two quartics with 50 lines and one +node each. Besides, there are quite a few quartics with non-empty singular locus +and 48 lines, suggesting once again that 48 is a reasonable threshold to cut the +classification. + +LINES ON K3-QUARTICS VIA TRIANGULAR SETS +15 +4. Smooth quartics +In this section we temporarily assume that the polarized lattice S ∋ h contains +no exceptional divisors; this will be used later in our study of line configurations +on smooth quartic surfaces X4 ⊂ P3. +In this case, we need to change the notion of admissible lattice/graph. Namely, +a polarized lattice S ∋ h is called smooth, or s-admissible, if it contains neither +exceptional divisors (e2 = −2, e · h = 0) nor 2-isotropic vectors (e2 = 0, e · h = 2). +A graph Γ is smooth, or s-admissible, if so is the lattice F(Γ). +If S ∋ h is smooth, then rt(S, h) = ∅; hence, automatically, ∆ = ∅ in (2.2) and +we have a well-defined Fano graph +(4.1) +Fn(S, h) := Fn∅(S, h) = root1(S, h). +An crucial consequence of (4.1) is the fact that Fn is monotonous: +(4.2) +if S′ ⊃ S ∋ h are smooth, then Fn(S′, h) ⊃ Fn(S, h). +Lemma 4.3. Let Γ be a smooth graph and Σ = (c1, c2, c3) ⊂ Γ a triangle. Then: +(1) Σ has a unique 3-section c0 := h − c1 − c2 − c3 ∈ Γ; +(2) the pencil Π as in (2.11) and △-sets seci as in (2.13) have no connected +components of types A2 or A3. +Proof. Clearly, c0 as in Statement (1) is a 3-section in the lattice spanned by h +and Σ. By (4.2), it remains a 3-section in any larger smooth lattice/graph. +If the pencil Π has a pair (s1, s2) ∼= A2, by (4.2) it also has s3 := κΣ − s1 − s2, +so that (s1, s2, s3) ∼= ˜A2. If Π has (s1, s2, s3) ∼= A3, then κΣ − s1 − s2 − s3 is an +exceptional divisor. The same argument applies to each set seci, i = 1, 2, 3, except +that we replace κΣ with (c0 + . . . + c3) − ci = κΣ + c0 − ci. +□ +In view of Lemma 4.3(1), each type ˜A2 fiber Σ0 := Σ = (c1, c2, c3) gives rise to +three more, viz. Σi := (c0, . . . , ˆci, . . .), i = 1, 2, 3 (where, as usual, ˆci indicates that +ci has been omitted). Thus, we can shift the paradigm and, instead of considering +a pencil Π and three sets seci, we can speak about “blending” four pencils +Π0 := Π = Π(Γ ⊃ Σ0), +Πi := Σi ⊔ seci = Π(Γ ⊃ Σi), +i = 1, 2, 3. +Assuming, as above, that Π is a maximal pencil in Γ, we can replace (3.5)–(3.7) +with a stronger set of compatibility conditions: +π ⊢ θ1 +if +3|θ1| + |π| ⩾ 48 and θ′ +1 ≼ π; +(4.4) +(π ⊢ θ1) ⊢ θ2 +if +2|θ2| + |θ1| + |π| ⩾ 48 and θ′ +2 ≼ θ′ +1; +(4.5) +(π ⊢ θ1 ⊢ θ2) ⊢ θ3 +if +|θ3| + |θ2| + |θ1| + |π| ⩾ 48 and θ′ +3 ≼ θ′ +2. +(4.6) +Here, ′ stands for the operator Θ �→ Θ′ := ˜A2 ⊔Θ and its natural descent to the set +of patterns. (Recall also that, for Theorem 1.2, we need to change the threshold to +|Γ| ⩾ 49.) In particular, from (4.4)–(4.6) we immediately conclude that +(4.7) +|Π| ⩾ 15, +cf. Lemma 3.11. Indeed, taking into account the 3-section given by Lemma 4.3, we +can rewrite (4.6) in the form |θ′ +3| + |θ′ +2| + |θ′ +1| + |π| ⩾ 57 = 48 + 9, and it remains +to observe that the assumption θ′ +3 ≼ θ′ +2 ≼ θ′ +1 ≼ π implies |θ′ +3| ⩽ |θ′ +2| ⩽ |θ′ +1| ⩽ |π|. +Thus, unlike Proposition 3.12, we do not need to introduce an analogue Ps +15 of the +set P14: the lower bound (4.7) would follow from the compatibility assumptions. + +16 +ALEX DEGTYAREV AND S�LAWOMIR RAMS +Now, a computation similar to (but much faster than) that of §3 yields the +following result (cf. Proposition 3.12; we retain the terminology of §3 and denote +by Ps ⊂ P and T s ⊂ T the sets of patterns appearing in smooth graphs). +Proposition 4.8 (cf. §3.6). Each compatible collection π ⊢ θ1 ⊢ θ2 ⊢ θ3, as in +(4.4)–(4.6), with π ∈ Ps and θ1, θ2, θ3 ∈ T s, either is ruled out or has one of the +five graphs +(4.9) +Z49, +(50), +Z50, +(52), +Z52 +(see Table 1) as a geometric representative. +⊳ +Similar to Proposition 3.12, this statement means that, with the five exceptions +listed in (4.9), any geometric representative Γ of a collection as in the hypotheses +has rk Γ = 20 and, moreover, all such representatives of rank 20 are encountered +and discarded in the course of the proof. As in §3, prior to discarding a graph +we compute all its geometric finite index extensions, thus arriving at the following +complete list of geometric rank 20 Fano graphs Γ with |Γ| ⩾ 48 + 1 (the extra 1 +standing for the 3-section given by Lemma 4.3). +Addendum 4.10. Let Γ be a geometric representative of a compatible collection +π ⊢ θ1 ⊢ θ2 ⊢ θ3 as in (4.4)–(4.6), where π ∈ Ps, θ1, θ2, θ3 ∈ T s, and rk Γ = 20. +Then Γ is one of the 21 smooth rank 20 configurations found in Table 1. +⊳ +Remark 4.11. Due to the lower threshold |Γ| ⩾ 49, occasionally we do have to run +the algorithm till the very last step fixed = {c3}, cf. Remark 3.22. This last step is +quite expensive (as the set of sections to begin with is quite large), but fortunately +it has to be done for four graphs only. This is yet another indication of the fact +that taking the classification down to 48 or fewer lines is hardly feasible. +Remark 4.12. In the smooth case, we can further reduce the overcounting by a +number of tricks based on the monotonicity property (4.2), using all lines present in +the lattice rather than only those added explicitly. Most notably, before switching +to a next step fixed = {ci}, i ⩽ 3, we can replace each graph Γ with the union +Π ∪ sec1 ∪ . . . ∪ seci−1 +in +Fn(F(Γ)) +and then retain a single representative of each isomorphism class of the list obtained. +Furthermore, in the subsequent computation we can impose an extra condition that +the above union should remain fixed. We refer to [5] for further details. +5. Quadrangular graphs +In this section the K3-quartic X4 is allowed to have singular points as in §3. +Consider an ˜A3-graph Γ and fix a quadrangle Σ := (c1, c2, c3, c4) ⊂ Γ. We assume +the edges ci ∈ Σ numbered cyclically and ordered consecutively: +ci+4 = ci, +ci · ci±1 = 1, +ci · ci±2 = 0. +In addition to (2.11)–(2.13), we consider the sets +sec∗ +ij := sec∗ +i ∪ sec∗ +j, +i = j ± 2. +The assumption that girth(Γ) = 4 implies that +sec∗ +13 ∩ sec∗ +24 = ∅ +and +each graph sec∗ +i is discrete. + +LINES ON K3-QUARTICS VIA TRIANGULAR SETS +17 +The elements of each seci are simple sections of Π, and those of +sec∗ +i ∖ seci = sec∗ +j ∖ secj = sec∗ +i ∩ sec∗ +j, +j = i ± 2, +are double sections; the set of all (multi-)sections of Π is sec∗ = sec∗ +13 ∪ sec∗ +24. +From now on, we state our results for geometric rather than subgeometric graphs; +in other words, we assume that Γ admits a triangle free geometric saturation. +Lemma 5.1. In a geometric quadrangular graph Γ ⊃ Σ as above, one has +(1) |seci| ⩽ 10 and |sec∗ +i | ⩽ 10 (sharp bounds), +(2) |sec∗ +ij| ⩽ 20 (a sharp bound), and +(3) |sec∗| ⩽ 32 (the best known example being 30). +Proof. Statement (1) is a direct computation: since each sec∗ +i is discrete, the union +Σ ∪ sec∗ +i depends on just two parameters, which can take the following values: +(5.2) +pi := |seci| += 0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +qi := |sec∗ +i ∖ Θi| ⩽ 8 +7 +6 +5 +5 +4 +4 +3 +2 +1 +0 +The bound in (2) follows from (1) (assuming pi ⩾ pj, we have |sec∗ +ij| ⩽ 2pi + qi), +and its sharpness is established by an explicit construction (obtained in the next +computation). Finally, statement (3) is also obtained by a computation similar to +[2, § B.5]: assuming that +(5.3) +|sec1| ⩾ |sec3|, +|sec2| ⩾ |sec4|, +|sec∗ +13| ⩾ |sec∗ +24|, +we start with a “standard” graph Σ ∪ sec∗ +1, letting (p1, q1) = (10, 0), (9, 1), (9, 0), +(8, 2), (8, 1), or (7, 3), and build all possible consecutive extensions +(5.4) +Γ := (Σ ∪ sec∗ +1) ∪ sec3 ∪ sec2 ∪ (sec∗ +2 ∖ sec2) ∪ sec4 +via discrete sets; the minimal size of each set to be added is determined using (5.2), +(5.3), and the goal |Γ| ⩾ 37. In most cases, this algorithm terminates (meaning +that each sufficiently large graph admitting a geometric triangle free saturation is +unacceptable, cf. Remark 2.10) at the very first nontrivial step sec3; in very few +cases we also need to use sec2. +□ +Remark 5.5. In the proof of Lemma 5.1 and Proposition 5.6 below, the essential +difference from [2] is that we do not limit the number of lines intersecting both given +ones (the presence of biquadrangles and longer “polyquadrangles”); this makes the +computation slightly more involved. +Proposition 5.6. For a geometric quadrangular graph Γ one has |Γ| ⩽ 48. +Remark 5.7. It is unlikely that the bound given by Proposition 5.6 is sharp: the +best example that we found has 39 lines (see also [2, Proposition 6.14]). +Proof of Proposition 5.6. The proof is a computation similar to [2, § C.3]. In view +of Lemma 5.1, it suffices to consider all quadrangular pencils Π of size |Π| ⩾ 17; +the list of such pencils admitting a polarization is compiled using [21]. Then, under +the assumptions of (5.3), we must have 4p1 + 2q1 + |Π| ⩾ 49. Similar to (5.4), we +start from a pencil Π and build a list of consecutive acceptable extensions +Γ := Π ∪ sec1 ∪ (sec∗ +1 ∖ sec1) ∪ sec3 ∪ sec2 ∪ (sec∗ +2 ∖ sec2) ∪ sec4. +Due to our modest goal |Γ| ⩾ 49, de facto the algorithm terminates at the first or, +occasionally, second step, so that we never need to consider even sec3. +□ + +18 +ALEX DEGTYAREV AND S�LAWOMIR RAMS +6. Other types of graphs +For the other types (in the sense of §2.4) of graph, the computation runs exactly +as in [2], and we merely state the updated results below. Remarkably, the upper +bounds obtained are exactly the same as in the smooth case (see [5]); furthermore, +unlike the case of octics (see [2]), all extremal configurations are smooth. +6.1. Pentagonal graphs. Recall that the assumption girth(Γ) = 5 implies that, +for a fiber ˜A4 ∼= Σ ⊂ Γ, one has +• sec∗ = sec1 ∪ . . . ∪ sec5, i.e., all sections are simple; +• each graph sec∗ +i = seci, i = 1, . . . , 5, is discrete. +The following bounds are sharp, and there are but two geometric pentagonal +graphs with 30 vertices, viz. Φ′ +30 and Φ′′ +30 (see [5]). Both represent configurations +of lines on smooth quartic surfaces only. +Lemma 6.1 (cf. [2, Lemma 7.6]). Let Γ be a geometric pentagonal graph, and let +Σ ⊂ Γ be a type ˜A4 subgraph. Then: +(1) one has |sec∗| ⩽ 16; +(2) if |sec∗| ⩾ 14, then |Γ| ⩽ 29. +⊳ +Proposition 6.2. One has |Γ| ⩽ 30 for any geometric pentagonal graph Π. +Proof. In view of Lemma 6.1, it suffices to consider pentagonal pencils Π such that +|Γ| ⩾ 17; they can be found using [21]. +□ +6.2. Astral graphs. We number the vertices (c1, . . . , c5) of a type D4 fiber Σ so +that the central vertex c1 is the one of valency 4 in Σ. Recall that the assumption +girth(Γ) ⩾ 6 implies that, for a fiber ˜D4 ∼= Σ ⊂ Γ one has +• sec∗ = sec1 ∪ . . . ∪ sec5, and +• the graph sec∗ is discrete. +Note though that it is not true that all sections are simple: the elements of sec1 +are double sections. (Recall that κΣ = 2c1 + c2 + . . . + c5.) +The following bounds are sharp, and the only geometric astral graph with 27 +vertices is ∆′ +27 (see [5]); it is represented by a unique smooth quartic surface. +Lemma 6.3 (cf. [2, Lemma 7.3]). Let Γ be a geometric astral graph and Σ ⊂ Γ a +type ˜D4 subgraph whose central vertex c1 has maximal valency in Γ. Then: +(1) one has |sec∗| ⩽ 12; +(2) if |sec∗| ⩾ 11, then |Γ| ⩽ 27. +⊳ +Proposition 6.4. One has |Γ| ⩽ 27 for any geometric astral graph Π. +Proof. In view of Lemma 6.3, it suffices to consider astral pencils Π with |Γ| ⩾ 18; +they can be found using [21]. +□ +6.3. Locally elliptic graphs. Let us recall, that the case of locally elliptic graphs +was considered in [2]. We have the inequality +(6.5) +|Γ| ⩽ 29 +for all geometric locally elliptic graphs Γ (see [2, (7.1)]). Machine-aided experi- +ments suggest that the sharp bound is |Γ| ⩽ 25 with a unique graph Λ25 that +attains the maximum , but such considerations are of no importance for the proof +of Theorem 1.1. + +LINES ON K3-QUARTICS VIA TRIANGULAR SETS +19 +7. Proofs +In order to render our exposition self-contained, we recall certain results from [2] +in §7.1, before presenting the proofs of the main results of the paper in §7.2, §7.3. +7.1. Fano graphs of K3-quartics. Let X4 ⊂ P3 be a complex degree-4 surface +with at worst Du Val (aka A–D–E, or simple) singularities and let π: ˜X4 → X4 +be the minimal resolution of its singularities. Denote h := π∗OX4(1) ∈ NS( ˜X4). +Recall that ˜X4 is a K3-surface. In particular, given an irreducible curve C ⊂ ˜X4, +we have +(7.1) +C · h = 1 and C2 = −2 if and only if π(C) is a degree one curve on X4. +The curves that satisfy (7.1) are called lines on ˜X4; they are obviously smooth and +rational. We follow [5] and define the (plain) Fano graph of the quartic X4 as the +loop free graph with vertices +(7.2) +Fn(X4) := +� +(−2)-curves C ⊂ ˜X4 with C · h = 1 +� +and each pair of vertices v, w ∈ Fn(X4) connected by an edge of multiplicity v · w. +(Here and below, we always consider the intersection form ”·” on NS( ˜X4).) +Recall that, by [2, (4.5)], +(7.3) +the graph Fn(X4) of a K3-quartic with at least 25 lines is hyperbolic. +General theory of lattice-polarized K3-surfaces (Nikulin [15], Saint-Donat [19]; +cf. also [8, Theorem 3.11] and [6, Theorem 7.3]) yields the following statement. (As +in [2, Convention 1.4], we say that the lattice NS( ˜X4) is spanned by lines if it is a +finite index extension of its sublattice generated by the classes of lines on ˜X4 and +the quasi-polarization h, i.e., it is spanned by lines and h over Q.) +Theorem 7.4 (see [2, Theorem 3.9]). A graph Γ is geometric if and only if one +has Γ ∼= Fn(X4) for a quartic X4 such that NS( ˜X4) is spanned by lines. +⊳ +Remark 7.5. As in the case of K3-octics, by [2, Lemma 2.8], in order to study the +maximal number of lines we can restrict our attention to the case when the lattice +NS( ˜X4) is spanned by lines and exceptional divisors (see also [2, § 8]) +Obviously, for a quartic X4 with non-empty singular locus the graph Fn(X4) +does not completely describe the configuration of lines on X4. The latter can be +inferred from the bi-colored extended Fano graph +(7.6) +Fnex(X4) := +� +(−2)-curves C ⊂ ˜X4 with C · h ⩽ 1 +� +, +with the colour of each vertex C defined as C · h. For such graphs we have a more +general statement. +Theorem 7.7 (see [2, Theorem 3.10]). A bi-colored graph Γ′ is geometric if and +only if Γ′ ∼= Fnex(X4) for a K3-quartic X4 ⊂ P3. +⊳ +7.2. Proof of Theorem 1.1. By Theorem 7.7, the assertion of Theorem 1.1 is +equivalent to the statement that there are no +(7.8) +geometric bi-colored graphs Γ′ such with |sp1 Γ′| ⩾ 53 and |sp0 Γ′| ̸= 0, +where spj Γ′ stands for the induced subgraph of Γ′ given by all its vertices of color +j = 0, 1. Moreover, by (7.3), the (plain) graph Γ := sp1 Γ′ is a Σ-graph for a certain +affine Dynkin diagram Σ ⩾ ˜A2. + +20 +ALEX DEGTYAREV AND S�LAWOMIR RAMS +The monotonicity given by [2, Lemma 2.8] combined with the consideration of +§6 implies that the graph Γ is neither pentagonal (see Proposition 6.2), nor astral +(see Proposition 6.4), nor locally elliptic (see (6.5)). Finally, Proposition 5.6 shows +that +Γ is triangular. +Then, by Lemma 3.11 and Proposition 3.12, we necessarily have rk(Γ) = 20, upon +which Addendum 3.23 implies that Γ is one of the eight smooth configurations +found in [8]. For each graph Γ obtained we compute its extended saturation(s) +satex Γ and check that none of the resulting bi-colored graphs has a vertex of color +zero (exceptional divisor), completing the proof of Theorem 1.1. +□ +7.3. Proof of Theorem 1.2. When dealing with smooth quartics, we can confine +ourselves to the case where the lattice NS(X4) is spanned by lines, see Remark 7.5 +and (4.2). Theorem 7.4 reduces the proof to the classification of all graphs Γ such +that +Γ is geometric and |Γ| ⩾ 49; +then, by (7.3), the (plain) graph Γ is a Σ-graph for a certain affine Dynkin diagram +Σ ⩾ ˜A2. As in the proof of Theorem 1.1, we infer that Γ is neither quadrangular +(Proposition 5.6), nor pentagonal (Proposition 6.2), nor astral (Proposition 6.4), +nor locally elliptic (see (6.5)). +For a geometric triangular graph Γ with at least 49 vertices and rk(Γ) < 20, we +apply Proposition 4.8 to show that Γ is one of the five graphs (4.9). Otherwise, by +Addendum 4.10, the graph Γ is one of the rank 20 graphs that appear in Table 1. +To complete the deformation classification, let Γ be one of the graphs in Table 1. +As part of our study of the saturation lists, we observe that the only geometric +finite index extension of F(Γ) is the trivial one; hence, one has +(7.9) +NS(X4) = F(Γ) +and +Oh(NS(X4)) = Aut Γ +for any smooth quartic X4 ⊂ P2 with Fn X4 ∼= Γ, and the latter group is found +using the digraph package in GAP [9]. According to [8, Theorem 3.9], the equilinear +deformation families of such quartics are in a bijection with the primitive isometric +embeddings +(7.10) +S := F(Γ) ֒→ L = 2E8 ⊕ 3U +regarded up to polarized autoisometry of S ∋ h and autoisometry of L preserving a +coherent orientation of maximal positive definite subspaces of L ⊗ R (the so-called +positive sign structure); such a family is real if and only if (7.10) admits a polarized +autoisometry reversing the positive sign structure. Hence, to complete the proof, +we classify embeddings (7.10) using Nikulin’s [16] theory of discriminant forms and +either Gauss [10] theory of binary quadratic forms (in the definite case rk T = 2) +or Miranda–Morrison [13] theory (in the indefinite case rk T ⩾ 3). +□ +Remark 7.11. The groups Sym X4 and Aut(X4, h) in Table 1 are computed as +the subgroups of (7.9) that, respectively, act identically on discr S or extend to an +appropriate autoisometry of L: the latter is required to preserve the positive sign +structure (if rk T = 2) or act on T by ±1 (if rk T ⩾ 3). + +LINES ON K3-QUARTICS VIA TRIANGULAR SETS +21 +7.4. Proof of Addendum 1.3. As stated in [8, Addendum 1.4], the number of +real lines does take all values in the range {0, . . ., 48}. Next, we recall that, when +counting the number of real lines on smooth quartics, it suffices to consider only +those quartics X4 whose all lines are real (with respect to a certain real structure +σ: X4 → X4, see [8, Proposition 3.10] or [5, Theorem 2.7]), and the latter is the +case if and only if the generic transcendental lattice T has a sublattice isomorphic +to [2] or U(2), see [8, Lemma 3.8]. Hence, the statement of the addendum follows +from Table 1 which lists all configurations of more than 48 lines and their respective +transcendental lattices. +□ +References +1. J. H. Conway and N. J. A. Sloane, Sphere packings, lattices and groups, Grundlehren der +Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 290, +Springer-Verlag, New York, 1988, With contributions by E. Bannai, J. Leech, S. P. Norton, +A. M. Odlyzko, R. A. Parker, L. Queen and B. B. Venkov. MR 920369 (89a:11067) +2. A. Degtyarev and S. Rams, Counting lines with Vinberg’s algorithm, 2021, To appear, +arXiv:2104.04583. +3. +, Lines on K3-sextics, in preparation, 2021. +4. +, Ancillary files for the paper: Lines on K3-quartics via triangular sets, 2022, available +on the arXiv as ancillary files for this preprint. +5. Alex Degtyarev, Lines on Smooth Polarized K3-Surfaces, Discrete Comput. Geom. 62 (2019), +no. 3, 601–648. MR 3996938 +6. +, Smooth models of singular K3-surfaces, Rev. Mat. Iberoam. 35 (2019), no. 1, 125– +172. MR 3914542 +7. +, Lines in supersingular quartics, J. Math. Soc. Japan 74 (2022), no. 3, 973–1019. +MR 4484237 +8. Alex Degtyarev, Ilia Itenberg, and Ali Sinan Sert¨oz, Lines on quartic surfaces, Math. Ann. +368 (2017), no. 1-2, 753–809. MR 3651588 +9. GAP – Groups, Algorithms, and Programming, Version 4.10.1, https://www.gap-system.org, +Feb 2019. +10. Carl Friedrich Gauss, Disquisitiones arithmeticae, Springer-Verlag, New York, 1986, Trans- +lated and with a preface by Arthur A. Clarke, Revised by William C. Waterhouse, Cornelius +Greither and A. W. Grootendorst and with a preface by Waterhouse. MR 837656 (87f:01105) +11. V´ıctor Gonz´alez-Alonso and S�lawomir Rams, Counting lines on quartic surfaces, Taiwanese +J. Math. 20 (2016), no. 4, 769–785. MR 3535673 +12. C.M. Jessop, Quartic surfaces with singular points, Cambridge University Press, Cambridge, +1916. +13. Rick Miranda and David R. Morrison, Embeddings of integral quadratic forms, Electronic, +http://www.math.ucsb.edu/~drm/manuscripts/eiqf.pdf, 2009. +14. Yoichi Miyaoka, Counting lines and conics on a surface, Publ. Res. Inst. Math. Sci. 45 (2009), +no. 3, 919–923. MR 2569571 +15. V. V. Nikulin, Finite groups of automorphisms of K¨ahlerian K3 surfaces, Trudy Moskov. +Mat. Obshch. 38 (1979), 75–137. MR 544937 +16. +, Integer symmetric bilinear forms and some of their geometric applications, Izv. Akad. +Nauk SSSR Ser. Mat. 43 (1979), no. 1, 111–177, 238, English translation: Math USSR-Izv. +14 (1979), no. 1, 103–167 (1980). MR 525944 (80j:10031) +17. S�lawomir Rams and Matthias Sch¨utt, 112 lines on smooth quartic surfaces (characteristic 3), +Q. J. Math. 66 (2015), no. 3, 941–951. MR 3396099 +18. +, 64 lines on smooth quartic surfaces, Math. Ann. 362 (2015), no. 1-2, 679–698. +MR 3343894 +19. B. Saint-Donat, Projective models of K-3 surfaces, Amer. J. Math. 96 (1974), 602–639. +MR 0364263 (51 #518) +20. B. Segre, The maximum number of lines lying on a quartic surface, Quart. J. Math., Oxford +Ser. 14 (1943), 86–96. MR 0010431 (6,16g) +21. Ichiro Shimada, Connected components of the moduli of elliptic K3 surfaces, Michigan Math. +J. 67 (2018), no. 3, 511–559. MR 3835563 + +22 +ALEX DEGTYAREV AND S�LAWOMIR RAMS +22. Davide Cesare Veniani, Lines on K3 quartic surfaces in characteristic 2, Q. J. Math. 68 +(2017), no. 2, 551–581. MR 3667213 +23. +, The maximum number of lines lying on a K3 quartic surface, Math. Z. 285 (2017), +no. 3-4, 1141–1166. MR 3623744 +24. +, Symmetries and equations of smooth quartic surfaces with many lines, Rev. Mat. +Iberoam. 36 (2020), no. 1, 233–256. MR 4061988 +25. +, Lines on K3 quartic surfaces in characteristic 3, Manuscripta Math. 167 (2022), +no. 3-4, 675–701. MR 4385387 +Department of Mathematics, Bilkent University, 06800 Ankara, TURKEY +Email address: +degt@fen.bilkent.edu.tr +Institute of Mathematics, Jagiellonian University, ul. �Lojasiewicza 6, 30-348 Krak´ow, +Poland +Email address: slawomir.rams@uj.edu.pl + diff --git a/mdE2T4oBgHgl3EQfzAi5/content/tmp_files/load_file.txt b/mdE2T4oBgHgl3EQfzAi5/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b5de89cbec22726d24f21a9b5437c15d26e33edd --- /dev/null +++ b/mdE2T4oBgHgl3EQfzAi5/content/tmp_files/load_file.txt @@ -0,0 +1,1352 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf,len=1351 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='04127v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='AG] 10 Jan 2023 LINES ON K3-QUARTICS VIA TRIANGULAR SETS ALEX DEGTYAREV AND S�LAWOMIR RAMS Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' We prove the sharp upper bound of at most 52 lines on a complex K3-surface of degree 4 with a non-empty singular locus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' We also classify the configurations of more than 48 lines on smooth complex quartics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Introduction Our main goal is to present an approach to study large line configurations on complex projective K3-quartics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' In particular, we prove the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='1 (see §7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Let X4 ⊂ P3(C) be a degree 4 K3-surface with non-empty singular locus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Then X4 contains at most 52 lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Moreover, each K3-quartic with at least 49 lines contains four coplanar lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' The above bound is sharp: the existence of a complex K3-quartic with 52 lines and non-empty singular locus (two simple nodes) was shown by the first named author in 2016 (via Torelli’s theorem, see [6, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='10]) and the equation of the surface in question was found by D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Veniani, see [25, Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' We conjecture that the quartic surface discovered in [6, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='10] is the only quartic that attains the bound of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='1, but the proof of this fact is beyond the scope of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' It is well-known that the complexity of large line configurations on projective K3-surfaces decreases as the degree d of the polarization grows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' In particular, a complete classification of close to maximal configurations is known for octics (see [2, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='1]) and sextics (to appear in [3]): the respective upper bounds are 32 and 36 in the presence of a singularity vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' 36 and 42 in the smooth case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' In contrast, even though quartic surfaces with singular points have been a subject of intensive study ever since the 19-th century (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=', the classical treatise [12]), hardly anything is known about large line configurations on such surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' The main reason is the existence of the so-called triangular configurations on quartics (see §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='4 for the definition) — a property that drastically increases the complexity of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Here, we circumvent this difficulty with the help of the so-called triangular sets introduced in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' One can easily check that the degree-d Fermat surface (over C) contains exactly 3d2 lines for d > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Moreover, for almost all integers d the Fermat surface is the best known example of a smooth complex projective surface with many lines, and the question whether smooth degree-d surfaces with more lines exist remains open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' To illustrate the power of our approach, we refine the results of [8] and classify 2000 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Primary: 14J28;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Secondary: 14J27, 14N25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' K3-surface, quartic, elliptic pencil, integral lattice, discriminant form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' was partially supported by the T¨UB˙ITAK grant 118F413.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' was partially supported by the National Science Centre, Poland, Opus grant no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' 2018/31/B/ST1/02857.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' 1 2 ALEX DEGTYAREV AND S�LAWOMIR RAMS Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Smooth complex quartics with at least 49 lines Γ |Aut Γ| |Sym X4| (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' c) NS(X4)⊥ X64 4608 (192,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' 1493)6 (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' 0) [8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' 8] X′ 60 480 (60,' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' 1) (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' 0) [4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' 24] X′′′ 50 16 (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' 1)2 (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' 1) [4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' 24] Z50 160 (10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' 1)2 (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' 0) U(5) ⊕ [4] (50) 96 (8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' 5) (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' 0) ∗U(2) ⊕ [28] Z49 36 (3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' 1) (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' 0) ∗U(2) ⊕ [28] all configurations of at least 49 lines on smooth quartics (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=', the configurations that are larger than the one on the Fermat quartic).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Remarkably, compared to [8], we found but three new configurations: one of rank 20 (Q′′′ 52 previously found in [6]) and two of rank 19 (designated as (52) and (50) in Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' On the other hand, there are at least 28 configurations of 48 lines on smooth quartics, giving yet another reason why 48 is a reasonable threshold (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' also Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='6 and Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='11 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='2 (see §7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Up to isomorphism, there are 26 configurations of at least 49 lines on smooth quartic surfaces, see Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' They are realized by 34 singular (aka projectively rigid) surfaces (18 real and 8 pairs of complex conjugate) and five connected 1-parameter families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' As a consequence, we answer a question left open in [8, Addendum 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' LINES ON K3-QUARTICS VIA TRIANGULAR SETS 3 Addendum 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='3 (see §7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' The complete list of values taken by the number of real lines on a real smooth quartic is {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=', 49, 50, 52, 56}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' The configurations of more than 48 real lines on a real smooth quartic are those marked with a ∗ in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Listed in Table 1 are: the name of the configuration Γ (mostly following [8]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' the subscript always refers to the number of lines (vertices of Γ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' the size of the group Aut Γ of abstract graph automorphisms of Γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' the group Sym X4 of simplectic automorphisms of a generic quartic X4 with the given Fano graph, in the form (size, index), referring to the SmallGroup library in GAP [9];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' the superscript is the index of Sym X4 in the full group Aut(X4, h) of projective automorphisms of X4 (if greater than 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' the numbers (r, c) of, respectively, real and pairs of complex conjugate components of the equilinear moduli space;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' the (generic, if rk ⩾ 3) transcendental lattice T := NS(X4)⊥;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' it is marked with a ∗ if the corresponding deformation family has a real quartic with all lines real (see [8, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' If T is not determined by Γ, each lattice is listed in a separate row (following the main entry), and the numbers (r, c) of components are itemized accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' As in [5], we use the following notation for common integral lattices: [a] := Zu is the lattice of rank 1 given by the condition u2 = a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' [a, b, c] := Zu + Zv, u2 = a, u · v = b, v2 = c, is a lattice of rank 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' when it is positive definite, we assume that 0 < a ⩽ c and 0 ⩽ 2b ⩽ a: then, u is a shortest vector, v is a next shortest one, and the triple (a, b, c) is unique;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' U := [0, 1, 0] is the unimodular even lattice of rank 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' L(n) denotes the lattice obtained by the scaling of a given lattice L by a fixed integer n ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' In general, we maintain the standard notation for various objects associated to a lattice L (the determinant, discriminant group, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=') —see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=', [1, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' The inertia indices of the quadratic form L ⊗ R are denoted by σ±,0(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Contents of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Roughly, the paper consists of two parts: the dis- crete one (§2–§6) and the geometric one (§7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Our approach is a refinement of the technique developed in [8, 2], and we recall the necessary facts and introduce certain technical terms (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=', acceptable graphs) in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Then, in §3, we define the main technical tool, viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' the triangular set, and discuss methods of extending a given graph by a collection of triangular sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Finally, after those preparations, we present the proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='12, which is the discrete counterpart of the most difficult case of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' §4 is a digression: we restrict our attention to the case of smooth lattices (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=', we assume that the lattice contains no exceptional divisors) and apply triangular sets to classify geometric Fano graphs with at least 49 vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' In §5 and §6, we turn back to the general case (with exceptional divisors allowed) and study the properties of triangular free Fano graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Finally, in §7 we recall the definition of the Fano graph (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' extended Fano graph) of a surface and its relation to the geometricity of the Fano graph of a 4 ALEX DEGTYAREV AND S�LAWOMIR RAMS lattice, see Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='4 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='7) and prove the principal results of the paper, viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='1 and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' History of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' As mentioned, configurations of lines (or, more generally, smooth rational curves) on quartic surfaces in P3 have been a subject of intensive study ever since the 19-th century.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Still, the methods of Italian school were not efficient enough to deal with the classification of large line configurations on such surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' It was not until the last decade that the theory of elliptic fibrations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Mordell–Weil groups,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Torelli’s theorem and progress in algorithmic methods in the theory of lattices led to a substantial progress in the case of smooth quartics: sharp bound for the number of lines over fields of characteristic p ̸= 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' 3 (see [20,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' 18,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' 8]),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' p = 3 (see [17]),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' p = 2 (see [7]),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' the classification of large configurations (see [8]),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' explicit equations of surfaces with many lines (see [24] and the bibliography therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Strangely enough, the Bogomolov–Miyaoka–Yau inequality yields no bounds in the case of quartics (see [14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' In contrast, in spite of long interest (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=', the classical text [12]), far less is known in the case of quartic surfaces with singular points — essentially, it was only shown that, over a field of characteristic p ̸= 2, the number of lines on a quartic with singularities cannot exceed the maximal number of lines on a smooth quartics — see [23, 25, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' For p = 2 the maximal number of lines on a quartic with singularities is 68 (vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' 60 in the smooth case, see [22]) and we do know projective models of surfaces that attain this maximum (see [17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' A refinement of the method pioneered in [8] led to the complete picture of large line configurations on smooth degree-d K3-surfaces for d > 2 in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Vinberg’s algorithms combined with the above methods yield a means to classify the large configurations of lines on degree-d K3-surfaces with at worst Du Val singularities for d > 4 (see [2, 3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' The methods of [2] are not sufficient to deal with the case of quartics (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=', d = 4): the existence of triangles (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=', ˜A2-configurations) of lines and the fact that, on quartic surfaces, said triangles may interlace lead to numerous configurations that are excluded on degree-d K3-surfaces for d > 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' In the present paper, we discuss an approach to deal with such configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' However, in order to keep our exposition compact, we apply our method to find the maximal number of lines on a complex K3-quartics with non-empty singular locus, but we do not try to classify all configurations of 52 lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' This paper was mostly written during our research stay at the Max-Planck-Institut f¨ur Mathematik, Bonn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' We are grateful to MPIM for creating perfect working conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' thanks IM PAN (Cracow, Poland) for the support that enabled him to complete this project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Preliminaries In this section we recall the main technical tools that we use in our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' To shorten the exposition, we focus on the case of 4-polarized 2-admissible lattices and graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' The details and more general statements can be found in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' To keep the exposition continuous,we assume the reader familiar with the basics of the theory of K3-surfaces, (−2)-curves, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' and adopt a formal graph-theoretical language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' The relation of graphs considered in §2–§6 to the problem at hand, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=', lines on quartic surfaces, is briefly discussed in §7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='1 below, right before the proofs of the principal results of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' LINES ON K3-QUARTICS VIA TRIANGULAR SETS 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Polarized lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Recall that a nondegenerate lattice S is called hyperbolic if σ+S = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' A polarized lattice S ∋ h is a hyperbolic lattice S equipped with a distinguished vector h of positive square;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' the square h2 is called the degree of the polarization and S is said to be h2-polarized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Here we assume h2 = 4, so whenever we speak of a polarized lattice we mean a 4-polarized lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Furthermore, we confine ourselves to lines and exceptional divisors (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' only lines in §4), leaving out smooth rational curves of higher degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' We make frequent use of the following obvious observation: if S is a hyperbolic lattice, then any sublattice N ⊂ S is either semidefinite (and then one has rk ker N = 1) or nondegenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' For a polarized lattice and n = 0, 1, one defines the sets rootn(S, h) := � r ∈ S �� r2 = −2, r · h = n � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' As in [2, § 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='2], we put rt(S, h) ⊂ h⊥ ⊂ S (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' C+(S, h)) to denote the sublattice generated by root0(S, h) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' the positive cone).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Recall that every connected component ∆♯ of C+(S, h) ∖ � r2=−2 r⊥ is a fundamental polyhedron for the group generated by reflections of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' By definition, rt(S, h) is a root lattice and each fixed Weyl chamber ∆ for (the group generated by reflections of) rt(S, h) gives rise to a distinguished fundamental polyhedron ∆♯.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' We put {∆} to denote the ”outward” roots orthogonal to the walls of ∆ and define the (plain) Fano graph of the polarized lattice (S, h) with a distinguished Weyl chamber ∆ for rt(S, h) as the set of vertices (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='2) Fn∆(S, h) := {∆♯}1 := � l ∈ root1(S, h) �� l · e ⩾ 0 for all e ∈ {∆} � , with two vertices l1 ̸= l2 connected by an edge of multiplicity l1 · l2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' The bi-colored extended Fano graph is defined as (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='3) Fnex ∆ (S, h) := {∆♯}1 ∪ {∆}, with the same convention about the multiplicities of the edges and vertices v colored according to the value v · h ∈ {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Let S ∋ h be a polarized lattice and let Γ be a subset of root1(S, h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' (1) A Weyl chamber ∆ is called compatible with Γ if Γ ⊂ Fn∆(S, h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' (2) A root r ∈ root0(S, h) is called separating with respect to Γ if there is a pair of vertices u, v ∈ Γ separated by r, so that r · u > 0 and r · v < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Finally, in order to use general theory of K3-surfaces in the sequel we need the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' A polarized lattice S ∋ h is called: (1) admissible, if there is no vector p ∈ S such that p2 = 0 and p · h = 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' (2) geometric, if it is admissible and there exists a primitive isometry S ֒→ L := 2E8 ⊕ 3U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' 6 ALEX DEGTYAREV AND S�LAWOMIR RAMS 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Subgeometric and geometric graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Let Γ be a (plain) graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' To Γ we associate the polarized lattice (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='6) F(Γ) := (ZΓ + Zh)/ ker, h2 = 4, h · v = 1 for v ∈ Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' where ZΓ is the lattice freely generated by the vertices v ∈ Γ, so that u · v = n when u ̸= v are connected by an n-fold edge, and v2 = −2 for each v ∈ Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Convention 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' As in [2, § 4], we speak of polarized graphs Γ (omitting the degree which is fixed to equal 4), and we apply to Γ the lattice theoretic terminology such as the rank rk Γ := rk F(Γ) etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Furthermore, we treat the vertices of Γ as vectors in F(Γ): e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=', u·v ∈ Z stands for the multiplicity of the edge [u, v], and we say that u, v ∈ Γ intersect if u · v = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' The only exception from this rule is the classification of graphs according to the inertia indices of ZΓ rather than F(Γ) (which is always assumed hyperbolic): thus, we say that Γ is elliptic, if σ+(ZΓ) = σ0(ZΓ) = 0, parabolic, if σ+(ZΓ) = 0 and σ0(ZΓ) > 0, and hyperbolic, if σ+(ZΓ) = 1 (no assumption on σ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Recall that any connected elliptic (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' parabolic) graph is a Dynkin diagram (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' affine Dynkin diagram);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' as in [5], we order the isomorphism classes of affine Dynkin diagrams according to their Milnor number, followed by A < D < E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Recall also that, for each connected parabolic subgraph Σ, there is a unique positive minimal generator κΣ ∈ ker ZΣ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' it has the form κΣ = � mcc, c ∈ Σ, with all mc > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' We define the perturbation order on the set of (isomorphism classes of) elliptic and parabolic graphs: Γ′ ⊳ Γ′′ if Γ′ is isomorphic to an induced subgraph of Γ′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Given an isotropic subgroup K ⊂ discr F(Γ) (aka kernel), we consider the fi- nite index extension F(Γ, K) of F(Γ) by K (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' [16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' The pair (Γ, K) is said to be extensible if it admits a compatible Weyl chamber ∆ for rt(F(Γ, K), h) (see Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='4): Γ ⊂ Fn∆ F(Γ, K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Recall that, by [2, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='4], we have (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='8) (Γ, K) is extensible if and only if F(Γ, K) has no separating roots (see Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='4);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' moreover, if this is the case, the compatible Weyl chamber ∆ ⊂ rt F(Γ, K) is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Therefore, for an extensible pair (Γ, K) one can define its saturation and extended saturation sat(Γ, K) := Fn∆ F(Γ, K), satex(Γ, K) := Fnex ∆ F(Γ, K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' A graph Γ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' pair (Γ, K)) is called admissible if it is extensible and the lattice F(Γ) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' F(Γ, K)) is admissible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Then, an isotropic subgroup K ⊂ discr F(Γ) is called a geometric kernel if the lattice F(Γ, K) is geometric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' We follow [2] and put G(Γ) := � K ⊂ discr F(Γ) �� K is geometric � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' After those preparations we can recall the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Let Γ be a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' (1) We call Γ subgeometric if the set G(Γ) is non-empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' (2) A subgeometric graph Γ is called geometric if Γ ∼= sat(Γ, K) for a certain kernel K ∈ G(Γ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' (3) A bi-colored graph Γ′ is geometric if Γ′ ∼= Fnex ∆ (S, h) for some geometric polarized lattice S ∋ h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' LINES ON K3-QUARTICS VIA TRIANGULAR SETS 7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' In many arguments we use computer-aided test to check that a given graph satisfies certain conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' All algorithms are found in [2, Appendix].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Given a graph Γ, one can check whether: (1) F(Γ) is hyperbolic, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=', σ+F(Γ) = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' straight as it is, this can often be done fast in bulk, see [2, Lemmas A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='3, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='4];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' (2) Γ is extensible and admissible, see the master test in [2, § A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' (3) Γ is (sub-)geometric, see [2, § A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' The principal tool of [2] was starting from a sufficiently large initial graph Γ0 and extending it by adding one vertex at a time, see § A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='4 in loc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' This is what is done in this paper, too, see §5 and §6, where, without much explanation, we merely state the updated results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' The principal novelty of this paper is §3, where, due to the more complicated geometric nature of the problem, extra vertices have to be added in groups of up to three.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' The details are discussed in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' A technical, but crucial part of our (as well as any lattice-based) approach is the fact that a geometric graph Γ has rk Γ ⩽ 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' It follows that, if rk Γ′ = 20, any geometric overgraph Γ ⊃ Γ′ would be of the form sat(Γ′, K) for some K ∈ G(Γ′), and all such finite index extensions of F(Γ′) can easily be found using [16] (see the saturation lists in [2, § A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Therefore, we introduce another technical term: Γ is acceptable if it is subgeometric and rk Γ ⩽ 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' It is understood that, after each step of every algorithm, only acceptable graphs are left for the further processing (and, thus, we do not need to add dozens of vertices to reach line counts over 50), whereas each intermediate graph Γ of rank rk Γ = 20 is excluded upon computing its saturation list and recording all “interesting” (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='24 below) geometric overgraphs to a global master list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Classification of graphs in terms of girth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Following [5, 8], we subdivide parabolic and hyperbolic graphs Γ according to the type of the minimal (in the sense of Convention 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='7) affine Dynkin diagram Σ ⊂ Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' The most important classes can also be characterised in terms of the girth girth(Γ) (the length of a shortest cycle in Γ, with the convention that the girth of a forest is ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Thus, Γ is called triangular, or ˜A2-, if girth(Γ) = 3, quadrangular, or ˜A3-, if girth(Γ) = 4, pentagonal, or ˜A4-, if girth(Γ) = 5, astral, or ˜D4-, if girth(Γ) ⩾ 6 and Γ has a vertex of valency ⩾ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' All other graphs are locally elliptic, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=', one has val v ⩽ 3 for each vertex v ∈ Γ (and we assume girth(Γ) ⩾ 6 to exclude a few trivial cases).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Given a graph Γ and a distinguished connected parabolic subgraph Σ ⊂ Γ, the pencil Π induced by Σ is defined as (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='11) Π := Π(Γ ⊃ Σ) := Σ ∪ � l ∈ Γ �� l · c = 0 for all vertices c ∈ Σ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' This graph Π ⊃ Σ is parabolic as it is orthogonal to κΣ (see Convention 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='7 and Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' We have Γ = Π ∪ sec∗, where (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='12) sec∗ := sec∗(Γ ⊃ Σ) := � l ∈ Γ ∖ Σ �� l · c = 1 for a vertex c ∈ Σ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' The elements of sec∗ are called the (multi-)sections, or m-sections of Π, where the integer m := l · κΣ is the multiplicity of a section l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' If m = 1, the section is called 8 ALEX DEGTYAREV AND S�LAWOMIR RAMS simple, otherwise, multiple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Fixing an order Σ = (c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' , cn), we also consider (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='13) seci := sec(Γ ⊃ Σ ∋ ci) := � l ∈ Γ ∖ Σ �� l · cj = δij for cj ∈ Σ � , sec∗ i := sec∗(Γ ⊃ Σ ∋ ci) := � l ∈ Γ ∖ Σ �� l · ci = 1 � ⊃ seci, where 1 ⩽ i ⩽ n and δij is the Kronecker symbol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' We have (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='14) each set sec∗ i ⊃ seci, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' , n, is either elliptic or parabolic, as it is orthogonal to the isotropic vector h − ci ̸= 0 (see Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' In the future, we almost never use the correct, but long notation referring to the full flag, as Γ ⊃ Σ = (c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' , cn) are always assumed fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Triangular sets A triangular set, or △-set, is an induced subgraph of an admissible graph whose all connected components are of type ˜A2, A3, A2, or A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Clearly, each triangular pencil is a △-set, but not vice versa: a △-set may also be elliptic, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=', have no type ˜A2 components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Combinatorially, a △-set Θ is uniquely of the form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='1) ˜a2 ˜A2 ⊕ a3A3 ⊕ a2A2 ⊕ a1A1, (˜a2, a3, a2, a1) ∈ N4, and the coefficient quadruple (˜a2, a3, a2, a1) determines Θ up to isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' The isomorphism classes of △-sets (or coefficient quadruples) are called patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' We define the cardinality |θ| of a pattern θ as that of any of its representatives and introduce the following order on the set of patterns: (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='2) θ′ ≺ θ′′ iff � |θ′| < |θ′′| or |θ′| = |θ′′| and (˜a′ 2, a′ 3, a′ 2, a′ 1) > (˜a′′ 2, a′′ 3, a′′ 2, a′′ 1), where the coefficient quadruples are compared lexicographically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' (Pay attention to the reverse lexicographic order!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=') This order is not to be mixed with the perturbation order defined in Convention 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' The latter, for △-sets, is easily described in terms of the coefficient quadruples: θ′ ⊳ θ′′ if and only if (˜a′ 2, a′ 3, a′ 2, a′ 1) is obtained from (˜a′′ 2, a′′ 3, a′′ 2, a′′ 1) by a finite sequence of elementary perturbations of the form (˜a2, a3, a2, a1) �→ (˜a2, a3, a2, a1) + δ, δ ∈ � (0, 0, −1, 1), (0, −1, 1, 0), (0, −1, 0, 2), (−1, 0, 1, 0) � provided that, at each step, the quadruple remains in N4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Constructing triangular graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Let Γ be a triangular admissible graph, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Fix a type ˜A2 fiber Σ = (c1, c2, c3) ⊂ Γ and consider the pencil Π and sets seci, i = 1, 2, 3, see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='11) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='13), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' A triangular pencil Π has at most one multiple section;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' hence, 0 ⩽ |Γ| − |Π ∪ sec1 ∪ sec2 ∪ sec3| ⩽ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Furthermore, if a multiple section s exists, it is disjoint from each seci, i = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Consider a multiple section s and let e := h − c1 − c2 − c3 − s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' If s · c1 = s · c2 = s · c3 = 1, then e is orthogonal to h and each ci, i = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Hence, e = 0 (see Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='1) and any other line l intersects exactly one of c1, c2, c3, s, implying both statements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' LINES ON K3-QUARTICS VIA TRIANGULAR SETS 9 Likewise, if s · c1 = s · c2 = 1 and s · c3 = 0, then e is an exceptional divisor and e · s = e · c3 = 1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Hence, any other line l intersects at most one of c1, c2, c3, s, as otherwise l · e < 0 and e would separate s and l, see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Each set seci, i = 1, 2, 3, is a △-set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' In view of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='14), it suffices to rule out the connected components of seci containing ˜A3, A4, or D4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Each of the offending graphs has a chain s1•−− s2•−− s3• and another vertex l adjacent to at least one of s1, s2, s3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Then s := s1 + s2 + s3 is a root, s · h = s · ci = 3, and l · s > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' hence, e := −h + ci + s is an exceptional divisor separating s1 and l, see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' □ Now, our goal is to describe all geometric graphs Γ of size |Γ| ⩾ 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' In view of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='3, it suffices to consider trigonal pencils Σ ⊂ Π ⊂ Γ such that |Π| + |sec1| + |sec2| + |sec3| ⩾ 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Clearly, we can also assume that Π ⊂ Γ is maximal with respect to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='2) and the edges c1, c2, c3 of Σ are ordered so that sec1 ≽ sec2 ≽ sec3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' These assumptions give rise to the following compatibility conditions on the patterns π ∋ Π and θi ∋ seci: π ⊢ θ1 if 3|θ1| + |π| ⩾ 52 and � θ1 ≼ π or θ1 is elliptic � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='5) (π ⊢ θ1) ⊢ θ2 if 2|θ2| + |θ1| + |π| ⩾ 52 and θ2 ≼ θ1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='6) (π ⊢ θ1 ⊢ θ2) ⊢ θ3 if |θ3| + |θ2| + |θ1| + |π| ⩾ 52 and θ3 ≼ θ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='7) In (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='5), we assume one of the terms, π or θ1, fixed and treat the condition as a restriction on the other term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' The other two conditions are restrictions on the last term provided that the parenthesized part is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Since △-sets appear as sets of sections, for an ordered fiber ˜A2 ∼= Σ = (c1, c2, c3) and △-set Θ we define Σ ⊔i Θ, where i = 1, 2, 3, as the graph obtained from the disjoint union of Σ and Θ by connecting ci to each vertex v ∈ Θ by a simple edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' This construction extends to patterns, producing an isomorphism class of graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Checking the parameter quadruples (˜a2, a3, a2, a1) one-by-one, it is fairly easy to compute the sets of patterns P := {subgeometric triangular pencils}/∼=, and T := {△-sets Θ such that ˜A2 ⊔i Θ is subgeometric}/∼=.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' (To simplify the computation, for P one can start from Shimada’s list [21] of Jaco- bian elliptic K3-surfaces, and for T one can take into account the bound val ci ⩽ 20 found in [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=') Then, condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='5) becomes a binary relation from P to T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' The other conditions also descend to patterns, as do the rank functions: (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='8) rk(π) = |Π| − ˜a2(Π) + 2, Π ∈ π ∈ P rk( ˜A2 ⊔i θ) = |Θ| − ˜a2(Θ) + 4, Θ ∈ θ ∈ T , where ˜a2(·) stands for the number of parabolic components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' The next lemma is an immediate consequence of this computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' For the last statement, we merely list all geometric extensions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' using algorithms from [2, Appendix A]) of the four graphs of rank 18 or 19, see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='8);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' in fact, the sharp bound in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='9(3) is |Γ| ⩽ 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' In a geometric graph Γ, for any type ˜A2 fiber Σ one has: 10 ALEX DEGTYAREV AND S�LAWOMIR RAMS (1) |seci| ⩽ 18 for each i = 1, 2, 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' (2) if seci is elliptic, then |seci| ⩽ 15;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' (3) if seci is elliptic and |seci| > 13, then |Γ| ⩽ 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' ⊳ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' In view of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='9, the compatibility condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='5) simplifies to the form π ⊢ θ1 if 3|θ1| + |π| ⩾ 52 and θ1 ≼ π more consistent with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='6) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Indeed, if θ1 is elliptic, we can assume that |θ1| ⩽ 13;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' then necessarily |π| ⩾ 13, see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='5), and θ1 ≼ π holds automatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' If Π ⊂ Γ is a maximal, with respect to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='2), triangular pencil and |Γ| > 52, then |Π| ⩾ 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' As explained in Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='10, in view of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='9 we have |Π| ⩾ 13 and |sec3| ⩽ |sec2| ⩽ |sec1| ⩽ |Π|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' If |Π| = 13, then |sec3| = |sec2| = |sec1| = 13 by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='5) and, moreover, Π must have a multiple section s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Assuming that s · c1 = s · c2 = 1, so that c1, c2, s constitute a triangle, the union (c1, c2, s) ⊔ sec3 ⊂ Γ is a triangular pencil (see Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='3) with 16 vertices, contradicting the maximality of Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Abundant collections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Let Γ0 ⊃ Σ = (c1, c2, c3) ∼= ˜A2 be a triangular graph, i = ∅, 1, 2, 3 a parameter, and θ ∈ P (if i = ∅) or θ ∈ T (otherwise) a pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' An overgraph Γ ⊃ Γ0 is said to represent (Γ0, θ)i if sec(Γ ⊃ Σ ∋ ci) ∈ θ and Γ ∖ seci = Γ0, if i = 1, 2, 3 Π(Γ ⊃ Σ) ∈ θ and Γ ∖ Π = Γ0 ∖ Σ, if i = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' A pair (Γ0, θ)i is called abundant if it cannot be represented by an acceptable graph, see Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Both notions extend to a collection G of graphs: Γ represents (G, θ)i if it represents (Γ0, θ)i for a graph Γ0 ∈ G, and (G, θ)i is abundant if so is each (Γ0, θ)i, Γ0 ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Iterating, we extend both notions to a compatible (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=', satisfying the compatibility conditions from §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='1) collection of patterns π ∈ P, θi ∈ T , i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' , n ⩽ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Clearly, if π ⊢ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' ⊢ θn is abundant, so is any π′ ⊢ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' ⊢ θ′ n with π ⊳ π′, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' , θn ⊳ θ′ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Our proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='1 essentially reduces to applying the algorithm in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='3 below to show that any compatible collection π ⊢ θ1 ⊢ θ2 ⊢ θ3 is abundant: indeed, conditions (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='5)–(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='7) guarantee that on the way we will encounter all subgeometric graphs Γ such that either |Γ| > 52 or rk Γ = 20, and in the latter case it would suffice to analyze all geometric saturations of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' To this end, we introduce the inductive notion of a ruled out collection: any abundant compatible collection is considered ruled out;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' in general, a compatible collection π ⊢ θ1 ⊢ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' ⊢ θn, n ⩽ 2, is ruled out if so is any compatible extension (π ⊢ θ1 ⊢ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' ⊢ θn) ⊢ θn+1, θn+1 ∈ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' By a machine aided computation, we establish the following statement;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' its proof is given in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='6, after we collect all the necessary facts in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='3, §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='4 and §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='12 (see §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Each compatible pair π ⊢ θ1, where π ∈ P14 := � π ∈ P �� |π| ⩾ 14 � and θ ∈ T , is ruled out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' LINES ON K3-QUARTICS VIA TRIANGULAR SETS 11 By the very definition, the assertion of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='12 means that, for each representative Γ of any compatible collection π ⊢ θ1 ⊢ θ2 ⊢ θ3, π ∈ P14, one has either |Γ| = 52 or rk Γ = 20, and, moreover, all such representatives are encountered in the course of the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' The latter fact enables us to obtain the complete list of representatives Γ of compatible collections π ⊢ θ1 ⊢ θ2 ⊢ θ3, π ∈ P14, such that |Γ| > 52 and rk Γ = 20 (see Addendum 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Extending a graph by a triangular set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' The heart of the computation is an algorithm extending a given subgeometric graph Γ0 by a given pattern θ ∈ T , the goal being listing all subgeometric overgraphs Γ ⊃ Γ0 representing (Γ0, θ)i (where i = ∅, 1, 2, 3 is also fixed, see §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' The elements of Γ ∖ Γ0 are referred to as sections, whereas the connected components of Γ ∖ Γ0 ∈ θ are polysections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' The algorithm is similar to that of [2], except that we can no longer guarantee that the sections are pairwise disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Therefore, instead of adding to Γ one section at time, we fix θ in advance and add whole polysections, in the order ˜A2, A3, A2, A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Convention 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' From now on, following [2, § A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='3], we identify a section v with its support supp v := � u ∈ Γ0 �� v · u = 1 � and thus treat it as a subset of Γ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' We also keep the notation Γ0 ⊔ v and Γ0 ⊔ v(m) for a multiset v (which we no longer assume sorted) and (|v| × |v|)-matrix m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' As explained in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='3, only acceptable graphs are retained after each step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' In practice, we start with computing the group G0 := Aut Γ0 and set (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='14) S(Γ0) := � s ⊂ Γ0 �� s ∩ Σ = fixed � of sections of Γ0 satisfying extra conditions imposed by the problem at hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' (Here, fixed ⊂ Σ is a certain subset fixed in advance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' We can also take into account a few obvious geometric restrictions, but this is not crucial: “wrong” sections are immediately ruled out by the preliminary tests in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='3(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' We omit many other technical tweaks, referring to the code [4] as the ultimate source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=') Then, running the tests cited in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='3(2) and (3), we compute the sets (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='15) A1(Γ0) := � v ∈ S(Γ0) �� Γ0 ⊔ v is acceptable � , m(Γ0) := � v ∈ A1(Γ0)dim m �� Γ0 ⊔ v(m) is acceptable � , where m = A2, ˜A2, 2A1, A3 (in this order).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Certainly, the tests are applied to a single representative of each G0-orbit;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' in what follows (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=', e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=', Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='19) this convention is taken for granted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' This computation is aborted if a “required” list is empty (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=', if A2(Γ0) = ∅ whereas θ contains ˜A2 or A3, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' the next remark).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='16 (a technical detail).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' The set A2(Γ0) is used in the computation of ˜A2(Γ0): we consider only those triples (v1, v2, v3) for which (vi, vj) ∈ A2(Γ0) for all 1 ⩽ i < j ⩽ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Likewise, both A2(Γ0) and 2A1(Γ0) are used in the computation of A3(Γ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Furthermore, 2A1(Γ0) is used at all subsequent steps: when iterating Γn−1 ⊔ v(mn) = � Γ0 ⊔ u(·) � ⊔ v(mn), in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='17) below, we check first that (u, v) ∈ 2A1(Γ0) for all u ∈ u, v ∈ v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' 12 ALEX DEGTYAREV AND S�LAWOMIR RAMS Now, let θ = ˜a2 ˜A2 + a3A3 + a2A2 + a1A1, so that N := ˜a2 + a3 + a2 + a1 is the number of components, and let m1 ⩾ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' ⩾ mN be the types of the components of θ ordered via ˜A2 > A3 > A2 > A1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Then, we start from S0 := {Γ0} and run the computation in up to N steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Step n ⩾ 1: for each graph Γn−1 ∈ Sn−1, we pick a single representative v of each (Aut Γn−1)-orbit on mn(Γ0) and use the tests of §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='3(2) and (3) to compute (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='17) Sn(Γn−1) := � Γv := Γn−1 ⊔ v(mn) �� Γv is acceptable � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' The step concludes by uniting all sets Sn(Γn−1), Γn−1 ∈ Sn−1, obtained followed by retaining a single representative of each graph isomorphism class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' The algorithm terminates either upon the completion of Step N (resulting in a list SN(Γ0) to be processed by other means) or when one of the previous steps results in an empty list Sn(Γ0) = ∅, implying that (Γ0, θ) is abundant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Both auto- and isomorphisms of graphs are computed using the digraph package in GAP [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' All morphisms are restricted: we assume the fiber Σ fixed as a set and, typically, one or two edges ci ∈ Σ fixed pointwise, so that the set S(Γ0) in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='14) be invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='19 (a technical detail).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Each time the matrix m changes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=', whenever mn > mn+1, we recompute the (relevant) sets m(Γn) for each graph Γn ∈ Sn and use these new lists in the subsequent steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Instead of starting from scratch, as in the case of Γ0, we merely run the tests on the ready lists m(Γm) for the last subgraph Γm ⊂ Γn for which they have been computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Processing several patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' The material of this section is of a purely technical nature;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' however, it is the tweak described here that makes the computa- tion much faster and eventually helps it to terminate reasonably fast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Typically, we fix a subgeometric graph Γ0 and try to rule out a whole collection of patterns T (Γ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Since patterns tend to have similar initial sequences, processing them all one-by-one would force us to repeat the same steps of the computation over and over again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' To avoid the repetition and remove a number of redundant steps, we sort the patterns in the direct lexicographic order and process them simultaneously, organizing the computation into four layers: the outermost ˜A2, A3, A2, and the innermost A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Each inner layer starts from a certain intermediate graph Γ and processes a collection of patterns T (Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' If the algorithm terminates prematurely, at a certain pattern θ, we conclude that (Γ, θ) is abundant, and hence so is (Γ, θ′) whenever θ ⊳ θ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' This fact is reported to the previous layer, where the information is consol- idated and often results in excluding the graph Γ and/or some patterns from the further consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' We refer to the code [4] for the precise details (we implement each next layer as a hook within the previous one, where it is used to modify the intermediate lists);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' here, we merely illustrate the paradigm by the following simple example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Assume that the patterns to be considered are T (Γ0) = � 2A2 ⊕ 6A1, 3A2 ⊕ 4A1, 4A2 ⊕ 2A1 � , LINES ON K3-QUARTICS VIA TRIANGULAR SETS 13 so that only two layers of computation are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' We run the first two steps of the A2-layer, resulting, say, in a list S2 = {Γ′ 2, Γ′′ 2}, and switch to the A1-layer for each of the two graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Assume that this inner layer terminates at step 4 for Γ′ 2 ⇒ (Γ′ 2, 4A1), (Γ′ 2, A2 ⊕ 3A1), (Γ′ 2, 2A2 ⊕ 2A1) are abundant, step 5 for Γ′′ 2 ⇒ (Γ′′ 2, 5A1), (Γ′′ 2, A2 ⊕ 4A1) are abundant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' (Obviously, 4A1 ⊳ A2 ⊕ 3A1 ⊳ 2A2 ⊕ 2A1 and 5A1 ⊳ A2 ⊕ 4A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=') We conclude that Γ′ 2 can be excluded from S2 and that both 2A2 ⊕ 6A1 and 3A2 ⊕ 4A1 can be excluded from T (Γ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Therefore, we can run two more steps of the A2-layer on the new reduced list {Γ′′ 2}, followed by the A1-layer on the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' (The A1-layer after Step 3 can be skipped as θ = 3A2 ⊕ 4A1 has already been ruled out!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=') If it were not for Γ′′ 2 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=', if the A1-layer terminated at a step ⩽ 4 for each of the two graphs), we would have stopped immediately, as all elements of T (Γ0) would have been ruled out by the A1-layer after Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' The aggressive version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' In certain cases, one can argue that, in order to achieve the goal |Γ| ⩾ 53, the overgraph Γ ⊃ Γ0 must be spanned over Γ0 by a few pairwise disjoint vertices independent over Γ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' (Precisely, this condition means that F(Γ) ⊗ Q is generated over F(Γ0) ⊗ Q by (rk Γ − rk Γ0) pairwise disjoint vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=') In this case, we switch to the aggressive version of the algorithm, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=', we add disjoint vertices only (the A1-layer), disregard the extra vertices that do not increase rank, and check the saturation lists of all intermediate graphs, including Γ0, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' the progressive mode in [2, § A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Precisely, this approach is used in the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='9(3): we add up to two disjoint vertices, and at the steps fixed = {c2} or {c3} in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='6 below (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' also Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='11), when extending a graph Γ0 of the submaximal rank rk Γ0 = 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' As stated, the proof is an explicit machine aided computation using the algorithm described in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='3 and §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' It runs in two steps: first, for each pattern θ1 ∈ T , we rule out almost all compatible patterns π ∈ P14;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' the set of these patterns is denoted by P− 14(θ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Then, for each π ∈ P14, we rule out the remaining patterns θ1 ∈ T such that π ⊢ θ1 and π /∈ P− 14(θ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' The choice of the set P− 14(θ1) at the first step looks quite arbitrary, and indeed so it is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' As a rule, we let π ∈ P− 14(θ1) if π ⊢ θ1 and rk π ⩽ rk(Σ ⊔1 θ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' However, a few border cases are subject to further manual tweaking, which is based on experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' More precisely,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' depending on the values r := rk(Σ⊔1 θ1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' n := |θ1|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' the following coefficient quadruples (˜a2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' a3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' a2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' a1) are excluded from P− 14(θ1): (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' n) = (12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' 11): (∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' ∗),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' (12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' 12): (∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' ∗),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' (13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' 11): (∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' ∗),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' (13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' 12): (3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' ∗),' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' ∗) (where,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' as usual,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' ∗ means any value).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' At the first step, we start with the graph Γ0 := Σ ⊔1 Θ1, Θ1 ∈ θ1 ∈ T , and use §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='4 to find all graphs Γ representing (Γ0, π)∅, π ∈ P− 14(θ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Technically, we extend Γ0 by a pattern θ such that ˜A2 ⊕ θ = π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Thus, we let fixed = ∅ in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='14) and use graph auto-/isomorphisms preserving Σ ∋ c1 (see Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' For each graph Γ on the resulting list SN, we consider the full set T (Γ) of patterns θ2 ∈ T satisfying (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='6), and run the same algorithm with fixed = {c2} and graph morphisms preserving c1 and c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Strictly speaking, we should have run the algorithm once more, using fixed = {c3} and patterns θ3 satisfying (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' However, our thresholds are chosen so that the list SN resulting from the first run consists of relatively few graphs of rank 19 (for which the aggressive version is used, see §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='5) and very few graphs of rank 18, for which the algorithm terminates fast and rules everything out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' At the second step, we start with a pencil Γ0 := Π ∈ π ∈ P14 and use §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='4 to find all graphs Γ representing (Γ0, θ1)1, π ⊢ θ1 and π /∈ P− 14(θ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' We let fixed = {c1} in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='14) and use graph morphisms preserving Σ ∋ c1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' As above, the relatively few graphs obtained, all of rank 19 or 18, are ruled out by the next run, using fixed = {c2} and θ2 ∈ T compatible in the sense of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' This completes the proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='12, as well as of Addendum 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='23 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' □ As explained right after Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='12, before discarding a graph Γ of rank 20, we analyze its geometric finite index extensions and record those of size greater than 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' The result of this analysis is stated below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Addendum 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Let Γ be a geometric representative of a compatible collection π ⊢ θ1 ⊢ θ2 ⊢ θ3, π ∈ P14, such that rk Γ = 20 and |Γ| > 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Then Γ is one of the eight smooth configurations found in [8], see the first eight rows of Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' ⊳ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' In order to produce a plethora of examples of large configurations of lines, when discarding the graphs of rank 20 (see Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='10) we collected all extended graphs with at least 48 lines or at least six exceptional divisors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' the results are found in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' In particular, in addition to the surfaces listed in Table 1, we found but one quartic with 52 lines and two nodes and two quartics with 50 lines and one node each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Besides, there are quite a few quartics with non-empty singular locus and 48 lines, suggesting once again that 48 is a reasonable threshold to cut the classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' LINES ON K3-QUARTICS VIA TRIANGULAR SETS 15 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Smooth quartics In this section we temporarily assume that the polarized lattice S ∋ h contains no exceptional divisors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' this will be used later in our study of line configurations on smooth quartic surfaces X4 ⊂ P3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' In this case, we need to change the notion of admissible lattice/graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Namely, a polarized lattice S ∋ h is called smooth, or s-admissible, if it contains neither exceptional divisors (e2 = −2, e · h = 0) nor 2-isotropic vectors (e2 = 0, e · h = 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' A graph Γ is smooth, or s-admissible, if so is the lattice F(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' If S ∋ h is smooth, then rt(S, h) = ∅;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' hence, automatically, ∆ = ∅ in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='2) and we have a well-defined Fano graph (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='1) Fn(S, h) := Fn∅(S, h) = root1(S, h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' An crucial consequence of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='1) is the fact that Fn is monotonous: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='2) if S′ ⊃ S ∋ h are smooth, then Fn(S′, h) ⊃ Fn(S, h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Let Γ be a smooth graph and Σ = (c1, c2, c3) ⊂ Γ a triangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Then: (1) Σ has a unique 3-section c0 := h − c1 − c2 − c3 ∈ Γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' (2) the pencil Π as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='11) and △-sets seci as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='13) have no connected components of types A2 or A3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Clearly, c0 as in Statement (1) is a 3-section in the lattice spanned by h and Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='2), it remains a 3-section in any larger smooth lattice/graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' If the pencil Π has a pair (s1, s2) ∼= A2, by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='2) it also has s3 := κΣ − s1 − s2, so that (s1, s2, s3) ∼= ˜A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' If Π has (s1, s2, s3) ∼= A3, then κΣ − s1 − s2 − s3 is an exceptional divisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' The same argument applies to each set seci, i = 1, 2, 3, except that we replace κΣ with (c0 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' + c3) − ci = κΣ + c0 − ci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' □ In view of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='3(1), each type ˜A2 fiber Σ0 := Σ = (c1, c2, c3) gives rise to three more, viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Σi := (c0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' , ˆci, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' ), i = 1, 2, 3 (where, as usual, ˆci indicates that ci has been omitted).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Thus, we can shift the paradigm and, instead of considering a pencil Π and three sets seci, we can speak about “blending” four pencils Π0 := Π = Π(Γ ⊃ Σ0), Πi := Σi ⊔ seci = Π(Γ ⊃ Σi), i = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Assuming, as above, that Π is a maximal pencil in Γ, we can replace (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='5)–(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='7) with a stronger set of compatibility conditions: π ⊢ θ1 if 3|θ1| + |π| ⩾ 48 and θ′ 1 ≼ π;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='4) (π ⊢ θ1) ⊢ θ2 if 2|θ2| + |θ1| + |π| ⩾ 48 and θ′ 2 ≼ θ′ 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='5) (π ⊢ θ1 ⊢ θ2) ⊢ θ3 if |θ3| + |θ2| + |θ1| + |π| ⩾ 48 and θ′ 3 ≼ θ′ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='6) Here, ′ stands for the operator Θ �→ Θ′ := ˜A2 ⊔Θ and its natural descent to the set of patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' (Recall also that, for Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='2, we need to change the threshold to |Γ| ⩾ 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=') In particular, from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='4)–(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='6) we immediately conclude that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='7) |Π| ⩾ 15, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Indeed, taking into account the 3-section given by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='3, we can rewrite (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='6) in the form |θ′ 3| + |θ′ 2| + |θ′ 1| + |π| ⩾ 57 = 48 + 9, and it remains to observe that the assumption θ′ 3 ≼ θ′ 2 ≼ θ′ 1 ≼ π implies |θ′ 3| ⩽ |θ′ 2| ⩽ |θ′ 1| ⩽ |π|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Thus, unlike Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='12, we do not need to introduce an analogue Ps 15 of the set P14: the lower bound (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='7) would follow from the compatibility assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' 16 ALEX DEGTYAREV AND S�LAWOMIR RAMS Now, a computation similar to (but much faster than) that of §3 yields the following result (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='12;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' we retain the terminology of §3 and denote by Ps ⊂ P and T s ⊂ T the sets of patterns appearing in smooth graphs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='8 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Each compatible collection π ⊢ θ1 ⊢ θ2 ⊢ θ3, as in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='4)–(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='6), with π ∈ Ps and θ1, θ2, θ3 ∈ T s, either is ruled out or has one of the five graphs (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='9) Z49, (50), Z50, (52), Z52 (see Table 1) as a geometric representative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' ⊳ Similar to Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='12, this statement means that, with the five exceptions listed in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='9), any geometric representative Γ of a collection as in the hypotheses has rk Γ = 20 and, moreover, all such representatives of rank 20 are encountered and discarded in the course of the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' As in §3, prior to discarding a graph we compute all its geometric finite index extensions, thus arriving at the following complete list of geometric rank 20 Fano graphs Γ with |Γ| ⩾ 48 + 1 (the extra 1 standing for the 3-section given by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Addendum 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Let Γ be a geometric representative of a compatible collection π ⊢ θ1 ⊢ θ2 ⊢ θ3 as in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='4)–(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='6), where π ∈ Ps, θ1, θ2, θ3 ∈ T s, and rk Γ = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Then Γ is one of the 21 smooth rank 20 configurations found in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' ⊳ Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Due to the lower threshold |Γ| ⩾ 49, occasionally we do have to run the algorithm till the very last step fixed = {c3}, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' This last step is quite expensive (as the set of sections to begin with is quite large), but fortunately it has to be done for four graphs only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' This is yet another indication of the fact that taking the classification down to 48 or fewer lines is hardly feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' In the smooth case, we can further reduce the overcounting by a number of tricks based on the monotonicity property (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='2), using all lines present in the lattice rather than only those added explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Most notably, before switching to a next step fixed = {ci}, i ⩽ 3, we can replace each graph Γ with the union Π ∪ sec1 ∪ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' ∪ seci−1 in Fn(F(Γ)) and then retain a single representative of each isomorphism class of the list obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Furthermore, in the subsequent computation we can impose an extra condition that the above union should remain fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' We refer to [5] for further details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Quadrangular graphs In this section the K3-quartic X4 is allowed to have singular points as in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Consider an ˜A3-graph Γ and fix a quadrangle Σ := (c1, c2, c3, c4) ⊂ Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' We assume the edges ci ∈ Σ numbered cyclically and ordered consecutively: ci+4 = ci, ci · ci±1 = 1, ci · ci±2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' In addition to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='11)–(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='13), we consider the sets sec∗ ij := sec∗ i ∪ sec∗ j, i = j ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' The assumption that girth(Γ) = 4 implies that sec∗ 13 ∩ sec∗ 24 = ∅ and each graph sec∗ i is discrete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' LINES ON K3-QUARTICS VIA TRIANGULAR SETS 17 The elements of each seci are simple sections of Π, and those of sec∗ i ∖ seci = sec∗ j ∖ secj = sec∗ i ∩ sec∗ j, j = i ± 2, are double sections;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' the set of all (multi-)sections of Π is sec∗ = sec∗ 13 ∪ sec∗ 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' From now on, we state our results for geometric rather than subgeometric graphs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' in other words, we assume that Γ admits a triangle free geometric saturation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' In a geometric quadrangular graph Γ ⊃ Σ as above, one has (1) |seci| ⩽ 10 and |sec∗ i | ⩽ 10 (sharp bounds), (2) |sec∗ ij| ⩽ 20 (a sharp bound), and (3) |sec∗| ⩽ 32 (the best known example being 30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Statement (1) is a direct computation: since each sec∗ i is discrete, the union Σ ∪ sec∗ i depends on just two parameters, which can take the following values: (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='2) pi := |seci| = 0 1 2 3 4 5 6 7 8 9 10 qi := |sec∗ i ∖ Θi| ⩽ 8 7 6 5 5 4 4 3 2 1 0 The bound in (2) follows from (1) (assuming pi ⩾ pj, we have |sec∗ ij| ⩽ 2pi + qi), and its sharpness is established by an explicit construction (obtained in the next computation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Finally, statement (3) is also obtained by a computation similar to [2, § B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='5]: assuming that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='3) |sec1| ⩾ |sec3|, |sec2| ⩾ |sec4|, |sec∗ 13| ⩾ |sec∗ 24|, we start with a “standard” graph Σ ∪ sec∗ 1, letting (p1, q1) = (10, 0), (9, 1), (9, 0), (8, 2), (8, 1), or (7, 3), and build all possible consecutive extensions (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='4) Γ := (Σ ∪ sec∗ 1) ∪ sec3 ∪ sec2 ∪ (sec∗ 2 ∖ sec2) ∪ sec4 via discrete sets;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' the minimal size of each set to be added is determined using (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='2), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='3), and the goal |Γ| ⩾ 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' In most cases, this algorithm terminates (meaning that each sufficiently large graph admitting a geometric triangle free saturation is unacceptable, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='10) at the very first nontrivial step sec3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' in very few cases we also need to use sec2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' □ Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' In the proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='1 and Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='6 below, the essential difference from [2] is that we do not limit the number of lines intersecting both given ones (the presence of biquadrangles and longer “polyquadrangles”);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' this makes the computation slightly more involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' For a geometric quadrangular graph Γ one has |Γ| ⩽ 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' It is unlikely that the bound given by Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='6 is sharp: the best example that we found has 39 lines (see also [2, Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' The proof is a computation similar to [2, § C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' In view of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='1, it suffices to consider all quadrangular pencils Π of size |Π| ⩾ 17;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' the list of such pencils admitting a polarization is compiled using [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Then, under the assumptions of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='3), we must have 4p1 + 2q1 + |Π| ⩾ 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Similar to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='4), we start from a pencil Π and build a list of consecutive acceptable extensions Γ := Π ∪ sec1 ∪ (sec∗ 1 ∖ sec1) ∪ sec3 ∪ sec2 ∪ (sec∗ 2 ∖ sec2) ∪ sec4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Due to our modest goal |Γ| ⩾ 49, de facto the algorithm terminates at the first or, occasionally, second step, so that we never need to consider even sec3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' □ 18 ALEX DEGTYAREV AND S�LAWOMIR RAMS 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Other types of graphs For the other types (in the sense of §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='4) of graph, the computation runs exactly as in [2], and we merely state the updated results below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Remarkably, the upper bounds obtained are exactly the same as in the smooth case (see [5]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' furthermore, unlike the case of octics (see [2]), all extremal configurations are smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Pentagonal graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Recall that the assumption girth(Γ) = 5 implies that, for a fiber ˜A4 ∼= Σ ⊂ Γ, one has sec∗ = sec1 ∪ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' ∪ sec5, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=', all sections are simple;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' each graph sec∗ i = seci, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' , 5, is discrete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' The following bounds are sharp, and there are but two geometric pentagonal graphs with 30 vertices, viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Φ′ 30 and Φ′′ 30 (see [5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Both represent configurations of lines on smooth quartic surfaces only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='1 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' [2, Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Let Γ be a geometric pentagonal graph, and let Σ ⊂ Γ be a type ˜A4 subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Then: (1) one has |sec∗| ⩽ 16;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' (2) if |sec∗| ⩾ 14, then |Γ| ⩽ 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' ⊳ Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' One has |Γ| ⩽ 30 for any geometric pentagonal graph Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' In view of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='1, it suffices to consider pentagonal pencils Π such that |Γ| ⩾ 17;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' they can be found using [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Astral graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' We number the vertices (c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' , c5) of a type D4 fiber Σ so that the central vertex c1 is the one of valency 4 in Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Recall that the assumption girth(Γ) ⩾ 6 implies that, for a fiber ˜D4 ∼= Σ ⊂ Γ one has sec∗ = sec1 ∪ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' ∪ sec5, and the graph sec∗ is discrete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Note though that it is not true that all sections are simple: the elements of sec1 are double sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' (Recall that κΣ = 2c1 + c2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' + c5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=') The following bounds are sharp, and the only geometric astral graph with 27 vertices is ∆′ 27 (see [5]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' it is represented by a unique smooth quartic surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='3 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' [2, Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Let Γ be a geometric astral graph and Σ ⊂ Γ a type ˜D4 subgraph whose central vertex c1 has maximal valency in Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Then: (1) one has |sec∗| ⩽ 12;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' (2) if |sec∗| ⩾ 11, then |Γ| ⩽ 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' ⊳ Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' One has |Γ| ⩽ 27 for any geometric astral graph Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' In view of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='3, it suffices to consider astral pencils Π with |Γ| ⩾ 18;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' they can be found using [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Locally elliptic graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Let us recall, that the case of locally elliptic graphs was considered in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' We have the inequality (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='5) |Γ| ⩽ 29 for all geometric locally elliptic graphs Γ (see [2, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='1)]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Machine-aided experi- ments suggest that the sharp bound is |Γ| ⩽ 25 with a unique graph Λ25 that attains the maximum , but such considerations are of no importance for the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' LINES ON K3-QUARTICS VIA TRIANGULAR SETS 19 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Proofs In order to render our exposition self-contained, we recall certain results from [2] in §7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='1, before presenting the proofs of the main results of the paper in §7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='2, §7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Fano graphs of K3-quartics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Let X4 ⊂ P3 be a complex degree-4 surface with at worst Du Val (aka A–D–E, or simple) singularities and let π: ˜X4 → X4 be the minimal resolution of its singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Denote h := π∗OX4(1) ∈ NS( ˜X4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Recall that ˜X4 is a K3-surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' In particular, given an irreducible curve C ⊂ ˜X4, we have (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='1) C · h = 1 and C2 = −2 if and only if π(C) is a degree one curve on X4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' The curves that satisfy (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='1) are called lines on ˜X4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' they are obviously smooth and rational.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' We follow [5] and define the (plain) Fano graph of the quartic X4 as the loop free graph with vertices (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='2) Fn(X4) := � (−2)-curves C ⊂ ˜X4 with C · h = 1 � and each pair of vertices v, w ∈ Fn(X4) connected by an edge of multiplicity v · w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' (Here and below, we always consider the intersection form ”·” on NS( ˜X4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=') Recall that, by [2, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='5)], (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='3) the graph Fn(X4) of a K3-quartic with at least 25 lines is hyperbolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' General theory of lattice-polarized K3-surfaces (Nikulin [15], Saint-Donat [19];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' also [8, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='11] and [6, Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='3]) yields the following statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' (As in [2, Convention 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='4], we say that the lattice NS( ˜X4) is spanned by lines if it is a finite index extension of its sublattice generated by the classes of lines on ˜X4 and the quasi-polarization h, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=', it is spanned by lines and h over Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=') Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='4 (see [2, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' A graph Γ is geometric if and only if one has Γ ∼= Fn(X4) for a quartic X4 such that NS( ˜X4) is spanned by lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' ⊳ Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' As in the case of K3-octics, by [2, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='8], in order to study the maximal number of lines we can restrict our attention to the case when the lattice NS( ˜X4) is spanned by lines and exceptional divisors (see also [2, § 8]) Obviously, for a quartic X4 with non-empty singular locus the graph Fn(X4) does not completely describe the configuration of lines on X4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' The latter can be inferred from the bi-colored extended Fano graph (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='6) Fnex(X4) := � (−2)-curves C ⊂ ˜X4 with C · h ⩽ 1 � , with the colour of each vertex C defined as C · h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' For such graphs we have a more general statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='7 (see [2, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' A bi-colored graph Γ′ is geometric if and only if Γ′ ∼= Fnex(X4) for a K3-quartic X4 ⊂ P3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' ⊳ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' By Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='7, the assertion of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='1 is equivalent to the statement that there are no (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='8) geometric bi-colored graphs Γ′ such with |sp1 Γ′| ⩾ 53 and |sp0 Γ′| ̸= 0, where spj Γ′ stands for the induced subgraph of Γ′ given by all its vertices of color j = 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Moreover, by (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='3), the (plain) graph Γ := sp1 Γ′ is a Σ-graph for a certain affine Dynkin diagram Σ ⩾ ˜A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' 20 ALEX DEGTYAREV AND S�LAWOMIR RAMS The monotonicity given by [2, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='8] combined with the consideration of §6 implies that the graph Γ is neither pentagonal (see Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='2), nor astral (see Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='4), nor locally elliptic (see (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='5)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Finally, Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='6 shows that Γ is triangular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Then, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='11 and Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='12, we necessarily have rk(Γ) = 20, upon which Addendum 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='23 implies that Γ is one of the eight smooth configurations found in [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' For each graph Γ obtained we compute its extended saturation(s) satex Γ and check that none of the resulting bi-colored graphs has a vertex of color zero (exceptional divisor), completing the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' □ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' When dealing with smooth quartics, we can confine ourselves to the case where the lattice NS(X4) is spanned by lines, see Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='5 and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='4 reduces the proof to the classification of all graphs Γ such that Γ is geometric and |Γ| ⩾ 49;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' then, by (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='3), the (plain) graph Γ is a Σ-graph for a certain affine Dynkin diagram Σ ⩾ ˜A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' As in the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='1, we infer that Γ is neither quadrangular (Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='6), nor pentagonal (Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='2), nor astral (Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='4), nor locally elliptic (see (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='5)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' For a geometric triangular graph Γ with at least 49 vertices and rk(Γ) < 20, we apply Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='8 to show that Γ is one of the five graphs (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Otherwise, by Addendum 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='10, the graph Γ is one of the rank 20 graphs that appear in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' To complete the deformation classification, let Γ be one of the graphs in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' As part of our study of the saturation lists, we observe that the only geometric finite index extension of F(Γ) is the trivial one;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' hence, one has (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='9) NS(X4) = F(Γ) and Oh(NS(X4)) = Aut Γ for any smooth quartic X4 ⊂ P2 with Fn X4 ∼= Γ, and the latter group is found using the digraph package in GAP [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' According to [8, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='9], the equilinear deformation families of such quartics are in a bijection with the primitive isometric embeddings (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='10) S := F(Γ) ֒→ L = 2E8 ⊕ 3U regarded up to polarized autoisometry of S ∋ h and autoisometry of L preserving a coherent orientation of maximal positive definite subspaces of L ⊗ R (the so-called positive sign structure);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' such a family is real if and only if (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='10) admits a polarized autoisometry reversing the positive sign structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Hence, to complete the proof, we classify embeddings (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='10) using Nikulin’s [16] theory of discriminant forms and either Gauss [10] theory of binary quadratic forms (in the definite case rk T = 2) or Miranda–Morrison [13] theory (in the indefinite case rk T ⩾ 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' □ Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' The groups Sym X4 and Aut(X4, h) in Table 1 are computed as the subgroups of (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='9) that, respectively, act identically on discr S or extend to an appropriate autoisometry of L: the latter is required to preserve the positive sign structure (if rk T = 2) or act on T by ±1 (if rk T ⩾ 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' LINES ON K3-QUARTICS VIA TRIANGULAR SETS 21 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Proof of Addendum 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' As stated in [8, Addendum 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='4], the number of real lines does take all values in the range {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=', 48}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Next, we recall that, when counting the number of real lines on smooth quartics, it suffices to consider only those quartics X4 whose all lines are real (with respect to a certain real structure σ: X4 → X4, see [8, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='10] or [5, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='7]), and the latter is the case if and only if the generic transcendental lattice T has a sublattice isomorphic to [2] or U(2), see [8, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Hence, the statement of the addendum follows from Table 1 which lists all configurations of more than 48 lines and their respective transcendental lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' □ References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Conway and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Sloane, Sphere packings, lattices and groups, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' 290, Springer-Verlag, New York, 1988, With contributions by E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Bannai, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Leech, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Norton, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Odlyzko, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Parker, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Queen and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Venkov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' MR 920369 (89a:11067) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Degtyarev and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Rams, Counting lines with Vinberg’s algorithm, 2021, To appear, arXiv:2104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='04583.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' , Lines on K3-sextics, in preparation, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' , Ancillary files for the paper: Lines on K3-quartics via triangular sets, 2022, available on the arXiv as ancillary files for this preprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Alex Degtyarev, Lines on Smooth Polarized K3-Surfaces, Discrete Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' 62 (2019), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' 3, 601–648.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' MR 3996938 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' , Smooth models of singular K3-surfaces, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Iberoam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' 35 (2019), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' 1, 125– 172.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' MR 3914542 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' , Lines in supersingular quartics, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Japan 74 (2022), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' 3, 973–1019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' MR 4484237 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Alex Degtyarev, Ilia Itenberg, and Ali Sinan Sert¨oz, Lines on quartic surfaces, Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' 368 (2017), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' 1-2, 753–809.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' MR 3651588 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' GAP – Groups, Algorithms, and Programming, Version 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='1, https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='gap-system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='org, Feb 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Carl Friedrich Gauss, Disquisitiones arithmeticae, Springer-Verlag, New York, 1986, Trans- lated and with a preface by Arthur A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Clarke, Revised by William C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Waterhouse, Cornelius Greither and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Grootendorst and with a preface by Waterhouse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' MR 837656 (87f:01105) 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' V´ıctor Gonz´alez-Alonso and S�lawomir Rams, Counting lines on quartic surfaces, Taiwanese J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' 20 (2016), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' 4, 769–785.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' MR 3535673 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Jessop, Quartic surfaces with singular points, Cambridge University Press, Cambridge, 1916.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Rick Miranda and David R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Morrison, Embeddings of integral quadratic forms, Electronic, http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='ucsb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='edu/~drm/manuscripts/eiqf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='pdf, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Yoichi Miyaoka, Counting lines and conics on a surface, Publ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' 45 (2009), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' 3, 919–923.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' MR 2569571 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Nikulin, Finite groups of automorphisms of K¨ahlerian K3 surfaces, Trudy Moskov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Obshch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' 38 (1979), 75–137.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' MR 544937 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' , Integer symmetric bilinear forms and some of their geometric applications, Izv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Akad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Nauk SSSR Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' 43 (1979), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' 1, 111–177, 238, English translation: Math USSR-Izv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' 14 (1979), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' 1, 103–167 (1980).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' MR 525944 (80j:10031) 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' S�lawomir Rams and Matthias Sch¨utt, 112 lines on smooth quartic surfaces (characteristic 3), Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' 66 (2015), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' 3, 941–951.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' MR 3396099 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' , 64 lines on smooth quartic surfaces, Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' 362 (2015), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' 1-2, 679–698.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' MR 3343894 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Saint-Donat, Projective models of K-3 surfaces, Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' 96 (1974), 602–639.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' MR 0364263 (51 #518) 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Segre, The maximum number of lines lying on a quartic surface, Quart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=', Oxford Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' 14 (1943), 86–96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' MR 0010431 (6,16g) 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Ichiro Shimada, Connected components of the moduli of elliptic K3 surfaces, Michigan Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' 67 (2018), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' 3, 511–559.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' MR 3835563 22 ALEX DEGTYAREV AND S�LAWOMIR RAMS 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Davide Cesare Veniani, Lines on K3 quartic surfaces in characteristic 2, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' 68 (2017), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' 2, 551–581.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' MR 3667213 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' , The maximum number of lines lying on a K3 quartic surface, Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' 285 (2017), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' 3-4, 1141–1166.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' MR 3623744 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' , Symmetries and equations of smooth quartic surfaces with many lines, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' Iberoam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' 36 (2020), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' 1, 233–256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' MR 4061988 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' , Lines on K3 quartic surfaces in characteristic 3, Manuscripta Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' 167 (2022), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' 3-4, 675–701.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' MR 4385387 Department of Mathematics, Bilkent University, 06800 Ankara, TURKEY Email address: degt@fen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='bilkent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='tr Institute of Mathematics, Jagiellonian University, ul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content=' �Lojasiewicza 6, 30-348 Krak´ow, Poland Email address: slawomir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='rams@uj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} +page_content='pl' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdE2T4oBgHgl3EQfzAi5/content/2301.04127v1.pdf'} diff --git a/mtFLT4oBgHgl3EQffC-V/content/tmp_files/2301.12093v1.pdf.txt b/mtFLT4oBgHgl3EQffC-V/content/tmp_files/2301.12093v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..aa37ea853b832a5187c39b5704c6e6397d130f5d --- /dev/null +++ b/mtFLT4oBgHgl3EQffC-V/content/tmp_files/2301.12093v1.pdf.txt @@ -0,0 +1,1133 @@ +Local Contrast and Global Contextual Information Make Infrared Small Object +Salient Again +Chenyi Wang +Huan Wang +Peiwen Pan +Nanjing University of Science and Technology +{wcyjerry, Nanjing}@njust.edu.cn +Abstract +Infrared small object detection (ISOS) aims to segment +small objects only covered with several pixels from clut- +ter background in infrared images. It’s of great challenge +due to: 1) small objects lack of sufficient intensity, shape +and texture information; 2) small objects are easily lost in +the process where detection models, say deep neural net- +works, obtain high-level semantic features and image-level +receptive fields through successive downsampling. This pa- +per proposes a reliable detection model for ISOS, dubbed +UCFNet, which can handle well the two issues. It builds +upon central difference convolution (CDC) and fast Fourier +convolution (FFC). On one hand, CDC can effectively guide +the network to learn the contrast information between small +objects and the background, as the contrast information is +very essential in human visual system dealing with the ISOS +task. On the other hand, FFC can gain image-level receptive +fields and extract global information while preventing small +objects from being overwhelmed. Experiments on several +public datasets demonstrate that our method significantly +outperforms the state-of-the-art ISOS models, and can pro- +vide useful guidelines for designing better ISOS deep models. +Codes will be available soon. +1. Introduction +Infrared small object detection (ISOS) is a key technique +broadly used in early warning systems, night navigation, mar- +itime surveillance, UAV search and tracking and the like, due +to its all-weather working, long-range detection and conceal- +ment characteristics. Therefore, improving its performance +is of great significance. Researches on ISOS have been con- +ducted for over several decades. Many methods are proposed +which can be roughly categorized into (1) traditional meth- +ods focusing on signal processing and prior knowledge and +(2) deep learning models relying on Convolutional Neural +Networks (CNNs) and Visual Transformers(ViTs). +Traditional methods consist of three representative subcat- +egories: background-oriented methods, object-oriented meth- +ods, and low-rank decomposition methods. Background- +Figure 1. Examples of ISOS with the object indicated by the red +bounding boxes and a close-up is shown in the top left corner. Left: +a airplane with recognizable shape in a remote distance. Middle: a +car whose shape is almost lost and only can be identified by estima- +tion. Right: A small dim object drowned in a cloud background. +oriented methods like Max-mean/Max-medium [12] and Top- +Hat [37] separate object from complex background by using +all kinds of filters to estimate the scene background. Object- +oriented methods segment small object by designing differ- +ent measure methods. For instance, LCM [3], ILCM [17] +and TLLCM [16] use the contrast measure between cen- +tral point and its surroundings and PatchSim [1] applies +patch similarities to suppress false alarms. Low-rank decom- +position methods are frequently based on robust principal +component analysis(RPCA), by inductively treating the in- +put as a superposition of low-rank background and sparse +objects and solving such detection issues via optimization +techniques. Take the infrared patch-image (IPI) [14] model +as an example. It suggests a patch-sliding design that ex- +ploits better non-local self-correlation properties of images +via RPCA. Subsequently, recent works put efforts into de- +signing sound low-rank and prior constraints [24, 30], ex- +ploiting spatio-temporal and multi-mode correlation [7], and +applying advanced optimization schemes [9]. Though tradi- +tional methods have achieved some results in experimental +scenarios, they are sensitive to hyper-parameter setting, lack +of generalization and suffer from low performance under +complex real scenes. +As deep learning has become the mainstream in many +computer vision tasks, many pioneers have achieved great +improvement in ISOS using deep neural networks. Wang +et al. [31] used two generators to focus on two different +1 +arXiv:2301.12093v1 [cs.CV] 28 Jan 2023 + +tasks miss detection and false alarm and one discriminator +to get the balanced result of two generators. Dai et al. [10] +proposed a novel feature fusion method named asymmetric +contextual modulation (ACM), and proposed the first public +ISOS dataset in real scenes. Then they proposed ALC [11] +which included an unlearnable conditional local contrast +module. Though existing deep learning methods have got +great results, they mostly focus on the feature fusion. Liu et +al. [19] first introduced transformer block into ISOS task and +got great results. Zhang et al. introduced attention-guided +context module to help AGPCNet [41] focus on small object. +Zhang et al. proposed ISNet [40] which take shape into +account to achieve better performance. +However, the above models more or less neglect two na- +tive problems in ISOS: the infrared small object lacks of +sufficient common information like color and shape and +small objects would be drown in the excessive downsam- +pling process. This paper aims to provide a better solution +to the problems of information lacking and objects being +drown. Specifically, we design a U-net network behaving +as our ISOS backbone. Then we employ central difference +convolution (CDC), a human visual system based convolu- +tion operator, to extract essential contrast information. As +for vanish problem, we leverage fast Fourier convolution +(FFC) which uses convolution operation on frequency do- +main, thus being easy to obtain global information while +avoiding object disappearance. We demonstrate the effec- +tiveness of CDC with other advanced convolution operators +and we also compare FFC with other global information +extracting methods. Our final model, coined as UCFNet, +achieves new state-of-the-art performance over competitive +ISOS methods on existing public datasets. The contributions +of our work can be summarized as: +• We reveal two native problems in infrared small object +detection through analysing experiments with common +segmentation methods and propose a baseline network +structure. +• We further propose CDC to extract local contrast infor- +mation based on human visual system which is essential +for infrared small object detection, meanwhile we use +FFC to obtain image-level receptive fields and global +context while maintaining high resolution. +• We comprehensively evaluate our proposed method on +two public datasets, our UCF achieves new state-of-the- +art performance over other methods. +2. Related Work +2.1. ISOS +There are two main types of ISOS methods, traditional +methods based on mathematical modeling and deep learning +methods based on neural networks. +Among the traditional methods, Max-mean-Max-medium +[12] and Top-Hat [37] used filters to separate target of back- +ground. LCM [3], ILCM [17], TLLCM [16] and MPCM [34] +segmented small object by designing salient measures. IPI +model [14] treated the input as a superposition of low-rank +background and sparse yet shaped target, and solved such +issues by using Low-rank decomposition, further methods +like sound low-rank [24] and prior constraints [30] were +proposed based on IPI. These methods suffered from low +performance under complex scenarios. +As for deep learning methods, Wang et al. [31] used condi- +tional GAN [23] with two generators and one discriminator +to gain a great balance between miss detection and false +alarm (MDvsFA). Dai et al. [10] proposed an asymmetric +contextual modulation to help network performance well and +introduced the first public ISOS dataset SIRST in real scenes, +Dai et al. [11] further applied a handcraft dilated local con- +trast measure into network. Liu et al. [19] firstly introduced +multi-head self-attention into ISOS tasks and got a good +result. Zhang et al. [41] proposed AGPCNet with attention- +guided context block and context pyramid module. Zhang et +al. [40] took shape into account and designed Taylor finite +difference edge block and two-orientation attention, and also +proposed a more challengable dataset IRSTD. Deep learning +methods have become more and more dominant in ISOS due +to their great ability of robustness and generalization. +2.2. Convolution Operators +The ordinary convolution operation is to divide the feature +map into patches of the same size as the convolution kernel +and then perform a weighted sum operation, this fixed oper- +ation at each position may be suboptimal for some specific +tasks. Therefore, researchers have proposed some advanced +convolution operators. In order to solve the problem of stan- +dard convolution treating all input pixels as valid ones in +image inpainting tasks, Liu et al. [20] proposed partial convo- +lution, where the convolution is masked and renormalized to +be conditioned on only valid pixels. Yu et al. [35] proposed +gated convolution to provide a learnable dynamic feature se- +lection mechanism for each channel at each spatial location +across all layers. While in modeling geometric transforma- +tions, vanilla convolution is limited due to its fixed geometric +structure, Dai et al. [6] proposed deformable convolution to +enhance this ability by adding additional offsets learned from +target task. Zhu et al. [43] then proposed deformable convo- +lution v2 to reduce the impact of irrelevant region by adding +additional weight and mimicking features from RCNN [25]. +Yu et al. [36] proposed central difference convolution in +face anti-spoofing task, which is able to capture intrinsic +detailed patterns via aggregating both intensity and gradient +information. +2 + +2.3. Global Information Extraction +One common concern is the ability of the network to +grasp the local and global information, local context usually +are easy to extract using convolution, while how global the in- +formation can get usually determined by the receptive fields +of the network. The most common strategy to improve the +receptive fields is stacking convolutions and downsamplings +constantly, continuous convolution can linearly improve re- +ceptive fields while downsamplings multiply, dilated (atrous) +convolution inserts holes between pixels in convolutional +kernels and can obtain larger receptive fields than standard +convolution, Chen et al. [4] proposed dilation atrous pyra- +mid pooling (ASPP) to capture multi-scale information and +Wang et al. [32] designed a hybrid dilated convolution to +enlarge receptive fields, they both got good improvements in +semantic segmentation tasks. Attention mechanism [29,33] +can gain global information by calculating the correlation +of each single pixel with other pixels. Fu et al. proposed +spatial attention and channel attention [13] and achieved +great results. Chi et al. [5] proposed fast Fourier convolution +which performs convolution operators in frequency domain +to conduct a global influence in spatial domain thus it can +extract image-level information. Suvorov et al. proposed +LAMA [28] which successfully applied it in large kernel +image inpainting tasks. Berenguel et al. [2] then used FFC +in monocular depth estimation and semantic segmentation. +3. Method +3.1. Inspiration +We draw inspiration from a series of experiments using +common segmentation networks, including FPN [18], U- +Net [26], PSPNet [42] and DeepLabv3 [4], the results in +Table 1 indicate two anomalies: one is the performance is +quite polarized, the models which fuse more low-level fea- +tures (FPN and U-Net) significantly outperforming others; +the other is their performance drops only a little as the net- +work width and depth increase. We further visualize the +Table 1. Results of common segmentation models in ISOS tasks. +Method +Backbone +Base width +IoU↑ +Params (M)↓ +U-Net [26] +- +64 +69.89 +31.04 +FPN [18] +ResNet-18 +64 +70.38 +14.12 +ResNet-50 +256 +70.12 +27.19 +ResNet-101 +256 +69.64 +46.18 +PSPNet [42] +ResNet-50 +256 +22.73 +46.65 +DeepLabv3 [4] +ResNet-50 +256 +39.71 +35.86 +attention maps of 4 stages using grad-cam [15,27] in FPN. +As we can see in Fig. 2, the small object has been completely +lost due to the excessive downsampling. Meanwhile directly +Figure 2. Visualization of 4 layers in FPN. Objects are completely +lost in layer 4 with a 32× downsampling. +increasing the width and depth of network still fails to mine +more information while may lead to redundancy of parame- +ters. Based on these observation, we summarize two essential +guidelines on ISOS: 1) Extracting global information while +maintaining high resolution may avoid small object being +overwhelmed. 2) Extracting additional essential information +using modules dedicated to ISOS could be more effective +than directly increasing the depth and width of a network. +3.2. Overview of UFCNet +Based on the enlightenment from Sec. 3.1, we proposed a +U-shape network with central difference convolution (CDC) +and fast Fourier convolution (FFC). The whole framework +can been seen in Fig. 3. The number of base channels is +32 and only 4 downsampling operations are performed in +the whole process, which satisfies the needs of ISOS tasks +while avoiding inefficient redundancy. Specially, during the +downsample process, we use central difference convolution +residual block which contains a standard convolution and +CDC with a residual convolution for channel alignment. We +use two cascaded FFC to form our fast Fourier convolu- +tion residual block which aims to extract global context in +high resolution. In what follows, we will briefly introduce +central difference convolution in Sec. 3.3 and fast Fourier +convolution in Sec. 3.4. +3.3. Central Difference Convolution +Convolution can effectively extract color, shape, texture +and other information, while standard convolution may be +limited because the lack of these information in ISOS tasks. +Inspired by human visual system sensitive to intensity differ- +ence and contrast, we utilize central difference convolution +3 + +input +layerl (4x) +layer2 (8x) +layer3 (16x) +layer4 (32x)Figure 3. The scheme of the proposed model UCFNet. UCFNet is based on central difference convolution residual blocks and fast Fourier +convolution residual blocks, can effectively extract both local contrast information and global context. +to guide our network. CDC can introduce additional contrast +information by computing the difference between the center +point and other points within the convolution window and +can be described as Eq. (1), +f(x, y) = +� +(i,j)∈R +w(i,j) · (F(x+i,y+j) − F(x,y)) +(1) +In addition, CDC is often combined with vanilla convolution +to retain some of the traditional feature extraction capabili- +ties, and the whole process can be expressed in the equation. +CDC(x, y) =θ · +� +(i,j)∈R +w(i,j)(F(x+i,y+j) − F(x,y)) ++ (1 − θ) · +� +(i,j)∈R +w(i,j)(F(x+i,y+j)) (2) +where the hyperparameter θ ∈ (0, 1) is used to determine +the contribution between CDC and vanilla convolution. The +window size R used to calculate the difference is equal to the +convolution kernel size while the receptive fields of it can +naturally increase during the forward process, thus making it +capable of extracting multi-level contrast information which +further help the network identify infrared small objects with +different size and effectively guide the network performs +well. +3.4. Fast Fourier Convolution +Another issue of ISOS is that the small objects are easily +overwhelmed in the excessive downsample layers which +is used to obtain global information. In order to solve this +contradiction, rather than using the Transformer model with +long range dependency, we employ fast Fourier convolution +(FFC) which can gain image-level receptive fields in high +resolution thus we can extract global context without losing +small objects. We follow the structure in LAMA [28] for +FFC. Specifically, we splits the channels into two parallel +branch, the local branch uses standard convolution to extract +local information while global branch applies the Fourier +Unit(FU) to gain global context, and there are two additional +short-cuts for information fusion. The whole process can +described as: +Yl = Convl−>l(xl) + Convl−>g(xl) +(3) +Yg = Convg−>l(xg) + FUg−>g(xg) +(4) +The Four Unit is key to get global context, because it trans- +forms input features from spatial domain to frequency do- +main where each single point corresponds to all points in the +spatial domain. Therefore, the convolution operation being +conducted within a small kernel size is able to influence +4 + +XN +Fast Fourier Conv +CDC +Residual Block +Conv +Central Difference Conv +Head +FFC +FFC +Residual Block +Conv +FFC +Conv +CDC +Fourier Unit + Local + Global +Conv-BN-Relu +Real FFT2d +Con +CDC +Conv-BN-Relu + BN-Relu +BN-Relu + Inv Real FFT2d +Conv + Local + Global +Conv +Convthe all image in spatial domain and obtain global contextual +information. The FU makes following steps: +1) Transform the input feature map from spatial domain to +frequency domain with Real FFT2d and concatenates +the real and imaginary parts +Real FFT2d : RH×W ×C → CH× W +2 ×C +ComplexToReal : CH× W +2 ×C → RH× W +2 ×2C +2) Applie convolution, normalization and activation func- +tion in the frequency domain +Conv ◦ Norm ◦ Act : RH× W +2 ×2C → RH× W +2 ×2C +3) Transform inversely from frequency domain to spatial +domain +RealToComplex : RH× W +2 ×2C → CH× W +2 ×C +Inverse Real FFT2d : CH× W +2 ×C → RH×W ×C +We concatenate the local branch and global branch into one +in the end, and it contains rich local and global information. +4. Experiment +4.1. Datasets and Metrics +Datasets. We evaluate our methods on two widely-used +datasets in ISOS: SIRST [10] and IRSTD [40]. SIRST con- +tains 427 images in real IR scenes with half of the objects +in SIRST only contains 0.1% pixels of whole image. Larger +than SIRST, IRSTD contains 1001 images with more chal- +lenging object and complex backgrounds. Both SIRST and +IRSTD are separated into training set and test set with a ratio +of 8:2. +Metrics. Following privious works, we use pixel-level +metrics (IoU and nIoU) and target-level metrics (Pd and Fa) +to measure our method. Intersection over Union (IoU) and +normalized Intersection over Union (nIoU) can described as: +IoU = 1 +n · +�n +i=0 tpi +�n +i=0(fpi + fni − tpi) +(5) +nIoU = 1 +n · +n +� +i=0 +tpi +fpi + fni − tpi +(6) +Where n means to the total number of samples, tp denotes +the true positive, fp denotes false positive and fn denotes +false negative. While target-level metrics probability of de- +tection (Pd) and false alarm rate (Fa) can be described as: +Pd = 1 +n · +n +� +i=0 +N i +pred +N i +all +(7) +Fa = 1 +n · +n +� +i=0 +P i +false +P i +all +(8) +where Npred, Nall denotes the number of correct detected +objects and the number of total objects and Pfalse, Pall +denotes the pixels of false detected objects and the pixels of +total objects. We regard that the detection is correct when +the distance between the centers of the predict result and the +ground truth is less than 4. +In addition, we use F-score to evaluate the balance of +the model’s Precision and Recall and we use receiver op- +erating characteristic (ROC) curve to evaluate the dynamic +relationship between true positive ratio and false positive +ratio. The equations of Precision and Recall are shown in +Eqs. (9) and (10). +Precision = +�n +i=0 tpi +�n +i=0(tpi + fpi) +(9) +Recall = +�n +i=0 tpi +�n +i=0(tpi + fni) +(10) +and F-score is the harmonic mean of Precision and Recall. +F − score = 2 · Precision · Recall +Precision + Recall +(11) +4.2. Implementation Details +We conduct experiments on a computer with 2.50GHz +CPU, 16GB RAM and GeForce RTX 3090 based on Pytorch. +For more details, we use AdamW optimizer with an initial +learning rate of 0.001 and decayed by Cosine-Anneling- +LR [22] schduler and we use binary cross entropy loss and +soft IoU loss as our criterion. Each experiment is trained for +300 epochs with a batch size of 8. Our UCFNet achieves the +best performance with CDC ratio θ of 0.7 while using 7 FFC +residual blocks. +4.3. Quantitative Results +We select some traditinal methods: Top-Hat [37], LCM +[3], WLDM [21], NARM [38], PSTNN [39], IPI [14], +RIPT [8], NIPPS [9] and several open source deep learning +SOTA methods including MDvsFA [], ACM [10], ALC [11], +AGCP [41] and Transformer [19] which for comparison. The +results are shown in Table 2, deep learning methods basi- +cally perform better than traditional ones due to their great +power of feature extraction and generalization. Our proposed +method (UCF) shows great superior over other deep learning +methods in terms of all metrics on both datasets. Especially +on SIRST, UCF gets the performance of 80.89 of IoU and +78.72 of nIoU about 8% improvements over other meth- +ods and impressively obtains 100% probability of detection +while only obtain 2.22×10−6 false alarm. While on IRSTD, +we also outperform 3 ∼ 8 in IoU and nIoU over other SOTA +deep learning methods result in 68.92 and 69.26. +We further study the dynamic relationship between Preci- +sion and Recall with LCM [3], IPI [14], ACM [10], AGCP +[41] and our method UCF. The ROC curve are shown in +5 + +SIRST (TR=0.5) +IRSTD (TR=0.5) +Pixel Level +Object Level +Pixel Level +Object Level +Method +IoU ↑ +nIoU ↑ +Pd ↑ +Fa ↓ +IoU ↑ +nIoU ↑ +Pd ↑ +Fa ↓ +Top-Hat [37] +5.86 +25.42 +78.90 +1397.12 +4.26 +15.08 +67.00 +422.25 +LCM [3] +6.84 +8.96 +77.06 +183.15 +4.45 +4.73 +57.58 +66.56 +WLDM [21] +22.28 +28.62 +87.16 +98.34 +9.77 +16.07 +63.97 +177.35 +NARM [38] +25.95 +32.23 +79.82 +19.74 +7.77 +12.24 +61.96 +12.24 +PSTNN [39] +39.44 +47.72 +83.49 +41.07 +16.44 +25.91 +65.32 +76.92 +IPI [14] +40.48 +50.95 +91.74 +148.37 +14.40 +31.29 +86.35 +450.36 +RIPT [8] +25.49 +33.01 +85.32 +24.75 +8.15 +16.12 +68.35 +26.36 +NIPPS [9] +33.16 +40.91 +80.73 +23.64 +16.38 +27.10 +70.37 +63.27 +MDvsFA [31] +56.17 +59.84 +90.88 +177.90 +50.85 +45.97 +81.48 +23.01 +ACM [10] +72.45 +72.15 +93.52 +12.39 +63.38 +60.80 +91.58 +15.31 +ALC [11] +73.32 +73.24 +99.08 +24.51 +63.91 +62.65 +92.58 +16.07 +Transformer [19] +72.82 +71.22 +98.15 +27.15 +61.89 +60.64 +90.91 +12.64 +AGPC [41] +73.69 +72.60 +98.17 +16.99 +66.29 +65.23 +92.83 +13.12 +UCF (Ours) +80.89 +78.72 +100.00 +2.26 +68.92 +69.26 +93.60 +11.01 +Table 2. Quantitative evaluation of ISOS on SIRST and IRSTD datasets. We report pixel level metric IoU (%) and nIoU (%) and object level +metric Pd (%) and Fa (10−6). All the deep learning methods outperform traditional methods and our UCF achieves the best performance in +all terms of metrics on both datasets. +Fig. 4, the area under the curve (AUC) is a key metric for +quantitative evaluation of ROC, our UCF also gets the largest +area and highest F-score on both datasets, as shown in Ta- +ble 3. +Figure 4. ROC curve on SIRST (solid line) and IRSTD (dotted line) +of different methods. +Table 3. F-score and the area under ROC curve on both datasets +with different methods. +Method +SIRST +IRSTD +F-score↑ +Auc↑ +F-score↑ +Auc↑ +LCM [3] +12.80 +0.058 +8.52 +0.099 +IPI [14] +57.63 +0.448 +25.17 +0.248 +ACM [10] +84.02 +0.684 +77.59 +0.719 +AGPC [41] +84.85 +0.765 +79.73 +0.734 +UCF +89.43 +0.843 +81.60 +0.745 +4.4. Qualitative Comparisons +Some visualization results of different methods are shown +in Fig. 5. The first four rows show scenes with quite compli- +cated backgrounds, where our method UCF can achieve less +false alarm and high detection rate, significantly outperform +others on object level metric because FFC can bring global +contextual information in high resolution feature maps. As to +relatively simple scenarios (the last two rows), the advantage +of UCF lies in providing more details about object shape +6 + +1.0 +0.8 +UCF +AGPC +ACM +0.6 +IPI +LCM +UCF +0.4 +AGPC +ACM +IPI +LCM +0.2 +0.0 +0.00 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +1.75 +2.00 +False positive rate +1e-4Figure 5. Qualitative results of different methods. Red bounding boxes indicate true objects and yellow bounding boxes indicate false alarms +which predicted. As seen in the first four rows, in complicated backgrounds, our UCF can get less false alarm and correctly detect more +targets in target level. From the last two rows, we can see that UCF describe the shape and texture in more detail in relatively simple scenes. +(Best viewed by zooming in) +and structure due to the contrast information mined by CDC, +thus favoring the pixel-level metric. +4.5. Ablation Experiments +We investigate the effectiveness of CDC and FFC with a +series of ablation experiments on SIRST, we first generally +shows the effectiveness of CDC and FFC in Table 4, CDC +improves the performance with local contrast information +while FFC helps a lot with global information. The combina- +tion of CDC and FFC achieve the best result in IoU, nIoU +and Pd while drop a little bit in Fa compares to only use FFC, +this is because of CDC is tended to fix the shape and texture +in detail which inevitably results in more false pixels. +Effectiveness of CDC. We conduct experiments on dif- +ferent hyperparameter θ which determines the ratio between +Table 4. Ablation study of CDC and FFC in IoU(%), nIoU(%), +Pd(%), Fa(10−6) +Method +IoU↑ +nIoU↑ +Pd↑ +Fa↓ +UCF (vanilla) +74.14 +74.89 +96.33 +4.83 +UCF + CDC +75.95 +76.29 +97.25 +3.46 +UCF + FFC +79.55 +77.23 +100.00 +1.82 +UCF + CDC + FFC +80.89 +78.72 +100.00 +2.22 +CDC and vanilla convolution. As shown in Fig. 6a, CDC +always achieves better performance than vanilla covolution +by mining essential contrast information and we achieve the +7 + +AGPCNet +Tophat +IPI +UCF +Infrared image +ACM +ALCNet +Ground truth +回 +口 +口 +口 +.1 +国 +0 +国 +国 +国 +国口 +， +. +14. +口 +口 +国 +M(a) IoU curves on different paramater θ in CDC, the best performance +is obtained when θ = 0.7. +(b) IoU curves using different number of FFC residual blocks, we +achieve our best performance with 5 FFC residual blocks. +Figure 6. IoU curves on SIRST with different parameter settings. +best performance when θ = 0.7. We further compare de- +formable convolution and gated convolution with CDC. As +show in Table 5, gated convolution only has little improve- +ment than vanilla convolution while deformable convolution +drops a lot, this is because the sharp information is quite in- +sufficient in ISOS tasks and deformable convolution fails to +learn the offsets according to the target’s shape. All these in- +dicates that our CDC has superiorities over other convolution +operators in ISOS. +Table 5. Ablation study of CDC with other convolution operators +in metrics of IoU (%), nIoU (%), Pd (%), Fa (10−6). +Conv Operator +IoU↑ +nIoU↑ +Pd↑ +Fa↓ +Vanilla +74.14 +74.89 +96.33 +4.83 +Gated [35] +74.31 +74.57 +97.25 +23.20 +DeformableV1 [6] +68.00 +69.60 +93.58 +12.60 +DeformableV2 [43] +69.72 +72.67 +96.33 +29.90 +CDC (θ = 0.7) +75.95 +76.29 +97.25 +3.46 +Effectiveness of FFC. As for FFC block, we first study +the inner parameter of n which determined the number of +FFC blocks we use in our method. As shown in Fig. 6b, +FFC residual block which designed to extract global context +can effectively improve the performance effectively and we +achieve the best performance when using 5 FFC residual +blocks. We further explore other global information extract- +ing methods mentioned in Sec. 2.3, from Table 6 we can +see dilated convolution and double attention also get some +improvements due to enlarging the receptive fields and ex- +tracting global context, but dilated convolution enlarges the +receptive fields step by step, it only get image-level informa- +Table 6. Ablation study of FFC with other global information ex- +tracting method (vanilla conv blocks, multi-dilated conv blocks and +double attention block in metrics of IoU (%), nIoU (%), Pd (%), +Fa (10−6). +Method +IoU↑ +nIoU↑ Pd↑ +Fa↓ +Vanilla Conv blocks +75.31 73.91 +98.17 +12.29 +Dilated Conv blocks [32] 77.34 75.51 +99.08 +6.56 +Double attention [13] +77.87 75.84 +100.00 12.51 +FFC blocks +79.55 77.23 +100.00 1.82 +tion in the deep layers, attention mechanism computes the +correlation in the whole image but lack of local inductive +bias. Thus they are inferior to FFC in ISOS tasks. +5. Conclusion +In this study, we have indicated and analyzed the two im- +portant issues on ISOS which affects the model performance. +Inspired from these issues, we propose a simple but effective +method. Specifically, to handel the first issue of insufficient +information, we use central difference convolution to guide +the network to focus on local contrast information. To deal +with the second one, we employ the fast Fourier convolution +to extract global context in high-resolution feature maps, +preventing small objects from being overwhelmed. Exten- +sive experiments have validated that our model UCF shows +great superiority over other state-of-the-art methods on both +two public datasets. The proposed model can also serve as a +guideline for further investigations on ISOS tasks. +8 + +75 +70 +65 +IoU +60 +vanilla +0.3 +0.5 +55 +0.7 +1.0 +50 +50 +100 +150 +200 +250 +300 +epoch75 +70 +IoU +65 +60 +55 +50 +50 +100 +150 +200 +250 +300 +epochReferences +[1] Kun Bai, Yuehuang Wang, and Qiong Song. Patch similar- +ity based edge-preserving background estimation for single +frame infrared small target detection. 2016 IEEE Interna- +tional Conference on Image Processing (ICIP), pages 181– +185, 2016. 1 +[2] Bruno Berenguel-Baeta, Jesus Bermudez-Cameo, and Jose J +Guerrero. Fredsnet: Joint monocular depth and semantic +segmentation with fast fourier convolutions. arXiv preprint +arXiv:2210.01595, 2022. 3 +[3] CL Philip Chen, Hong Li, Yantao Wei, Tian Xia, and +Yuan Yan Tang. A local contrast method for small infrared +target detection. IEEE transactions on geoscience and remote +sensing, 52(1):574–581, 2013. 1, 2, 5, 6 +[4] Liang-Chieh Chen, George Papandreou, Iasonas Kokkinos, +Kevin Murphy, and Alan L Yuille. Deeplab: Semantic image +segmentation with deep convolutional nets, atrous convolu- +tion, and fully connected crfs. IEEE transactions on pattern +analysis and machine intelligence, 40(4):834–848, 2017. 3 +[5] Lu Chi, Borui Jiang, and Yadong Mu. Fast fourier convolution. +Advances in Neural Information Processing Systems, 33:4479– +4488, 2020. 3 +[6] Jifeng Dai, Haozhi Qi, Yuwen Xiong, Yi Li, Guodong Zhang, +Han Hu, and Yichen Wei. Deformable convolutional net- +works. In Proceedings of the IEEE international conference +on computer vision, pages 764–773, 2017. 2, 8 +[7] Yimian Dai and Yiquan Wu. Reweighted infrared patch-tensor +model with both nonlocal and local priors for single-frame +small target detection. IEEE journal of selected topics in +applied earth observations and remote sensing, 10(8):3752– +3767, 2017. 1 +[8] Yimian Dai and Yiquan Wu. Reweighted infrared patch-tensor +model with both nonlocal and local priors for single-frame +small target detection. IEEE journal of selected topics in +applied earth observations and remote sensing, 10(8):3752– +3767, 2017. 5, 6 +[9] Yimian Dai, Yiquan Wu, and Yu Song. Infrared small target +and background separation via column-wise weighted robust +principal component analysis. Infrared Physics & Technology, +77:421–430, 2016. 1, 5, 6 +[10] Yimian Dai, Yiquan Wu, Fei Zhou, and Kobus Barnard. +Asymmetric contextual modulation for infrared small target +detection. In Proceedings of the IEEE/CVF Winter Confer- +ence on Applications of Computer Vision, pages 950–959, +2021. 2, 5, 6 +[11] Yimian Dai, Yiquan Wu, Fei Zhou, and Kobus Barnard. At- +tentional Local Contrast Networks for Infrared Small Target +Detection. IEEE Transactions on Geoscience and Remote +Sensing, pages 1–12, 2021. 2, 5, 6 +[12] Suyog D. Deshpande, Meng Hwa Er, Ronda Venkateswarlu, +and Philip Chan. Max-mean and max-median filters for de- +tection of small targets. In Optics & Photonics, 1999. 1, +2 +[13] Jun Fu, Jing Liu, Haijie Tian, Yong Li, Yongjun Bao, Zhiwei +Fang, and Hanqing Lu. Dual attention network for scene +segmentation. In Proceedings of the IEEE/CVF conference on +computer vision and pattern recognition, pages 3146–3154, +2019. 3, 8 +[14] Chenqiang Gao, Deyu Meng, Yi Yang, Yongtao Wang, Xiao- +fang Zhou, and Alexander Hauptmann. Infrared patch-image +model for small target detection in a single image. IEEE +Transactions on Image Processing, 22:4996–5009, 2013. 1, +2, 5, 6 +[15] Jacob Gildenblat and contributors. +Pytorch library for +cam methods. +https://github.com/jacobgil/ +pytorch-grad-cam, 2021. 3 +[16] Jinhui Han, Sibang Liu, Gang Qin, Qian Zhao, Honghui +Zhang, and Nana Li. A local contrast method combined +with adaptive background estimation for infrared small target +detection. IEEE Geoscience and Remote Sensing Letters, +16(9):1442–1446, 2019. 1, 2 +[17] Jinhui Han, Yong Ma, Bo Zhou, Fan Fan, Kun Liang, and +Yu Fang. A robust infrared small target detection algorithm +based on human visual system. IEEE Geoscience and Remote +Sensing Letters, 11(12):2168–2172, 2014. 1, 2 +[18] Tsung-Yi Lin, Piotr Doll´ar, Ross Girshick, Kaiming He, +Bharath Hariharan, and Serge Belongie. Feature pyramid +networks for object detection. In Proceedings of the IEEE +conference on computer vision and pattern recognition, pages +2117–2125, 2017. 3 +[19] Fangcen Liu, Chenqiang Gao, Fang Chen, Deyu Meng, Wang- +meng Zuo, and Xinbo Gao. Infrared small-dim target detec- +tion with transformer under complex backgrounds. arXiv +preprint arXiv:2109.14379, 2021. 2, 5, 6 +[20] Guilin Liu, Fitsum A Reda, Kevin J Shih, Ting-Chun Wang, +Andrew Tao, and Bryan Catanzaro. Image inpainting for +irregular holes using partial convolutions. In Proceedings of +the European conference on computer vision (ECCV), pages +85–100, 2018. 2 +[21] Jie Liu, Ziqing He, Zuolong Chen, and Lei Shao. Tiny and +dim infrared target detection based on weighted local contrast. +IEEE Geoscience and Remote Sensing Letters, 15(11):1780– +1784, 2018. 5, 6 +[22] Ilya Loshchilov and Frank Hutter. Sgdr: Stochastic gradient +descent with warm restarts. arXiv preprint arXiv:1608.03983, +2016. 5 +[23] Mehdi Mirza and Simon Osindero. Conditional generative +adversarial nets. arXiv preprint arXiv:1411.1784, 2014. 2 +[24] Sur Singh Rawat, Sashi Kant Verma, and Yatindra Kumar. +Reweighted infrared patch image model for small target de- +tection based on non-convex lp-norm minimisation and tv +regularisation. IET image processing, 14(9):1937–1947, 2020. +1, 2 +[25] Shaoqing Ren, Kaiming He, Ross Girshick, and Jian Sun. +Faster r-cnn: Towards real-time object detection with region +proposal networks. Advances in neural information process- +ing systems, 28, 2015. 2 +[26] Olaf Ronneberger, Philipp Fischer, and Thomas Brox. U-net: +Convolutional networks for biomedical image segmentation. +In International Conference on Medical image computing +and computer-assisted intervention, pages 234–241. Springer, +2015. 3 +9 + +[27] Ramprasaath R Selvaraju, Michael Cogswell, Abhishek Das, +Ramakrishna Vedantam, Devi Parikh, and Dhruv Batra. Grad- +cam: Visual explanations from deep networks via gradient- +based localization. In Proceedings of the IEEE international +conference on computer vision, pages 618–626, 2017. 3 +[28] Roman Suvorov, Elizaveta Logacheva, Anton Mashikhin, +Anastasia Remizova, Arsenii Ashukha, Aleksei Silvestrov, +Naejin Kong, Harshith Goka, Kiwoong Park, and Victor Lem- +pitsky. Resolution-robust large mask inpainting with fourier +convolutions. In Proceedings of the IEEE/CVF Winter Confer- +ence on Applications of Computer Vision, pages 2149–2159, +2022. 3, 4 +[29] Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszko- +reit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, and Illia +Polosukhin. Attention is all you need. Advances in neural +information processing systems, 30, 2017. 3 +[30] Minjie Wan, Guohua Gu, Yunkai Xu, Weixian Qian, Kan +Ren, and Qian Chen. Total variation-based interframe in- +frared patch-image model for small target detection. IEEE +Geoscience and Remote Sensing Letters, 19:1–5, 2021. 1, 2 +[31] Huan Wang, Luping Zhou, and Lei Wang. Miss detection +vs. false alarm: Adversarial learning for small object segmen- +tation in infrared images. In Proceedings of the IEEE/CVF +International Conference on Computer Vision, pages 8509– +8518, 2019. 1, 2, 6 +[32] Panqu Wang, Pengfei Chen, Ye Yuan, Ding Liu, Zehua Huang, +Xiaodi Hou, and Garrison Cottrell. Understanding convolu- +tion for semantic segmentation. In 2018 IEEE winter con- +ference on applications of computer vision (WACV), pages +1451–1460. Ieee, 2018. 3, 8 +[33] Xiaolong Wang, Ross Girshick, Abhinav Gupta, and Kaiming +He. Non-local neural networks. In Proceedings of the IEEE +conference on computer vision and pattern recognition, pages +7794–7803, 2018. 3 +[34] Yantao Wei, Xinge You, and Hong Li. Multiscale patch-based +contrast measure for small infrared target detection. Pattern +Recognition, 58:216–226, 2016. 2 +[35] Jiahui Yu, Zhe Lin, Jimei Yang, Xiaohui Shen, Xin Lu, and +Thomas S Huang. Free-form image inpainting with gated +convolution. In Proceedings of the IEEE/CVF international +conference on computer vision, pages 4471–4480, 2019. 2, 8 +[36] Zitong Yu, Chenxu Zhao, Zezheng Wang, Yunxiao Qin, Zhuo +Su, Xiaobai Li, Feng Zhou, and Guoying Zhao. Searching cen- +tral difference convolutional networks for face anti-spoofing. +In Proceedings of the IEEE/CVF Conference on Computer +Vision and Pattern Recognition, pages 5295–5305, 2020. 2 +[37] Ming Zeng, Jian xun Li, and Zhang xiao Peng. The design of +top-hat morphological filter and application to infrared target +detection. Infrared Physics & Technology, 48:67–76, 2006. +1, 2, 5, 6 +[38] Landan Zhang, Lingbing Peng, Tianfang Zhang, Siying Cao, +and Zhenming Peng. Infrared small target detection via non- +convex rank approximation minimization joint l 2, 1 norm. +Remote Sensing, 10(11):1821, 2018. 5, 6 +[39] Landan Zhang and Zhenming Peng. Infrared small target +detection based on partial sum of the tensor nuclear norm. +Remote Sensing, 11(4):382, 2019. 5, 6 +[40] Mingjin Zhang, Rui Zhang, Yuxiang Yang, Haichen Bai, Jing +Zhang, and Jie Guo. Isnet: Shape matters for infrared small +target detection. In Proceedings of the IEEE/CVF Conference +on Computer Vision and Pattern Recognition, pages 877–886, +2022. 2, 5 +[41] Tianfang Zhang, Siying Cao, Tian Pu, and Zhenming Peng. +Agpcnet: Attention-guided pyramid context networks for in- +frared small target detection, 2021. 2, 5, 6 +[42] Hengshuang Zhao, Jianping Shi, Xiaojuan Qi, Xiaogang +Wang, and Jiaya Jia. Pyramid scene parsing network. In +Proceedings of the IEEE conference on computer vision and +pattern recognition, pages 2881–2890, 2017. 3 +[43] Xizhou Zhu, Han Hu, Stephen Lin, and Jifeng Dai. +De- +formable convnets v2: More deformable, better results. In +Proceedings of the IEEE/CVF conference on computer vision +and pattern recognition, pages 9308–9316, 2019. 2, 8 +10 + diff --git a/mtFLT4oBgHgl3EQffC-V/content/tmp_files/load_file.txt b/mtFLT4oBgHgl3EQffC-V/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..0d78335bc21a7c655db4190eb77a469eb2d1f8e2 --- /dev/null +++ b/mtFLT4oBgHgl3EQffC-V/content/tmp_files/load_file.txt @@ -0,0 +1,615 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf,len=614 +page_content='Local Contrast and Global Contextual Information Make Infrared Small Object Salient Again Chenyi Wang Huan Wang Peiwen Pan Nanjing University of Science and Technology {wcyjerry, Nanjing}@njust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='cn Abstract Infrared small object detection (ISOS) aims to segment small objects only covered with several pixels from clut- ter background in infrared images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' It’s of great challenge due to: 1) small objects lack of sufficient intensity, shape and texture information;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' 2) small objects are easily lost in the process where detection models, say deep neural net- works, obtain high-level semantic features and image-level receptive fields through successive downsampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' This pa- per proposes a reliable detection model for ISOS, dubbed UCFNet, which can handle well the two issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' It builds upon central difference convolution (CDC) and fast Fourier convolution (FFC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' On one hand, CDC can effectively guide the network to learn the contrast information between small objects and the background, as the contrast information is very essential in human visual system dealing with the ISOS task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' On the other hand, FFC can gain image-level receptive fields and extract global information while preventing small objects from being overwhelmed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Experiments on several public datasets demonstrate that our method significantly outperforms the state-of-the-art ISOS models, and can pro- vide useful guidelines for designing better ISOS deep models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Codes will be available soon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Introduction Infrared small object detection (ISOS) is a key technique broadly used in early warning systems, night navigation, mar- itime surveillance, UAV search and tracking and the like, due to its all-weather working, long-range detection and conceal- ment characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Therefore, improving its performance is of great significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Researches on ISOS have been con- ducted for over several decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Many methods are proposed which can be roughly categorized into (1) traditional meth- ods focusing on signal processing and prior knowledge and (2) deep learning models relying on Convolutional Neural Networks (CNNs) and Visual Transformers(ViTs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Traditional methods consist of three representative subcat- egories: background-oriented methods, object-oriented meth- ods, and low-rank decomposition methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Background- Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Examples of ISOS with the object indicated by the red bounding boxes and a close-up is shown in the top left corner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Left: a airplane with recognizable shape in a remote distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Middle: a car whose shape is almost lost and only can be identified by estima- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Right: A small dim object drowned in a cloud background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' oriented methods like Max-mean/Max-medium [12] and Top- Hat [37] separate object from complex background by using all kinds of filters to estimate the scene background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Object- oriented methods segment small object by designing differ- ent measure methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' For instance, LCM [3], ILCM [17] and TLLCM [16] use the contrast measure between cen- tral point and its surroundings and PatchSim [1] applies patch similarities to suppress false alarms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Low-rank decom- position methods are frequently based on robust principal component analysis(RPCA), by inductively treating the in- put as a superposition of low-rank background and sparse objects and solving such detection issues via optimization techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Take the infrared patch-image (IPI) [14] model as an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' It suggests a patch-sliding design that ex- ploits better non-local self-correlation properties of images via RPCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Subsequently, recent works put efforts into de- signing sound low-rank and prior constraints [24, 30], ex- ploiting spatio-temporal and multi-mode correlation [7], and applying advanced optimization schemes [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Though tradi- tional methods have achieved some results in experimental scenarios, they are sensitive to hyper-parameter setting, lack of generalization and suffer from low performance under complex real scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' As deep learning has become the mainstream in many computer vision tasks, many pioneers have achieved great improvement in ISOS using deep neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' [31] used two generators to focus on two different 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='12093v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='CV] 28 Jan 2023 tasks miss detection and false alarm and one discriminator to get the balanced result of two generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Dai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' [10] proposed a novel feature fusion method named asymmetric contextual modulation (ACM), and proposed the first public ISOS dataset in real scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Then they proposed ALC [11] which included an unlearnable conditional local contrast module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Though existing deep learning methods have got great results, they mostly focus on the feature fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' [19] first introduced transformer block into ISOS task and got great results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' introduced attention-guided context module to help AGPCNet [41] focus on small object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' proposed ISNet [40] which take shape into account to achieve better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' However, the above models more or less neglect two na- tive problems in ISOS: the infrared small object lacks of sufficient common information like color and shape and small objects would be drown in the excessive downsam- pling process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' This paper aims to provide a better solution to the problems of information lacking and objects being drown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Specifically, we design a U-net network behaving as our ISOS backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Then we employ central difference convolution (CDC), a human visual system based convolu- tion operator, to extract essential contrast information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' As for vanish problem, we leverage fast Fourier convolution (FFC) which uses convolution operation on frequency do- main, thus being easy to obtain global information while avoiding object disappearance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' We demonstrate the effec- tiveness of CDC with other advanced convolution operators and we also compare FFC with other global information extracting methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Our final model, coined as UCFNet, achieves new state-of-the-art performance over competitive ISOS methods on existing public datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' The contributions of our work can be summarized as: We reveal two native problems in infrared small object detection through analysing experiments with common segmentation methods and propose a baseline network structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' We further propose CDC to extract local contrast infor- mation based on human visual system which is essential for infrared small object detection, meanwhile we use FFC to obtain image-level receptive fields and global context while maintaining high resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' We comprehensively evaluate our proposed method on two public datasets, our UCF achieves new state-of-the- art performance over other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Related Work 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' ISOS There are two main types of ISOS methods, traditional methods based on mathematical modeling and deep learning methods based on neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Among the traditional methods, Max-mean-Max-medium [12] and Top-Hat [37] used filters to separate target of back- ground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' LCM [3], ILCM [17], TLLCM [16] and MPCM [34] segmented small object by designing salient measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' IPI model [14] treated the input as a superposition of low-rank background and sparse yet shaped target, and solved such issues by using Low-rank decomposition, further methods like sound low-rank [24] and prior constraints [30] were proposed based on IPI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' These methods suffered from low performance under complex scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' As for deep learning methods, Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' [31] used condi- tional GAN [23] with two generators and one discriminator to gain a great balance between miss detection and false alarm (MDvsFA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Dai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' [10] proposed an asymmetric contextual modulation to help network performance well and introduced the first public ISOS dataset SIRST in real scenes, Dai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' [11] further applied a handcraft dilated local con- trast measure into network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' [19] firstly introduced multi-head self-attention into ISOS tasks and got a good result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' [41] proposed AGPCNet with attention- guided context block and context pyramid module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' [40] took shape into account and designed Taylor finite difference edge block and two-orientation attention, and also proposed a more challengable dataset IRSTD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Deep learning methods have become more and more dominant in ISOS due to their great ability of robustness and generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Convolution Operators The ordinary convolution operation is to divide the feature map into patches of the same size as the convolution kernel and then perform a weighted sum operation, this fixed oper- ation at each position may be suboptimal for some specific tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Therefore, researchers have proposed some advanced convolution operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' In order to solve the problem of stan- dard convolution treating all input pixels as valid ones in image inpainting tasks, Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' [20] proposed partial convo- lution, where the convolution is masked and renormalized to be conditioned on only valid pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' [35] proposed gated convolution to provide a learnable dynamic feature se- lection mechanism for each channel at each spatial location across all layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' While in modeling geometric transforma- tions, vanilla convolution is limited due to its fixed geometric structure, Dai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' [6] proposed deformable convolution to enhance this ability by adding additional offsets learned from target task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' [43] then proposed deformable convo- lution v2 to reduce the impact of irrelevant region by adding additional weight and mimicking features from RCNN [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' [36] proposed central difference convolution in face anti-spoofing task, which is able to capture intrinsic detailed patterns via aggregating both intensity and gradient information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Global Information Extraction One common concern is the ability of the network to grasp the local and global information, local context usually are easy to extract using convolution, while how global the in- formation can get usually determined by the receptive fields of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' The most common strategy to improve the receptive fields is stacking convolutions and downsamplings constantly, continuous convolution can linearly improve re- ceptive fields while downsamplings multiply, dilated (atrous) convolution inserts holes between pixels in convolutional kernels and can obtain larger receptive fields than standard convolution, Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' [4] proposed dilation atrous pyra- mid pooling (ASPP) to capture multi-scale information and Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' [32] designed a hybrid dilated convolution to enlarge receptive fields, they both got good improvements in semantic segmentation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Attention mechanism [29,33] can gain global information by calculating the correlation of each single pixel with other pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Fu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' proposed spatial attention and channel attention [13] and achieved great results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Chi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' [5] proposed fast Fourier convolution which performs convolution operators in frequency domain to conduct a global influence in spatial domain thus it can extract image-level information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Suvorov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' proposed LAMA [28] which successfully applied it in large kernel image inpainting tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Berenguel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' [2] then used FFC in monocular depth estimation and semantic segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Method 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Inspiration We draw inspiration from a series of experiments using common segmentation networks, including FPN [18], U- Net [26], PSPNet [42] and DeepLabv3 [4], the results in Table 1 indicate two anomalies: one is the performance is quite polarized, the models which fuse more low-level fea- tures (FPN and U-Net) significantly outperforming others;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' the other is their performance drops only a little as the net- work width and depth increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' We further visualize the Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Results of common segmentation models in ISOS tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Method Backbone Base width IoU↑ Params (M)↓ U-Net [26] 64 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='89 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='04 FPN [18] ResNet-18 64 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='38 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='12 ResNet-50 256 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='12 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='19 ResNet-101 256 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='64 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='18 PSPNet [42] ResNet-50 256 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='73 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='65 DeepLabv3 [4] ResNet-50 256 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='71 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='86 attention maps of 4 stages using grad-cam [15,27] in FPN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' As we can see in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' 2, the small object has been completely lost due to the excessive downsampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Meanwhile directly Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Visualization of 4 layers in FPN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Objects are completely lost in layer 4 with a 32× downsampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' increasing the width and depth of network still fails to mine more information while may lead to redundancy of parame- ters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Based on these observation, we summarize two essential guidelines on ISOS: 1) Extracting global information while maintaining high resolution may avoid small object being overwhelmed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' 2) Extracting additional essential information using modules dedicated to ISOS could be more effective than directly increasing the depth and width of a network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Overview of UFCNet Based on the enlightenment from Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='1, we proposed a U-shape network with central difference convolution (CDC) and fast Fourier convolution (FFC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' The whole framework can been seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' The number of base channels is 32 and only 4 downsampling operations are performed in the whole process, which satisfies the needs of ISOS tasks while avoiding inefficient redundancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Specially, during the downsample process, we use central difference convolution residual block which contains a standard convolution and CDC with a residual convolution for channel alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' We use two cascaded FFC to form our fast Fourier convolu- tion residual block which aims to extract global context in high resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' In what follows, we will briefly introduce central difference convolution in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='3 and fast Fourier convolution in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Central Difference Convolution Convolution can effectively extract color, shape, texture and other information, while standard convolution may be limited because the lack of these information in ISOS tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Inspired by human visual system sensitive to intensity differ- ence and contrast, we utilize central difference convolution 3 input layerl (4x) layer2 (8x) layer3 (16x) layer4 (32x)Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' The scheme of the proposed model UCFNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' UCFNet is based on central difference convolution residual blocks and fast Fourier convolution residual blocks, can effectively extract both local contrast information and global context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' to guide our network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' CDC can introduce additional contrast information by computing the difference between the center point and other points within the convolution window and can be described as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' (1), f(x, y) = � (i,j)∈R w(i,j) · (F(x+i,y+j) − F(x,y)) (1) In addition, CDC is often combined with vanilla convolution to retain some of the traditional feature extraction capabili- ties, and the whole process can be expressed in the equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' CDC(x, y) =θ · � (i,j)∈R w(i,j)(F(x+i,y+j) − F(x,y)) + (1 − θ) · � (i,j)∈R w(i,j)(F(x+i,y+j)) (2) where the hyperparameter θ ∈ (0, 1) is used to determine the contribution between CDC and vanilla convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' The window size R used to calculate the difference is equal to the convolution kernel size while the receptive fields of it can naturally increase during the forward process, thus making it capable of extracting multi-level contrast information which further help the network identify infrared small objects with different size and effectively guide the network performs well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Fast Fourier Convolution Another issue of ISOS is that the small objects are easily overwhelmed in the excessive downsample layers which is used to obtain global information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' In order to solve this contradiction, rather than using the Transformer model with long range dependency, we employ fast Fourier convolution (FFC) which can gain image-level receptive fields in high resolution thus we can extract global context without losing small objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' We follow the structure in LAMA [28] for FFC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Specifically, we splits the channels into two parallel branch, the local branch uses standard convolution to extract local information while global branch applies the Fourier Unit(FU) to gain global context, and there are two additional short-cuts for information fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' The whole process can described as: Yl = Convl−>l(xl) + Convl−>g(xl) (3) Yg = Convg−>l(xg) + FUg−>g(xg) (4) The Four Unit is key to get global context, because it trans- forms input features from spatial domain to frequency do- main where each single point corresponds to all points in the spatial domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Therefore, the convolution operation being conducted within a small kernel size is able to influence 4 XN Fast Fourier Conv CDC Residual Block Conv Central Difference Conv Head FFC FFC Residual Block Conv FFC Conv CDC Fourier Unit Local Global Conv-BN-Relu Real FFT2d Con CDC Conv-BN-Relu BN-Relu BN-Relu Inv Real FFT2d Conv Local Global Conv Convthe all image in spatial domain and obtain global contextual information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' The FU makes following steps: 1) Transform the input feature map from spatial domain to frequency domain with Real FFT2d and concatenates the real and imaginary parts Real FFT2d : RH×W ×C → CH× W 2 ×C ComplexToReal : CH× W 2 ×C → RH× W 2 ×2C 2) Applie convolution,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' normalization and activation func- tion in the frequency domain Conv ◦ Norm ◦ Act : RH× W 2 ×2C → RH× W 2 ×2C 3) Transform inversely from frequency domain to spatial domain RealToComplex : RH× W 2 ×2C → CH× W 2 ×C Inverse Real FFT2d : CH× W 2 ×C → RH×W ×C We concatenate the local branch and global branch into one in the end,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' and it contains rich local and global information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Experiment 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Datasets and Metrics Datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' We evaluate our methods on two widely-used datasets in ISOS: SIRST [10] and IRSTD [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' SIRST con- tains 427 images in real IR scenes with half of the objects in SIRST only contains 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='1% pixels of whole image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Larger than SIRST, IRSTD contains 1001 images with more chal- lenging object and complex backgrounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Both SIRST and IRSTD are separated into training set and test set with a ratio of 8:2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Following privious works, we use pixel-level metrics (IoU and nIoU) and target-level metrics (Pd and Fa) to measure our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Intersection over Union (IoU) and normalized Intersection over Union (nIoU) can described as: IoU = 1 n · �n i=0 tpi �n i=0(fpi + fni − tpi) (5) nIoU = 1 n · n � i=0 tpi fpi + fni − tpi (6) Where n means to the total number of samples, tp denotes the true positive, fp denotes false positive and fn denotes false negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' While target-level metrics probability of de- tection (Pd) and false alarm rate (Fa) can be described as: Pd = 1 n · n � i=0 N i pred N i all (7) Fa = 1 n · n � i=0 P i false P i all (8) where Npred, Nall denotes the number of correct detected objects and the number of total objects and Pfalse, Pall denotes the pixels of false detected objects and the pixels of total objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' We regard that the detection is correct when the distance between the centers of the predict result and the ground truth is less than 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' In addition, we use F-score to evaluate the balance of the model’s Precision and Recall and we use receiver op- erating characteristic (ROC) curve to evaluate the dynamic relationship between true positive ratio and false positive ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' The equations of Precision and Recall are shown in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' (9) and (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Precision = �n i=0 tpi �n i=0(tpi + fpi) (9) Recall = �n i=0 tpi �n i=0(tpi + fni) (10) and F-score is the harmonic mean of Precision and Recall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' F − score = 2 · Precision · Recall Precision + Recall (11) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Implementation Details We conduct experiments on a computer with 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='50GHz CPU, 16GB RAM and GeForce RTX 3090 based on Pytorch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' For more details, we use AdamW optimizer with an initial learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='001 and decayed by Cosine-Anneling- LR [22] schduler and we use binary cross entropy loss and soft IoU loss as our criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Each experiment is trained for 300 epochs with a batch size of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Our UCFNet achieves the best performance with CDC ratio θ of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='7 while using 7 FFC residual blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Quantitative Results We select some traditinal methods: Top-Hat [37], LCM [3], WLDM [21], NARM [38], PSTNN [39], IPI [14], RIPT [8], NIPPS [9] and several open source deep learning SOTA methods including MDvsFA [], ACM [10], ALC [11], AGCP [41] and Transformer [19] which for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' The results are shown in Table 2, deep learning methods basi- cally perform better than traditional ones due to their great power of feature extraction and generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Our proposed method (UCF) shows great superior over other deep learning methods in terms of all metrics on both datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Especially on SIRST, UCF gets the performance of 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='89 of IoU and 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='72 of nIoU about 8% improvements over other meth- ods and impressively obtains 100% probability of detection while only obtain 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='22×10−6 false alarm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' While on IRSTD, we also outperform 3 ∼ 8 in IoU and nIoU over other SOTA deep learning methods result in 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='92 and 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' We further study the dynamic relationship between Preci- sion and Recall with LCM [3], IPI [14], ACM [10], AGCP [41] and our method UCF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' The ROC curve are shown in 5 SIRST (TR=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='5) IRSTD (TR=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='5) Pixel Level Object Level Pixel Level Object Level Method IoU ↑ nIoU ↑ Pd ↑ Fa ↓ IoU ↑ nIoU ↑ Pd ↑ Fa ↓ Top-Hat [37] 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='86 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='42 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='90 1397.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='12 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='26 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='08 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='00 422.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='25 LCM [3] 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='84 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='96 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='06 183.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='15 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='45 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='73 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='58 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='56 WLDM [21] 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='28 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='62 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='16 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='34 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='77 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='07 63.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='01 ACM [10] 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='45 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='15 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='52 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='39 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='38 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='80 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='58 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='31 ALC [11] 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='32 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='24 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='08 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='51 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='91 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='65 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='58 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='07 Transformer [19] 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='82 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='22 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='15 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='15 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='89 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='64 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='91 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='64 AGPC [41] 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='69 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='60 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='17 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='99 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='29 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='23 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='83 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='12 UCF (Ours) 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='89 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='72 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='00 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='26 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='92 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='26 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='60 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='01 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Quantitative evaluation of ISOS on SIRST and IRSTD datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' We report pixel level metric IoU (%) and nIoU (%) and object level metric Pd (%) and Fa (10−6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' All the deep learning methods outperform traditional methods and our UCF achieves the best performance in all terms of metrics on both datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' 4, the area under the curve (AUC) is a key metric for quantitative evaluation of ROC, our UCF also gets the largest area and highest F-score on both datasets, as shown in Ta- ble 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' ROC curve on SIRST (solid line) and IRSTD (dotted line) of different methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' F-score and the area under ROC curve on both datasets with different methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Method SIRST IRSTD F-score↑ Auc↑ F-score↑ Auc↑ LCM [3] 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='058 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='52 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='099 IPI [14] 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='63 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='448 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='248 ACM [10] 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='684 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='59 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='719 AGPC [41] 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='765 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='73 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='734 UCF 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='43 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='843 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='745 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Qualitative Comparisons Some visualization results of different methods are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' The first four rows show scenes with quite compli- cated backgrounds, where our method UCF can achieve less false alarm and high detection rate, significantly outperform others on object level metric because FFC can bring global contextual information in high resolution feature maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' As to relatively simple scenarios (the last two rows), the advantage of UCF lies in providing more details about object shape 6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='8 UCF AGPC ACM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='6 IPI LCM UCF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='4 AGPC ACM IPI LCM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='00 False positive rate 1e-4Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Qualitative results of different methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Red bounding boxes indicate true objects and yellow bounding boxes indicate false alarms which predicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' As seen in the first four rows, in complicated backgrounds, our UCF can get less false alarm and correctly detect more targets in target level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' From the last two rows, we can see that UCF describe the shape and texture in more detail in relatively simple scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' (Best viewed by zooming in) and structure due to the contrast information mined by CDC, thus favoring the pixel-level metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Ablation Experiments We investigate the effectiveness of CDC and FFC with a series of ablation experiments on SIRST, we first generally shows the effectiveness of CDC and FFC in Table 4, CDC improves the performance with local contrast information while FFC helps a lot with global information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' The combina- tion of CDC and FFC achieve the best result in IoU, nIoU and Pd while drop a little bit in Fa compares to only use FFC, this is because of CDC is tended to fix the shape and texture in detail which inevitably results in more false pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Effectiveness of CDC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' We conduct experiments on dif- ferent hyperparameter θ which determines the ratio between Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Ablation study of CDC and FFC in IoU(%), nIoU(%), Pd(%), Fa(10−6) Method IoU↑ nIoU↑ Pd↑ Fa↓ UCF (vanilla) 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='14 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='89 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='33 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='83 UCF + CDC 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='95 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='29 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='25 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='46 UCF + FFC 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='55 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='23 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='82 UCF + CDC + FFC 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='89 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='72 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='00 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='22 CDC and vanilla convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' 6a, CDC always achieves better performance than vanilla covolution by mining essential contrast information and we achieve the 7 AGPCNet Tophat IPI UCF Infrared image ACM ALCNet Ground truth 回 口 口 口 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='1 国 0 国 国 国 国口 ， .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' 口 口 国 M(a) IoU curves on different paramater θ in CDC, the best performance is obtained when θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' (b) IoU curves using different number of FFC residual blocks, we achieve our best performance with 5 FFC residual blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' IoU curves on SIRST with different parameter settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' best performance when θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' We further compare de- formable convolution and gated convolution with CDC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' As show in Table 5, gated convolution only has little improve- ment than vanilla convolution while deformable convolution drops a lot, this is because the sharp information is quite in- sufficient in ISOS tasks and deformable convolution fails to learn the offsets according to the target’s shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' All these in- dicates that our CDC has superiorities over other convolution operators in ISOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Ablation study of CDC with other convolution operators in metrics of IoU (%), nIoU (%), Pd (%), Fa (10−6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Conv Operator IoU↑ nIoU↑ Pd↑ Fa↓ Vanilla 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='14 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='89 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='33 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='83 Gated [35] 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='31 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='57 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='25 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='20 DeformableV1 [6] 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='00 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='60 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='58 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='60 DeformableV2 [43] 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='72 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='67 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='33 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='90 CDC (θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='7) 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='95 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='29 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='25 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='46 Effectiveness of FFC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' As for FFC block, we first study the inner parameter of n which determined the number of FFC blocks we use in our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' 6b, FFC residual block which designed to extract global context can effectively improve the performance effectively and we achieve the best performance when using 5 FFC residual blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' We further explore other global information extract- ing methods mentioned in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='3, from Table 6 we can see dilated convolution and double attention also get some improvements due to enlarging the receptive fields and ex- tracting global context, but dilated convolution enlarges the receptive fields step by step, it only get image-level informa- Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Ablation study of FFC with other global information ex- tracting method (vanilla conv blocks, multi-dilated conv blocks and double attention block in metrics of IoU (%), nIoU (%), Pd (%), Fa (10−6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Method IoU↑ nIoU↑ Pd↑ Fa↓ Vanilla Conv blocks 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='31 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='91 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='17 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='29 Dilated Conv blocks [32] 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='34 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='51 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='08 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='56 Double attention [13] 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='87 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='84 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='00 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='51 FFC blocks 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='55 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='23 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='82 tion in the deep layers, attention mechanism computes the correlation in the whole image but lack of local inductive bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Thus they are inferior to FFC in ISOS tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Conclusion In this study, we have indicated and analyzed the two im- portant issues on ISOS which affects the model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Inspired from these issues, we propose a simple but effective method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Specifically, to handel the first issue of insufficient information, we use central difference convolution to guide the network to focus on local contrast information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' To deal with the second one, we employ the fast Fourier convolution to extract global context in high-resolution feature maps, preventing small objects from being overwhelmed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Exten- sive experiments have validated that our model UCF shows great superiority over other state-of-the-art methods on both two public datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' The proposed model can also serve as a guideline for further investigations on ISOS tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' 8 75 70 65 IoU 60 vanilla 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='5 55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='0 50 50 100 150 200 250 300 epoch75 70 IoU 65 60 55 50 50 100 150 200 250 300 epochReferences [1] Kun Bai, Yuehuang Wang, and Qiong Song.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Patch similar- ity based edge-preserving background estimation for single frame infrared small target detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' 2016 IEEE Interna- tional Conference on Image Processing (ICIP), pages 181– 185, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' 1 [2] Bruno Berenguel-Baeta, Jesus Bermudez-Cameo, and Jose J Guerrero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Fredsnet: Joint monocular depth and semantic segmentation with fast fourier convolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' arXiv preprint arXiv:2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='01595, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' 3 [3] CL Philip Chen, Hong Li, Yantao Wei, Tian Xia, and Yuan Yan Tang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' A local contrast method for small infrared target detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' IEEE transactions on geoscience and remote sensing, 52(1):574–581, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' 1, 2, 5, 6 [4] Liang-Chieh Chen, George Papandreou, Iasonas Kokkinos, Kevin Murphy, and Alan L Yuille.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Deeplab: Semantic image segmentation with deep convolutional nets, atrous convolu- tion, and fully connected crfs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' IEEE transactions on pattern analysis and machine intelligence, 40(4):834–848, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' 3 [5] Lu Chi, Borui Jiang, and Yadong Mu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Fast fourier convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Advances in Neural Information Processing Systems, 33:4479– 4488, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' 3 [6] Jifeng Dai, Haozhi Qi, Yuwen Xiong, Yi Li, Guodong Zhang, Han Hu, and Yichen Wei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Deformable convolutional net- works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' In Proceedings of the IEEE international conference on computer vision, pages 764–773, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' 2, 8 [7] Yimian Dai and Yiquan Wu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Reweighted infrared patch-tensor model with both nonlocal and local priors for single-frame small target detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' IEEE journal of selected topics in applied earth observations and remote sensing, 10(8):3752– 3767, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' 1 [8] Yimian Dai and Yiquan Wu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Reweighted infrared patch-tensor model with both nonlocal and local priors for single-frame small target detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' IEEE journal of selected topics in applied earth observations and remote sensing, 10(8):3752– 3767, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' 5, 6 [9] Yimian Dai, Yiquan Wu, and Yu Song.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Infrared small target and background separation via column-wise weighted robust principal component analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Infrared Physics & Technology, 77:421–430, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' 1, 5, 6 [10] Yimian Dai, Yiquan Wu, Fei Zhou, and Kobus Barnard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Asymmetric contextual modulation for infrared small target detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' In Proceedings of the IEEE/CVF Winter Confer- ence on Applications of Computer Vision, pages 950–959, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' 2, 5, 6 [11] Yimian Dai, Yiquan Wu, Fei Zhou, and Kobus Barnard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' At- tentional Local Contrast Networks for Infrared Small Target Detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' IEEE Transactions on Geoscience and Remote Sensing, pages 1–12, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' 2, 5, 6 [12] Suyog D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Deshpande, Meng Hwa Er, Ronda Venkateswarlu, and Philip Chan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Max-mean and max-median filters for de- tection of small targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' In Optics & Photonics, 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' 1, 2 [13] Jun Fu, Jing Liu, Haijie Tian, Yong Li, Yongjun Bao, Zhiwei Fang, and Hanqing Lu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Dual attention network for scene segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pages 3146–3154, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' 3, 8 [14] Chenqiang Gao, Deyu Meng, Yi Yang, Yongtao Wang, Xiao- fang Zhou, and Alexander Hauptmann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Infrared patch-image model for small target detection in a single image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' IEEE Transactions on Image Processing, 22:4996–5009, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' 1, 2, 5, 6 [15] Jacob Gildenblat and contributors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Pytorch library for cam methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='com/jacobgil/ pytorch-grad-cam, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' 3 [16] Jinhui Han, Sibang Liu, Gang Qin, Qian Zhao, Honghui Zhang, and Nana Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' A local contrast method combined with adaptive background estimation for infrared small target detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' IEEE Geoscience and Remote Sensing Letters, 16(9):1442–1446, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' 1, 2 [17] Jinhui Han, Yong Ma, Bo Zhou, Fan Fan, Kun Liang, and Yu Fang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' A robust infrared small target detection algorithm based on human visual system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' IEEE Geoscience and Remote Sensing Letters, 11(12):2168–2172, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' 1, 2 [18] Tsung-Yi Lin, Piotr Doll´ar, Ross Girshick, Kaiming He, Bharath Hariharan, and Serge Belongie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Feature pyramid networks for object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 2117–2125, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' 3 [19] Fangcen Liu, Chenqiang Gao, Fang Chen, Deyu Meng, Wang- meng Zuo, and Xinbo Gao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Infrared small-dim target detec- tion with transformer under complex backgrounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' arXiv preprint arXiv:2109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='14379, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' 2, 5, 6 [20] Guilin Liu, Fitsum A Reda, Kevin J Shih, Ting-Chun Wang, Andrew Tao, and Bryan Catanzaro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Image inpainting for irregular holes using partial convolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' In Proceedings of the European conference on computer vision (ECCV), pages 85–100, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' 2 [21] Jie Liu, Ziqing He, Zuolong Chen, and Lei Shao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Tiny and dim infrared target detection based on weighted local contrast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' IEEE Geoscience and Remote Sensing Letters, 15(11):1780– 1784, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' 5, 6 [22] Ilya Loshchilov and Frank Hutter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Sgdr: Stochastic gradient descent with warm restarts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' arXiv preprint arXiv:1608.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='03983, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' 5 [23] Mehdi Mirza and Simon Osindero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Conditional generative adversarial nets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' arXiv preprint arXiv:1411.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content='1784, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' 2 [24] Sur Singh Rawat, Sashi Kant Verma, and Yatindra Kumar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Reweighted infrared patch image model for small target de- tection based on non-convex lp-norm minimisation and tv regularisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' IET image processing, 14(9):1937–1947, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' 1, 2 [25] Shaoqing Ren, Kaiming He, Ross Girshick, and Jian Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Faster r-cnn: Towards real-time object detection with region proposal networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Advances in neural information process- ing systems, 28, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' 2 [26] Olaf Ronneberger, Philipp Fischer, and Thomas Brox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' U-net: Convolutional networks for biomedical image segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' In International Conference on Medical image computing and computer-assisted intervention, pages 234–241.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Springer, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' 3 9 [27] Ramprasaath R Selvaraju, Michael Cogswell, Abhishek Das, Ramakrishna Vedantam, Devi Parikh, and Dhruv Batra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Grad- cam: Visual explanations from deep networks via gradient- based localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' In Proceedings of the IEEE international conference on computer vision, pages 618–626, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' 3 [28] Roman Suvorov, Elizaveta Logacheva, Anton Mashikhin, Anastasia Remizova, Arsenii Ashukha, Aleksei Silvestrov, Naejin Kong, Harshith Goka, Kiwoong Park, and Victor Lem- pitsky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Resolution-robust large mask inpainting with fourier convolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' In Proceedings of the IEEE/CVF Winter Confer- ence on Applications of Computer Vision, pages 2149–2159, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' 3, 4 [29] Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszko- reit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, and Illia Polosukhin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Attention is all you need.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Advances in neural information processing systems, 30, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' 3 [30] Minjie Wan, Guohua Gu, Yunkai Xu, Weixian Qian, Kan Ren, and Qian Chen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Total variation-based interframe in- frared patch-image model for small target detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' IEEE Geoscience and Remote Sensing Letters, 19:1–5, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' 1, 2 [31] Huan Wang, Luping Zhou, and Lei Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Miss detection vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' false alarm: Adversarial learning for small object segmen- tation in infrared images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' In Proceedings of the IEEE/CVF International Conference on Computer Vision, pages 8509– 8518, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' 1, 2, 6 [32] Panqu Wang, Pengfei Chen, Ye Yuan, Ding Liu, Zehua Huang, Xiaodi Hou, and Garrison Cottrell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Understanding convolu- tion for semantic segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' In 2018 IEEE winter con- ference on applications of computer vision (WACV), pages 1451–1460.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Ieee, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' 3, 8 [33] Xiaolong Wang, Ross Girshick, Abhinav Gupta, and Kaiming He.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Non-local neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 7794–7803, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' 3 [34] Yantao Wei, Xinge You, and Hong Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Multiscale patch-based contrast measure for small infrared target detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Pattern Recognition, 58:216–226, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' 2 [35] Jiahui Yu, Zhe Lin, Jimei Yang, Xiaohui Shen, Xin Lu, and Thomas S Huang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Free-form image inpainting with gated convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' In Proceedings of the IEEE/CVF international conference on computer vision, pages 4471–4480, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' 2, 8 [36] Zitong Yu, Chenxu Zhao, Zezheng Wang, Yunxiao Qin, Zhuo Su, Xiaobai Li, Feng Zhou, and Guoying Zhao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Searching cen- tral difference convolutional networks for face anti-spoofing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 5295–5305, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' 2 [37] Ming Zeng, Jian xun Li, and Zhang xiao Peng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' The design of top-hat morphological filter and application to infrared target detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Infrared Physics & Technology, 48:67–76, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' 1, 2, 5, 6 [38] Landan Zhang, Lingbing Peng, Tianfang Zhang, Siying Cao, and Zhenming Peng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Infrared small target detection via non- convex rank approximation minimization joint l 2, 1 norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Remote Sensing, 10(11):1821, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' 5, 6 [39] Landan Zhang and Zhenming Peng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Infrared small target detection based on partial sum of the tensor nuclear norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Remote Sensing, 11(4):382, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' 5, 6 [40] Mingjin Zhang, Rui Zhang, Yuxiang Yang, Haichen Bai, Jing Zhang, and Jie Guo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Isnet: Shape matters for infrared small target detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 877–886, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' 2, 5 [41] Tianfang Zhang, Siying Cao, Tian Pu, and Zhenming Peng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Agpcnet: Attention-guided pyramid context networks for in- frared small target detection, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' 2, 5, 6 [42] Hengshuang Zhao, Jianping Shi, Xiaojuan Qi, Xiaogang Wang, and Jiaya Jia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' Pyramid scene parsing network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 2881–2890, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' 3 [43] Xizhou Zhu, Han Hu, Stephen Lin, and Jifeng Dai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' De- formable convnets v2: More deformable, better results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pages 9308–9316, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} +page_content=' 2, 8 10' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFLT4oBgHgl3EQffC-V/content/2301.12093v1.pdf'} diff --git a/n9FQT4oBgHgl3EQfqTb3/content/tmp_files/2301.13380v1.pdf.txt b/n9FQT4oBgHgl3EQfqTb3/content/tmp_files/2301.13380v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..b742691ac5685f4942d0bdc3110ac567d0232cc3 --- /dev/null +++ b/n9FQT4oBgHgl3EQfqTb3/content/tmp_files/2301.13380v1.pdf.txt @@ -0,0 +1,199 @@ +Extended abstract for Late-Breaking/Demo +ISMIR 2019 +AUTOMATED TIME-FREQUENCY DOMAIN AUDIO +CROSSFADES USING GRAPH CUTS +Kyle Robinson +University of Waterloo +kyle.robinson@uwaterloo.ca +Dan Brown +University of Waterloo +0 +5 +10 +15 +20 +25 +30 +Time +0 +64 +128 +256 +512 +1024 +2048 +4096 +8192 +Hz +Figure 1: This spectrogram shows overlapped segments of two music tracks after being combined and re- +constructed along a per-frequency seam (bright yellow). The tracks were beat and tempo matched, then +overlapped by 64 beats. +EXTENDED ABSTRACT +The problem of transitioning smoothly from one audio clip to another arises in many music consumption +scenarios; especially as music consumption has moved from professionally curated and live-streamed radios +to personal playback devices and services. Classically, transitioning from one song to another has been reliant +on either pre-mixed transitions on recorded digital or physical media, hardware or software crossfading on +the playback device, or professional transitions by a host or disk jockey (DJ). While options for software +crossfading are ubiquitous on music streaming platforms and media players alike, these transitions pale in +quality when compared to those manually applied by an audio engineer or DJ who can harmonically and +rhythmically align tracks—and importantly—manually apply equalizer (EQ) filters during transitions. The +application of EQ filters specifically allow for different transitions in different audio spectrums. For example, +the bass register of one track can be made to replace the bass register of another track before transitioning +the higher frequencies. Typically the task of deciding how and where to apply transitions in the frequency +domain has been completed manually using a limited number of EQ filters. +There is much research on creating, sorting, and extending playlists so as to have tracks naturally flow into +each other, as well as on determining optimal times to transition between similar tracks [1, 3, 4, 6, 8]. Both +of these research areas play a key role in synthesising a human DJ. To our knowledge, however, all of +these approaches still rely on classical methods of transitioning tracks using amplitude in the time domain +(crossfading). +Through the application of an existing visual texture extension algorithm borrowed from computer vision, +we present the first steps toward a new method of automatically transitioning from one audio clip to another +by discretizing the frequency spectrum into bins and then finding transition times for each bin. [5]. +We begin by phrasing the problem of transitioning from one song to another as a graph optimization problem: +the graph represents the two songs in the transition range, and a cut happens when we transition from one +song to the other at a particular time point. To obtain these representations we first apply a short term fourier +transform (STFT) to each song, and then convert the resulting complex time-frequency mapped amplitude +values into real decibel values. In order to align the tracks we apply rudimentary tempo matching and beat +alignment using the libROSA Python library, and overlap the tracks by a number of beats [7]. We call the +resulting STFT transformed data of the first and second song’s overlapped segments matrix A and matrix B +© Kyle Robinson, Dan Brown. Licensed under a Creative Commons Attribution 4.0 International License (CC BY 4.0). +Attribution: +Kyle Robinson, Dan Brown. “Automated Time-frequency Domain Audio Crossfades using Graph Cuts”, Late Break- +ing/Demo at the 20th International Society for Music Information Retrieval, Delft, The Netherlands, 2019. +1 +arXiv:2301.13380v1 [cs.SD] 31 Jan 2023 + +Extended abstract for Late-Breaking/Demo +ISMIR 2019 +0 +10 +20 +30 +Time +0 +64 +128 +256 +512 +1024 +2048 +4096 +8192 +Hz +A= +B= +0 +10 +20 +30 +Time +0 +64 +128 +256 +512 +1024 +2048 +4096 +8192 +Hz +Figure 2: Overlapping song segment spectro- +gram’s used to compute graph weights using +eq. (1). Each segment represents 64 beats. +0 Hz +H Hz +0 s +T s +Source +Sink +0,0 +w +0,1 +0,0 +w +1,0 +0,0 +w +0,1 +1,0 +w +1,1 +h-1,0 +w +h-1,1 +h-1,0 +w +h,0 +h,0 +w +h,1 +h-1,1 +w +h,1 +0,t-1 +w +0,t +0,t-1 +w +1,t-1 +0,t +w +1,t +1,t-1 +w +1,t +h-1,t-1 +w +h-1,t +h-1,t-1 +w +h,t-1 +h,t-1 +w +h,t +h,t-1 +w +h,t +Figure 3: Flow graph model showing adja- +cent weight calculations. The orange and +green nodes are fixed to the first and second +songs respectfully. Nodes represent time- +frequency bins. +respectfully, as seen in Figure 2. Next, we define a simple loss function: +wk,l +i,j (A, B) = ||Ai,j − Bi,j|| + ||Ak,l − Bk,l|| +(1) +We apply the loss function to each adjacent time-frequency bin and use the resulting values to assign weights +to edges in an undirected graph with dimensions equal to A and B. Finally, the resulting graphs left-most +nodes are anchored to a source node, and the right-most nodes are anchored to a sink node. Figure 3 shows a +representation of the completed flow graph. In order to obtain a min-cut, we apply the Boykov-Kolmogorov +algorithm [2]. The indices of this min-cut are the seam where each frequency bin transitions. +In order to apply the found transition to the audio tracks, we concatenate the complex time-frequency song +representations found earlier along the seam, and apply an inverse STFT to obtain the final audio transition +seen in Figure 1. +The work here presents an initial foray into automatically transitioning between songs in the time-frequency +domain. The loss function described in 1 does not well characterize the inherent qualities found in music, +but there is good reason to believe such a cost function can be found through further development. On +harmonically similar tracks with similar tempi, the current implementation produces acoustically pleasing +results. +REFERENCES +[1] Rachel M Bittner, Minwei Gu, Gandalf Hernandez, Eric J Humphrey, Tristan Jehan, P. Hunter McCurry, and Nicola Montecchio. +Automatic Playlist Sequencing and Transitions. Proceedings of the 18th International Society for Music Information Retrieval +Conference (ISMIR), pages 442–448, 2017. +[2] Yuri Boykov and Vladimir Kolmogorov. An Experimental Comparison of Min-Cut/Max-Flow Algorithms for Energy Minimization +in Vision 1 Introduction. Technical Report 9, 2004. +[3] Roman B. Gebhardt, Matthew E.P. Davies, and Bernhard U. Seeber. Psychoacoustic approaches for harmonic music mixing. Applied +Sciences, 6(5), 2016. +[4] Tatsunori Hirai, Hironori Doi, and Shigeo Morishima. MusicMixer: Computer-Aided DJ system based on an automatic song mixing. +In ACM International Conference Proceeding Series, volume 16-19-Nove. Association for Computing Machinery, nov 2015. +[5] Vivek Kwatra, Arno Schödl, Irfan Essa, Greg Turk, and Aaron Bobick. Graphcut textures: Image and video synthesis using graph +cuts. In ACM SIGGRAPH 2003 Papers, SIGGRAPH ’03, pages 277–286, 2003. +[6] Heng Yi Lin, Yin Tzu Lin, Ming Chun Tien, and Ja Ling Wu. Music paste: Concatenating music clips based on chroma and rhythm +features. In Proceedings of the 10th International Society for Music Information Retrieval Conference, ISMIR 2009, pages 213–218, +2009. +[7] Brian McFee, Colin Raffel, Dawen Liang, Daniel Ellis, Matt McVicar, Eric Battenberg, and Oriol Nieto. librosa: Audio and Music +Signal Analysis in Python. In Proceedings of the 14th Python in Science Conference, pages 18–24, 2015. +[8] Jaume Parera. Dj codo nudo: a novel method for seamless transition between songs for electronic music. Master’s thesis, Universitat +Pompeu Fabra, Barcelona, 2016. +2 + diff --git a/n9FQT4oBgHgl3EQfqTb3/content/tmp_files/load_file.txt b/n9FQT4oBgHgl3EQfqTb3/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8c9bf8668b1041cc271f8b4de6448c0448720103 --- /dev/null +++ b/n9FQT4oBgHgl3EQfqTb3/content/tmp_files/load_file.txt @@ -0,0 +1,75 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FQT4oBgHgl3EQfqTb3/content/2301.13380v1.pdf,len=74 +page_content='Extended abstract for Late-Breaking/Demo ISMIR 2019 AUTOMATED TIME-FREQUENCY DOMAIN AUDIO CROSSFADES USING GRAPH CUTS Kyle Robinson University of Waterloo kyle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FQT4oBgHgl3EQfqTb3/content/2301.13380v1.pdf'} +page_content='robinson@uwaterloo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FQT4oBgHgl3EQfqTb3/content/2301.13380v1.pdf'} +page_content='ca Dan Brown University of Waterloo 0 5 10 15 20 25 30 Time 0 64 128 256 512 1024 2048 4096 8192 Hz Figure 1: This spectrogram shows overlapped segments of two music tracks after being combined and re- constructed along a per-frequency seam (bright yellow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FQT4oBgHgl3EQfqTb3/content/2301.13380v1.pdf'} +page_content=' The tracks were beat and tempo matched, then overlapped by 64 beats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FQT4oBgHgl3EQfqTb3/content/2301.13380v1.pdf'} +page_content=' EXTENDED ABSTRACT The problem of transitioning smoothly from one audio clip to another arises in many music consumption scenarios;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FQT4oBgHgl3EQfqTb3/content/2301.13380v1.pdf'} +page_content=' especially as music consumption has moved from professionally curated and live-streamed radios to personal playback devices and services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FQT4oBgHgl3EQfqTb3/content/2301.13380v1.pdf'} +page_content=' Classically, transitioning from one song to another has been reliant on either pre-mixed transitions on recorded digital or physical media, hardware or software crossfading on the playback device, or professional transitions by a host or disk jockey (DJ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FQT4oBgHgl3EQfqTb3/content/2301.13380v1.pdf'} +page_content=' While options for software crossfading are ubiquitous on music streaming platforms and media players alike, these transitions pale in quality when compared to those manually applied by an audio engineer or DJ who can harmonically and rhythmically align tracks—and importantly—manually apply equalizer (EQ) filters during transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FQT4oBgHgl3EQfqTb3/content/2301.13380v1.pdf'} +page_content=' The application of EQ filters specifically allow for different transitions in different audio spectrums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FQT4oBgHgl3EQfqTb3/content/2301.13380v1.pdf'} +page_content=' For example, the bass register of one track can be made to replace the bass register of another track before transitioning the higher frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FQT4oBgHgl3EQfqTb3/content/2301.13380v1.pdf'} +page_content=' Typically the task of deciding how and where to apply transitions in the frequency domain has been completed manually using a limited number of EQ filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FQT4oBgHgl3EQfqTb3/content/2301.13380v1.pdf'} +page_content=' There is much research on creating, sorting, and extending playlists so as to have tracks naturally flow into each other, as well as on determining optimal times to transition between similar tracks [1, 3, 4, 6, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FQT4oBgHgl3EQfqTb3/content/2301.13380v1.pdf'} +page_content=' Both of these research areas play a key role in synthesising a human DJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FQT4oBgHgl3EQfqTb3/content/2301.13380v1.pdf'} +page_content=' To our knowledge, however, all of these approaches still rely on classical methods of transitioning tracks using amplitude in the time domain (crossfading).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FQT4oBgHgl3EQfqTb3/content/2301.13380v1.pdf'} +page_content=' Through the application of an existing visual texture extension algorithm borrowed from computer vision, we present the first steps toward a new method of automatically transitioning from one audio clip to another by discretizing the frequency spectrum into bins and then finding transition times for each bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FQT4oBgHgl3EQfqTb3/content/2301.13380v1.pdf'} +page_content=' [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FQT4oBgHgl3EQfqTb3/content/2301.13380v1.pdf'} +page_content=' We begin by phrasing the problem of transitioning from one song to another as a graph optimization problem: the graph represents the two songs in the transition range, and a cut happens when we transition from one song to the other at a particular time point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FQT4oBgHgl3EQfqTb3/content/2301.13380v1.pdf'} +page_content=' To obtain these representations we first apply a short term fourier transform (STFT) to each song, and then convert the resulting complex time-frequency mapped amplitude values into real decibel values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FQT4oBgHgl3EQfqTb3/content/2301.13380v1.pdf'} +page_content=' In order to align the tracks we apply rudimentary tempo matching and beat alignment using the libROSA Python library, and overlap the tracks by a number of beats [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FQT4oBgHgl3EQfqTb3/content/2301.13380v1.pdf'} +page_content=' We call the resulting STFT transformed data of the first and second song’s overlapped segments matrix A and matrix B © Kyle Robinson, Dan Brown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FQT4oBgHgl3EQfqTb3/content/2301.13380v1.pdf'} +page_content=' Licensed under a Creative Commons Attribution 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FQT4oBgHgl3EQfqTb3/content/2301.13380v1.pdf'} +page_content='0 International License (CC BY 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FQT4oBgHgl3EQfqTb3/content/2301.13380v1.pdf'} +page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FQT4oBgHgl3EQfqTb3/content/2301.13380v1.pdf'} +page_content=' Attribution: Kyle Robinson, Dan Brown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FQT4oBgHgl3EQfqTb3/content/2301.13380v1.pdf'} +page_content=' “Automated Time-frequency Domain Audio Crossfades using Graph Cuts”, Late Break- ing/Demo at the 20th International Society for Music Information Retrieval, Delft, The Netherlands, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FQT4oBgHgl3EQfqTb3/content/2301.13380v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FQT4oBgHgl3EQfqTb3/content/2301.13380v1.pdf'} +page_content='13380v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FQT4oBgHgl3EQfqTb3/content/2301.13380v1.pdf'} +page_content='SD] 31 Jan 2023 Extended abstract for Late-Breaking/Demo ISMIR 2019 0 10 20 30 Time 0 64 128 256 512 1024 2048 4096 8192 Hz A= B= 0 10 20 30 Time 0 64 128 256 512 1024 2048 4096 8192 Hz Figure 2: Overlapping song segment spectro- gram’s used to compute graph weights using eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FQT4oBgHgl3EQfqTb3/content/2301.13380v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FQT4oBgHgl3EQfqTb3/content/2301.13380v1.pdf'} +page_content=' Each segment represents 64 beats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FQT4oBgHgl3EQfqTb3/content/2301.13380v1.pdf'} +page_content=' 0 Hz H Hz 0 s T s Source Sink 0,0 w 0,1 0,0 w 1,0 0,0 w 0,1 1,0 w 1,1 h-1,0 w h-1,1 h-1,0 w h,0 h,0 w h,1 h-1,1 w h,1 0,t-1 w 0,t 0,t-1 w 1,t-1 0,t w 1,t 1,t-1 w 1,t h-1,t-1 w h-1,t h-1,t-1 w h,t-1 h,t-1 w h,t h,t-1 w h,t Figure 3: Flow graph model showing adja- cent weight calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FQT4oBgHgl3EQfqTb3/content/2301.13380v1.pdf'} +page_content=' The orange and green nodes are fixed to the first and second songs respectfully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FQT4oBgHgl3EQfqTb3/content/2301.13380v1.pdf'} +page_content=' Nodes represent time- frequency bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FQT4oBgHgl3EQfqTb3/content/2301.13380v1.pdf'} +page_content=' respectfully, as seen in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FQT4oBgHgl3EQfqTb3/content/2301.13380v1.pdf'} +page_content=' Next, we define a simple loss function: wk,l i,j (A, B) = ||Ai,j − Bi,j|| + ||Ak,l − Bk,l|| (1) We apply the loss function to each adjacent time-frequency bin and use the resulting values to assign weights to edges in an undirected graph with dimensions equal to A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FQT4oBgHgl3EQfqTb3/content/2301.13380v1.pdf'} +page_content=' Finally, the resulting graphs left-most nodes are anchored to a source node, and the right-most nodes are anchored to a sink node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FQT4oBgHgl3EQfqTb3/content/2301.13380v1.pdf'} +page_content=' Figure 3 shows a representation of the completed flow graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FQT4oBgHgl3EQfqTb3/content/2301.13380v1.pdf'} +page_content=' In order to obtain a min-cut, we apply the Boykov-Kolmogorov algorithm [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FQT4oBgHgl3EQfqTb3/content/2301.13380v1.pdf'} +page_content=' The indices of this min-cut are the seam where each frequency bin transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FQT4oBgHgl3EQfqTb3/content/2301.13380v1.pdf'} +page_content=' In order to apply the found transition to the audio tracks, we concatenate the complex time-frequency song representations found earlier along the seam, and apply an inverse STFT to obtain the final audio transition seen in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FQT4oBgHgl3EQfqTb3/content/2301.13380v1.pdf'} +page_content=' The work here presents an initial foray into automatically transitioning between songs in the time-frequency domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FQT4oBgHgl3EQfqTb3/content/2301.13380v1.pdf'} +page_content=' The loss function described in 1 does not well characterize the inherent qualities found in music, but there is good reason to believe such a cost function can be found through further development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FQT4oBgHgl3EQfqTb3/content/2301.13380v1.pdf'} +page_content=' On harmonically similar tracks with similar tempi, the current implementation produces acoustically pleasing results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FQT4oBgHgl3EQfqTb3/content/2301.13380v1.pdf'} +page_content=' REFERENCES [1] Rachel M Bittner, Minwei Gu, Gandalf Hernandez, Eric J Humphrey, Tristan Jehan, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FQT4oBgHgl3EQfqTb3/content/2301.13380v1.pdf'} +page_content=' Hunter McCurry, and Nicola Montecchio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FQT4oBgHgl3EQfqTb3/content/2301.13380v1.pdf'} +page_content=' Automatic Playlist Sequencing and Transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FQT4oBgHgl3EQfqTb3/content/2301.13380v1.pdf'} +page_content=' Proceedings of the 18th International Society for Music Information Retrieval Conference (ISMIR), pages 442–448, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FQT4oBgHgl3EQfqTb3/content/2301.13380v1.pdf'} +page_content=' [2] Yuri Boykov and Vladimir Kolmogorov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FQT4oBgHgl3EQfqTb3/content/2301.13380v1.pdf'} +page_content=' An Experimental Comparison of Min-Cut/Max-Flow Algorithms for Energy Minimization in Vision 1 Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FQT4oBgHgl3EQfqTb3/content/2301.13380v1.pdf'} +page_content=' Technical Report 9, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FQT4oBgHgl3EQfqTb3/content/2301.13380v1.pdf'} +page_content=' [3] Roman B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FQT4oBgHgl3EQfqTb3/content/2301.13380v1.pdf'} +page_content=' Gebhardt, Matthew E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FQT4oBgHgl3EQfqTb3/content/2301.13380v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FQT4oBgHgl3EQfqTb3/content/2301.13380v1.pdf'} +page_content=' Davies, and Bernhard U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FQT4oBgHgl3EQfqTb3/content/2301.13380v1.pdf'} +page_content=' Seeber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FQT4oBgHgl3EQfqTb3/content/2301.13380v1.pdf'} +page_content=' Psychoacoustic approaches for harmonic music mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FQT4oBgHgl3EQfqTb3/content/2301.13380v1.pdf'} +page_content=' Applied Sciences, 6(5), 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FQT4oBgHgl3EQfqTb3/content/2301.13380v1.pdf'} +page_content=' [4] Tatsunori Hirai, Hironori Doi, and Shigeo Morishima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FQT4oBgHgl3EQfqTb3/content/2301.13380v1.pdf'} +page_content=' MusicMixer: Computer-Aided DJ system based on an automatic song mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FQT4oBgHgl3EQfqTb3/content/2301.13380v1.pdf'} +page_content=' In ACM International Conference Proceeding Series, volume 16-19-Nove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FQT4oBgHgl3EQfqTb3/content/2301.13380v1.pdf'} +page_content=' Association for Computing Machinery, nov 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FQT4oBgHgl3EQfqTb3/content/2301.13380v1.pdf'} +page_content=' [5] Vivek Kwatra, Arno Schödl, Irfan Essa, Greg Turk, and Aaron Bobick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FQT4oBgHgl3EQfqTb3/content/2301.13380v1.pdf'} +page_content=' Graphcut textures: Image and video synthesis using graph cuts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FQT4oBgHgl3EQfqTb3/content/2301.13380v1.pdf'} +page_content=' In ACM SIGGRAPH 2003 Papers, SIGGRAPH ’03, pages 277–286, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FQT4oBgHgl3EQfqTb3/content/2301.13380v1.pdf'} +page_content=' [6] Heng Yi Lin, Yin Tzu Lin, Ming Chun Tien, and Ja Ling Wu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FQT4oBgHgl3EQfqTb3/content/2301.13380v1.pdf'} +page_content=' Music paste: Concatenating music clips based on chroma and rhythm features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FQT4oBgHgl3EQfqTb3/content/2301.13380v1.pdf'} +page_content=' In Proceedings of the 10th International Society for Music Information Retrieval Conference, ISMIR 2009, pages 213–218, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FQT4oBgHgl3EQfqTb3/content/2301.13380v1.pdf'} +page_content=' [7] Brian McFee, Colin Raffel, Dawen Liang, Daniel Ellis, Matt McVicar, Eric Battenberg, and Oriol Nieto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FQT4oBgHgl3EQfqTb3/content/2301.13380v1.pdf'} +page_content=' librosa: Audio and Music Signal Analysis in Python.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FQT4oBgHgl3EQfqTb3/content/2301.13380v1.pdf'} +page_content=' In Proceedings of the 14th Python in Science Conference, pages 18–24, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FQT4oBgHgl3EQfqTb3/content/2301.13380v1.pdf'} +page_content=' [8] Jaume Parera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FQT4oBgHgl3EQfqTb3/content/2301.13380v1.pdf'} +page_content=' Dj codo nudo: a novel method for seamless transition between songs for electronic music.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9FQT4oBgHgl3EQfqTb3/content/2301.13380v1.pdf'} +page_content=' Master’s thesis, Universitat Pompeu Fabra, Barcelona, 2016.' metadata={'source': 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a/pNFIT4oBgHgl3EQfvyuP/content/tmp_files/2301.11349v1.pdf.txt b/pNFIT4oBgHgl3EQfvyuP/content/tmp_files/2301.11349v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..0dc53980af2d55ef39c2dd5087856b4a3fa1ccf4 --- /dev/null +++ b/pNFIT4oBgHgl3EQfvyuP/content/tmp_files/2301.11349v1.pdf.txt @@ -0,0 +1,2514 @@ +arXiv:2301.11349v1 [hep-ph] 26 Jan 2023 +An EFT hunter’s guide to two-to-two scattering: +HEFT and SMEFT on-shell amplitudes +Hongkai Liu a, Teng Ma a, b, Yael Shadmi a, Michael Waterbury a +a Physics Department, Technion – Israel Institute of Technology, +Technion city, Haifa 3200003, Israel +b IFAE and BIST, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona +We derive the contact terms contributing to the four-point amplitudes of the +standard-model particles, keeping terms with up to quartic energy growth. Imposing +just the unbroken low-energy symmetry, and treating the electroweak gauge bosons +and the Higgs as independent degrees of freedom, we obtain the most general +four-point contact-term amplitudes, corresponding to the HEFT framework. The +contact terms are spanned by a basis of Stripped Contact Terms (SCTs), which carry +the polarization information, multiplied by polynomials in the Mandelstam invariants. +For terms with quadratic energy growth, we also derive the low-energy SMEFT +predictions, via on-shell Higgsing of the massless SMEFT contact terms. We discuss +several aspects of bottom-up versus top-down on-shell derivations of the HEFT and +SMEFT amplitudes, highlighting in particular the simple counting of HEFT +dimensions in the on-shell approach and the transparent relation between +perturbative unitarity and gauge-invariance in the little-group covariant massive +spinor formalism. Our results provide a formulation of EFT analyses directly in terms +of observable quantities. For terms with quadratic energy growth, we also provide the +mapping to the Warsaw basis. + +Contents +1 +Introduction +2 +2 +Preliminaries +6 +3 +Four-point contact terms at O(E2) +8 +3.1 +HEFT contact terms +8 +3.2 +SMEFT contact terms +10 +4 +Four-point contact ters at E3 and E4 +14 +4.1 +Bosonic Amplitudes with All Massive Particles +15 +4.1.1 +hhhh +15 +4.1.2 +Zhhh +15 +4.1.3 +ZZhh +15 +4.1.4 +W +W −hh +15 +4.1.5 +W +W −Zh +16 +4.1.6 +ZZZh +16 +4.1.7 +W +W −ZZ +16 +4.1.8 +W +W +W −W − +17 +4.1.9 +ZZZZ +17 +4.2 +Fermionic Amplitudes with All Massive Particles +17 +4.2.1 +f cfhh +17 +4.2.2 +f cfZh and f cf ′Wh +17 +4.2.3 +f cf cff +18 +4.2.4 +W +W −f cf and WZf cf ′ +19 +4.2.5 +ZZf cf +19 +4.3 +Bosonic Amplitudes with Massless Vectors +20 +4.3.1 +γhhh +20 +4.3.2 +γZhh +20 +4.3.3 +gghh and γγhh +20 +4.3.4 +gggh +20 +4.3.5 +ggγh and γγγh +21 +4.3.6 +ggZh and γγZh +21 +4.3.7 +γZZh +21 +4.3.8 +γWWh +21 +1 + +4.3.9 +γγγγ +21 +4.3.10 gggZ, gggγ and γγγZ +22 +4.3.11 ggZZ and γγZZ +22 +4.3.12 ggWW and γγWW +22 +4.3.13 γZWW +22 +4.3.14 γZZZ +23 +4.3.15 ggγγ +23 +4.3.16 ggγZ +23 +4.3.17 gggg +23 +4.4 +Fermionic Amplitudes with Massless Vectors +24 +4.4.1 +f cfγh and f cfgh +24 +4.4.2 +ggf cf and γγf cf +24 +4.4.3 +γgf cf +24 +4.4.4 +γZf cf, γWf cf ′, gZf cf and gWf cf ′ +25 +5 +Conclusions +25 +A W W hh: On-shell construction of the HEFT and SMEFT amplitudes +and on-shell Higgsing +26 +A.1 The structure of the full amplitude +26 +A.2 Four-point contact terms from on-shell Higgsing +29 +A.3 The WWh coupling from on-shell Higgsing +30 +B Energy-growth of factorizable amplitudes +33 +1 +Introduction +Precise measurements of the interactions of the standard-model (SM) particles, and in +particular, the electroweak bosons and the top, will be a focus of the LHC program in +the coming decade. These interactions can be systematically parameterized in terms +of Effective Field Theory (EFT) Lagrangians, which in principle provide a model- +independent framework for indirect searches for new physics. +Much of the collider +EFT program has been guided by the Standard Model EFT (SMEFT), whose starting +point is the unbroken SU(3)×SU(2)×U(1) theory with a single Higgs doublet, focusing +in particular on dimension-six operators [1–3]. +Even in the SMEFT framework, it +is plausible however that a given set of heavy fields couple differently to different SM +fields. Different SMEFT operator bases are therefore better suited to describe the effect +2 + +of different UV models [3–6], and truncating the EFT at dimension-six may moreover +leave out important effects (see for example [7–9]). Furthermore, some extensions of +the SM are not captured by the SMEFT (at least at low-dimensions), and would lead at +low energies to the framework known as the Higgs EFT (HEFT) [10–23]. These include +for example models featuring fields which get their masses from electroweak symmetry +breaking (EWSB), or fields which provide additional sources of this breaking [15, 24– +26]. +A natural alternative for EFT constructions is provided by the on-shell bootstrap +(see for example [27, 28]). EFT extensions of the SM can be formulated in this ap- +proach directly in terms of the physical observables of interest, namely, the scattering +amplitudes of the known SM particles, with a one-to-one mapping of EFT operators +and contact-term amplitudes [29–31]. Furthermore, since they are not obscured by field +redefinitions and operator redundancies, questions such as the distinctions between dif- +ferent EFT extensions, or the assignments of EFT dimensions, are concretely phrased +in terms of physical quantities. +In this paper, we derive the full set of four-point contact-term amplitudes, featur- +ing the SM massive and massless particles, keeping terms with up to quartic energy +growth. Together with the three-point amplitudes listed in [31], these determine the +EFT predictions for four-point amplitudes with this energy growth. These amplitudes +are the most interesting objects for phenomenological purposes, since two-to-two scat- +tering processes, followed by two- and three-particle decays, are the ones where we +can hope to get the most data. Our results are collected in Tables 1 and 3 and Sec- +tion 4. The low-energy E2 HEFT contact terms are given in Table 1. The low-energy +E2 SMEFT contact terms appear in Table 3 and are mapped to the massless SMEFT +contact terms [32] collected in Table 2, where we also give the relations to Warsaw basis +operators. Section 4 contains the low-energy E4 HEFT contact terms. +Indeed, on-shell methods have emerged in recent years as a powerful alternative +to EFT Lagrangian constructions [29–40]. Bottom-up constructions of SM, or SM like +amplitudes were discussed in [41, 42, 31, 43–47]. The emergence of symmetry from the +amplitude bootstrap, and its relation to the geometry of field space was studied for +instance in [48–50]. +We employ two types of on-shell constructions. +The first is purely bottom-up +and gives HEFT amplitudes. The second is top-down and starts from the massless +amplitudes of the unbroken theory, yielding the SMEFT low-energy contact terms. +We now sketch these in turn. Various methods for constructing generic contact-term +bases for massless amplitudes were described in [32, 51, 24, 36, 52]. The construction +of generic massive contact terms using little-group covariant spinors was discussed +in [41, 31, 53–55, 46]. +3 + +We first derive the most general four-point contact terms involving the SM particles +consistent with SU(3)×U(1)EM symmetry and baryon and lepton number conservation. +Since they are built in terms of the broken-phase electroweak sector, with the physical +Higgs h and the massive W and Z treated as independent degrees of freedom, the +resulting amplitudes are valid beyond the SMEFT. In particular, any tree-amplitude +featuring the SM particles with Wilson coefficients determined by the running to the +energy scale of interest can be spanned by these contact terms. +Thus, the contact +terms derived in this way correspond to HEFT amplitudes. Our analysis extends [31], +which derived the three-point SM amplitudes and one four-point example, to include +the complete set of four-point contact terms. +To construct the independent contact terms, we use the strategy of [51, 53]: working +with the little-group covariant massive spinor formalism [41], the basic building blocks +of the basis are Stripped Contact Terms (SCTs), which are massive spinor structures +with no additional factors of Mandelstam invariants. To get the full set of contact +terms, each SCT is then multiplied by an expansion in these invariants. Note that the +SCTs carry the little-group weights of the external particles, and encode the information +on their polarizations. The expansion in the Mandelstam invariants on the other hand +only depends on the scattering angles, and corresponds to the derivative expansion of +EFT Lagrangians. Generic four-point SCT bases for spins 0, 1/2, and 1 were given +in [53]. Starting from these, we specify to the SM particle content, impose the low- +energy symmetry and (anti)symmetrize over identical particles. Partial results on the +electroweak sector contact terms were derived in [53], and our analysis extends these to +the full set of massive and massless SM four-points. For each four-point contact term, +we indicate the dimension of the corresponding HEFT operator, namely, the operator +which generates this contact term at leading order, and the dimension at which this +contact term can be generated in the SMEFT. +Turning to the construction of SMEFT contact terms, one way to proceed, which +relies on low-energy input only, is to start from the HEFT contact terms and impose +perturbative unitarity [31]. As we will see in Section 2, the equivalence of perturbative +unitarity and gauge invariance is clearly exposed when the amplitudes are written using +the little-group covariant massive spinor formalism. +To recover all the SMEFT relations from this bottom-up approach, however, one +needs to consider a sufficiently large set of amplitudes, including in particular higher- +point amplitudes. Instead, one can start from the massless SMEFT contact terms and +“Higgs” these to obtain the massive contact terms [45]. In the little group-covariant +massive spinor formalism, massless SCTs featuring just fermions and vectors are simply +bolded into massive SCTs. SCTs featuring an external scalar line give rise to two types +of massive contact terms. Directly bolding the massless SCT gives a massive SCT with +4 + +an external scalar line—a physical Higgs. Massless SCTs featuring a scalar momentum +p bold into a massive vector line, with p → p⟩[p. This Higgsing relies on Lorentz +symmetry, specifically the little group transformations of the SCTs in the massless +and massive theory, and exploits the simple relations between the two in the massive +formalism [41]. +Two examples of four-point dimension ≤8 SMEFT amplitudes, namely WWhh and +¯udWh were derived in [45]. Here we extend these results to all the SM particles, but +only include dimension-six contributions. Our starting point is thus the dimension-six +massless SMEFT contact terms derived in [32]. Higgsing these as described above, we +obtain the massive SMEFT contact terms, recovering all the E2 HEFT contact terms, +with Wilson coefficients dictated by the SMEFT. In the bosonic sector, the number of +dimension-six independent Wilson coefficients is reduced from eight in the HEFT to +six in the SMEFT. Additional relations appear in the fermionic amplitudes. +Bottom-up derivations of HEFT operators appeared recently in [39, 56–58]. Ref. [58] +presented the list of HEFT operators corresponding to four-point SCTs with Higgs ex- +ternal legs. (These operators are referred to as primary operators in [58].) Where they +overlap, our results agree with the operator counting of [57, 58]. Hilbert series methods +for counting independent EFT operators [2, 59, 60] were also extended recently to the +case of massive theories and in particular to the HEFT [61]. +This paper is organized as follows. In Section 2, we review some elements of the +little-group covariant massive spinors and SCT construction. We explain our normal- +ization of SCTs with inverse powers of the mass and the cutoff, and the implications +for identifying the operator dimensions and perturbative unitarity. We also discuss +the equivalence of perturbative unitarity and gauge invariance, and comment on the +differences between SMEFT and HEFT amplitudes from the point of view of locality +and analyticity. In Section 3.1 we derive the generic—or HEFT—SM amplitudes with +up to E2 growth. These are shown in Table 1. In Section 3, we derive the analogous +SMEFT amplitudes. The unbroken SMEFT contact terms are reviewed in Table 2, +where we also relate their Wilson coefficients to those of the Warsaw basis. We then +list the massive SMEFT contact terms in Table 3. Thus, each kinematic structure in the +physical amplitudes can be associated with a specific operator in the Warsaw basis. In +Section 4, we derive the remaining HEFT amplitudes featuring cubic or quartic energy +growth. For completeness, we flesh out the details of on-shell Higgsing in Appendix A +using the WWhh amplitude as an example. We discuss the general structure of the +low-energy amplitude, explain the derivation of four-point contact terms, and derive +the dimension-six correction to the WWh amplitude from the relevant massless fac- +torizable six-point amplitude. Finally, in Appendix B, we list the leading high-energy +behavior of the generic low-energy factorizable four-point amplitudes. +5 + +2 +Preliminaries +Each four-point amplitude consists of a factorizable part, which depends on the three- +point couplings; and a non-factorizable part, which is purely local and contains the +four-point contact terms. +The independent parameters entering the amplitude are +thus the renormalizable and non-renormalizable three-point couplings, as well as the +coefficients of independent four-point contact terms. +Together with the three-point +couplings given in Ref. [31], the four-point couplings we will list here parameterize the +most general SM EFT amplitudes, and allow for the construction of the full four-point +amplitude. +We derive the contact terms of the massive and massless SM particles below the +electroweak breaking scale. For the most part, we assume baryon- and lepton number +conservation, but we will comment on the modifications to the contact terms in the +presence of Majorana neutrinos. The low energy theory features several dimension- +ful parameters, namely the particle masses, and the cutoff, which we denote by ¯Λ. +Neglecting fermion masses apart from the top, the masses are parametrically of the +same order, and the contact terms can be written as a double expansion in m/E and +E/¯Λ, where m denotes the common mass scale and E is the energy. If we only impose +SU(3)×U(1), the contact terms we derive apriori describe HEFT amplitudes. To obtain +the low-energy SMEFT contact terms, we start from the SMEFT contact terms at high +energies, with SU(2)×U(1) broken by the vacuum expectation value (VEV) v of a single +Higgs doublet. The low-energy amplitudes then involve two dimensionful parameters, +namely v and the SMEFT cutoff, which we denote by Λ. A large hierarchy between v +and the cutoff is only possible in the SMEFT, where SU(2)×U(1) is linearly realized +at Λ. Thus, for HEFT amplitudes it is appropriate to set ¯Λ = v. On the other hand, +in the SMEFT, with lepton-number conservation, the massless amplitudes feature only +even powers of Λ, and typically 1/¯Λ2 in the low-energy SMEFT amplitudes maps to +1/Λ2, while 1/¯Λ maps to v/Λ2. +The amplitudes are written in terms of spinor variables [62], using the little-group- +covariant spinor formalism of [41] for massive particles. An external massive particle i +of momentum pi is described by a pair of massless spinors, i)M=1,2. Here and in the fol- +lowing, i) stands for either i] or i⟩. For each external fermion i, the amplitude contains +one factor of i)M=1,2 with M = 1, 2 corresponding to positive and negative helicity re- +spectively for square spinors, and conversely for angle spinors. For each external vector +i, the amplitude contains i){Mi)N}. For square spinors, (MN) = 11, (MN) = 22, and +{12}, correspond to positive, negative, and zero polarizations, respectively. Boldface +indicates symmetrization over vector indices, but we use it for any massive spinor or +momentum to distinguish them from massless ones. Our conventions for the spinors and +6 + +their high energy limits follow Ref. [31]. The different spinor structures contributing +to a given amplitude can be classified according to their helicity category, namely, the +helicities of the external particles in the massless spinor structure obtained by naively +unbolding the massive structure [53]. +To obtain the HEFT contact terms, we follow [53]. The contact terms are deter- +mined by Lorentz symmetry, which dictates their little-group transformations, locality, +and the additional symmetries of the theory, in this case, SU(3)×U(1)EM and baryon +and lepton number. Manifestly-local contact terms can be constructed from the list of +independent SCTs, namely spinor structures with no additional powers of the Mandel- +stams, and then appending a polynomial in the independent Mandelstam invariants, +say s and t. Isolating the independent SCTs can be largely done by relying on the +massless limit. However, once they are multiplied by the Mandelstams, some terms +can become redundant. We refer the reader to [53] for more details. The relevant +SCT bases for four-point of spins 0, 1/2, and 1 were presented in [53] and provide the +basis for our analysis. Note that the SCTs carry all the polarization information of the +external particles. +Since the low-energy amplitude features several mass scales, the energy growth of +a certain contact term may not simply correlate with the dimension of the operator +which generates it at leading order. In particular, spinor structures in longitudinal- +vector helicity categories should be accompanied by an inverse factor of the mass, +namely i]⟨i/Mi, in order to correctly obtain the dimension at which these structures +first appear at the Lagrangian level [53]. This can be seen by noting that i]⟨i/Mi maps +to the massive vector polarization, or from the fact that the longitudinal vector arises +from a derivatively coupled Goldstone, with the mass in the denominator required to +correct the dimension from ∂G → ∂G/M. Note that i]⟨i/Mi is finite in the high-energy +limit for transverse vector polarizations, but scales as E/Mi for a longitudinal vector +polarization. In fact, the 1/MV “poles” appearing in these contact terms reflect their +non-local nature. These terms are required to cancel E/m growth in the factorizable +massive amplitude in order to obtain a well-behaved theory above v, and are therefore +associated with the factorizable part of the amplitudes. In contrast, the non-factorizable +parts of the amplitude consist of terms that are manifestly local, and which are therefore +suppressed purely by powers of ¯Λ. +The distinction between ¯Λ and MV suppression is only sharp in the SMEFT, where +these scales can be hierarchically separated. For the SMEFT amplitudes to be sensible +at high-energy, v ≪ E ≪ Λ, positive powers of E/m must cancel. We list the leading +power of E/m terms of the factorizable amplitudes in Appendix B. The equivalence of +perturbative unitarity and gauge invariance is very transparent in the massive spinor +formalism. The sources of E/m behavior are factors such as i]⟨i/M. For zero vector +7 + +polarization, this scales as E/M, and typically leads to amplitudes growing as a positive +power of E/M. Such terms violate perturbative unitarity, which requires En growth to +be suppressed by the same power of the cutoff. Choosing instead the vector polarization +to be positive, the factor i]i⟩/MV is finite, and can be written as i]ξi⟩ where i] is the +high-energy limit of i]I=1, and ξi⟩ = i⟩I=2/MV is finite. Requiring the high-energy +amplitude to be independent of the arbitrary spinor ξi⟩ is thus equivalent to requiring +perturbative unitarity. On the other hand, in the massless, high-energy theory, ξi⟩ is an +arbitrary spinor, which is nothing but the reference spinor associated with the vector +polarizations, and the condition that the amplitude is independent of ξi⟩ translates to +the condition that it is gauge invariant. We show one example of this type, namely the +WWhh amplitude in Appendix A. +As mentioned above, inverse mass behavior signals the non-locality of amplitudes, +associated with their factorizable parts. This is precisely the type of behavior we expect +to see in the HEFT. The fact that states getting their mass from EWSB are integrated +out, translates in the on-shell picture to 1/v non-analyticity of the amplitudes, which +implies a cutoff of order v. In contrast, in the SMEFT, the full amplitudes, including +factorizable and contact terms pieces, should be well behaved for v ≪ E ≪ Λ, with no +E/M pieces. +3 +Four-point contact terms at O(E2) +3.1 +HEFT contact terms +In this section, we list the independent four-point contact terms of SM particles with +E2 energy growth, imposing SU(3)×U(1)EM invariance and baryon- and lepton-number +conservation. Bases of independent contact terms for four-point massive amplitudes of +particles of spin ≤1 were derived in Ref. [53]. Here we apply these results to the case +of SM amplitudes. +We list contact terms with E2 growth here, and contact terms +with E4 growth in Section 4. Together with the three-point electroweak amplitudes +derived in Ref. [31]1, the four-point contact terms and their coefficients allow for a full +parametrization of general EFT amplitudes up to E4. +The generic dimension-six contact terms are listed in Table 1. The bolded products +(ij) stand for either square or angle brackets, as appropriate for the helicity category +in question. The Wilson coefficients of these structures are denoted by capital C’s, +with subscripts denoting the external particles, and superscripts denoting the helicity +1Three-point gluons were not included in Ref. [31], but can be obtained from the photon amplitudes +by simply adding a color factor. +8 + +category. Here and in the following, f denotes any SM fermion, V denotes the W or +the Z, and h denotes the physical Higgs. +Note that at this order, all the contact terms are given by spinor structures with +no additional powers of the Mandelstam invariants. Thus they correspond to SCTs. +In Section 4, when we consider also E4 terms, these expansions will appear. Recall +that the SCTs carry the little group weights associated with the external particles and +encode their polarization information. Amplitudes not shown in this Table have their +leading contributions from SCTs involving more than two spinor products. +Most of the contact terms in Table 1 are suppressed by two powers of the cutoff, +namely 1/¯Λ2, and correspond to independent dimension-six operators. The exceptions +are structures in longitudinal vector categories. As mentioned above, these are nor- +malized as ⟨12⟩[12]/M2 +V and ⟨13⟩[23]/(MV ¯Λ) (and similarly for 1 ↔ 2). With this +normalization, we can read off the dimension of the low-energy operator which first +generates these terms as 4 and 5 respectively. Indeed, ⟨12⟩[12] is first generated at +dimension-4, and corresponds to the operator V µVµh2. It is required to cancel the high- +energy growth of the massive SM factorizable amplitude 2. We can split the coefficient +of the contact term ⟨12⟩[12] as C00 +W W hh = C00,fac +W W hh + C00,CT +W W hh with the part C00,fac +W W hh can- +celing the E2/M2 +V part of the factorizable amplitude. Thus C00,fac +W W hh is determined by +three-point couplings, while the remaining C00,CT +W W hh constitutes an independent Wilson +coefficient. In the SMEFT, this split cleanly correlates with the counting of operator +dimensions in the high-energy theory. C00 +W W hh is an expansion in v2/Λ2, with the lead- +ing v0 piece corresponding to C00,fac +W W hh, and determined by the SM dimension-four gauge +coupling. At dimension-six, both the three-point couplings and C00,fac +W W hh are shifted by +v2/Λ2 corrections such that the cancelation still holds. On top of this, C00,CT +W W hh/Λ2 is +an independent 4-point Wilson coefficient. +In the HEFT, on the other hand, the various couplings are just numbers, and there +is no expansion in the VEV. Power counting can be done in various ways. Splitting +C00 +W W hh as before, C00,fac +W W hh is naturally treated as dimension-four, such that upon adding +it to the factorizable part, the full amplitude has no E growth. The coefficient C00,CT +W W hh +can be viewed as dimension-six, since it generates E2 terms. Alternatively, it can be +viewed as dimension-four, since it corresponds to the operator V 2h2. In any case, the +physical quantity is the numerical coefficient of each kinematic structure, and these +differences are just a matter of theory interpretation. +Moreover, there is no sharp +distinction in the HEFT between the cutoff ¯Λ and the electroweak mass scale v, with +¯Λ ∼ v. +In the following, when we refer to HEFT dimensions, we will refer to the +dimension of the corresponding operator. The contact terms ⟨12⟩[12] and ⟨13⟩[23] +2The leading high-energy behavior of each factorizable amplitude is shown in Table 4. +9 + +are then dimension-4 and 5 respectively. Furthermore, it is easy to read off the minimal +dimensions of these operators in the SMEFT. To leading order in the v expansion, +¯Λ−2 = Λ−2, and ¯Λ−1 = vΛ−2. Therefore, both of these contact terms can be first +generated at dimension-six in the SMEFT. This is consistent with the fact that the +factorizable fermion-fermion-vector-higgs amplitudes only feature E/M growth (see +Table 4), so C±∓0,fac +ffV h += 0. +Indeed, as was shown in [31], perturbative unitarity of +this amplitude only implies relations between SM couplings, specifically, the relation +between the fermion mass, the Yukawa coupling, and the Higgs VEV. +Massive amplitudes +E2 contact terms +M(WWhh) +C00 +W W hh⟨12⟩[12], C±± +W W hh(12)2 +M(ZZhh) +C00 +ZZhh⟨12⟩[12], C±± +ZZhh(12)2 +M(gghh) +C±± +gghh(12)2 +M(γγhh) +C±± +γγhh(12)2 +M(γZhh) +C± +γZhh(12)2 +M(hhhh) +Chhhh +M(f cfhh) +C±± +ffhh(12) +M(f cfWh) +C+−0 +ffW h[13]⟨23⟩ , C−+0 +ffW h⟨13⟩[23] , C±±± +ffW h(13)(23) +M(f cfZh) +C+−0 +ffZh[13]⟨23⟩ , C−+0 +ffZh⟨13⟩[23] , C±±± +ffZh(13)(23) +M(f cfγh) +C±±± +ffγh(13)(23) +M(qcqgh) +C±±± +qqgh (13)(23) +M(f cff cf) +C±±±±,1 +ffff +(12)(34), C−−++ +ffff ⟨12⟩[34], C−+−+ +ffff ⟨13⟩[24], C−++− +ffff ⟨14⟩[23] +C±±±±,2 +ffff +(13)(24), C++−− +ffff [12]⟨34⟩, C+−+− +ffff [13]⟨24⟩, C+−−+ +ffff [14]⟨23⟩ +Table 1: Contact terms with E2 growth. The C’s stand for independent HEFT coefficients, +and are mostly generated at ¯Λ−2, corresponding to d = 6 operators. The only exceptions +are C00 +W W hh and C±∓0 +ffV h which appear with M−2 +V +and (MV ¯Λ)−1 respectively, corresponding +to d = 4 and d = 5 operators (for details see text). Color structures and indices are not +shown but can be added unambiguously. For identical Majorana neutrinos, the structures +C±±± +ffZh(13)(23) and C±±± +ffγh(13)(23) do not appear. +3.2 +SMEFT contact terms +To obtain the SMEFT contact terms, we start with the massless dimension-six SMEFT +contact terms. These were derived in [32] and we list them for completeness in Table 2. +10 + +Amplitude +Contact term +Warsaw basis operator +Coefficient +A(Hc +i Hc +jHc +kHlHmHn) +T + lmn +ijk +OH/6 +c(H†H)3 +A(Hc +i Hc +jHlHk) +s12T + kl +ij +OHD/2 + OH □/4 +c(+) +(H†H)2 +A(Hc +i Hc +jHlHk) +(s13 − s23)T − kl +ij +OHD/2 − OH □/4 +c(−) +(H†H)2 +A(B±B±Hc +i Hj) +(12)2δj +i +(OHB ± iOH ˜B)/2 +c±± +BBHH +A(B±W I±Hc +i Hj) +(12)2(σI)j +i +OHW B ± iOH ˜ +W B +c±± +BW HH +A(W I+W J+Hc +i Hj) +(12)2δIJδj +i +(OHW ± iOH ˜ +W)/2 +c±± +W W HH +A(gA±gB±Hc +i Hj) +(12)2δABδj +i +(OHG ± iOH ˜G)/2 +c±± +GGHH +A(Lc +ieHc +jHkHl) +[12]T + kl +ij +OeH/2 +c++ +LeHHH +A(Qc +a,idbHc +jHkHl) +[12]T + kl +ij δb +a +OdH/2 +c++ +QdHHH +A(Qc +a,iubHc +jHc +kHl) +[12]εimT + ml +jk δb +a +OuH/2 +c++ +QuHHH +A(eceHc +i Hj) +[231⟩δj +i +OHe/2 +c−+ +eeHH +A(uc +aubHc +i Hj) +[231⟩δj +i δb +a +OHu/2 +c−+ +uuHH +A(dc +adbHc +i Hj) +[231⟩δj +i δb +a +OHd/2 +c−+ +ddHH +A(uc +adbHiHj) +[231⟩ǫijδb +a +OHud/2 +c−+ +udHH +A(Lc +iLjHc +kHl) +[142⟩T + jl +ik +� +O(1) +HL + O(3) +HL +� +/8 +c+−,(+) +LLHH +A(Lc +iLjHc +kHl) +[142⟩T − jl +ik +� +O(1) +HL − O(3) +HL +� +/8 +c+−,(−) +LLHH +A(Qc +a,iQb,jHc +kHl) +[142⟩T + jl +ik δb +a +� +3O(1) +HQ + O(3) +HQ +� +/8 +c+−,(+) +QQHH +A(Qc +a,iQb,jHc +kHl) +[142⟩T − jl +ik δb +a +(O(1) +HQ − O(3) +HQ)/8 +c+−,(−) +QQHH +A(Lc +ieB+Hj) +[13][23]δj +i +−iOeB/(2 +√ +2) +c+++ +LeBH +A(Qc +a,idbB+Hj) +[13][23]δj +i δb +a +−iOdB/(2 +√ +2) +c+++ +QdBH +A(Qc +a,iubB+Hc +j) +[13][23]ǫijδb +a +−iOuB/(2 +√ +2) +c+++ +QuBH +A(Lc +ieW I+Hj) +[13][23](σI)j +i +−iOeW /(2 +√ +2) +c+++ +LeW H +A(Qc +a,idbW I+Hj) +[13][23](σI)j +iδb +a +−iOdW/(2 +√ +2) +c+++ +QdW H +A(Qc +a,iubW I+Hc +j) +[13][23](σI)ikǫk +jδb +a +−iOuW /(2 +√ +2) +c+++ +QuW H +A(Qc +a,idbgA+Hj) +[13][23]δj +i (λA)b +a +−iOdG/(2 +√ +2) +c+++ +QdGH +A(Qc +a,iubgA+Hc +j) +[13][23]ǫij(λA)b +a +−iOuG/(2 +√ +2) +c+++ +QuGH +A(W I±W J±W K±) +(12)(23)(31)ǫIJK +(OW ± iO ˜ +W)/6 +c±±± +W W W +A(gA±gB±gC±) +(12)(23)(31)f ABC +(OG ± iO ˜G)/6 +c±±± +GGG +Table 2: Massless d = 6 SMEFT contact terms [32] and their relations to Warsaw basis +operators [3]. For each operator (or operator combination) O in the third column, c O gen- +erates the structure in the second column with the coefficient c given in the fourth column. +c-superscripts denote charge conjugation. +For each amplitude in Table 2, we show the kinematic and group theory structure. +11 + +We also list the Warsaw basis operator, or combination of operators, O, that generates +this structure, and the corresponding Wilson coefficient c. We use H to denote the +Higgs doublet, g, W and B for an SU(3), SU(2) or U(1) gauge boson respectively, Q +(L) for SU(2)-doublet quarks (leptons), and u, d (e) for SU(2)-singlet quarks (leptons). +The different group theory factors are denoted as follows: σI are the Pauli matrices, +λA are the Gell-Mann matrices, T ± +kl +ij ≡ 1/2(δi +kδj +l ± δj +kδi +l), T +ijk +lmn ≡ δi +lδj +mδk +n + δi +lδk +mδj +n + +δj +l δi +mδk +n + δj +l δk +mδi +n + δk +l δj +mδi +n + δk +l δi +mδj +n, and ǫIJK and f ABC are the SU(2)L and SU(3)c +structure constants respectively. +Parameterizing the Higgs doublet as +H = +� +G+, 1 +√ +2(v + h + iG0) +�T +, +(1) +we can obtain the high-energy amplitudes featuring the Goldstones G± and the radial +mode h on the external legs. Each one of the massless contact terms is then “Higgsed” +to obtain the corresponding massive contact term(s), as described in Ref. [45]. Massless +contact terms featuring only fermions and vectors are simply bolded to give massive +contact terms with fermions and vectors, in transverse vector helicity categories. Mass- +less contact terms featuring a Higgs leg give rise to contact terms with a massive scalar +leg, in which case they are simply bolded; or to contact terms with a massive vector +leg. Thus for example, based on kinematics alone, it is easy to see that at order E2, +the Q†QH†H contact term gives rise to a Q†QZh contact term, but does not generate +a contact term with two physical Higgses. The massless amplitude features [132⟩. We +can then read off the massive structure using, +[132⟩ = [13]⟨32⟩ → [13]⟨32⟩ , +(2) +which contributes to the Q†QZh amplitude. Note that only a structure with a mo- +mentum insertion p3 can give rise to a vector amplitude. Indeed [132⟩ is consistent +with being a Goldstone amplitude since it is derivatively coupled. On the other hand, +[132⟩ cannot contribute to a low-energy amplitude with two physical Higgses: Bose +symmetry would require [132⟩ → [1(3 + 4)2⟩ which is vanishing. +This procedure reproduces the full set of structures of Table 1, and relates their +coefficients to the massless SMEFT coefficients. We collect the massive SMEFT contact +terms and their coefficients in Table 3. +12 + +Massive d = 6 amplitudes +SMEFT Wilson coefficients +M(W + +L W − +L hh) = C00 +W W hh⟨12⟩[12] +C00 +W W hh = (c(+) +(H†H)2 − 3c(−) +(H†H)2)/2 +M(W + +± W − +± hh) = C±± +W W hh(12)2 +C±± +W W hh = 2c±± +W W HH +M(ZLZLhh) = C00 +ZZhh⟨12⟩[12] +C00 +ZZhh = −2c(+) +(H†H)2 +M(Z±Z±hh) = C±± +ZZhh(12)2 +C±± +ZZhh = c2 +Wc±± +W W HH + s2 +Wc±± +BBHH + cWsWc±± +BW HH +M(g±g±hh) = C±± +gghh(12)2 +C±± +gghh = c±± +GGHH +M(γ±γ±hh) = C±± +γγhh(12)2 +C±± +γγhh = s2 +Wc±± +W W HH + c2 +Wc±± +BBHH − cWsWc±± +BW HH +M(γ±Zhh) = C± +γZhh(12)2 +C± +γZhh = sWcWc±± +W W HH − sWcWc±± +BBHH + 1 +2(s2 +W − c2 +W)c±± +BW HH +M(hhhh) = Chhhh +Chhhh = −3c(H†H)2 + 45 v2c(H†H)3 +M(f c +±f±hh) = C±± +ffhh(12) +C±± +ffhh = 3c±± +ΨψHHHv/(2 +√ +2) +M(f c ++f ′ +−WLh) = C+−0 +ffW h[13]⟨23⟩ +C+−0 +ffW h = −(c+−,(+) +ΨΨHH − c+−,(−) +ΨΨHH)/2 +M(f c +−f ′ ++WLh) = C−+0 +ffW h⟨13⟩[23] +C−+0 +ffW h = −c−+ +ψRψ′ +RHH +M(f c +±f ′ +±W±h) = C±±± +ffW h(13)(23) +C±±± +ffW h = c±±± +ΨψW H/2 +M(f c ++f−ZLh) = C+−0 +ffZh[13]⟨23⟩ +C+−0 +eLeLZh = i +√ +2c+−,(+) +ΨΨHH, C+−0 +νLνLZh = i(c+−,(+) +ΨΨHH + c+−,(−) +ΨΨHH)/ +√ +2 +M(f c +−f+ZLh) = C−+0 +ffZh⟨13⟩[23] +C−+0,CT +ffZh += i +√ +2c−+ +ψψHH +M(f c +±f±Z±h) = C±±± +ffZh(13)(23) +C±±± +ffZh = −(sWc±±± +ΨψBH + cWc±±± +ΨψW H)/ +√ +2 +M(f c +±f±γ±h) = C±±± +ffγh(13)(23) +C±±± +ffγh = (−sWc±±± +ΨψW H + cWc±±± +ΨψBH)/ +√ +2 +M(qc +±q±gA +±h) = C±±± +qqgh λA(13)(23) +C±±± +qqgh = c±±± +ΨψGH/ +√ +2 +Table 3: The low-energy E2 contact terms (left column) and their d = 6 coefficients in +the SMEFT (right column). c(H†H)2 without a superscript is the renormalizable four-Higgs +coupling. The mapping for four fermion contact terms is trivial, so we do not include them +here. +Four-fermion contact terms are not shown here because their matching to the high- +energy amplitudes is straightforward. Each of the Wilson coefficients C in Table 3 is d = +6, and is suppressed by Λ2. As explained in Section 3.1, the low-energy amplitudes may +also contain mass-suppressed contact terms in longitudinal vector helicity categories, +which are associated with the factorizable part of the amplitude. Thus for example, +the structure ⟨12⟩[12] in the WWhh amplitude has two pieces: one comes with a +coefficient C00,fac +W W hh, which is determined by three-point couplings, and one which is an +independent SMEFT d = 6 four-point coupling, C00,CT +W W hh. Only the latter is given in +Table 3, but we omit the superscript CT for simplicity. +Note furthermore that high-energy four-point contact terms with Higgs legs may +also correct the three-point couplings. The d = 6 SMEFT corrections to the three- +points were derived in Ref. [31] by matching to the Feynman diagram result obtained +using Ref. [63]. These corrections can also be obtained by on-shell Higgsing. For an +13 + +explicit example, see Appendix A, where we calculate the v2/Λ2 correction to the WWh +coupling from the massless H2(H†)2WW amplitude. +For the d = 6 bosonic contact terms of Table 3, the only change compared to the +HEFT contact terms of Table 1 is in the ±± helicity categories of V V hh, where six +d = 6 SMEFT parameters control eight HEFT parameters. Additional relations appear +among the fermion SMEFT amplitudes, where the coefficients of up- and down-quark +(or antiquark) amplitudes featuring i⟩, (or i]) are equal, since they originate from the +same doublet (anti)-quark amplitude. The coefficients of lepton-doublet amplitudes are +similarly related. +4 +Four-point contact ters at E3 and E4 +In this section, we derive the remaining contact terms contributing to the SM am- +plitudes up to and including quartic energy growth. These include additional SCTs +beyond those listed in Table 2, as well as variations of the SCTs in Table 2 multiplied +by powers of the Mandelstam invariants. For generic four-point amplitudes with spins +≤ 1, the list of independent SCTs is exhausted at quartic energy growth. However, for +the SM particle content, some of these only contribute at higher orders, when multi- +plied by additional powers of the invariants, due to (anti)symmetrization over identical +particles. We comment on these additional contributions where relevant. +For each amplitude, we show the independent contact terms, and the dimension +of the corresponding HEFT operator, following the discussion in Section 3.1. Recall +that apart from longitudinal vector categories, all structures are suppressed by the +appropriate power of ¯Λ, namely ¯Λ3 or ¯Λ4 here. On the other hand, each longitudinal +vector i comes with a factor i⟩[i/Mi. In the HEFT, ¯Λ = v ∼ MV , but the MV factors +allow us to infer the dimension of the corresponding operator. We also show the lowest +dimension at which each structure may be generated in the SMEFT, using the fact +that any single power of ¯Λ can be written as 1/¯Λ = v/Λ2. Thus for example, in the +WWZh amplitude, the structure [13][12]⟨23⟩ is accompanied by 1/(MWMZ ¯Λ), and +its HEFT and possible SMEFT dimensions are given as (5, 6). +Where appropriate, we only show “half” the allowed structures, with the rest ob- +tained by a parity flip (PF), switching all angle and square brackets. The number of +independent structures is also given, following the HEFT and SMEFT operator dimen- +sions. In the HEFT, the coefficients of the terms listed here are all independent. In +the SMEFT, many of them are related. These relations can be derived by “Higgsing” +the massless amplitudes. This was done for ¯udWh and WWhh in Ref. [45]. We also +comment on how the contact terms are modified when Majorana neutrinos are involved. +14 + +4.1 +Bosonic Amplitudes with All Massive Particles +4.1.1 +hhhh +There is no E2 contact term due to the Bose symmetry of the Higgs legs. The first +contact term appears at E4 and is, +˜s2 +12 + ˜s2 +13 + ˜s2 +14 +(8, 8) − 1 +(3) +The numbers in the parenthesis indicate the dimension of the corresponding HEFT and +SMEFT operators respectively. The number 1 following the parenthesis is the number +of independent contact terms. +4.1.2 +Zhhh +Once we symmetrize over h legs, there is no E2 contact term. At E4 there is a single +structure, +0 : +˜s12[121⟩ + ˜s13[131⟩ + ˜s14[141⟩ +(7; 8) − 1 +(4) +The Mandelstams are necessary due to symmetrization over h. The symmetric sum of +˜s13[121⟩ is (˜s13 + ˜s14)[121⟩ + (˜s12 + ˜s14)[131⟩ + (˜s13 + ˜s14)[141⟩ which simplifies to the +above structure. Note that there is no LE factorizable amplitude. +There is an additional SCT in this case, which first contributes at dimension 13, +(˜s12 − ˜s13)(˜s12 − ˜s14)(˜s13 − ˜s14)([1231] − ⟨1231⟩). +4.1.3 +ZZhh +00 : +[131⟩[232⟩ + [141⟩[242⟩, ˜s12[12]⟨12⟩ +(6; 8) − 2 +++ : +˜s12[12]2; PF +(8; 8) − 2 ++− : +[1(3 − 4)2⟩2 + ⟨1(3 − 4)2]2 +(8; 8) − 1 +(5) +Since there is no E4/(M2¯Λ2) growth in the factorizable amplitude, there are no M2Λ2- +suppressed contact terms in the SMEFT. All independent vvss SCTs appear at E4 +order. +4.1.4 +W +W −hh +00 : +[131⟩[242⟩ + [141⟩[232⟩, ˜s12[12]⟨12⟩ +(6; 8) − 2 +++ : +˜s12[12]2; PF +(8; 8) − 2 ++− : +[1(3 − 4)2⟩2; PF +(8; 8) − 2 +(6) +All independent vvss SCTs appear at order E4. +15 + +4.1.5 +W +W −Zh +000 : +[12][343⟩⟨12⟩, (1 ↔ 3), (2 ↔ 3) +(5; 8) − 3 ++00 : +[12]⟨23⟩[31]; Perm(+00); PF +(5; 8) − 6 ++ + 0 : +{[12]2[313⟩, [12]2[323⟩}; Perm(+ + 0); PF +(7; 8) − 12 ++ − 0 : +[13][142⟩⟨23⟩, Perm(+ − 0) +(7; 8) − 6 ++ + + : +[12][13][23]; PF +(7, 8) − 2 +(7) +Above, “Perm” stands for the different possible helicity assignments, eg, (+00), (0 + 0), +(00+). For the (+ − 0) helicity category, two of the six structures can be exchanged +for other O(E4) SCTs times Mandelstams. Since the latter are beyond quartic order +and therefore not included in our counting, all six structures (+ − 0) are independent. +4.1.6 +ZZZh +000 : +[12][343⟩⟨12⟩ + Perm(123) +(5; 8) − 1 ++ + 0 : +[12]2[343⟩ + Perm(+ + 0); PF +(7; 8) − 2 ++ − 0 : +[13][142⟩⟨23⟩ + Perm(+ − 0) +(7; 8) − 1 +(8) +Here, Perm(123) means all permutations of the momenta. The remaining SCTs which +appear in WWZh require additional Mandelstams to satisfy the Bose symmetry of the +Z bosons. The (+00) helicity category first appears at E5 as (s12 − s13)[12]⟨23⟩[31]. +With the parity flipped structure, this introduces two independent coefficients. The +(+++) helicity category first appears at E9 from (s12−s13)(s13−s23)(s21−s23)⟨12⟩⟨13⟩⟨23⟩, +with an additional independent structure from parity. +4.1.7 +W +W −ZZ +0000 : +[12][34]⟨12⟩⟨34⟩, [13][24]⟨13⟩⟨24⟩ + (3 ↔ 4) +(4; 8) − 2 ++ + 00 : +[12]2[34]⟨34⟩; PF +(6; 8) − 2 ++0 + 0 : +{[12][34][13]⟨24⟩, [14][23][13]⟨24⟩} + (3 ↔ 4); (1 ↔ 2); PF (6; 8) − 8 +00 + + : +[34]2[12]⟨12⟩; PF +(6; 8) − 2 ++ − 00 : +[13][14]⟨23⟩⟨24⟩; PF +(6; 8) − 2 ++0 − 0 : +{[12][14]⟨23⟩⟨34⟩ + (3 ↔ 4), (1 ↔ 2)}; PF +(6; 8) − 4 +00 + − : +[13][23]⟨14⟩⟨24⟩ + (3 ↔ 4) +(6; 8) − 1 ++ + ++ : +{[12]2[34]2, [13]2[24]2 + (3 ↔ 4)}; PF +(8; 8) − 4 ++ + −− : +[12]2⟨34⟩2; PF +(8; 8) − 2 +− + −+ : +[14]2⟨23⟩2 + (3 ↔ 4); PF +(8; 8) − 2 +(9) +At order E5 several new vvvv SCTs become independent in the (+000), (+ + +0), and +(+ + −0) helicity categories. +16 + +4.1.8 +W +W +W −W − +0000 : +[12][34]⟨12⟩⟨34⟩, [13][24]⟨13⟩⟨24⟩ + (3 ↔ 4) +(4; 8) − 2 ++ + 00 : +[12]2[34]⟨34⟩; PF +(6; 8) − 2 ++0 + 0 : +{[12][34][13]⟨24⟩, [14][23][13]⟨24⟩} + (1 ↔ 2) + (3 ↔ 4); PF (6; 8) − 4 +00 + + : +[34]2[12]⟨12⟩; PF +(6; 8) − 2 ++ − 00 : +[13][14]⟨23⟩⟨24⟩ + (1 ↔ 2) +(6; 8) − 1 ++0 − 0 : +[12][14]⟨23⟩⟨34⟩ + (1 ↔ 2) + (3 ↔ 4); PF +(6; 8) − 2 +00 + − : +[13][23]⟨14⟩⟨24⟩ + (3 ↔ 4) +(6; 8) − 1 ++ + ++ : +{[13]2[24]2 + (1 ↔ 2), [13][14][23][24]}; PF +(8; 8) − 4 ++ + −− : +[12]2⟨34⟩2; PF +(8; 8) − 2 +− + −+ : +[24]2⟨13⟩2 + [14]2⟨23⟩2 + (3 ↔ 4) +(8; 8) − 1 +(10) +At E5 several new vvvv SCTs become independent in the (+000), (+ + +0), and +(+ + −0) helicity categories. +4.1.9 +ZZZZ +0000 : +[13][24]⟨13⟩⟨24⟩ + Perm(1234) +(4; 8) − 1 ++ + 00 : +[12]2[34]⟨34⟩ + Perm(1234); PF +(6; 8) − 2 ++ − 00 : +[13][14]⟨23⟩⟨24⟩ + Perm(1234) +(6; 8) − 1 ++ + ++ : +[12]2[34]2 + [13]2[24]2 + [14]2[23]2; PF (8; 8) − 2 ++ + −− : +[12]2⟨34⟩2 + Perm(1234) +(8; 8) − 1 +(11) +At E5 several new vvvv SCTs become independent in the (+000), (+ + +0), and +(+ + −0) helicity categories. +4.2 +Fermionic Amplitudes with All Massive Particles +4.2.1 +f cfhh +++ : +˜s12[12]; PF +(7; 8) − 2 ++− : +{˜s14[132⟩ + ˜s13[142⟩}; PF (8; 8) − 2 +(12) +All SCT bases are covered at E4. +For Majorana neutrinos, there is only a single +independent coefficient in the (+−) category. +4.2.2 +f cfZh and f cf ′W h ++ + 0 : +{[12][313⟩, [12][323⟩}; PF (6; 8) − 4 ++ − + : +{[13][312⟩, [23][321⟩}; PF (7; 8) − 4 ++ − 0 : +[13]⟨23⟩ × {˜s12, ˜s13}; PF (7; 8) − 4 ++ + + : +[13][23] × {˜s12, ˜s13}; PF +(8; 8) − 4 ++ + − : +[12]⟨3123⟩; PF +(8; 8) − 2 +(13) +17 + +All SCT bases are covered at E4. +For identical Majorana neutrinos, the ννZh structures are modified to, ++ + 0 : +[12][313⟩; PF +(6) − 2 ++ − + : +[13][312⟩; PF +(7) − 2 ++ − 0 : +([13]⟨23⟩ − (1 ↔ 2)) × ˜s12, ([13]⟨23⟩ + (1 ↔ 2)) × (˜s13 − ˜s23) (7) − 2 ++ + + : +[13][23] × (˜s13 − ˜s23); PF +(8) − 2 +(14) +where we only show the HEFT operator dimensions. The (+ + −) helicity category +only appears at E6, with the two independent E4 structures multiplied by s13 − s14. +4.2.3 +f cf cff +When the four fermions are distinguishable, the contact terms are, ++ + +− : +{[12][324⟩, Perm(+ + +−)}; PF +(7; 8) − 8 ++ + ++ : +{˜s13[13][24], ˜s13[14][23], ˜s14[14][23]}; PF (8; 8) − 6 ++ + −− : +{[12]⟨34⟩, Perm(+ + −−)} × {˜s12, ˜s13} +(8; 8) − 12 +(15) +All SCTs are covered at E4. For four Dirac fermions of the same flavor, f c +1f c +1f1f1, the +basis is to modified to, ++ + ++ : +{[12][34] × ˜s12, ([13][24] + (1 ↔ 2)) × (˜s13 − ˜s14)}; PF (8; 8) − 4 ++ + −− : +[12]⟨34⟩ × ˜s12; PF +(8; 8) − 2 ++ − +− : +[([13]⟨24⟩ − (3 ↔ 4)) − (1 ↔ 2)] × ˜s12, +[([13]⟨24⟩ + (3 ↔ 4)) + (1 ↔ 2)] × (˜s13 − ˜s14) +(8; 8) − 2 +(16) +For four identical Majorana neutrinos, one has ++ + ++ : +[12][34] × ˜s12 + Perm(1234); PF (8) − 2 ++ + −− : +[12]⟨34⟩ × ˜s12 + Perm(1234) +(8) − 1 +(17) +For the same flavor and Majorana neutrinos, the missing SCTs in the (+++−) helicity +category appear at E5. +18 + +4.2.4 +W +W −f cf and W Zf cf ′ +00 + + : +{⟨12⟩[12][34], ⟨12⟩[13][24]}; PF +(5; 8) − 4 +0 + +− : +⟨14⟩[12][23]; (1 ↔ 2); (3 ↔ 4); PF +(6; 8) − 8 +00 + − : +{⟨14⟩⟨231][23], (1 ↔ 2)}; PF +(6; 8) − 4 ++ + ++ : +{[12]2[34], [12][13][24]}; PF +(7; 8) − 4 ++ + −− : +[12]2⟨34⟩; PF +(7; 8) − 2 +0 − ++ : +{⟨12⟩[34]⟨241], (1 ↔ 2)}; PF +(7; 8) − 4 +0 + ++ : +{⟨132][12][34], ⟨132][13][24]}; (1 ↔ 2); PF (7; 8) − 8 ++ + +− : +{[12]2[314⟩, (3 ↔ 4)}; PF +(8; 8) − 4 ++ − −+ : +{[14][132⟩⟨23⟩, (1 ↔ 2)}; (3 ↔ 4) +(8; 8) − 4 +(18) +There is a non-trivial reduction of the spinor basis for the (0−++) helicity category, but +the reduction appears as a linear combination of terms with higher energy growth which +we have neglected. Thus all of the structures appear with independent coefficients in +our basis. All independent SCTs appear at E4 order for distinguishable fermions. +For identical Majorana neutrinos, W +W −νν, +00 + + : +⟨12⟩[12][34]; PF +(5) − 2 +0 + +− : +{⟨14⟩[12][23] − (3 ↔ 4), (1 ↔ 2)}; PF (6) − 4 +00 + − : +⟨14⟩⟨231][23] − (3 ↔ 4), (1 ↔ 2) +(6) − 2 ++ + ++ : +[12]2[34]; PF +(7) − 2 ++ + −− : +[12]2⟨34⟩; PF +(7) − 2 +0 + ++ : +{⟨132][13][24], (1 ↔ 2)}; PF +(7) − 4 ++ + +− : +[12]2[314⟩ − (3 ↔ 4); PF +(8) − 2 ++ − −+ : +[14][132⟩⟨23⟩ − (3 ↔ 4), (1 ↔ 2) +(8) − 2 +(19) +The missing (0−++) helicity category SCT first appears at E6 as (s13−s14)⟨12⟩[34]⟨2(3 − 4)1] +with four independent coefficients. +4.2.5 +ZZf cf +00 + + : +⟨12⟩[12][34]; PF +(5; 8) − 2 +0 + +− : +{⟨14⟩[12][23] + (1 ↔ 2), (3 ↔ 4)}; PF +(6; 7) − 4 +00 + − : +⟨14⟩⟨231][23] + (1 ↔ 2); PF +(6; 8) − 2 ++ + ++ : +[12]2[34]; PF +(7; 8) − 2 ++ + −− : +[12]2⟨34⟩; PF +(7; 8) − 2 +0 − ++ : +⟨12⟩[34]⟨241] + (1 ↔ 2); PF +(7; 8) − 2 +0 + ++ : +{⟨132][12][34], ⟨132][13][24]} + (1 ↔ 2); PF (7; 8) − 4 ++ − −+ : +[14][132⟩⟨23⟩ + (1 ↔ 2), (3 ↔ 4) +(8; 8) − 2 +(20) +All independent SCTs appear at E4. +19 + +For identical Majorana neutrinos, +00 + + : +⟨12⟩[12][34]; PF +(5) − 2 +0 + +− : +[⟨14⟩[12][23] + (1 ↔ 2)] − (3 ↔ 4); PF (6) − 2 +00 + − : +[⟨14⟩⟨231][23] + (1 ↔ 2)] − (3 ↔ 4) +(6) − 1 ++ + ++ : +[12]2[34]; PF +(7) − 2 ++ + −− : +[12]2⟨34⟩; PF +(7) − 2 +0 + ++ : +[⟨132][13][24] + (1 ↔ 2)] − (3 ↔ 4); PF (7) − 2 ++ − −+ : +[[14][132⟩⟨23⟩ + (1 ↔ 2)] − (3 ↔ 4) +(8) − 1 +(21) +The missing (0 − ++) helicity category SCT first appears at E6 from symmetrizing +(s13 − s14)⟨12⟩[34]⟨2(3 − 4)1] with two independent coefficients. +4.3 +Bosonic Amplitudes with Massless Vectors +4.3.1 +γhhh +There is a single structure appearing at dimension 13, using (˜s12 − ˜s13)(˜s12 − ˜s14)(˜s13 − +˜s14)([1231] − ⟨1231⟩). +4.3.2 +γZhh +++ : +˜s12[12]2; PF +(8; 8) − 2 ++− : +[132⟩2 + (3 ↔ 4); PF +(8; 8) − 2 +(22) +All independent vvss SCTs appear already at E4. +4.3.3 +gghh and γγhh +++ : +˜s12[12]2; PF +(8; 8) − 2 ++− : +[[132⟩2 + (3 ↔ 4)] + (1 ↔ 2) +(8; 8) − 1 +(23) +The δAB color factors are suppressed in gghh. +All independent vvss SCTs appear +already at E4. These amplitudes were derived in [29] up to dimension-10. +4.3.4 +gggh ++ + + : +f ABC[12][13][23]; PF +(7; 8) − 2 +(24) +The (+ + −) helicity amplitude is first generated at E5 as f ABC [12]3⟨13⟩⟨23⟩ with 2 +independent coefficients. The gggh amplitudes are given to dimension-13 in [29]. +20 + +4.3.5 +ggγh and γγγh +The only structures with En, n ≤ 4 are [12][13][23] which are manifestly antisymmetric +under 1 −2 exchange and therefore do not appear. This SCT structure first appears at +E9 by multiplying by (s12 − s13)(s21 − s23)(s31 − s32) with two independent coefficients. +The + + − helicity amplitude is first generated at E7 with [12]3⟨13⟩⟨23⟩)(s13 − s23). +The γγγh amplitudes can be easily obtained from the gggh amplitudes given in [29]. +4.3.6 +ggZh and γγZh ++ + 0 : +[12]2[313⟩ + (1 ↔ 2); PF +(7; 8) − 2 ++ − 0 : +[13][142⟩⟨23⟩ + (1 ↔ 2) +(7; 8) − 1 +(25) +We suppressed the δAB color factor in ggZh. For same-helicity gluons or photons, the +helicity category (± ± ±) first appears at E5 with two independent coefficients. For +opposite gluon or photon helicities, (± ∓ ±) first appears at E5 with two independent +coefficients. (Note that (± ± ∓) which could appear at E7, is reducible to a linear +combination of other structures multiplied by Mandelstams and the Z mass [53].) +4.3.7 +γZZh ++ + 0 : +{[12]2[313⟩ + (2 ↔ 3), [12]2[323⟩ + (2 ↔ 3)}; PF +(7; 8) − 4 ++ − 0 : +[13][142⟩⟨23⟩ + (2 ↔ 3); PF +(7; 8) − 2 +(26) +The helicity category (+ + +) first appears at E5 with two independent coefficients, +while (− + +) first appears at E7 with two independent coefficients. The (+ − +) +structure first appears at E7 but is reducible to a linear combination of other structures +multiplied by Mandelstams and the Z mass [53]. +4.3.8 +γW W h ++00 : +[12]⟨23⟩[31]; PF +(5; 8) − 2 ++ + 0 : +{[12]2[313⟩, [12]2[323⟩}; (2 ↔ 3); PF +(7; 8) − 8 ++ − 0 : +[13][142⟩⟨23⟩, (2 ↔ 3) +(7; 8) − 4 ++ + + : +[12][13][23]; PF +(7, 8) − 2 +(27) +The helicity category (− + +) first appears at E7 with four independent coefficients. +The (+ + −) structure first appears at E5 but is reducible to a linear combination of +other structures multiplied by Mandelstams and the W mass [53]. +4.3.9 +γγγγ ++ + ++ : +[12]2[34]2 + [13]2[24]2 + [14]2[23]2; PF +(8; 8) − 2 ++ + −− : +[12]2⟨34⟩2 + Perm(1234) +(8; 8) − 1 +(28) +There is an additional structure in the helicity category (+ + +−) which first appears +at E6. +21 + +4.3.10 +gggZ, gggγ and γγγZ ++ + −− : +[12]2⟨34⟩2 + Perm(123); PF +(8; 8) − 2 ++ + ++ : +[12]2[34]2 + [13]2[24]2 + [14]2[23]2; PF +(8; 8) − 2 +(29) +We suppressed the dABC color factor. There are no contact terms with an f ABC struc- +ture at this order. The gggZ amplitudes were worked out in [29] up to dimension-12. +There are additional structures in the (+ + +0), (+ + −0), and (+ + −+) helicity +categories which first appear at E5, E7, and E10 respectively. Note that the (+ + +0) +and (+ + −0) structures are not present for the gggγ contact term, and the (+ + +−) +helicity category structure is reducible. gggγ can be obtained by unbolding. The γγγZ +contact terms can be obtained from gggZ with f ABC = 0 and dABC = 1. +4.3.11 +ggZZ and γγZZ ++ + 00 : +{[12]2[34]⟨34⟩}; PF +(6; 8) − 2 ++ − 00 : +[13][14]⟨23⟩⟨24⟩ + (1 ↔ 2) +(6; 8) − 1 ++ + ++ : +{[13]2[24]2 + (1 ↔ 2), [13][14][23][24]; PF (8; 8) − 4 ++ + −− : +[12]2⟨34⟩2; PF +(8; 8) − 2 +− + −+ : +([24]2⟨13⟩2 + [14]2⟨23⟩2) + (3 ↔ 4) +(8; 8) − 1 +(30) +For ggZZ, there is a δAB color factor. There are additional structures in the (+ + +0), +(+ + −0), and (+ − ++) helicity categories which first appear at E5, E7, and E6 +respectively. Note that the (+ + +−) helicity category structures are reducible. +4.3.12 +ggW W and γγW W ++ + 00 : +{[12]2[34]⟨34⟩}; PF +(6; 8) − 2 ++ − 00 : +[13][14]⟨23⟩⟨24⟩ + (1 ↔ 2) +(6; 8) − 1 ++ + ++ : +{[12]2[34]2, [13]2[24]2 + (1 ↔ 2)}; PF (8; 8) − 4 ++ + −− : +{[12]2⟨34⟩2}; PF +(8; 8) − 2 +− + −+ : +{[14]2⟨23⟩2 + (1 ↔ 2)}; PF +(8; 8) − 2 +(31) +For ggWW there is a δAB color factor. There are additional structures in the (+++0), +(+ + −0), and (+ − ++) helicity categories which first appear at E5, E7, and E6 +respectively. Note that the (+ + +−) helicity category structures are reducible. +4.3.13 +γZW W ++ + 00 : +{[12]2[34]⟨34⟩, [12][13][24]⟨34⟩}; Perm(+00); PF +(6; 8) − 12 ++ − 00 : +{[13][14]⟨23⟩⟨24⟩, Perm(−00)}; PF +(6; 8) − 6 ++ + ++ : +{[12]2[34]2, [13]2[24]2, [14]2[23]2}; PF +(8; 8) − 6 ++ + −− : +{[12]2⟨34⟩2, Perm(+ − −)}; PF +(8; 8) − 6 +(32) +22 + +There are additional structures in the (+ + +0), (+ + −0), and (+ + +−) helicity +categories which first appear at E5, E5, and E6 respectively. Note that the (− + ++) +helicity category structures are reducible. +4.3.14 +γZZZ ++ + 00 : +[12]2[34]⟨34⟩ + [13]2[24]⟨24⟩ + [14]2[23]⟨23⟩; PF +(6; 8) − 2 ++ − 00 : +[13][14]⟨23⟩⟨24⟩ + Perm(234); PF +(6; 8) − 2 ++ + ++ : +[12]2[34]2 + [13]2[24]2 + [14]2[23]2; PF +(8; 8) − 2 ++ + −− : +[12]2⟨34⟩2 + Perm(234); PF +(8; 8) − 2 +(33) +There are additional structures in the (+ + +0), (+ + −0), and (+ + +−) helicity +categories which first appear at E5, E5, and E6 respectively. Note that the (− + ++) +helicity category structures are reducible. +4.3.15 +ggγγ ++ + ++ : +{[12]2[34]2, [13]2[24]2 + (1 ↔ 2)}; PF +(8; 8) − 4 ++ + −− : +{[12]2⟨34⟩2}; PF +(8; 8) − 2 ++ − +− : +[[13]2⟨24⟩2 + (1 ↔ 2)] + (3 ↔ 4) +(8; 8) − 1 +(34) +We suppressed the δAB group factor for the gluons. The (+ + +−) helicity category +structure first appears at E6. +4.3.16 +ggγZ ++ + ++ : +{[12]2[34]2, [13]2[24]2 + (1 ↔ 2)}; PF (8; 8) − 4 ++ + −− : +{[12]2⟨34⟩2}; PF +(8; 8) − 2 ++ − −+ : +{[14]2⟨23⟩2 + (1 ↔ 2)}; PF +(8; 8) − 2 +(35) +We suppressed the δAB group factor for the gluons. There are additional structures in +the (+++0), (++−0), and (++−+) helicity categories which first appear at E5, E7, +and E6 respectively. Note that the (+++−) helicity category structures are reducible. +4.3.17 +gggg ++ + ++ : +{G × [12]2[34]2 + Perm(1234), f ABEf CDE[13]2[24]2 + Perm(1234)} (8; 8) − 3 +− − −− : +{G × ⟨12⟩2⟨34⟩2 + Perm(1234), f ABEf CDE⟨13⟩2⟨24⟩2 + Perm(1234)} (8; 8) − 3 ++ + −− : +{G, f ACEf BDE + f BCEf ADE} × ([12]2⟨34⟩2 + Perm(1234)) +(8; 8) − 3 +(36) +Here G = {δABδCD, dABEdCDE} is a set of SU(3) structures. +There are additional +structures in the helicity category (+ + +−) which first appears at E6. +23 + +4.4 +Fermionic Amplitudes with Massless Vectors +4.4.1 +f cfγh and f cfgh ++ + + : +[13][23] × {s12, s13}; PF (8; 8) − 4 ++ − + : +{[13][312⟩, (1 ↔ 2)}; PF (7; 8) − 4 ++ + − : +[12]⟨3123⟩; PF +(8; 8) − 2 +(37) +The gluon amplitude is only nonzero when the fermions are quarks, in which case it +appears with the color factor (λA)b +a. All SCT bases are covered at E4. For identical +Majorana neutrinos, the independent structures are, ++ + + : +[13][23] × (s13 − s23); PF (8) − 2 ++ − + : +[13][312⟩ − (1 ↔ 2); PF (7) − 2 +(38) +In this case the (++−) helicity category only appears at E6 with the two independent +E4 SCTs multiplied by s13 − s14. +4.4.2 +ggf cf and γγf cf ++ + ++ : +[12]2[34]; PF +(7; 8) − 2 ++ + −− : +[12]2⟨34⟩; PF +(7; 8) − 2 ++ − −+ : +[14][132⟩⟨23⟩ + (1 ↔ 2); (3 ↔ 4) (8; 8) − 2 +(39) +The gluon contact terms are proportional to δAB. The fermions form an SU(3) singlet +in both cases. For identical Majorana neutrinos one has ++ + ++ : +[12]2[34]; PF +(7) − 2 ++ + −− : +[12]2⟨34⟩; PF +(7) − 2 ++ − −+ : +([14][132⟩⟨23⟩ + (1 ↔ 2)) − (3 ↔ 4) (8) − 1 +(40) +There are structures in the (+ + +−) and (− + ++) helicity categories which first +appear at E6 and E5 respectively. +4.4.3 +γgf cf ++ + ++ : +{[12]2[34], [12][13][24]}; PF +(7; 8) − 4 ++ + −− : +[12]2⟨34⟩; PF +(7; 8) − 2 ++ + +− : +{[12]2[314⟩, (3 ↔ 4)}; PF +(8; 8) − 4 ++ − −+ : +{[14][132⟩⟨23⟩, (1 ↔ 2)}; (3 ↔ 4) (8; 8) − 4 +(41) +This amplitude is only nonzero for quarks, and involves the color factor (λA)b +a. There +is an SCT in the (− + ++) helicity categories which first contributes at E5. +24 + +4.4.4 +γZf cf, γW f cf ′, gZf cf and gW f cf ′ ++0 + − : +{[12][13]⟨24⟩, (3 ↔ 4)}; PF +(6; 8) − 4 ++ + ++ : +{[12]2[34], [12][13][24]}; PF +(7; 8) − 4 ++ + −− : +[12]2⟨34⟩; PF +(7; 8) − 2 +−0 + + : +⟨12⟩[34]⟨142]; PF +(7; 8) − 2 ++0 + + : +{⟨231][12][34], ⟨231][23][14]}; PF (7; 8) − 4 ++ + +− : +{[12]2[314⟩, (3 ↔ 4)}; PF +(8; 8) − 4 ++ − −+ : +{[14][132⟩⟨23⟩, (3 ↔ 4)}; PF +(8; 8) − 4 +(42) +The gluon amplitude is only non-zero for quarks, and appears with (λA)b +a. There is an +additional structure in the (− + ++) helicity category which first appears at E5. +For identical Majorana neutrinos, γZνν, one has instead, ++0 + − : +[12][13]⟨24⟩ − (3 ↔ 4); PF (6) − 2 ++ + ++ : +[12]2[34]; PF +(7) − 2 ++ + −− : +[12]2⟨34⟩; PF +(7) − 2 ++0 + + : +⟨231][23][14] − (3 ↔ 4); PF (7) − 2 ++ + +− : +[12]2[314⟩ − (3 ↔ 4); PF +(8) − 2 ++ − −+ : +[14][132⟩⟨23⟩ − (3 ↔ 4); PF (8) − 2 +(43) +The (−0 + +) helicity-category SCT first appears at E6. +5 +Conclusions +The on-shell bootstrap is ideally suited to the derivation of low-energy effective am- +plitudes, since the latter are fully determined by the particle content and assumed +symmetry. In this paper, we have applied these methods to parametrize the four-point +local amplitudes of the known particles, keeping structures scaling with the energy as +En≤4. These are given by a set of independent contact terms, which can be organized +in terms of linear combinations of independent, manifestly-local spinor structures, or +SCTs, multiplied by expansions in the Mandelstam invariants. +Our results provide the basic building blocks for collider EFT searches. Two-to-two +EFT scattering amplitudes can be derived from the set of SM three-point amplitudes +obtained in [31], and the four-point contact terms derived here. Together, these con- +tact terms also parametrize two- and three-particle decay amplitudes. The resulting +EFT formulation involves just physical quantities. We also discuss the mapping of the +contact terms to the HEFT and the SMEFT. In particular, in Table 3, we derive the +low-energy SMEFT contact terms, and relate them to the Warsaw basis. These results +can be used to distinguish between the HEFT and SMEFT frameworks, and to identify +observables that are particularly sensitive to different types of UV models. +25 + +Acknowledgements +We thank Csaba Csaki, Gauthier Durieux, Christophe Grojean and Markus Luty for +discussions. YS and MW thank MITP Mainz, and YS thanks the Aspen Center for +physics, which is supported by the National Science Foundation (grant PHY-1607611), +where parts of this work were completed. Research supported in part by the Israel +Science Foundation (Grant No. 751/19), and by the NSF-BSF (Grant No. 2020-785). +T.M. is supported by “Study in Israel" Fellowship for Outstanding Post-Doctoral Re- +searchers from China and India by PBC of CHE. M.W. is supported at the Technion +by a Zuckerman Fellowship. H.L. is supported by ISF, BSF and Azrieli foundation. +A +W W hh: On-shell construction of the HEFT and SMEFT +amplitudes and on-shell Higgsing +In this Appendix we provide a detailed derivation of the WWhh HEFT and SMEFT +amplitudes. This illustrates several points: +• Mass-suppressed contact terms, which appear for longitudinal-vector helicity cat- +egories are associated with the factorizable part of the SMEFT amplitude. +• The coefficients of these terms are determined by the three-point couplings based +on the high-energy limit of the amplitude. This can be done in two equivalent +ways: requiring that the zero-polarization amplitude has no E/m growth, or re- +quiring that the transverse-polarization amplitude does not depend on spurious +spinors. This latter requirement is nothing but gauge invariance, so the equiva- +lence of gauge invariance and perturbative unitarity are manifest. +• The remaining contact terms can then be determined by “Higgsing” the massless +4 + nH contact terms. +A.1 +The structure of the full amplitude +The on-shell construction of the WWhh amplitude requires as inputs the WWh and +hhh 3-point couplings, as well as the 4-point WWhh contact terms. In the HEFT, the +3- and 4-point couplings are the most general ones consistent with the symmetry of the +low-energy theory. In the SMEFT, both the 3-points and the 4-points can be derived +by Higgsing the massless amplitudes in the unbroken phase. +The most general three points consistent with the symmetries of the low-energy +theory are [31], +M(W +, W −, h) = C00 +W W h +⟨12⟩[12] +MW ++ C++ +W W h +[12][12] +¯Λ ++ C−− +W W h +⟨12⟩⟨12⟩ +¯Λ +, +(44) +26 + +M(h, h, h) = mhChhh . +(45) +The three-point couplings CW W h and Chhh are numbers, since there is no kinematic +dependence in 3-point amplitudes. In the SMEFT, they are given as an expansion in +v/Λ. In this subsection, we neglect the terms with C±± +W W h because they will not affect +our discussion3. The WWhh amplitude is then, +M(W +, W −, h, h) = −mhMWC00 +W W hchhh +s12 − m2 +h +[12]⟨12⟩ +M2 +W ++ C00 +W W h +2 +s13 − M2 +W +�⟨131]⟨242] +2M2 +W +− [12]⟨12⟩ +� ++ C00 +W W h +2 +s14 − M2 +W +�⟨141]⟨232] +2M2 +W +− [12]⟨12⟩ +� ++ C00,fac +W W hh +[12]⟨12⟩ +M2 +W ++ C00,CT +W W hh +[12]⟨12⟩ +¯Λ2 ++ C′ 00,fac +W W hh +[12]⟨12⟩s12 +M2 +W ¯Λ2 ++ C′′ 00,fac +W W hh +⟨131]⟨242] + ⟨141]⟨232] +M2 +W ¯Λ2 +(46) +where we have kept terms up to E4. The first two lines of this expression are obtained +by gluing the 3-point amplitudes4. The last two lines of Eq. (46) contain the most +general manifestly-local structures, or contact terms5. As explained above, to read off +the operator dimension at which each longitudinal vector structure first appears, we +normalize it with a factor of 1/MW. In the SMEFT, each of the couplings CW W hh +entering the amplitude is given as an expansion in v/Λ, and in particular, each power +of ¯Λ is given in terms of Λ and potentially v. +Note that we have isolated the 1/M2 +W pieces of the longitudinal-vector contact +terms, and labeled them with the superscript fac: these pieces are associated with +the factorizable parts of the amplitudes, and in the SMEFT arise from the factorizable +parts of the massless amplitudes. +Thus, these pieces are determined by the three- +point couplings. The remaining pieces are labeled by CT for contact terms: at each +dimension, these are the novel inputs in the theory, which arise in the SMEFT from +massless contact terms. In the HEFT on the other hand, ¯Λ = v ∼ MW, there is no real +separation between the two types of terms. +3Our focus here is on the C00 +WWhh terms, and the relevant helicity amplitudes to consider are ±∓ +and 00. The C±± +WWh contributions are subleading in the high-energy limit of these amplitudes. One +can show, either by Higgsing the high-energy amplitudes or by requiring good high-energy behavior +of the amplitudes with the same W helicities that C±± +WWh ∝ v +4For details of this gluing, see Ref. [31]. Note that this simple gluing can only be done for massive +legs (see eg [47] for a recent discussion). +5In the notation of Ref. [31], the coefficients CWWhh of the spinor structures are expansions in the +Mandelstams, but here we expanded these out. +27 + +To determine the 1/M2 +W pieces, consider the expansion of the amplitude in terms +of sij/¯Λ2 and sij/M2 +W. For the longitudinal-W amplitude, this expansion starts as, +M(W +(0), W −(0), h, h) ∼ −C00,fac +W W hh +s12 +M2 +W +− C00,CT +W W hh +s12 +¯Λ2 − C00 +W W h +2 +2 +s12 +M2 +W +−C′ 00,fac +W W hh +s2 +12 +M2 +W ¯Λ2 + C′′ 00,fac +W W hh +s2 +13 + s2 +14 +M2 +W ¯Λ2 +(47) +where we only kept terms which grow with the energy. The superscripts (0) denote +the W polarization. At dimension-six, the amplitude should be unitary up to E2/¯Λ2 +terms. Thus, pieces with E2/M2 +W should vanish, fixing, +C00,fac +W W hh = −C00 +W W h +2 +2 +C′ 00,fac +W W hh = C′′ 00,fac +W W hh = 0 . +(48) +Therefore, at dimension-six, these coefficients are determined in terms of the three- +point couplings. At dim-8, this implies cancellations between terms in C00,fac +W W hh and +C′ 00,fac +W W hh. In the HEFT, one cannot take the high energy limit, since the cutoff cannot +be much higher than the the electroweak scale. Thus, the reasoning above only serves +to differentiate between dimension-4, 6, etc contributions to the Wilson coefficients. In +particular, since we can set ¯Λ = v, there are still independent dimension-six contact +terms suppressed by v, such as C00,CT +W W hh[12]⟨12⟩/v2. +In contrast, in the SMEFT, we can really take the high-energy limit as in Eq. (47), +with MW ∼ v ≪ E ≪ Λ. Furthermore, we can alternatively obtain the relations in +Eq. (48) by considering the high-energy limit of the amplitude with transverse W’s +(which are finite in this limit), +A(W +(+), W −(−), h, h) ∼ c00,fac +W W hh +[1k2q]⟨1q2k⟩ +M2 +W ++ c00 +W W h +2 +2M2 +W +[1k31q⟩[2q42k⟩ +s13 ++ c00 +W W h +2 +2M2 +W +[1k41q⟩[2q32k⟩ +s14 +. +(49) +The spinors 1q) and 2q) scale as the mass, and can be written as (see eg [45]) +iq) = MW +(ikξi) ξi) +(50) +where ξi) is an arbitrary constant spinor. For the amplitude to be well-defined at high +energies, it must be independent of the arbitrary spinors. Requiring that the amplitude +in Eq. (49) is independent of these arbitrary spinors, one recovers the relation +C00,fac +W W hh = −C00 +W W h +2 +2 +. +(51) +28 + +Indeed, in the SMEFT, the high energy EFT has the full unbroken gauge symmetry, and +the role of the arbitrary spinors ξi is clear—these are the reference spinors for the two +massless vector polarizations [45] (see also [64]). Thus the equivalence of perturbative +unitarity and the restoration of gauge symmetry is manifest in this example. +A.2 +Four-point contact terms from on-shell Higgsing +As we saw above, the independent couplings appearing in the amplitude are the three- +point couplings and the Λ-suppressed contact terms (with no MW suppression). In +the SMEFT, these can be determined by Higgsing the massless SU(3)×SU(2)×U(1)- +symmetric amplitudes [45]. Two examples, namely WWhh and ¯udWh were worked out +in detail in Ref. [45] up to dimension-8. Here we briefly repeat the WWhh derivation +for completeness, keeping only dimension-six terms. The results for all the four-points +at dimension-six appear in Table 3. +The dimension-six contributions to the three points were derived in Ref. [31] by +matching the amplitudes to the broken-phase SMEFT using [63]. We do not repeat +the derivation using on-shell Higgsing. Instead, we just show a few examples deriving +the three-points from on-shell Higgsing in the next subsection. +The low-energy WWhh contact terms originate from a number of high-energy con- +tact terms. The coefficient of each structure in a given helicity category is determined, +to leading order in v/Λ, by the corresponding helicity amplitude in the high-energy +theory. For instance, the massless WWH†H amplitudes give the leading-order con- +tribution to the ++ and −− helicity categories, while (H†H)2 is the leading-order +contribution to the 00 helicity category. +Sub-leading v2/Λ2 contributions originate +from higher-point contact terms such as (H†H)3. Here, we just focus on their leading +order pieces. It is worth noting that the high-energy contact terms fully determine the +coefficients of the low-energy massive contact terms. Each massive contact term is of +course a little-group tensor, and the derivation of its coefficient Clow essentially relies on +the matching of the leading-energy component to the contact term of the corresponding +high energy amplitude, with Wilson coefficient chigh. The sub-leading components of +the massive structure are generated by factorizable high-energy amplitude featuring +the coupling chigh. +Let us begin with the contribution from (H†H)2. The high-energy amplitude is +A(HiH† +kHjH† +l ) ⊃ c+ +HHHH +s13 +Λ2 T + ij +kl + c− +HHHH +s12 − s14 +Λ2 +T − ij +kl +(52) +where T ± ij +kl = (δi +kδj +l ± δi +lδj +k)/2 are the symmetric and anti-symmetric SU(2) structures. +29 + +Parameterizing the Higgs doublet as in 1, we find that +A(G+G−hh) = 1 +2 +� +A(H1H† +1H2H† +2) + A(H1H† +1H† +2H2) +� += − +c+ +(H†H)2 − 3c− +(H†H)2 +2 +s12 +2Λ2 +(53) +which bolds into +− +c+ +(H†H)2 − 3c− +(H†H)2 +2 +s12 +2Λ2 −→ +c+ +(H†H)2 − 3c− +(H†H)2 +2 +[12]⟨12⟩ +Λ2 +. +(54) +Thus there is a contact term in the massive EFT of the form, +C00,CT +W W hh +[12]⟨12⟩ +Λ2 += +c+ +(H†H)2 − 3c− +(H†H)2 +2 +[12]⟨12⟩ +Λ2 +. +(55) +We now turn to the contribution of the massless WWH†H contact terms, +A(W a,±W b,±H† +i Hj) = c±± +W W HH +(12)2 +Λ2 +� +T ab�j +i . +(56) +Here ± are the helicites of the W’s, (12) = [12] for ++, (12) = ⟨12⟩ for −−, and +� +T ab�j +i = δabδj +i is required by the symmetry of the spinor structure. The kinematics +bold trivially (12) → (12), and one finds the low-energy contact terms +C±± +W W hh +(12) +Λ2 += 2c±± +W W HH +(12) +Λ2 . +(57) +A.3 +The W W h coupling from on-shell Higgsing +The WWh coupling C00 +W W h in Eq. (44) is given by the gauge coupling to leading order, +with a v2/Λ2 correction at dimension-six. Here we show how both of these contributions +can be obtained from on-shell Higgsing. The relevant massless amplitudes to consider +are WH†H or WWH†H to determine the dimension-4 part, and H2(H†)2WW to +determine the dimension-6 shift. +Our discussion closely parallels the derivation of three-points from the massless +amplitudes of a toy model with higgsed U(1) symmetry in Ref. [45], and we refer the +reader to that paper for more details. +To determine C00 +W W h one can start from either of its components, which map to +different high-energy amplitudes. Apriori, the obvious amplitude to start from is the +massless three point WH†H, which for positive W helicity is, +A((W +)+G−h) = g +√ +2 +[12][13] +[23] +(58) +30 + +where we parametrized the Higgs doublet as in Eq. (1). Bolding this amplitude is not +entirely straightforward because it is non-local, and indeed, its non-locality translates +in the massive amplitude to the 1/MW “pole”. However, we can proceed by multiplying +and dividing Eq. (58) by ⟨3ξ⟩ for some arbitrary ξ. After some manipulations using +momentum conservation, we get +A((W +)+G−h) = g +√ +2 +[12]⟨ξ2⟩ +⟨1ξ⟩ +. +(59) +Identifying 1q] = ξ], this maps to the longitudinal W component of +C00 +W W h +⟨12⟩[12] +MW +(60) +with C00 +W W h = g. Alternatively, a simpler way to get this coupling is to start from +the 4-point amplitude H2(H†)2, which matches the all-longitudinal component of the +spinor structure. At the renormalizable level, this amplitude, +ASM((W −)−(W +)+hh) = −g2 +2 +⟨132] +⟨231] . +(61) +Identifying iq) = (MW/(ik3)) 3) for i = 1, 2 and taking p4 → 0 gives, +Am +SM(hW +W −) = g2v +2 +[23]⟨23⟩ +M2 +W +(62) +where we also used ⟨ikiq⟩ = [iqik] = MW. Therefore, at the leading order, C00 +W W h = +g2v/MW. Note that the arbitray ξ spinor that we identified with the q spinors above +naturally arises in this case from the soft higgs leg, with ˆ1q and ˆ2q along ˆ4 (see also +Ref. [64]). +Turning to the dimension-six correction to C00 +W W h, this originates, to leading or- +der, from the 6-point H2(H†)2WW factorizable amplitude with a single insertion of a +H2(H†)2 contact term. The massless amplitude can be written as, +Atot +d=6((H0)4(W +)+(W −)−) = A(3)((H0)4(W +)+(W −)−) + A(4)((H0)4(W +)+(W −)−) , +with +A(3)((H0)4(W +)+(W −)−) = 8g2C +4 +� [142⟩ +⟨12⟩s41 +s63 + [162⟩ +⟨12⟩s61 +s43 + [132⟩ +⟨12⟩s31 +s64 +� ⟨251] +[21]s52 ++ (5 ↔ 4, 6, 3) , +A(4)((H0)4(W +)+(W −)−) = 8g2C +4 +⟨25⟩[15] +⟨15⟩[25] + (5 ↔ 3, 4, 6) , +(63) +31 + +4 +5 +6 +3 +1 +2 +1 +2 +3 +4 +6 +5 +W+ +W− +W+ +W+ +W− +Figure 1. Feynman-diagram of H2(H†)2WW factorizable amplitude. +and +C = (c+ +(HH†)2 − 3c− +(HH†)2)/2 . +(64) +The piece A(3)((H0)4(W +)+(W −)−) is the sum over Feynman diagrams in which the +vector legs attach to different scalar legs (left panel of Fig. 1), and A(4)((H0)4(W +)+(W −)−) +is the piece with the vector legs attached to the same scalar legs (see the right figure +in Fig. 1). (Of course, only the sum is gauge invariant.) Taking the momenta of three +H0 legs-4,5,6 to be soft, only A(4) ((H0)4(W +)+(W −)−) survives, +lim +q4,5,6→0 A(4) � +H0(q4)H0(q5)H0(q6)H0(3)(W +)+(1)(W −)−(2) +� += 6g2v3C [1kξ2k⟩ +[2kξ1k⟩ , (65) +where at the last stage we set qi ∝ ξ for some arbitrary ξ as before. This bolds to +the same massive structure as above. Altogether, after adding in the renormalizable +contribution we have, +Mm +d=6(h(W +)+(W −)−) = g(1 + v2C)[12]⟨12⟩ +MW +. +(66) +Note that this correction is nothing but the correction to the Higgs wave-function +renormalization induced by the four-Higgs contact term C. +One can derive the SMEFT corrections to the remaining three-point low-energy +amplitudes along the same lines. However, since the most general 3-point couplings +were listed in Ref. [31] based on Feynman diagrams matching, we do not do so here. +32 + +B +Energy-growth of factorizable amplitudes +Massive amplitudes +E growth +M(Zhhh) +no factorizable contribution +M(V V hh) +E2/M2 +M(WWZh) +E2/M2, E3/(M2¯Λ) +M(ZZZh) +no factorizable contribution +M(WWWW/ZZWW) +E3/(M2¯Λ), E4/M4, E4/(M2¯Λ2) +M(ZZZZ) +E2/M2, E2/(M ¯Λ) +M(ffV h) +E/M +M(ffWW) +E2/M2, E3/(M2¯Λ) +M(ffZZ) +E/M, E2/(M ¯Λ) +Table 4: Leading energy growth in the massive factorizable amplitudes. +References +[1] W. Buchmuller and D. Wyler, Effective Lagrangian Analysis of New Interactions and +Flavor Conservation, Nucl. Phys. B 268 (1986) 621. +[2] E. E. Jenkins and A. V. Manohar, Algebraic Structure of Lepton and Quark Flavor +Invariants and CP Violation, JHEP 10 (2009) 094, arXiv:0907.4763 [hep-ph]. +[3] B. Grzadkowski, M. Iskrzynski, M. Misiak, and J. Rosiek, Dimension-Six Terms in the +Standard Model Lagrangian, JHEP 10 (2010) 085, arXiv:1008.4884 [hep-ph]. +[4] G. F. Giudice, C. Grojean, A. Pomarol, and R. Rattazzi, The Strongly-Interacting Light +Higgs, JHEP 06 (2007) 045, arXiv:hep-ph/0703164. +[5] R. S. Gupta, A. Pomarol, and F. Riva, BSM Primary Effects, Phys. Rev. +D 91 (2015) 035001, arXiv:1405.0181 [hep-ph]. +[6] A. Falkowski, B. Fuks, K. Mawatari, K. Mimasu, F. Riva, and V. Sanz, Rosetta: an +operator basis translator for Standard Model effective field theory, Eur. Phys. J. +C 75 (2015) 583, arXiv:1508.05895 [hep-ph]. +[7] C. Hays, A. Martin, V. Sanz, and J. Setford, On the impact of dimension-eight SMEFT +operators on Higgs measurements, JHEP 02 (2019) 123, arXiv:1808.00442 [hep-ph]. +[8] S. Alioli, R. Boughezal, E. Mereghetti, and F. Petriello, Novel angular dependence in +Drell-Yan lepton production via dimension-8 operators, Phys. Lett. +B 809 (2020) 135703, arXiv:2003.11615 [hep-ph]. +33 + +[9] S. Dawson, S. Homiller, and M. Sullivan, Impact of dimension-eight SMEFT +contributions: A case study, Phys. Rev. D 104 (2021) 115013, arXiv:2110.06929 [hep-ph]. +[10] F. Feruglio, The Chiral approach to the electroweak interactions, Int. J. Mod. Phys. +A 8 (1993) 4937, arXiv:hep-ph/9301281. +[11] J. Bagger, V. D. Barger, K.-m. Cheung, J. F. Gunion, T. Han, G. A. Ladinsky, +R. Rosenfeld, and C. P. Yuan, The Strongly interacting W W system: Gold plated +modes, Phys. Rev. D 49 (1994) 1246, arXiv:hep-ph/9306256. +[12] V. Koulovassilopoulos and R. S. Chivukula, The Phenomenology of a nonstandard Higgs +boson in W(L) W(L) scattering, Phys. Rev. D 50 (1994) 3218, arXiv:hep-ph/9312317. +[13] C. P. Burgess, J. Matias, and M. Pospelov, A Higgs or not a Higgs? What to do if you +discover a new scalar particle, Int. J. Mod. Phys. A 17 (2002) 1841, +arXiv:hep-ph/9912459. +[14] B. Grinstein and M. Trott, A Higgs-Higgs bound state due to new physics at a TeV, +Phys. Rev. D 76 (2007) 073002, arXiv:0704.1505 [hep-ph]. +[15] R. Alonso, M. B. Gavela, L. Merlo, S. Rigolin, and J. Yepes, The Effective Chiral +Lagrangian for a Light Dynamical ”Higgs Particle”, Phys. Lett. B 722 (2013) 330, +[Erratum: Phys.Lett.B 726, 926 (2013)], arXiv:1212.3305 [hep-ph]. +[16] D. Espriu, F. Mescia, and B. Yencho, Radiative corrections to WL WL scattering in +composite Higgs models, Phys. Rev. D 88 (2013) 055002, arXiv:1307.2400 [hep-ph]. +[17] G. Buchalla, O. Catà, and C. Krause, Complete Electroweak Chiral Lagrangian with a +Light Higgs at NLO, Nucl. Phys. B 880 (2014) 552, [Erratum: Nucl.Phys.B 913, 475–478 +(2016)], arXiv:1307.5017 [hep-ph]. +[18] I. Brivio, T. Corbett, O. J. P. Éboli, M. B. Gavela, J. Gonzalez-Fraile, M. C. +Gonzalez-Garcia, L. Merlo, and S. Rigolin, Disentangling a dynamical Higgs, +JHEP 03 (2014) 024, arXiv:1311.1823 [hep-ph]. +[19] R. Alonso, E. E. Jenkins, and A. V. Manohar, A Geometric Formulation of Higgs +Effective Field Theory: Measuring the Curvature of Scalar Field Space, Phys. Lett. +B 754 (2016) 335, arXiv:1511.00724 [hep-ph]. +[20] R. Alonso, E. E. Jenkins, and A. V. Manohar, Geometry of the Scalar Sector, +JHEP 08 (2016) 101, arXiv:1605.03602 [hep-ph]. +[21] G. Buchalla, O. Cata, A. Celis, M. Knecht, and C. Krause, Complete One-Loop +Renormalization of the Higgs-Electroweak Chiral Lagrangian, Nucl. Phys. +B 928 (2018) 93, arXiv:1710.06412 [hep-ph]. +[22] R. Alonso, K. Kanshin, and S. Saa, Renormalization group evolution of Higgs effective +field theory, Phys. Rev. D 97 (2018) 035010, arXiv:1710.06848 [hep-ph]. +34 + +[23] J. de Blas, O. Eberhardt, and C. Krause, Current and Future Constraints on Higgs +Couplings in the Nonlinear Effective Theory, JHEP 07 (2018) 048, arXiv:1803.00939 +[hep-ph]. +[24] A. Falkowski and R. Rattazzi, Which EFT, JHEP 10 (2019) 255, arXiv:1902.05936 +[hep-ph]. +[25] T. Cohen, N. Craig, X. Lu, and D. Sutherland, Is SMEFT Enough?, JHEP 03 (2021) 237, +arXiv:2008.08597 [hep-ph]. +[26] I. Banta, T. Cohen, N. Craig, X. Lu, and D. Sutherland, Non-decoupling new particles, +JHEP 02 (2022) 029, arXiv:2110.02967 [hep-ph]. +[27] H. Elvang, D. Z. Freedman, and M. Kiermaier, A simple approach to counterterms in +N=8 supergravity, JHEP 11 (2010) 016, arXiv:1003.5018 [hep-th]. +[28] C. Cheung, K. Kampf, J. Novotny, C.-H. Shen, and J. Trnka, A Periodic Table of +Effective Field Theories, JHEP 02 (2017) 020, arXiv:1611.03137 [hep-th]. +[29] Y. Shadmi and Y. Weiss, Effective Field Theory Amplitudes the On-Shell Way: Scalar +and Vector Couplings to Gluons, JHEP 02 (2019) 165, arXiv:1809.09644 [hep-ph]. +[30] R. Aoude and C. S. Machado, The Rise of SMEFT On-shell Amplitudes, +JHEP 12 (2019) 058, arXiv:1905.11433 [hep-ph]. +[31] G. Durieux, T. Kitahara, Y. Shadmi, and Y. Weiss, The electroweak effective field theory +from on-shell amplitudes, JHEP 01 (2020) 119, arXiv:1909.10551 [hep-ph]. +[32] T. Ma, J. Shu, and M.-L. Xiao, Standard Model Effective Field Theory from On-shell +Amplitudes, Chin. Phys. C 47 (2023) 023105, arXiv:1902.06752 [hep-ph]. +[33] I. Low and Z. Yin, Soft Bootstrap and Effective Field Theories, JHEP 11 (2019) 078, +arXiv:1904.12859 [hep-th]. +[34] I. Low and Z. Yin, New Flavor-Kinematics Dualities and Extensions of Nonlinear Sigma +Models, Phys. Lett. B 807 (2020) 135544, arXiv:1911.08490 [hep-th]. +[35] A. Falkowski, G. Isabella, and C. S. Machado, On-shell effective theory for higher-spin +dark matter, SciPost Phys. 10 (2021) 101, arXiv:2011.05339 [hep-ph]. +[36] M. Accettulli Huber and S. De Angelis, Standard Model EFTs via on-shell methods, +JHEP 11 (2021) 221, arXiv:2108.03669 [hep-th]. +[37] H.-L. Li, Z. Ren, J. Shu, M.-L. Xiao, J.-H. Yu, and Y.-H. Zheng, Complete set of +dimension-eight operators in the standard model effective field theory, Phys. Rev. +D 104 (2021) 015026, arXiv:2005.00008 [hep-ph]. +[38] H.-L. Li, Z. Ren, M.-L. Xiao, J.-H. Yu, and Y.-H. Zheng, Complete set of +dimension-nine operators in the standard model effective field theory, Phys. Rev. +D 104 (2021) 015025, arXiv:2007.07899 [hep-ph]. +35 + +[39] H. Sun, M.-L. Xiao, and J.-H. Yu, Complete NLO Operators in the Higgs Effective Field +Theory, (2022), arXiv:2206.07722 [hep-ph]. +[40] I. Low, J. Shu, M.-L. Xiao, and Y.-H. Zheng, Amplitude/operator basis in chiral +perturbation theory, JHEP 01 (2023) 031, arXiv:2209.00198 [hep-ph]. +[41] N. Arkani-Hamed, T.-C. Huang, and Y.-t. Huang, Scattering amplitudes for all masses +and spins, JHEP 11 (2021) 070, arXiv:1709.04891 [hep-th]. +[42] N. Christensen and B. Field, Constructive standard model, Phys. Rev. +D 98 (2018) 016014, arXiv:1802.00448 [hep-ph]. +[43] B. Bachu and A. Yelleshpur, On-Shell Electroweak Sector and the Higgs Mechanism, +JHEP 08 (2020) 039, arXiv:1912.04334 [hep-th]. +[44] N. Christensen, B. Field, A. Moore, and S. Pinto, Two-, three-, and four-body decays in +the constructive standard model, Phys. Rev. D 101 (2020) 065019, arXiv:1909.09164 +[hep-ph]. +[45] R. Balkin, G. Durieux, T. Kitahara, Y. Shadmi, and Y. Weiss, On-shell Higgsing for +EFTs, JHEP 03 (2022) 129, arXiv:2112.09688 [hep-ph]. +[46] S. De Angelis, Amplitude bases in generic EFTs, JHEP 08 (2022) 299, arXiv:2202.02681 +[hep-th]. +[47] N. Christensen, H. Diaz-Quiroz, B. Field, J. Hayward, and J. Miles, Challenges with +Internal Photons in Constructive QED, (2022), arXiv:2209.15018 [hep-ph]. +[48] C. Cheung and Z. Moss, Symmetry and Unification from Soft Theorems and Unitarity, +JHEP 05 (2021) 161, arXiv:2012.13076 [hep-th]. +[49] D. Liu and Z. Yin, Gauge invariance from on-shell massive amplitudes and tree-level +unitarity, Phys. Rev. D 106 (2022) 076003, arXiv:2204.13119 [hep-th]. +[50] C. Cheung, A. Helset, and J. Parra-Martinez, Geometric soft theorems, +JHEP 04 (2022) 011, arXiv:2111.03045 [hep-th]. +[51] G. Durieux and C. S. Machado, Enumerating higher-dimensional operators with on-shell +amplitudes, Phys. Rev. D 101 (2020) 095021, arXiv:1912.08827 [hep-ph]. +[52] B. Henning and T. Melia, Constructing effective field theories via their harmonics, Phys. +Rev. D 100 (2019) 016015, arXiv:1902.06754 [hep-ph]. +[53] G. Durieux, T. Kitahara, C. S. Machado, Y. Shadmi, and Y. Weiss, Constructing +massive on-shell contact terms, JHEP 12 (2020) 175, arXiv:2008.09652 [hep-ph]. +[54] Z.-Y. Dong, T. Ma, and J. Shu, Constructing on-shell operator basis for all masses and +spins, (2021), arXiv:2103.15837 [hep-ph]. +[55] Z.-Y. Dong, T. Ma, J. Shu, and Y.-H. Zheng, Constructing generic effective field theory +for all masses and spins, Phys. Rev. D 106 (2022) 116010, arXiv:2202.08350 [hep-ph]. +36 + +[56] H. Sun, M.-L. Xiao, and J.-H. Yu, Complete NNLO Operator Bases in Higgs Effective +Field Theory, (2022), arXiv:2210.14939 [hep-ph]. +[57] Z.-Y. Dong, T. Ma, J. Shu, and Z.-Z. Zhou, The New Formulation of Higgs Effective +Field Theory, (2022), arXiv:2211.16515 [hep-ph]. +[58] S. Chang, M. Chen, D. Liu, and M. A. Luty, Primary Observables for Indirect Searches +at Colliders, (2022), arXiv:2212.06215 [hep-ph]. +[59] L. Lehman and A. Martin, Low-derivative operators of the Standard Model effective field +theory via Hilbert series methods, JHEP 02 (2016) 081, arXiv:1510.00372 [hep-ph]. +[60] B. Henning, X. Lu, T. Melia, and H. Murayama, 2, 84, 30, 993, 560, 15456, 11962, +261485, ...: Higher dimension operators in the SM EFT, JHEP 08 (2017) 016, +[Erratum: JHEP 09, 019 (2019)], arXiv:1512.03433 [hep-ph]. +[61] L. Gráf, B. Henning, X. Lu, T. Melia, and H. Murayama, Hilbert Series, the Higgs +Mechanism, and HEFT, (2022), arXiv:2211.06275 [hep-ph]. +[62] F. A. Berends, R. Kleiss, P. De Causmaecker, R. Gastmans, and T. T. Wu, Single +Bremsstrahlung Processes in Gauge Theories, Phys. Lett. B 103 (1981) 124. +[63] A. Dedes, W. Materkowska, M. Paraskevas, J. Rosiek, and K. Suxho, Feynman rules for +the Standard Model Effective Field Theory in Rξ -gauges, JHEP 06 (2017) 143, +arXiv:1704.03888 [hep-ph]. +[64] N. Craig, H. Elvang, M. Kiermaier, and T. Slatyer, Massive amplitudes on the Coulomb +branch of N=4 SYM, JHEP 12 (2011) 097, arXiv:1104.2050 [hep-th]. +37 + diff --git a/pNFIT4oBgHgl3EQfvyuP/content/tmp_files/load_file.txt b/pNFIT4oBgHgl3EQfvyuP/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..fbe2a43200415bb68afc8e1395848452bd67e931 --- /dev/null +++ b/pNFIT4oBgHgl3EQfvyuP/content/tmp_files/load_file.txt @@ -0,0 +1,1548 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf,len=1547 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='11349v1 [hep-ph] 26 Jan 2023 An EFT hunter’s guide to two-to-two scattering: HEFT and SMEFT on-shell amplitudes Hongkai Liu a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Teng Ma a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Yael Shadmi a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Michael Waterbury a a Physics Department,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Technion – Israel Institute of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Technion city,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Haifa 3200003,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Israel b IFAE and BIST,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Universitat Autònoma de Barcelona,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 08193 Bellaterra,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Barcelona We derive the contact terms contributing to the four-point amplitudes of the standard-model particles,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' keeping terms with up to quartic energy growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Imposing just the unbroken low-energy symmetry, and treating the electroweak gauge bosons and the Higgs as independent degrees of freedom, we obtain the most general four-point contact-term amplitudes, corresponding to the HEFT framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' The contact terms are spanned by a basis of Stripped Contact Terms (SCTs), which carry the polarization information, multiplied by polynomials in the Mandelstam invariants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' For terms with quadratic energy growth, we also derive the low-energy SMEFT predictions, via on-shell Higgsing of the massless SMEFT contact terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' We discuss several aspects of bottom-up versus top-down on-shell derivations of the HEFT and SMEFT amplitudes, highlighting in particular the simple counting of HEFT dimensions in the on-shell approach and the transparent relation between perturbative unitarity and gauge-invariance in the little-group covariant massive spinor formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Our results provide a formulation of EFT analyses directly in terms of observable quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' For terms with quadratic energy growth, we also provide the mapping to the Warsaw basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Contents 1 Introduction 2 2 Preliminaries 6 3 Four-point contact terms at O(E2) 8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='1 HEFT contact terms 8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='2 SMEFT contact terms 10 4 Four-point contact ters at E3 and E4 14 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='1 Bosonic Amplitudes with All Massive Particles 15 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='1 hhhh 15 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='2 Zhhh 15 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='3 ZZhh 15 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='4 W +W −hh 15 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='5 W +W −Zh 16 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='6 ZZZh 16 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='7 W +W −ZZ 16 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='8 W +W +W −W − 17 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='9 ZZZZ 17 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='2 Fermionic Amplitudes with All Massive Particles 17 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='1 f cfhh 17 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='2.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='3 gghh and γγhh 20 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='4 gggh 20 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='5 ggγh and γγγh 21 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='6 ggZh and γγZh 21 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='7 γZZh 21 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='8 γWWh 21 1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='9 γγγγ 21 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='10 gggZ, gggγ and γγγZ 22 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='11 ggZZ and γγZZ 22 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='12 ggWW and γγWW 22 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='13 γZWW 22 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='14 γZZZ 23 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='15 ggγγ 23 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='16 ggγZ 23 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='17 gggg 23 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='4 Fermionic Amplitudes with Massless Vectors 24 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='1 f cfγh and f cfgh 24 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='2 ggf cf and γγf cf 24 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='3 γgf cf 24 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='4 γZf cf, γWf cf ′, gZf cf and gWf cf ′ 25 5 Conclusions 25 A W W hh: On-shell construction of the HEFT and SMEFT amplitudes and on-shell Higgsing 26 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='1 The structure of the full amplitude 26 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='2 Four-point contact terms from on-shell Higgsing 29 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='3 The WWh coupling from on-shell Higgsing 30 B Energy-growth of factorizable amplitudes 33 1 Introduction Precise measurements of the interactions of the standard-model (SM) particles, and in particular, the electroweak bosons and the top, will be a focus of the LHC program in the coming decade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' These interactions can be systematically parameterized in terms of Effective Field Theory (EFT) Lagrangians, which in principle provide a model- independent framework for indirect searches for new physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Much of the collider EFT program has been guided by the Standard Model EFT (SMEFT), whose starting point is the unbroken SU(3)×SU(2)×U(1) theory with a single Higgs doublet, focusing in particular on dimension-six operators [1–3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Even in the SMEFT framework, it is plausible however that a given set of heavy fields couple differently to different SM fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Different SMEFT operator bases are therefore better suited to describe the effect 2 of different UV models [3–6], and truncating the EFT at dimension-six may moreover leave out important effects (see for example [7–9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Furthermore, some extensions of the SM are not captured by the SMEFT (at least at low-dimensions), and would lead at low energies to the framework known as the Higgs EFT (HEFT) [10–23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' These include for example models featuring fields which get their masses from electroweak symmetry breaking (EWSB), or fields which provide additional sources of this breaking [15, 24– 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' A natural alternative for EFT constructions is provided by the on-shell bootstrap (see for example [27, 28]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' EFT extensions of the SM can be formulated in this ap- proach directly in terms of the physical observables of interest, namely, the scattering amplitudes of the known SM particles, with a one-to-one mapping of EFT operators and contact-term amplitudes [29–31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Furthermore, since they are not obscured by field redefinitions and operator redundancies, questions such as the distinctions between dif- ferent EFT extensions, or the assignments of EFT dimensions, are concretely phrased in terms of physical quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' In this paper, we derive the full set of four-point contact-term amplitudes, featur- ing the SM massive and massless particles, keeping terms with up to quartic energy growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Together with the three-point amplitudes listed in [31], these determine the EFT predictions for four-point amplitudes with this energy growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' These amplitudes are the most interesting objects for phenomenological purposes, since two-to-two scat- tering processes, followed by two- and three-particle decays, are the ones where we can hope to get the most data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Our results are collected in Tables 1 and 3 and Sec- tion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' The low-energy E2 HEFT contact terms are given in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' The low-energy E2 SMEFT contact terms appear in Table 3 and are mapped to the massless SMEFT contact terms [32] collected in Table 2, where we also give the relations to Warsaw basis operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Section 4 contains the low-energy E4 HEFT contact terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Indeed, on-shell methods have emerged in recent years as a powerful alternative to EFT Lagrangian constructions [29–40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Bottom-up constructions of SM, or SM like amplitudes were discussed in [41, 42, 31, 43–47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' The emergence of symmetry from the amplitude bootstrap, and its relation to the geometry of field space was studied for instance in [48–50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' We employ two types of on-shell constructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' The first is purely bottom-up and gives HEFT amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' The second is top-down and starts from the massless amplitudes of the unbroken theory, yielding the SMEFT low-energy contact terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' We now sketch these in turn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Various methods for constructing generic contact-term bases for massless amplitudes were described in [32, 51, 24, 36, 52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' The construction of generic massive contact terms using little-group covariant spinors was discussed in [41, 31, 53–55, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 3 We first derive the most general four-point contact terms involving the SM particles consistent with SU(3)×U(1)EM symmetry and baryon and lepton number conservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Since they are built in terms of the broken-phase electroweak sector, with the physical Higgs h and the massive W and Z treated as independent degrees of freedom, the resulting amplitudes are valid beyond the SMEFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' In particular, any tree-amplitude featuring the SM particles with Wilson coefficients determined by the running to the energy scale of interest can be spanned by these contact terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Thus, the contact terms derived in this way correspond to HEFT amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Our analysis extends [31], which derived the three-point SM amplitudes and one four-point example, to include the complete set of four-point contact terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' To construct the independent contact terms, we use the strategy of [51, 53]: working with the little-group covariant massive spinor formalism [41], the basic building blocks of the basis are Stripped Contact Terms (SCTs), which are massive spinor structures with no additional factors of Mandelstam invariants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' To get the full set of contact terms, each SCT is then multiplied by an expansion in these invariants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Note that the SCTs carry the little-group weights of the external particles, and encode the information on their polarizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' The expansion in the Mandelstam invariants on the other hand only depends on the scattering angles, and corresponds to the derivative expansion of EFT Lagrangians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Generic four-point SCT bases for spins 0, 1/2, and 1 were given in [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Starting from these, we specify to the SM particle content, impose the low- energy symmetry and (anti)symmetrize over identical particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Partial results on the electroweak sector contact terms were derived in [53], and our analysis extends these to the full set of massive and massless SM four-points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' For each four-point contact term, we indicate the dimension of the corresponding HEFT operator, namely, the operator which generates this contact term at leading order, and the dimension at which this contact term can be generated in the SMEFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Turning to the construction of SMEFT contact terms, one way to proceed, which relies on low-energy input only, is to start from the HEFT contact terms and impose perturbative unitarity [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' As we will see in Section 2, the equivalence of perturbative unitarity and gauge invariance is clearly exposed when the amplitudes are written using the little-group covariant massive spinor formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' To recover all the SMEFT relations from this bottom-up approach, however, one needs to consider a sufficiently large set of amplitudes, including in particular higher- point amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Instead, one can start from the massless SMEFT contact terms and “Higgs” these to obtain the massive contact terms [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' In the little group-covariant massive spinor formalism, massless SCTs featuring just fermions and vectors are simply bolded into massive SCTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' SCTs featuring an external scalar line give rise to two types of massive contact terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Directly bolding the massless SCT gives a massive SCT with 4 an external scalar line—a physical Higgs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Massless SCTs featuring a scalar momentum p bold into a massive vector line, with p → p⟩[p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' This Higgsing relies on Lorentz symmetry, specifically the little group transformations of the SCTs in the massless and massive theory, and exploits the simple relations between the two in the massive formalism [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Two examples of four-point dimension ≤8 SMEFT amplitudes, namely WWhh and ¯udWh were derived in [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Here we extend these results to all the SM particles, but only include dimension-six contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Our starting point is thus the dimension-six massless SMEFT contact terms derived in [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Higgsing these as described above, we obtain the massive SMEFT contact terms, recovering all the E2 HEFT contact terms, with Wilson coefficients dictated by the SMEFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' In the bosonic sector, the number of dimension-six independent Wilson coefficients is reduced from eight in the HEFT to six in the SMEFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Additional relations appear in the fermionic amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Bottom-up derivations of HEFT operators appeared recently in [39, 56–58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' [58] presented the list of HEFT operators corresponding to four-point SCTs with Higgs ex- ternal legs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' (These operators are referred to as primary operators in [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=') Where they overlap, our results agree with the operator counting of [57, 58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Hilbert series methods for counting independent EFT operators [2, 59, 60] were also extended recently to the case of massive theories and in particular to the HEFT [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' In Section 2, we review some elements of the little-group covariant massive spinors and SCT construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' We explain our normal- ization of SCTs with inverse powers of the mass and the cutoff, and the implications for identifying the operator dimensions and perturbative unitarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' We also discuss the equivalence of perturbative unitarity and gauge invariance, and comment on the differences between SMEFT and HEFT amplitudes from the point of view of locality and analyticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' In Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='1 we derive the generic—or HEFT—SM amplitudes with up to E2 growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' These are shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' In Section 3, we derive the analogous SMEFT amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' The unbroken SMEFT contact terms are reviewed in Table 2, where we also relate their Wilson coefficients to those of the Warsaw basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' We then list the massive SMEFT contact terms in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Thus, each kinematic structure in the physical amplitudes can be associated with a specific operator in the Warsaw basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' In Section 4, we derive the remaining HEFT amplitudes featuring cubic or quartic energy growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' For completeness, we flesh out the details of on-shell Higgsing in Appendix A using the WWhh amplitude as an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' We discuss the general structure of the low-energy amplitude, explain the derivation of four-point contact terms, and derive the dimension-six correction to the WWh amplitude from the relevant massless fac- torizable six-point amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Finally, in Appendix B, we list the leading high-energy behavior of the generic low-energy factorizable four-point amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 5 2 Preliminaries Each four-point amplitude consists of a factorizable part, which depends on the three- point couplings;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' and a non-factorizable part, which is purely local and contains the four-point contact terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' The independent parameters entering the amplitude are thus the renormalizable and non-renormalizable three-point couplings, as well as the coefficients of independent four-point contact terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Together with the three-point couplings given in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' [31], the four-point couplings we will list here parameterize the most general SM EFT amplitudes, and allow for the construction of the full four-point amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' We derive the contact terms of the massive and massless SM particles below the electroweak breaking scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' For the most part, we assume baryon- and lepton number conservation, but we will comment on the modifications to the contact terms in the presence of Majorana neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' The low energy theory features several dimension- ful parameters, namely the particle masses, and the cutoff, which we denote by ¯Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Neglecting fermion masses apart from the top, the masses are parametrically of the same order, and the contact terms can be written as a double expansion in m/E and E/¯Λ, where m denotes the common mass scale and E is the energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' If we only impose SU(3)×U(1), the contact terms we derive apriori describe HEFT amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' To obtain the low-energy SMEFT contact terms, we start from the SMEFT contact terms at high energies, with SU(2)×U(1) broken by the vacuum expectation value (VEV) v of a single Higgs doublet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' The low-energy amplitudes then involve two dimensionful parameters, namely v and the SMEFT cutoff, which we denote by Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' A large hierarchy between v and the cutoff is only possible in the SMEFT, where SU(2)×U(1) is linearly realized at Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Thus, for HEFT amplitudes it is appropriate to set ¯Λ = v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' On the other hand, in the SMEFT, with lepton-number conservation, the massless amplitudes feature only even powers of Λ, and typically 1/¯Λ2 in the low-energy SMEFT amplitudes maps to 1/Λ2, while 1/¯Λ maps to v/Λ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' The amplitudes are written in terms of spinor variables [62], using the little-group- covariant spinor formalism of [41] for massive particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' An external massive particle i of momentum pi is described by a pair of massless spinors, i)M=1,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Here and in the fol- lowing, i) stands for either i] or i⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' For each external fermion i, the amplitude contains one factor of i)M=1,2 with M = 1, 2 corresponding to positive and negative helicity re- spectively for square spinors, and conversely for angle spinors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' For each external vector i, the amplitude contains i){Mi)N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' For square spinors, (MN) = 11, (MN) = 22, and {12}, correspond to positive, negative, and zero polarizations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Boldface indicates symmetrization over vector indices, but we use it for any massive spinor or momentum to distinguish them from massless ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Our conventions for the spinors and 6 their high energy limits follow Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' The different spinor structures contributing to a given amplitude can be classified according to their helicity category, namely, the helicities of the external particles in the massless spinor structure obtained by naively unbolding the massive structure [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' To obtain the HEFT contact terms, we follow [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' The contact terms are deter- mined by Lorentz symmetry, which dictates their little-group transformations, locality, and the additional symmetries of the theory, in this case, SU(3)×U(1)EM and baryon and lepton number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Manifestly-local contact terms can be constructed from the list of independent SCTs, namely spinor structures with no additional powers of the Mandel- stams, and then appending a polynomial in the independent Mandelstam invariants, say s and t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Isolating the independent SCTs can be largely done by relying on the massless limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' However, once they are multiplied by the Mandelstams, some terms can become redundant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' We refer the reader to [53] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' The relevant SCT bases for four-point of spins 0, 1/2, and 1 were presented in [53] and provide the basis for our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Note that the SCTs carry all the polarization information of the external particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Since the low-energy amplitude features several mass scales, the energy growth of a certain contact term may not simply correlate with the dimension of the operator which generates it at leading order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' In particular, spinor structures in longitudinal- vector helicity categories should be accompanied by an inverse factor of the mass, namely i]⟨i/Mi, in order to correctly obtain the dimension at which these structures first appear at the Lagrangian level [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' This can be seen by noting that i]⟨i/Mi maps to the massive vector polarization, or from the fact that the longitudinal vector arises from a derivatively coupled Goldstone, with the mass in the denominator required to correct the dimension from ∂G → ∂G/M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Note that i]⟨i/Mi is finite in the high-energy limit for transverse vector polarizations, but scales as E/Mi for a longitudinal vector polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' In fact, the 1/MV “poles” appearing in these contact terms reflect their non-local nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' These terms are required to cancel E/m growth in the factorizable massive amplitude in order to obtain a well-behaved theory above v, and are therefore associated with the factorizable part of the amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' In contrast, the non-factorizable parts of the amplitude consist of terms that are manifestly local, and which are therefore suppressed purely by powers of ¯Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' The distinction between ¯Λ and MV suppression is only sharp in the SMEFT, where these scales can be hierarchically separated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' For the SMEFT amplitudes to be sensible at high-energy, v ≪ E ≪ Λ, positive powers of E/m must cancel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' We list the leading power of E/m terms of the factorizable amplitudes in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' The equivalence of perturbative unitarity and gauge invariance is very transparent in the massive spinor formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' The sources of E/m behavior are factors such as i]⟨i/M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' For zero vector 7 polarization, this scales as E/M, and typically leads to amplitudes growing as a positive power of E/M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Such terms violate perturbative unitarity, which requires En growth to be suppressed by the same power of the cutoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Choosing instead the vector polarization to be positive, the factor i]i⟩/MV is finite, and can be written as i]ξi⟩ where i] is the high-energy limit of i]I=1, and ξi⟩ = i⟩I=2/MV is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Requiring the high-energy amplitude to be independent of the arbitrary spinor ξi⟩ is thus equivalent to requiring perturbative unitarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' On the other hand, in the massless, high-energy theory, ξi⟩ is an arbitrary spinor, which is nothing but the reference spinor associated with the vector polarizations, and the condition that the amplitude is independent of ξi⟩ translates to the condition that it is gauge invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' We show one example of this type, namely the WWhh amplitude in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' As mentioned above, inverse mass behavior signals the non-locality of amplitudes, associated with their factorizable parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' This is precisely the type of behavior we expect to see in the HEFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' The fact that states getting their mass from EWSB are integrated out, translates in the on-shell picture to 1/v non-analyticity of the amplitudes, which implies a cutoff of order v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' In contrast, in the SMEFT, the full amplitudes, including factorizable and contact terms pieces, should be well behaved for v ≪ E ≪ Λ, with no E/M pieces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 3 Four-point contact terms at O(E2) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='1 HEFT contact terms In this section, we list the independent four-point contact terms of SM particles with E2 energy growth, imposing SU(3)×U(1)EM invariance and baryon- and lepton-number conservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Bases of independent contact terms for four-point massive amplitudes of particles of spin ≤1 were derived in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Here we apply these results to the case of SM amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' We list contact terms with E2 growth here, and contact terms with E4 growth in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Together with the three-point electroweak amplitudes derived in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' [31]1, the four-point contact terms and their coefficients allow for a full parametrization of general EFT amplitudes up to E4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' The generic dimension-six contact terms are listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' The bolded products (ij) stand for either square or angle brackets, as appropriate for the helicity category in question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' The Wilson coefficients of these structures are denoted by capital C’s, with subscripts denoting the external particles, and superscripts denoting the helicity 1Three-point gluons were not included in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' [31], but can be obtained from the photon amplitudes by simply adding a color factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8 category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Here and in the following, f denotes any SM fermion, V denotes the W or the Z, and h denotes the physical Higgs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Note that at this order, all the contact terms are given by spinor structures with no additional powers of the Mandelstam invariants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Thus they correspond to SCTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' In Section 4, when we consider also E4 terms, these expansions will appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Recall that the SCTs carry the little group weights associated with the external particles and encode their polarization information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Amplitudes not shown in this Table have their leading contributions from SCTs involving more than two spinor products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Most of the contact terms in Table 1 are suppressed by two powers of the cutoff, namely 1/¯Λ2, and correspond to independent dimension-six operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' The exceptions are structures in longitudinal vector categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' As mentioned above, these are nor- malized as ⟨12⟩[12]/M2 V and ⟨13⟩[23]/(MV ¯Λ) (and similarly for 1 ↔ 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' With this normalization, we can read off the dimension of the low-energy operator which first generates these terms as 4 and 5 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Indeed, ⟨12⟩[12] is first generated at dimension-4, and corresponds to the operator V µVµh2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' It is required to cancel the high- energy growth of the massive SM factorizable amplitude 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' We can split the coefficient of the contact term ⟨12⟩[12] as C00 W W hh = C00,fac W W hh + C00,CT W W hh with the part C00,fac W W hh can- celing the E2/M2 V part of the factorizable amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Thus C00,fac W W hh is determined by three-point couplings, while the remaining C00,CT W W hh constitutes an independent Wilson coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' In the SMEFT, this split cleanly correlates with the counting of operator dimensions in the high-energy theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' C00 W W hh is an expansion in v2/Λ2, with the lead- ing v0 piece corresponding to C00,fac W W hh, and determined by the SM dimension-four gauge coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' At dimension-six, both the three-point couplings and C00,fac W W hh are shifted by v2/Λ2 corrections such that the cancelation still holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' On top of this, C00,CT W W hh/Λ2 is an independent 4-point Wilson coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' In the HEFT, on the other hand, the various couplings are just numbers, and there is no expansion in the VEV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Power counting can be done in various ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Splitting C00 W W hh as before, C00,fac W W hh is naturally treated as dimension-four, such that upon adding it to the factorizable part, the full amplitude has no E growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' The coefficient C00,CT W W hh can be viewed as dimension-six, since it generates E2 terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Alternatively, it can be viewed as dimension-four, since it corresponds to the operator V 2h2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' In any case, the physical quantity is the numerical coefficient of each kinematic structure, and these differences are just a matter of theory interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Moreover, there is no sharp distinction in the HEFT between the cutoff ¯Λ and the electroweak mass scale v, with ¯Λ ∼ v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' In the following, when we refer to HEFT dimensions, we will refer to the dimension of the corresponding operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' The contact terms ⟨12⟩[12] and ⟨13⟩[23] 2The leading high-energy behavior of each factorizable amplitude is shown in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 9 are then dimension-4 and 5 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Furthermore, it is easy to read off the minimal dimensions of these operators in the SMEFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' To leading order in the v expansion, ¯Λ−2 = Λ−2, and ¯Λ−1 = vΛ−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Therefore, both of these contact terms can be first generated at dimension-six in the SMEFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' This is consistent with the fact that the factorizable fermion-fermion-vector-higgs amplitudes only feature E/M growth (see Table 4), so C±∓0,fac ffV h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Indeed, as was shown in [31], perturbative unitarity of this amplitude only implies relations between SM couplings, specifically, the relation between the fermion mass, the Yukawa coupling, and the Higgs VEV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Massive amplitudes E2 contact terms M(WWhh) C00 W W hh⟨12⟩[12],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' C±± W W hh(12)2 M(ZZhh) C00 ZZhh⟨12⟩[12],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' C±± ZZhh(12)2 M(gghh) C±± gghh(12)2 M(γγhh) C±± γγhh(12)2 M(γZhh) C± γZhh(12)2 M(hhhh) Chhhh M(f cfhh) C±± ffhh(12) M(f cfWh) C+−0 ffW h[13]⟨23⟩ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' C−+0 ffW h⟨13⟩[23] ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' C±±± ffW h(13)(23) M(f cfZh) C+−0 ffZh[13]⟨23⟩ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' C−+0 ffZh⟨13⟩[23] ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' C±±± ffZh(13)(23) M(f cfγh) C±±± ffγh(13)(23) M(qcqgh) C±±± qqgh (13)(23) M(f cff cf) C±±±±,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='1 ffff (12)(34),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' C−−++ ffff ⟨12⟩[34],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' C−+−+ ffff ⟨13⟩[24],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' C−++− ffff ⟨14⟩[23] C±±±±,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='2 ffff (13)(24),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' C++−− ffff [12]⟨34⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' C+−+− ffff [13]⟨24⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' C+−−+ ffff [14]⟨23⟩ Table 1: Contact terms with E2 growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' The C’s stand for independent HEFT coefficients, and are mostly generated at ¯Λ−2, corresponding to d = 6 operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' The only exceptions are C00 W W hh and C±∓0 ffV h which appear with M−2 V and (MV ¯Λ)−1 respectively, corresponding to d = 4 and d = 5 operators (for details see text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Color structures and indices are not shown but can be added unambiguously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' For identical Majorana neutrinos, the structures C±±± ffZh(13)(23) and C±±± ffγh(13)(23) do not appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='2 SMEFT contact terms To obtain the SMEFT contact terms, we start with the massless dimension-six SMEFT contact terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' These were derived in [32] and we list them for completeness in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='Amplitude ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='Contact term ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='Warsaw basis operator ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='Coefficient ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='A(Hc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='i Hc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='jHc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='kHlHmHn) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='T + lmn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='ijk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='OH/6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='c(H†H)3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='A(Hc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='i Hc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='jHlHk) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='s12T + kl ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='ij ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='OHD/2 + OH □/4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='c(+) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='(H†H)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='A(Hc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='i Hc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='jHlHk) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='(s13 − s23)T − kl ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='ij ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='OHD/2 − OH □/4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='c(−) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='(H†H)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='A(B±B±Hc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='i Hj) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='(12)2δj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='(OHB ± iOH ˜B)/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='c±± ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='BBHH ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='A(B±W I±Hc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='i Hj) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='(12)2(σI)j ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='OHW B ± iOH ˜ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='W B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='c±± ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='BW HH ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='A(W I+W J+Hc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='i Hj) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='(12)2δIJδj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='(OHW ± iOH ˜ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='W)/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='c±± ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='W W HH ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='A(gA±gB±Hc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='i Hj) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='(12)2δABδj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='(OHG ± iOH ˜G)/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='c±± ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='GGHH ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='A(Lc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='ieHc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='jHkHl) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='[12]T + kl ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='ij ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='OeH/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='c++ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='LeHHH ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='A(Qc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='idbHc jHkHl) [12]T + kl ij δb a OdH/2 c++ QdHHH A(Qc a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='iubHc jHc kHl) [12]εimT + ml jk δb a OuH/2 c++ QuHHH A(eceHc i Hj) [231⟩δj i OHe/2 c−+ eeHH A(uc aubHc i Hj) [231⟩δj i δb a OHu/2 c−+ uuHH A(dc adbHc i Hj) [231⟩δj i δb a OHd/2 c−+ ddHH A(uc adbHiHj) [231⟩ǫijδb a OHud/2 c−+ udHH A(Lc iLjHc kHl) [142⟩T + jl ik � O(1) HL + O(3) HL � /8 c+−,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='(+) LLHH A(Lc iLjHc kHl) [142⟩T − jl ik � O(1) HL − O(3) HL � /8 c+−,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='(−) LLHH A(Qc a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='iQb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='jHc kHl) [142⟩T + jl ik δb a � 3O(1) HQ + O(3) HQ � /8 c+−,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='(+) QQHH A(Qc a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='iQb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='jHc kHl) [142⟩T − jl ik δb a (O(1) HQ − O(3) HQ)/8 c+−,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='(−) QQHH A(Lc ieB+Hj) [13][23]δj i −iOeB/(2 √ 2) c+++ LeBH A(Qc a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='idbB+Hj) [13][23]δj i δb a −iOdB/(2 √ 2) c+++ QdBH A(Qc a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='iubB+Hc j) [13][23]ǫijδb a −iOuB/(2 √ 2) c+++ QuBH A(Lc ieW I+Hj) [13][23](σI)j i −iOeW /(2 √ 2) c+++ LeW H A(Qc a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='idbW I+Hj) [13][23](σI)j iδb a −iOdW/(2 √ 2) c+++ QdW H A(Qc a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='iubW I+Hc j) [13][23](σI)ikǫk jδb a −iOuW /(2 √ 2) c+++ QuW H A(Qc a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='idbgA+Hj) [13][23]δj i (λA)b a −iOdG/(2 √ 2) c+++ QdGH A(Qc a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='iubgA+Hc j) [13][23]ǫij(λA)b a −iOuG/(2 √ 2) c+++ QuGH A(W I±W J±W K±) (12)(23)(31)ǫIJK (OW ± iO ˜ W)/6 c±±± W W W A(gA±gB±gC±) (12)(23)(31)f ABC (OG ± iO ˜G)/6 c±±± GGG Table 2: Massless d = 6 SMEFT contact terms [32] and their relations to Warsaw basis operators [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' For each operator (or operator combination) O in the third column, c O gen- erates the structure in the second column with the coefficient c given in the fourth column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' c-superscripts denote charge conjugation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' For each amplitude in Table 2, we show the kinematic and group theory structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 11 We also list the Warsaw basis operator, or combination of operators, O, that generates this structure, and the corresponding Wilson coefficient c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' We use H to denote the Higgs doublet, g, W and B for an SU(3), SU(2) or U(1) gauge boson respectively, Q (L) for SU(2)-doublet quarks (leptons), and u, d (e) for SU(2)-singlet quarks (leptons).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' The different group theory factors are denoted as follows: σI are the Pauli matrices, λA are the Gell-Mann matrices, T ± kl ij ≡ 1/2(δi kδj l ± δj kδi l), T +ijk lmn ≡ δi lδj mδk n + δi lδk mδj n + δj l δi mδk n + δj l δk mδi n + δk l δj mδi n + δk l δi mδj n, and ǫIJK and f ABC are the SU(2)L and SU(3)c structure constants respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Parameterizing the Higgs doublet as H = � G+, 1 √ 2(v + h + iG0) �T , (1) we can obtain the high-energy amplitudes featuring the Goldstones G± and the radial mode h on the external legs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Each one of the massless contact terms is then “Higgsed” to obtain the corresponding massive contact term(s), as described in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Massless contact terms featuring only fermions and vectors are simply bolded to give massive contact terms with fermions and vectors, in transverse vector helicity categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Mass- less contact terms featuring a Higgs leg give rise to contact terms with a massive scalar leg, in which case they are simply bolded;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' or to contact terms with a massive vector leg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Thus for example, based on kinematics alone, it is easy to see that at order E2, the Q†QH†H contact term gives rise to a Q†QZh contact term, but does not generate a contact term with two physical Higgses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' The massless amplitude features [132⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' We can then read off the massive structure using, [132⟩ = [13]⟨32⟩ → [13]⟨32⟩ , (2) which contributes to the Q†QZh amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Note that only a structure with a mo- mentum insertion p3 can give rise to a vector amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Indeed [132⟩ is consistent with being a Goldstone amplitude since it is derivatively coupled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' On the other hand, [132⟩ cannot contribute to a low-energy amplitude with two physical Higgses: Bose symmetry would require [132⟩ → [1(3 + 4)2⟩ which is vanishing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' This procedure reproduces the full set of structures of Table 1, and relates their coefficients to the massless SMEFT coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' We collect the massive SMEFT contact terms and their coefficients in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='Massive d = 6 amplitudes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='SMEFT Wilson coefficients ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='M(W + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='L W − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='L hh) = C00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='W W hh⟨12⟩[12] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='C00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='W W hh = (c(+) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='(H†H)2 − 3c(−) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='(H†H)2)/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='M(W + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='± W − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='± hh) = C±± ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='W W hh(12)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='C±± ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='W W hh = 2c±± ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='W W HH ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='M(ZLZLhh) = C00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='ZZhh⟨12⟩[12] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='C00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='ZZhh = −2c(+) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='(H†H)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='M(Z±Z±hh) = C±± ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='ZZhh(12)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='C±± ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='ZZhh = c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='Wc±± ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='W W HH + s2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='Wc±± ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='BBHH + cWsWc±± ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='BW HH ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='M(g±g±hh) = C±± ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='gghh(12)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='C±± ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='gghh = c±± ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='GGHH ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='M(γ±γ±hh) = C±± ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='γγhh(12)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='C±± ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='γγhh = s2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='Wc±± ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='W W HH + c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='Wc±± ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='BBHH − cWsWc±± ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='BW HH ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='M(γ±Zhh) = C± ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='γZhh(12)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='C± ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='γZhh = sWcWc±± ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='W W HH − sWcWc±± ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='BBHH + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='2(s2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='W − c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='W)c±± ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='BW HH ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='M(hhhh) = Chhhh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='Chhhh = −3c(H†H)2 + 45 v2c(H†H)3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='M(f c ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='±f±hh) = C±± ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='ffhh(12) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='C±± ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='ffhh = 3c±± ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='ΨψHHHv/(2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='M(f c ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='+f ′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='−WLh) = C+−0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='ffW h[13]⟨23⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='C+−0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='ffW h = −(c+−,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='(+) ΨΨHH − c+−,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='(−) ΨΨHH)/2 M(f c −f ′ +WLh) = C−+0 ffW h⟨13⟩[23] C−+0 ffW h = −c−+ ψRψ′ RHH M(f c ±f ′ ±W±h) = C±±± ffW h(13)(23) C±±± ffW h = c±±± ΨψW H/2 M(f c +f−ZLh) = C+−0 ffZh[13]⟨23⟩ C+−0 eLeLZh = i √ 2c+−,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='(+) ΨΨHH,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' C+−0 νLνLZh = i(c+−,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='(+) ΨΨHH + c+−,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='(−) ΨΨHH)/ √ 2 M(f c −f+ZLh) = C−+0 ffZh⟨13⟩[23] C−+0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='CT ffZh = i √ 2c−+ ψψHH M(f c ±f±Z±h) = C±±± ffZh(13)(23) C±±± ffZh = −(sWc±±± ΨψBH + cWc±±± ΨψW H)/ √ 2 M(f c ±f±γ±h) = C±±± ffγh(13)(23) C±±± ffγh = (−sWc±±± ΨψW H + cWc±±± ΨψBH)/ √ 2 M(qc ±q±gA ±h) = C±±± qqgh λA(13)(23) C±±± qqgh = c±±± ΨψGH/ √ 2 Table 3: The low-energy E2 contact terms (left column) and their d = 6 coefficients in the SMEFT (right column).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' c(H†H)2 without a superscript is the renormalizable four-Higgs coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' The mapping for four fermion contact terms is trivial, so we do not include them here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Four-fermion contact terms are not shown here because their matching to the high- energy amplitudes is straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Each of the Wilson coefficients C in Table 3 is d = 6, and is suppressed by Λ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' As explained in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='1, the low-energy amplitudes may also contain mass-suppressed contact terms in longitudinal vector helicity categories, which are associated with the factorizable part of the amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Thus for example, the structure ⟨12⟩[12] in the WWhh amplitude has two pieces: one comes with a coefficient C00,fac W W hh, which is determined by three-point couplings, and one which is an independent SMEFT d = 6 four-point coupling, C00,CT W W hh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Only the latter is given in Table 3, but we omit the superscript CT for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Note furthermore that high-energy four-point contact terms with Higgs legs may also correct the three-point couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' The d = 6 SMEFT corrections to the three- points were derived in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' [31] by matching to the Feynman diagram result obtained using Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' These corrections can also be obtained by on-shell Higgsing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' For an 13 explicit example, see Appendix A, where we calculate the v2/Λ2 correction to the WWh coupling from the massless H2(H†)2WW amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' For the d = 6 bosonic contact terms of Table 3, the only change compared to the HEFT contact terms of Table 1 is in the ±± helicity categories of V V hh, where six d = 6 SMEFT parameters control eight HEFT parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Additional relations appear among the fermion SMEFT amplitudes, where the coefficients of up- and down-quark (or antiquark) amplitudes featuring i⟩, (or i]) are equal, since they originate from the same doublet (anti)-quark amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' The coefficients of lepton-doublet amplitudes are similarly related.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 4 Four-point contact ters at E3 and E4 In this section, we derive the remaining contact terms contributing to the SM am- plitudes up to and including quartic energy growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' These include additional SCTs beyond those listed in Table 2, as well as variations of the SCTs in Table 2 multiplied by powers of the Mandelstam invariants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' For generic four-point amplitudes with spins ≤ 1, the list of independent SCTs is exhausted at quartic energy growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' However, for the SM particle content, some of these only contribute at higher orders, when multi- plied by additional powers of the invariants, due to (anti)symmetrization over identical particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' We comment on these additional contributions where relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' For each amplitude, we show the independent contact terms, and the dimension of the corresponding HEFT operator, following the discussion in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Recall that apart from longitudinal vector categories, all structures are suppressed by the appropriate power of ¯Λ, namely ¯Λ3 or ¯Λ4 here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' On the other hand, each longitudinal vector i comes with a factor i⟩[i/Mi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' In the HEFT, ¯Λ = v ∼ MV , but the MV factors allow us to infer the dimension of the corresponding operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' We also show the lowest dimension at which each structure may be generated in the SMEFT, using the fact that any single power of ¯Λ can be written as 1/¯Λ = v/Λ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Thus for example, in the WWZh amplitude, the structure [13][12]⟨23⟩ is accompanied by 1/(MWMZ ¯Λ), and its HEFT and possible SMEFT dimensions are given as (5, 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Where appropriate, we only show “half” the allowed structures, with the rest ob- tained by a parity flip (PF), switching all angle and square brackets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' The number of independent structures is also given, following the HEFT and SMEFT operator dimen- sions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' In the HEFT, the coefficients of the terms listed here are all independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' In the SMEFT, many of them are related.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' These relations can be derived by “Higgsing” the massless amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' This was done for ¯udWh and WWhh in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' We also comment on how the contact terms are modified when Majorana neutrinos are involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 14 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='1 Bosonic Amplitudes with All Massive Particles 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='1 hhhh There is no E2 contact term due to the Bose symmetry of the Higgs legs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' The first contact term appears at E4 and is, ˜s2 12 + ˜s2 13 + ˜s2 14 (8, 8) − 1 (3) The numbers in the parenthesis indicate the dimension of the corresponding HEFT and SMEFT operators respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' The number 1 following the parenthesis is the number of independent contact terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='2 Zhhh Once we symmetrize over h legs, there is no E2 contact term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' At E4 there is a single structure, 0 : ˜s12[121⟩ + ˜s13[131⟩ + ˜s14[141⟩ (7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 1 (4) The Mandelstams are necessary due to symmetrization over h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' The symmetric sum of ˜s13[121⟩ is (˜s13 + ˜s14)[121⟩ + (˜s12 + ˜s14)[131⟩ + (˜s13 + ˜s14)[141⟩ which simplifies to the above structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Note that there is no LE factorizable amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' There is an additional SCT in this case, which first contributes at dimension 13, (˜s12 − ˜s13)(˜s12 − ˜s14)(˜s13 − ˜s14)([1231] − ⟨1231⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='3 ZZhh 00 : [131⟩[232⟩ + [141⟩[242⟩, ˜s12[12]⟨12⟩ (6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 2 ++ : ˜s12[12]2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 2 +− : [1(3 − 4)2⟩2 + ⟨1(3 − 4)2]2 (8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 1 (5) Since there is no E4/(M2¯Λ2) growth in the factorizable amplitude, there are no M2Λ2- suppressed contact terms in the SMEFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' All independent vvss SCTs appear at E4 order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='4 W +W −hh 00 : [131⟩[242⟩ + [141⟩[232⟩, ˜s12[12]⟨12⟩ (6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 2 ++ : ˜s12[12]2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 2 +− : [1(3 − 4)2⟩2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 2 (6) All independent vvss SCTs appear at order E4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 15 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='5 W +W −Zh 000 : [12][343⟩⟨12⟩, (1 ↔ 3), (2 ↔ 3) (5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 3 +00 : [12]⟨23⟩[31];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Perm(+00);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 6 + + 0 : {[12]2[313⟩, [12]2[323⟩};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Perm(+ + 0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 12 + − 0 : [13][142⟩⟨23⟩, Perm(+ − 0) (7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 6 + + + : [12][13][23];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (7, 8) − 2 (7) Above, “Perm” stands for the different possible helicity assignments, eg, (+00), (0 + 0), (00+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' For the (+ − 0) helicity category, two of the six structures can be exchanged for other O(E4) SCTs times Mandelstams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Since the latter are beyond quartic order and therefore not included in our counting, all six structures (+ − 0) are independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='6 ZZZh 000 : [12][343⟩⟨12⟩ + Perm(123) (5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 1 + + 0 : [12]2[343⟩ + Perm(+ + 0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 2 + − 0 : [13][142⟩⟨23⟩ + Perm(+ − 0) (7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 1 (8) Here, Perm(123) means all permutations of the momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' The remaining SCTs which appear in WWZh require additional Mandelstams to satisfy the Bose symmetry of the Z bosons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' The (+00) helicity category first appears at E5 as (s12 − s13)[12]⟨23⟩[31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' With the parity flipped structure, this introduces two independent coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' The (+++) helicity category first appears at E9 from (s12−s13)(s13−s23)(s21−s23)⟨12⟩⟨13⟩⟨23⟩, with an additional independent structure from parity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='7 W +W −ZZ 0000 : [12][34]⟨12⟩⟨34⟩, [13][24]⟨13⟩⟨24⟩ + (3 ↔ 4) (4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 2 + + 00 : [12]2[34]⟨34⟩;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 2 +0 + 0 : {[12][34][13]⟨24⟩, [14][23][13]⟨24⟩} + (3 ↔ 4);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' (1 ↔ 2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 8 00 + + : [34]2[12]⟨12⟩;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 2 + − 00 : [13][14]⟨23⟩⟨24⟩;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 2 +0 − 0 : {[12][14]⟨23⟩⟨34⟩ + (3 ↔ 4), (1 ↔ 2)};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 4 00 + − : [13][23]⟨14⟩⟨24⟩ + (3 ↔ 4) (6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 1 + + ++ : {[12]2[34]2, [13]2[24]2 + (3 ↔ 4)};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 4 + + −− : [12]2⟨34⟩2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 2 − + −+ : [14]2⟨23⟩2 + (3 ↔ 4);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 2 (9) At order E5 several new vvvv SCTs become independent in the (+000), (+ + +0), and (+ + −0) helicity categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 16 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='8 W +W +W −W − 0000 : [12][34]⟨12⟩⟨34⟩, [13][24]⟨13⟩⟨24⟩ + (3 ↔ 4) (4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 2 + + 00 : [12]2[34]⟨34⟩;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 2 +0 + 0 : {[12][34][13]⟨24⟩, [14][23][13]⟨24⟩} + (1 ↔ 2) + (3 ↔ 4);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 4 00 + + : [34]2[12]⟨12⟩;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 2 + − 00 : [13][14]⟨23⟩⟨24⟩ + (1 ↔ 2) (6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 1 +0 − 0 : [12][14]⟨23⟩⟨34⟩ + (1 ↔ 2) + (3 ↔ 4);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 2 00 + − : [13][23]⟨14⟩⟨24⟩ + (3 ↔ 4) (6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 1 + + ++ : {[13]2[24]2 + (1 ↔ 2), [13][14][23][24]};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 4 + + −− : [12]2⟨34⟩2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 2 − + −+ : [24]2⟨13⟩2 + [14]2⟨23⟩2 + (3 ↔ 4) (8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 1 (10) At E5 several new vvvv SCTs become independent in the (+000), (+ + +0), and (+ + −0) helicity categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='9 ZZZZ 0000 : [13][24]⟨13⟩⟨24⟩ + Perm(1234) (4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 1 + + 00 : [12]2[34]⟨34⟩ + Perm(1234);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 2 + − 00 : [13][14]⟨23⟩⟨24⟩ + Perm(1234) (6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 1 + + ++ : [12]2[34]2 + [13]2[24]2 + [14]2[23]2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 2 + + −− : [12]2⟨34⟩2 + Perm(1234) (8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 1 (11) At E5 several new vvvv SCTs become independent in the (+000), (+ + +0), and (+ + −0) helicity categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='2 Fermionic Amplitudes with All Massive Particles 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='1 f cfhh ++ : ˜s12[12];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 2 +− : {˜s14[132⟩ + ˜s13[142⟩};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 2 (12) All SCT bases are covered at E4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' For Majorana neutrinos, there is only a single independent coefficient in the (+−) category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='2 f cfZh and f cf ′W h + + 0 : {[12][313⟩, [12][323⟩};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 4 + − + : {[13][312⟩, [23][321⟩};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 4 + − 0 : [13]⟨23⟩ × {˜s12, ˜s13};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 4 + + + : [13][23] × {˜s12, ˜s13};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 4 + + − : [12]⟨3123⟩;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 2 (13) 17 All SCT bases are covered at E4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' For identical Majorana neutrinos, the ννZh structures are modified to, + + 0 : [12][313⟩;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (6) − 2 + − + : [13][312⟩;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (7) − 2 + − 0 : ([13]⟨23⟩ − (1 ↔ 2)) × ˜s12, ([13]⟨23⟩ + (1 ↔ 2)) × (˜s13 − ˜s23) (7) − 2 + + + : [13][23] × (˜s13 − ˜s23);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (8) − 2 (14) where we only show the HEFT operator dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' The (+ + −) helicity category only appears at E6, with the two independent E4 structures multiplied by s13 − s14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='3 f cf cff When the four fermions are distinguishable, the contact terms are, + + +− : {[12][324⟩, Perm(+ + +−)};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 8 + + ++ : {˜s13[13][24], ˜s13[14][23], ˜s14[14][23]};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 6 + + −− : {[12]⟨34⟩, Perm(+ + −−)} × {˜s12, ˜s13} (8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 12 (15) All SCTs are covered at E4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' For four Dirac fermions of the same flavor, f c 1f c 1f1f1, the basis is to modified to, + + ++ : {[12][34] × ˜s12, ([13][24] + (1 ↔ 2)) × (˜s13 − ˜s14)};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 4 + + −− : [12]⟨34⟩ × ˜s12;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 2 + − +− : [([13]⟨24⟩ − (3 ↔ 4)) − (1 ↔ 2)] × ˜s12, [([13]⟨24⟩ + (3 ↔ 4)) + (1 ↔ 2)] × (˜s13 − ˜s14) (8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 2 (16) For four identical Majorana neutrinos, one has + + ++ : [12][34] × ˜s12 + Perm(1234);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (8) − 2 + + −− : [12]⟨34⟩ × ˜s12 + Perm(1234) (8) − 1 (17) For the same flavor and Majorana neutrinos, the missing SCTs in the (+++−) helicity category appear at E5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 18 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='4 W +W −f cf and W Zf cf ′ 00 + + : {⟨12⟩[12][34], ⟨12⟩[13][24]};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 4 0 + +− : ⟨14⟩[12][23];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' (1 ↔ 2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' (3 ↔ 4);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 8 00 + − : {⟨14⟩⟨231][23], (1 ↔ 2)};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 4 + + ++ : {[12]2[34], [12][13][24]};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 4 + + −− : [12]2⟨34⟩;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 2 0 − ++ : {⟨12⟩[34]⟨241], (1 ↔ 2)};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 4 0 + ++ : {⟨132][12][34], ⟨132][13][24]};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' (1 ↔ 2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 8 + + +− : {[12]2[314⟩, (3 ↔ 4)};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 4 + − −+ : {[14][132⟩⟨23⟩, (1 ↔ 2)};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' (3 ↔ 4) (8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 4 (18) There is a non-trivial reduction of the spinor basis for the (0−++) helicity category, but the reduction appears as a linear combination of terms with higher energy growth which we have neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Thus all of the structures appear with independent coefficients in our basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' All independent SCTs appear at E4 order for distinguishable fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' For identical Majorana neutrinos, W +W −νν, 00 + + : ⟨12⟩[12][34];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (5) − 2 0 + +− : {⟨14⟩[12][23] − (3 ↔ 4), (1 ↔ 2)};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (6) − 4 00 + − : ⟨14⟩⟨231][23] − (3 ↔ 4), (1 ↔ 2) (6) − 2 + + ++ : [12]2[34];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (7) − 2 + + −− : [12]2⟨34⟩;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (7) − 2 0 + ++ : {⟨132][13][24], (1 ↔ 2)};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (7) − 4 + + +− : [12]2[314⟩ − (3 ↔ 4);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (8) − 2 + − −+ : [14][132⟩⟨23⟩ − (3 ↔ 4), (1 ↔ 2) (8) − 2 (19) The missing (0−++) helicity category SCT first appears at E6 as (s13−s14)⟨12⟩[34]⟨2(3 − 4)1] with four independent coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='5 ZZf cf 00 + + : ⟨12⟩[12][34];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 2 0 + +− : {⟨14⟩[12][23] + (1 ↔ 2), (3 ↔ 4)};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 7) − 4 00 + − : ⟨14⟩⟨231][23] + (1 ↔ 2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 2 + + ++ : [12]2[34];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 2 + + −− : [12]2⟨34⟩;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 2 0 − ++ : ⟨12⟩[34]⟨241] + (1 ↔ 2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 2 0 + ++ : {⟨132][12][34], ⟨132][13][24]} + (1 ↔ 2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 4 + − −+ : [14][132⟩⟨23⟩ + (1 ↔ 2), (3 ↔ 4) (8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 2 (20) All independent SCTs appear at E4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 19 For identical Majorana neutrinos, 00 + + : ⟨12⟩[12][34];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (5) − 2 0 + +− : [⟨14⟩[12][23] + (1 ↔ 2)] − (3 ↔ 4);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (6) − 2 00 + − : [⟨14⟩⟨231][23] + (1 ↔ 2)] − (3 ↔ 4) (6) − 1 + + ++ : [12]2[34];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (7) − 2 + + −− : [12]2⟨34⟩;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (7) − 2 0 + ++ : [⟨132][13][24] + (1 ↔ 2)] − (3 ↔ 4);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (7) − 2 + − −+ : [[14][132⟩⟨23⟩ + (1 ↔ 2)] − (3 ↔ 4) (8) − 1 (21) The missing (0 − ++) helicity category SCT first appears at E6 from symmetrizing (s13 − s14)⟨12⟩[34]⟨2(3 − 4)1] with two independent coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='3 Bosonic Amplitudes with Massless Vectors 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='1 γhhh There is a single structure appearing at dimension 13, using (˜s12 − ˜s13)(˜s12 − ˜s14)(˜s13 − ˜s14)([1231] − ⟨1231⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='2 γZhh ++ : ˜s12[12]2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 2 +− : [132⟩2 + (3 ↔ 4);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 2 (22) All independent vvss SCTs appear already at E4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='3 gghh and γγhh ++ : ˜s12[12]2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 2 +− : [[132⟩2 + (3 ↔ 4)] + (1 ↔ 2) (8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 1 (23) The δAB color factors are suppressed in gghh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' All independent vvss SCTs appear already at E4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' These amplitudes were derived in [29] up to dimension-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='4 gggh + + + : f ABC[12][13][23];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 2 (24) The (+ + −) helicity amplitude is first generated at E5 as f ABC [12]3⟨13⟩⟨23⟩ with 2 independent coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' The gggh amplitudes are given to dimension-13 in [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 20 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='5 ggγh and γγγh The only structures with En, n ≤ 4 are [12][13][23] which are manifestly antisymmetric under 1 −2 exchange and therefore do not appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' This SCT structure first appears at E9 by multiplying by (s12 − s13)(s21 − s23)(s31 − s32) with two independent coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' The + + − helicity amplitude is first generated at E7 with [12]3⟨13⟩⟨23⟩)(s13 − s23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' The γγγh amplitudes can be easily obtained from the gggh amplitudes given in [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='6 ggZh and γγZh + + 0 : [12]2[313⟩ + (1 ↔ 2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 2 + − 0 : [13][142⟩⟨23⟩ + (1 ↔ 2) (7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 1 (25) We suppressed the δAB color factor in ggZh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' For same-helicity gluons or photons, the helicity category (± ± ±) first appears at E5 with two independent coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' For opposite gluon or photon helicities, (± ∓ ±) first appears at E5 with two independent coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' (Note that (± ± ∓) which could appear at E7, is reducible to a linear combination of other structures multiplied by Mandelstams and the Z mass [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=') 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='7 γZZh + + 0 : {[12]2[313⟩ + (2 ↔ 3), [12]2[323⟩ + (2 ↔ 3)};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 4 + − 0 : [13][142⟩⟨23⟩ + (2 ↔ 3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 2 (26) The helicity category (+ + +) first appears at E5 with two independent coefficients, while (− + +) first appears at E7 with two independent coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' The (+ − +) structure first appears at E7 but is reducible to a linear combination of other structures multiplied by Mandelstams and the Z mass [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='8 γW W h +00 : [12]⟨23⟩[31];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 2 + + 0 : {[12]2[313⟩, [12]2[323⟩};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' (2 ↔ 3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 8 + − 0 : [13][142⟩⟨23⟩, (2 ↔ 3) (7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 4 + + + : [12][13][23];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (7, 8) − 2 (27) The helicity category (− + +) first appears at E7 with four independent coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' The (+ + −) structure first appears at E5 but is reducible to a linear combination of other structures multiplied by Mandelstams and the W mass [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='9 γγγγ + + ++ : [12]2[34]2 + [13]2[24]2 + [14]2[23]2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 2 + + −− : [12]2⟨34⟩2 + Perm(1234) (8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 1 (28) There is an additional structure in the helicity category (+ + +−) which first appears at E6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 21 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='10 gggZ, gggγ and γγγZ + + −− : [12]2⟨34⟩2 + Perm(123);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 2 + + ++ : [12]2[34]2 + [13]2[24]2 + [14]2[23]2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 2 (29) We suppressed the dABC color factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' There are no contact terms with an f ABC struc- ture at this order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' The gggZ amplitudes were worked out in [29] up to dimension-12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' There are additional structures in the (+ + +0), (+ + −0), and (+ + −+) helicity categories which first appear at E5, E7, and E10 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Note that the (+ + +0) and (+ + −0) structures are not present for the gggγ contact term, and the (+ + +−) helicity category structure is reducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' gggγ can be obtained by unbolding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' The γγγZ contact terms can be obtained from gggZ with f ABC = 0 and dABC = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='11 ggZZ and γγZZ + + 00 : {[12]2[34]⟨34⟩};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 2 + − 00 : [13][14]⟨23⟩⟨24⟩ + (1 ↔ 2) (6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 1 + + ++ : {[13]2[24]2 + (1 ↔ 2), [13][14][23][24];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 4 + + −− : [12]2⟨34⟩2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 2 − + −+ : ([24]2⟨13⟩2 + [14]2⟨23⟩2) + (3 ↔ 4) (8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 1 (30) For ggZZ, there is a δAB color factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' There are additional structures in the (+ + +0), (+ + −0), and (+ − ++) helicity categories which first appear at E5, E7, and E6 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Note that the (+ + +−) helicity category structures are reducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='12 ggW W and γγW W + + 00 : {[12]2[34]⟨34⟩};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 2 + − 00 : [13][14]⟨23⟩⟨24⟩ + (1 ↔ 2) (6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 1 + + ++ : {[12]2[34]2, [13]2[24]2 + (1 ↔ 2)};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 4 + + −− : {[12]2⟨34⟩2};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 2 − + −+ : {[14]2⟨23⟩2 + (1 ↔ 2)};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 2 (31) For ggWW there is a δAB color factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' There are additional structures in the (+++0), (+ + −0), and (+ − ++) helicity categories which first appear at E5, E7, and E6 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Note that the (+ + +−) helicity category structures are reducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='13 γZW W + + 00 : {[12]2[34]⟨34⟩, [12][13][24]⟨34⟩};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Perm(+00);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 12 + − 00 : {[13][14]⟨23⟩⟨24⟩, Perm(−00)};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 6 + + ++ : {[12]2[34]2, [13]2[24]2, [14]2[23]2};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 6 + + −− : {[12]2⟨34⟩2, Perm(+ − −)};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 6 (32) 22 There are additional structures in the (+ + +0), (+ + −0), and (+ + +−) helicity categories which first appear at E5, E5, and E6 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Note that the (− + ++) helicity category structures are reducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='14 γZZZ + + 00 : [12]2[34]⟨34⟩ + [13]2[24]⟨24⟩ + [14]2[23]⟨23⟩;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 2 + − 00 : [13][14]⟨23⟩⟨24⟩ + Perm(234);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 2 + + ++ : [12]2[34]2 + [13]2[24]2 + [14]2[23]2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 2 + + −− : [12]2⟨34⟩2 + Perm(234);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 2 (33) There are additional structures in the (+ + +0), (+ + −0), and (+ + +−) helicity categories which first appear at E5, E5, and E6 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Note that the (− + ++) helicity category structures are reducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='15 ggγγ + + ++ : {[12]2[34]2, [13]2[24]2 + (1 ↔ 2)};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 4 + + −− : {[12]2⟨34⟩2};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 2 + − +− : [[13]2⟨24⟩2 + (1 ↔ 2)] + (3 ↔ 4) (8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 1 (34) We suppressed the δAB group factor for the gluons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' The (+ + +−) helicity category structure first appears at E6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='16 ggγZ + + ++ : {[12]2[34]2, [13]2[24]2 + (1 ↔ 2)};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 4 + + −− : {[12]2⟨34⟩2};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 2 + − −+ : {[14]2⟨23⟩2 + (1 ↔ 2)};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 2 (35) We suppressed the δAB group factor for the gluons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' There are additional structures in the (+++0), (++−0), and (++−+) helicity categories which first appear at E5, E7, and E6 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Note that the (+++−) helicity category structures are reducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='17 gggg + + ++ : {G × [12]2[34]2 + Perm(1234), f ABEf CDE[13]2[24]2 + Perm(1234)} (8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 3 − − −− : {G × ⟨12⟩2⟨34⟩2 + Perm(1234), f ABEf CDE⟨13⟩2⟨24⟩2 + Perm(1234)} (8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 3 + + −− : {G, f ACEf BDE + f BCEf ADE} × ([12]2⟨34⟩2 + Perm(1234)) (8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 3 (36) Here G = {δABδCD, dABEdCDE} is a set of SU(3) structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' There are additional structures in the helicity category (+ + +−) which first appears at E6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 23 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='4 Fermionic Amplitudes with Massless Vectors 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='1 f cfγh and f cfgh + + + : [13][23] × {s12, s13};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 4 + − + : {[13][312⟩, (1 ↔ 2)};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 4 + + − : [12]⟨3123⟩;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 2 (37) The gluon amplitude is only nonzero when the fermions are quarks, in which case it appears with the color factor (λA)b a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' All SCT bases are covered at E4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' For identical Majorana neutrinos, the independent structures are, + + + : [13][23] × (s13 − s23);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (8) − 2 + − + : [13][312⟩ − (1 ↔ 2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (7) − 2 (38) In this case the (++−) helicity category only appears at E6 with the two independent E4 SCTs multiplied by s13 − s14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='2 ggf cf and γγf cf + + ++ : [12]2[34];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 2 + + −− : [12]2⟨34⟩;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 2 + − −+ : [14][132⟩⟨23⟩ + (1 ↔ 2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' (3 ↔ 4) (8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 2 (39) The gluon contact terms are proportional to δAB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' The fermions form an SU(3) singlet in both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' For identical Majorana neutrinos one has + + ++ : [12]2[34];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (7) − 2 + + −− : [12]2⟨34⟩;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (7) − 2 + − −+ : ([14][132⟩⟨23⟩ + (1 ↔ 2)) − (3 ↔ 4) (8) − 1 (40) There are structures in the (+ + +−) and (− + ++) helicity categories which first appear at E6 and E5 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='3 γgf cf + + ++ : {[12]2[34], [12][13][24]};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 4 + + −− : [12]2⟨34⟩;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 2 + + +− : {[12]2[314⟩, (3 ↔ 4)};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 4 + − −+ : {[14][132⟩⟨23⟩, (1 ↔ 2)};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' (3 ↔ 4) (8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 4 (41) This amplitude is only nonzero for quarks, and involves the color factor (λA)b a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' There is an SCT in the (− + ++) helicity categories which first contributes at E5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 24 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='4 γZf cf, γW f cf ′, gZf cf and gW f cf ′ +0 + − : {[12][13]⟨24⟩, (3 ↔ 4)};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 4 + + ++ : {[12]2[34], [12][13][24]};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 4 + + −− : [12]2⟨34⟩;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 2 −0 + + : ⟨12⟩[34]⟨142];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 2 +0 + + : {⟨231][12][34], ⟨231][23][14]};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 4 + + +− : {[12]2[314⟩, (3 ↔ 4)};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 4 + − −+ : {[14][132⟩⟨23⟩, (3 ↔ 4)};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 8) − 4 (42) The gluon amplitude is only non-zero for quarks, and appears with (λA)b a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' There is an additional structure in the (− + ++) helicity category which first appears at E5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' For identical Majorana neutrinos, γZνν, one has instead, +0 + − : [12][13]⟨24⟩ − (3 ↔ 4);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (6) − 2 + + ++ : [12]2[34];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (7) − 2 + + −− : [12]2⟨34⟩;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (7) − 2 +0 + + : ⟨231][23][14] − (3 ↔ 4);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (7) − 2 + + +− : [12]2[314⟩ − (3 ↔ 4);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (8) − 2 + − −+ : [14][132⟩⟨23⟩ − (3 ↔ 4);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' PF (8) − 2 (43) The (−0 + +) helicity-category SCT first appears at E6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 5 Conclusions The on-shell bootstrap is ideally suited to the derivation of low-energy effective am- plitudes, since the latter are fully determined by the particle content and assumed symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' In this paper, we have applied these methods to parametrize the four-point local amplitudes of the known particles, keeping structures scaling with the energy as En≤4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' These are given by a set of independent contact terms, which can be organized in terms of linear combinations of independent, manifestly-local spinor structures, or SCTs, multiplied by expansions in the Mandelstam invariants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Our results provide the basic building blocks for collider EFT searches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Two-to-two EFT scattering amplitudes can be derived from the set of SM three-point amplitudes obtained in [31], and the four-point contact terms derived here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Together, these con- tact terms also parametrize two- and three-particle decay amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' The resulting EFT formulation involves just physical quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' We also discuss the mapping of the contact terms to the HEFT and the SMEFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' In particular, in Table 3, we derive the low-energy SMEFT contact terms, and relate them to the Warsaw basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' These results can be used to distinguish between the HEFT and SMEFT frameworks, and to identify observables that are particularly sensitive to different types of UV models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 25 Acknowledgements We thank Csaba Csaki, Gauthier Durieux, Christophe Grojean and Markus Luty for discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' YS and MW thank MITP Mainz, and YS thanks the Aspen Center for physics, which is supported by the National Science Foundation (grant PHY-1607611), where parts of this work were completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Research supported in part by the Israel Science Foundation (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 751/19), and by the NSF-BSF (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 2020-785).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' is supported by “Study in Israel" Fellowship for Outstanding Post-Doctoral Re- searchers from China and India by PBC of CHE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' is supported at the Technion by a Zuckerman Fellowship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' is supported by ISF, BSF and Azrieli foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' A W W hh: On-shell construction of the HEFT and SMEFT amplitudes and on-shell Higgsing In this Appendix we provide a detailed derivation of the WWhh HEFT and SMEFT amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' This illustrates several points: Mass-suppressed contact terms, which appear for longitudinal-vector helicity cat- egories are associated with the factorizable part of the SMEFT amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' The coefficients of these terms are determined by the three-point couplings based on the high-energy limit of the amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' This can be done in two equivalent ways: requiring that the zero-polarization amplitude has no E/m growth, or re- quiring that the transverse-polarization amplitude does not depend on spurious spinors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' This latter requirement is nothing but gauge invariance, so the equiva- lence of gauge invariance and perturbative unitarity are manifest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' The remaining contact terms can then be determined by “Higgsing” the massless 4 + nH contact terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='1 The structure of the full amplitude The on-shell construction of the WWhh amplitude requires as inputs the WWh and hhh 3-point couplings, as well as the 4-point WWhh contact terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' In the HEFT, the 3- and 4-point couplings are the most general ones consistent with the symmetry of the low-energy theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' In the SMEFT, both the 3-points and the 4-points can be derived by Higgsing the massless amplitudes in the unbroken phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' The most general three points consistent with the symmetries of the low-energy theory are [31], M(W +, W −, h) = C00 W W h ⟨12⟩[12] MW + C++ W W h [12][12] ¯Λ + C−− W W h ⟨12⟩⟨12⟩ ¯Λ , (44) 26 M(h, h, h) = mhChhh .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' (45) The three-point couplings CW W h and Chhh are numbers, since there is no kinematic dependence in 3-point amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' In the SMEFT, they are given as an expansion in v/Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' In this subsection, we neglect the terms with C±± W W h because they will not affect our discussion3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' The WWhh amplitude is then, M(W +, W −, h, h) = −mhMWC00 W W hchhh s12 − m2 h [12]⟨12⟩ M2 W + C00 W W h 2 s13 − M2 W �⟨131]⟨242] 2M2 W − [12]⟨12⟩ � + C00 W W h 2 s14 − M2 W �⟨141]⟨232] 2M2 W − [12]⟨12⟩ � + C00,fac W W hh [12]⟨12⟩ M2 W + C00,CT W W hh [12]⟨12⟩ ¯Λ2 + C′ 00,fac W W hh [12]⟨12⟩s12 M2 W ¯Λ2 + C′′ 00,fac W W hh ⟨131]⟨242] + ⟨141]⟨232] M2 W ¯Λ2 (46) where we have kept terms up to E4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' The first two lines of this expression are obtained by gluing the 3-point amplitudes4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' The last two lines of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' (46) contain the most general manifestly-local structures, or contact terms5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' As explained above, to read off the operator dimension at which each longitudinal vector structure first appears, we normalize it with a factor of 1/MW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' In the SMEFT, each of the couplings CW W hh entering the amplitude is given as an expansion in v/Λ, and in particular, each power of ¯Λ is given in terms of Λ and potentially v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Note that we have isolated the 1/M2 W pieces of the longitudinal-vector contact terms, and labeled them with the superscript fac: these pieces are associated with the factorizable parts of the amplitudes, and in the SMEFT arise from the factorizable parts of the massless amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Thus, these pieces are determined by the three- point couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' The remaining pieces are labeled by CT for contact terms: at each dimension, these are the novel inputs in the theory, which arise in the SMEFT from massless contact terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' In the HEFT on the other hand, ¯Λ = v ∼ MW, there is no real separation between the two types of terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 3Our focus here is on the C00 WWhh terms, and the relevant helicity amplitudes to consider are ±∓ and 00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' The C±± WWh contributions are subleading in the high-energy limit of these amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' One can show, either by Higgsing the high-energy amplitudes or by requiring good high-energy behavior of the amplitudes with the same W helicities that C±± WWh ∝ v 4For details of this gluing, see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Note that this simple gluing can only be done for massive legs (see eg [47] for a recent discussion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 5In the notation of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' [31], the coefficients CWWhh of the spinor structures are expansions in the Mandelstams, but here we expanded these out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 27 To determine the 1/M2 W pieces, consider the expansion of the amplitude in terms of sij/¯Λ2 and sij/M2 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' For the longitudinal-W amplitude, this expansion starts as, M(W +(0), W −(0), h, h) ∼ −C00,fac W W hh s12 M2 W − C00,CT W W hh s12 ¯Λ2 − C00 W W h 2 2 s12 M2 W −C′ 00,fac W W hh s2 12 M2 W ¯Λ2 + C′′ 00,fac W W hh s2 13 + s2 14 M2 W ¯Λ2 (47) where we only kept terms which grow with the energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' The superscripts (0) denote the W polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' At dimension-six, the amplitude should be unitary up to E2/¯Λ2 terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Thus, pieces with E2/M2 W should vanish, fixing, C00,fac W W hh = −C00 W W h 2 2 C′ 00,fac W W hh = C′′ 00,fac W W hh = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' (48) Therefore, at dimension-six, these coefficients are determined in terms of the three- point couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' At dim-8, this implies cancellations between terms in C00,fac W W hh and C′ 00,fac W W hh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' In the HEFT, one cannot take the high energy limit, since the cutoff cannot be much higher than the the electroweak scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Thus, the reasoning above only serves to differentiate between dimension-4, 6, etc contributions to the Wilson coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' In particular, since we can set ¯Λ = v, there are still independent dimension-six contact terms suppressed by v, such as C00,CT W W hh[12]⟨12⟩/v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' In contrast, in the SMEFT, we can really take the high-energy limit as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' (47), with MW ∼ v ≪ E ≪ Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Furthermore, we can alternatively obtain the relations in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' (48) by considering the high-energy limit of the amplitude with transverse W’s (which are finite in this limit), A(W +(+), W −(−), h, h) ∼ c00,fac W W hh [1k2q]⟨1q2k⟩ M2 W + c00 W W h 2 2M2 W [1k31q⟩[2q42k⟩ s13 + c00 W W h 2 2M2 W [1k41q⟩[2q32k⟩ s14 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' (49) The spinors 1q) and 2q) scale as the mass, and can be written as (see eg [45]) iq) = MW (ikξi) ξi) (50) where ξi) is an arbitrary constant spinor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' For the amplitude to be well-defined at high energies, it must be independent of the arbitrary spinors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Requiring that the amplitude in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' (49) is independent of these arbitrary spinors, one recovers the relation C00,fac W W hh = −C00 W W h 2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' (51) 28 Indeed, in the SMEFT, the high energy EFT has the full unbroken gauge symmetry, and the role of the arbitrary spinors ξi is clear—these are the reference spinors for the two massless vector polarizations [45] (see also [64]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Thus the equivalence of perturbative unitarity and the restoration of gauge symmetry is manifest in this example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='2 Four-point contact terms from on-shell Higgsing As we saw above, the independent couplings appearing in the amplitude are the three- point couplings and the Λ-suppressed contact terms (with no MW suppression).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' In the SMEFT, these can be determined by Higgsing the massless SU(3)×SU(2)×U(1)- symmetric amplitudes [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Two examples, namely WWhh and ¯udWh were worked out in detail in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' [45] up to dimension-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Here we briefly repeat the WWhh derivation for completeness, keeping only dimension-six terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' The results for all the four-points at dimension-six appear in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' The dimension-six contributions to the three points were derived in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' [31] by matching the amplitudes to the broken-phase SMEFT using [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' We do not repeat the derivation using on-shell Higgsing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Instead, we just show a few examples deriving the three-points from on-shell Higgsing in the next subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' The low-energy WWhh contact terms originate from a number of high-energy con- tact terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' The coefficient of each structure in a given helicity category is determined, to leading order in v/Λ, by the corresponding helicity amplitude in the high-energy theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' For instance, the massless WWH†H amplitudes give the leading-order con- tribution to the ++ and −− helicity categories, while (H†H)2 is the leading-order contribution to the 00 helicity category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Sub-leading v2/Λ2 contributions originate from higher-point contact terms such as (H†H)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Here, we just focus on their leading order pieces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' It is worth noting that the high-energy contact terms fully determine the coefficients of the low-energy massive contact terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Each massive contact term is of course a little-group tensor, and the derivation of its coefficient Clow essentially relies on the matching of the leading-energy component to the contact term of the corresponding high energy amplitude, with Wilson coefficient chigh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' The sub-leading components of the massive structure are generated by factorizable high-energy amplitude featuring the coupling chigh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Let us begin with the contribution from (H†H)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' The high-energy amplitude is A(HiH† kHjH† l ) ⊃ c+ HHHH s13 Λ2 T + ij kl + c− HHHH s12 − s14 Λ2 T − ij kl (52) where T ± ij kl = (δi kδj l ± δi lδj k)/2 are the symmetric and anti-symmetric SU(2) structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 29 Parameterizing the Higgs doublet as in 1, we find that A(G+G−hh) = 1 2 � A(H1H† 1H2H† 2) + A(H1H† 1H† 2H2) � = − c+ (H†H)2 − 3c− (H†H)2 2 s12 2Λ2 (53) which bolds into − c+ (H†H)2 − 3c− (H†H)2 2 s12 2Λ2 −→ c+ (H†H)2 − 3c− (H†H)2 2 [12]⟨12⟩ Λ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' (54) Thus there is a contact term in the massive EFT of the form, C00,CT W W hh [12]⟨12⟩ Λ2 = c+ (H†H)2 − 3c− (H†H)2 2 [12]⟨12⟩ Λ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' (55) We now turn to the contribution of the massless WWH†H contact terms, A(W a,±W b,±H† i Hj) = c±± W W HH (12)2 Λ2 � T ab�j i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' (56) Here ± are the helicites of the W’s, (12) = [12] for ++, (12) = ⟨12⟩ for −−, and � T ab�j i = δabδj i is required by the symmetry of the spinor structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' The kinematics bold trivially (12) → (12), and one finds the low-energy contact terms C±± W W hh (12) Λ2 = 2c±± W W HH (12) Λ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' (57) A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='3 The W W h coupling from on-shell Higgsing The WWh coupling C00 W W h in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' (44) is given by the gauge coupling to leading order, with a v2/Λ2 correction at dimension-six.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Here we show how both of these contributions can be obtained from on-shell Higgsing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' The relevant massless amplitudes to consider are WH†H or WWH†H to determine the dimension-4 part, and H2(H†)2WW to determine the dimension-6 shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Our discussion closely parallels the derivation of three-points from the massless amplitudes of a toy model with higgsed U(1) symmetry in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' [45], and we refer the reader to that paper for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' To determine C00 W W h one can start from either of its components, which map to different high-energy amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Apriori, the obvious amplitude to start from is the massless three point WH†H, which for positive W helicity is, A((W +)+G−h) = g √ 2 [12][13] [23] (58) 30 where we parametrized the Higgs doublet as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Bolding this amplitude is not entirely straightforward because it is non-local, and indeed, its non-locality translates in the massive amplitude to the 1/MW “pole”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' However, we can proceed by multiplying and dividing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' (58) by ⟨3ξ⟩ for some arbitrary ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' After some manipulations using momentum conservation, we get A((W +)+G−h) = g √ 2 [12]⟨ξ2⟩ ⟨1ξ⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' (59) Identifying 1q] = ξ], this maps to the longitudinal W component of C00 W W h ⟨12⟩[12] MW (60) with C00 W W h = g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Alternatively, a simpler way to get this coupling is to start from the 4-point amplitude H2(H†)2, which matches the all-longitudinal component of the spinor structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' At the renormalizable level, this amplitude, ASM((W −)−(W +)+hh) = −g2 2 ⟨132] ⟨231] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' (61) Identifying iq) = (MW/(ik3)) 3) for i = 1, 2 and taking p4 → 0 gives, Am SM(hW +W −) = g2v 2 [23]⟨23⟩ M2 W (62) where we also used ⟨ikiq⟩ = [iqik] = MW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Therefore, at the leading order, C00 W W h = g2v/MW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Note that the arbitray ξ spinor that we identified with the q spinors above naturally arises in this case from the soft higgs leg, with ˆ1q and ˆ2q along ˆ4 (see also Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' [64]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Turning to the dimension-six correction to C00 W W h, this originates, to leading or- der, from the 6-point H2(H†)2WW factorizable amplitude with a single insertion of a H2(H†)2 contact term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' The massless amplitude can be written as, Atot d=6((H0)4(W +)+(W −)−) = A(3)((H0)4(W +)+(W −)−) + A(4)((H0)4(W +)+(W −)−) , with A(3)((H0)4(W +)+(W −)−) = 8g2C 4 � [142⟩ ⟨12⟩s41 s63 + [162⟩ ⟨12⟩s61 s43 + [132⟩ ⟨12⟩s31 s64 � ⟨251] [21]s52 + (5 ↔ 4, 6, 3) , A(4)((H0)4(W +)+(W −)−) = 8g2C 4 ⟨25⟩[15] ⟨15⟩[25] + (5 ↔ 3, 4, 6) , (63) 31 4 5 6 3 1 2 1 2 3 4 6 5 W+ W− W+ W+ W− Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Feynman-diagram of H2(H†)2WW factorizable amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' and C = (c+ (HH†)2 − 3c− (HH†)2)/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' (64) The piece A(3)((H0)4(W +)+(W −)−) is the sum over Feynman diagrams in which the vector legs attach to different scalar legs (left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 1), and A(4)((H0)4(W +)+(W −)−) is the piece with the vector legs attached to the same scalar legs (see the right figure in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' (Of course, only the sum is gauge invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=') Taking the momenta of three H0 legs-4,5,6 to be soft, only A(4) ((H0)4(W +)+(W −)−) survives, lim q4,5,6→0 A(4) � H0(q4)H0(q5)H0(q6)H0(3)(W +)+(1)(W −)−(2) � = 6g2v3C [1kξ2k⟩ [2kξ1k⟩ , (65) where at the last stage we set qi ∝ ξ for some arbitrary ξ as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' This bolds to the same massive structure as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Altogether, after adding in the renormalizable contribution we have, Mm d=6(h(W +)+(W −)−) = g(1 + v2C)[12]⟨12⟩ MW .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' (66) Note that this correction is nothing but the correction to the Higgs wave-function renormalization induced by the four-Higgs contact term C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' One can derive the SMEFT corrections to the remaining three-point low-energy amplitudes along the same lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' However, since the most general 3-point couplings were listed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' [31] based on Feynman diagrams matching, we do not do so here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 32 B Energy-growth of factorizable amplitudes Massive amplitudes E growth M(Zhhh) no factorizable contribution M(V V hh) E2/M2 M(WWZh) E2/M2, E3/(M2¯Λ) M(ZZZh) no factorizable contribution M(WWWW/ZZWW) E3/(M2¯Λ), E4/M4, E4/(M2¯Λ2) M(ZZZZ) E2/M2, E2/(M ¯Λ) M(ffV h) E/M M(ffWW) E2/M2, E3/(M2¯Λ) M(ffZZ) E/M, E2/(M ¯Λ) Table 4: Leading energy growth in the massive factorizable amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' References [1] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Buchmuller and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Wyler, Effective Lagrangian Analysis of New Interactions and Flavor Conservation, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' B 268 (1986) 621.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' [2] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Jenkins and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Manohar, Algebraic Structure of Lepton and Quark Flavor Invariants and CP Violation, JHEP 10 (2009) 094, arXiv:0907.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='4763 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' [3] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Grzadkowski, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Iskrzynski, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Misiak, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Rosiek, Dimension-Six Terms in the Standard Model Lagrangian, JHEP 10 (2010) 085, arXiv:1008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='4884 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' [4] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Giudice, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Grojean, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Pomarol, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Rattazzi, The Strongly-Interacting Light Higgs, JHEP 06 (2007) 045, arXiv:hep-ph/0703164.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' [5] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Gupta, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Pomarol, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Riva, BSM Primary Effects, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' D 91 (2015) 035001, arXiv:1405.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='0181 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' [6] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Falkowski, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Fuks, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Mawatari, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Mimasu, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Riva, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Sanz, Rosetta: an operator basis translator for Standard Model effective field theory, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' C 75 (2015) 583, arXiv:1508.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='05895 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' [7] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Hays, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Martin, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Sanz, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Setford, On the impact of dimension-eight SMEFT operators on Higgs measurements, JHEP 02 (2019) 123, arXiv:1808.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='00442 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' [8] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Alioli, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Boughezal, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Mereghetti, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Petriello, Novel angular dependence in Drell-Yan lepton production via dimension-8 operators, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' B 809 (2020) 135703, arXiv:2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='11615 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 33 [9] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Dawson, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Homiller, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Sullivan, Impact of dimension-eight SMEFT contributions: A case study, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' D 104 (2021) 115013, arXiv:2110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='06929 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' [10] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Feruglio, The Chiral approach to the electroweak interactions, Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' A 8 (1993) 4937, arXiv:hep-ph/9301281.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' [11] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Bagger, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Barger, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='-m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Cheung, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Gunion, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Han, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Ladinsky, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Rosenfeld, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Yuan, The Strongly interacting W W system: Gold plated modes, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' D 49 (1994) 1246, arXiv:hep-ph/9306256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' [12] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Koulovassilopoulos and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Chivukula, The Phenomenology of a nonstandard Higgs boson in W(L) W(L) scattering, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' D 50 (1994) 3218, arXiv:hep-ph/9312317.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' [13] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Burgess, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Matias, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Pospelov, A Higgs or not a Higgs?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' What to do if you discover a new scalar particle, Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' A 17 (2002) 1841, arXiv:hep-ph/9912459.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' [14] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Grinstein and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Trott, A Higgs-Higgs bound state due to new physics at a TeV, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' D 76 (2007) 073002, arXiv:0704.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='1505 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' [15] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Alonso, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Gavela, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Merlo, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Rigolin, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Yepes, The Effective Chiral Lagrangian for a Light Dynamical ”Higgs Particle”, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' B 722 (2013) 330, [Erratum: Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='B 726, 926 (2013)], arXiv:1212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='3305 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' [16] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Espriu, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Mescia, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Yencho, Radiative corrections to WL WL scattering in composite Higgs models, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' D 88 (2013) 055002, arXiv:1307.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='2400 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' [17] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Buchalla, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Catà, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Krause, Complete Electroweak Chiral Lagrangian with a Light Higgs at NLO, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' B 880 (2014) 552, [Erratum: Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='B 913, 475–478 (2016)], arXiv:1307.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='5017 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' [18] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Brivio, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Corbett, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Éboli, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Gavela, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Gonzalez-Fraile, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Gonzalez-Garcia, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Merlo, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Rigolin, Disentangling a dynamical Higgs, JHEP 03 (2014) 024, arXiv:1311.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='1823 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' [19] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Alonso, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Jenkins, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Manohar, A Geometric Formulation of Higgs Effective Field Theory: Measuring the Curvature of Scalar Field Space, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' B 754 (2016) 335, arXiv:1511.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='00724 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' [20] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Alonso, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Jenkins, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Manohar, Geometry of the Scalar Sector, JHEP 08 (2016) 101, arXiv:1605.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='03602 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' [21] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Buchalla, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Cata, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Celis, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Knecht, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Krause, Complete One-Loop Renormalization of the Higgs-Electroweak Chiral Lagrangian, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' B 928 (2018) 93, arXiv:1710.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='06412 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' [22] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Alonso, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Kanshin, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Saa, Renormalization group evolution of Higgs effective field theory, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' D 97 (2018) 035010, arXiv:1710.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='06848 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 34 [23] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' de Blas, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Eberhardt, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Krause, Current and Future Constraints on Higgs Couplings in the Nonlinear Effective Theory, JHEP 07 (2018) 048, arXiv:1803.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='00939 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' [24] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Falkowski and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Rattazzi, Which EFT, JHEP 10 (2019) 255, arXiv:1902.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='05936 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' [25] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Cohen, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Craig, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Lu, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Sutherland, Is SMEFT Enough?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=', JHEP 03 (2021) 237, arXiv:2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='08597 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' [26] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Banta, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Cohen, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Craig, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Lu, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Sutherland, Non-decoupling new particles, JHEP 02 (2022) 029, arXiv:2110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='02967 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' [27] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Elvang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Freedman, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Kiermaier, A simple approach to counterterms in N=8 supergravity, JHEP 11 (2010) 016, arXiv:1003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='5018 [hep-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' [28] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Cheung, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Kampf, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Novotny, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Shen, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Trnka, A Periodic Table of Effective Field Theories, JHEP 02 (2017) 020, arXiv:1611.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='03137 [hep-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' [29] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Shadmi and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Weiss, Effective Field Theory Amplitudes the On-Shell Way: Scalar and Vector Couplings to Gluons, JHEP 02 (2019) 165, arXiv:1809.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='09644 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' [30] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Aoude and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Machado, The Rise of SMEFT On-shell Amplitudes, JHEP 12 (2019) 058, arXiv:1905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='11433 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' [31] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Durieux, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Kitahara, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Shadmi, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Weiss, The electroweak effective field theory from on-shell amplitudes, JHEP 01 (2020) 119, arXiv:1909.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='10551 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' [32] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Ma, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Shu, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Xiao, Standard Model Effective Field Theory from On-shell Amplitudes, Chin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' C 47 (2023) 023105, arXiv:1902.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='06752 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' [33] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Low and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Yin, Soft Bootstrap and Effective Field Theories, JHEP 11 (2019) 078, arXiv:1904.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='12859 [hep-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' [34] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Low and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Yin, New Flavor-Kinematics Dualities and Extensions of Nonlinear Sigma Models, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' B 807 (2020) 135544, arXiv:1911.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='08490 [hep-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' [35] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Falkowski, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Isabella, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Machado, On-shell effective theory for higher-spin dark matter, SciPost Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 10 (2021) 101, arXiv:2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='05339 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' [36] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Accettulli Huber and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' De Angelis, Standard Model EFTs via on-shell methods, JHEP 11 (2021) 221, arXiv:2108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='03669 [hep-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' [37] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Li, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Ren, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Shu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Xiao, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Yu, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Zheng, Complete set of dimension-eight operators in the standard model effective field theory, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' D 104 (2021) 015026, arXiv:2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='00008 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' [38] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Li, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Ren, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Xiao, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Yu, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Zheng, Complete set of dimension-nine operators in the standard model effective field theory, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' D 104 (2021) 015025, arXiv:2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='07899 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 35 [39] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Sun, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Xiao, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Yu, Complete NLO Operators in the Higgs Effective Field Theory, (2022), arXiv:2206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='07722 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' [40] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Low, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Shu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Xiao, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Zheng, Amplitude/operator basis in chiral perturbation theory, JHEP 01 (2023) 031, arXiv:2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='00198 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' [41] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Arkani-Hamed, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Huang, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='-t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Huang, Scattering amplitudes for all masses and spins, JHEP 11 (2021) 070, arXiv:1709.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='04891 [hep-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' [42] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Christensen and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Field, Constructive standard model, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' D 98 (2018) 016014, arXiv:1802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='00448 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' [43] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Bachu and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Yelleshpur, On-Shell Electroweak Sector and the Higgs Mechanism, JHEP 08 (2020) 039, arXiv:1912.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='04334 [hep-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' [44] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Christensen, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Field, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Moore, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Pinto, Two-, three-, and four-body decays in the constructive standard model, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' D 101 (2020) 065019, arXiv:1909.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='09164 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' [45] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Balkin, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Durieux, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Kitahara, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Shadmi, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Weiss, On-shell Higgsing for EFTs, JHEP 03 (2022) 129, arXiv:2112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='09688 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' [46] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' De Angelis, Amplitude bases in generic EFTs, JHEP 08 (2022) 299, arXiv:2202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='02681 [hep-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' [47] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Christensen, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Diaz-Quiroz, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Field, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Hayward, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Miles, Challenges with Internal Photons in Constructive QED, (2022), arXiv:2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='15018 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' [48] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Cheung and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Moss, Symmetry and Unification from Soft Theorems and Unitarity, JHEP 05 (2021) 161, arXiv:2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='13076 [hep-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' [49] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Liu and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Yin, Gauge invariance from on-shell massive amplitudes and tree-level unitarity, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' D 106 (2022) 076003, arXiv:2204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='13119 [hep-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' [50] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Cheung, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Helset, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Parra-Martinez, Geometric soft theorems, JHEP 04 (2022) 011, arXiv:2111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='03045 [hep-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' [51] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Durieux and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Machado, Enumerating higher-dimensional operators with on-shell amplitudes, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' D 101 (2020) 095021, arXiv:1912.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='08827 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' [52] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Henning and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Melia, Constructing effective field theories via their harmonics, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' D 100 (2019) 016015, arXiv:1902.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='06754 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' [53] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Durieux, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Kitahara, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Machado, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Shadmi, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Weiss, Constructing massive on-shell contact terms, JHEP 12 (2020) 175, arXiv:2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='09652 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' [54] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Dong, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Ma, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Shu, Constructing on-shell operator basis for all masses and spins, (2021), arXiv:2103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='15837 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' [55] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Dong, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Ma, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Shu, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Zheng, Constructing generic effective field theory for all masses and spins, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' D 106 (2022) 116010, arXiv:2202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='08350 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 36 [56] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Sun, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Xiao, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Yu, Complete NNLO Operator Bases in Higgs Effective Field Theory, (2022), arXiv:2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='14939 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' [57] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Dong, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Ma, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Shu, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='-Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Zhou, The New Formulation of Higgs Effective Field Theory, (2022), arXiv:2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='16515 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' [58] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Chang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Chen, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Liu, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Luty, Primary Observables for Indirect Searches at Colliders, (2022), arXiv:2212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='06215 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' [59] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Lehman and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Martin, Low-derivative operators of the Standard Model effective field theory via Hilbert series methods, JHEP 02 (2016) 081, arXiv:1510.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='00372 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' [60] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Henning, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Lu, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Melia, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Murayama, 2, 84, 30, 993, 560, 15456, 11962, 261485, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=': Higher dimension operators in the SM EFT, JHEP 08 (2017) 016, [Erratum: JHEP 09, 019 (2019)], arXiv:1512.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='03433 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' [61] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Gráf, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Henning, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Lu, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Melia, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Murayama, Hilbert Series, the Higgs Mechanism, and HEFT, (2022), arXiv:2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='06275 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' [62] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Berends, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Kleiss, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' De Causmaecker, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Gastmans, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Wu, Single Bremsstrahlung Processes in Gauge Theories, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' B 103 (1981) 124.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' [63] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Dedes, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Materkowska, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Paraskevas, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Rosiek, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Suxho, Feynman rules for the Standard Model Effective Field Theory in Rξ -gauges, JHEP 06 (2017) 143, arXiv:1704.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='03888 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' [64] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Craig, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Elvang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Kiermaier, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' Slatyer, Massive amplitudes on the Coulomb branch of N=4 SYM, JHEP 12 (2011) 097, arXiv:1104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content='2050 [hep-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} +page_content=' 37' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFIT4oBgHgl3EQfvyuP/content/2301.11349v1.pdf'} diff --git a/qdE3T4oBgHgl3EQfMAmP/content/2301.04369v1.pdf b/qdE3T4oBgHgl3EQfMAmP/content/2301.04369v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..ad1158b416cd8a28060c842abafd7bdb624841e1 --- /dev/null +++ b/qdE3T4oBgHgl3EQfMAmP/content/2301.04369v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:f6338cab55917b729ded9af1777376335d4c644c3577f2b03332fb4c1c3ba793 +size 126069 diff --git a/qdE3T4oBgHgl3EQfMAmP/vector_store/index.faiss b/qdE3T4oBgHgl3EQfMAmP/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..0ab964f1fa3d04061cb8234f210af34b82aa3984 --- /dev/null +++ b/qdE3T4oBgHgl3EQfMAmP/vector_store/index.faiss @@ -0,0 +1,3 @@ +version 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a/r9FKT4oBgHgl3EQfJS1Z/content/tmp_files/2301.11737v1.pdf.txt b/r9FKT4oBgHgl3EQfJS1Z/content/tmp_files/2301.11737v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..d176757d640ef9e222cf5fbc6d7219e3489da388 --- /dev/null +++ b/r9FKT4oBgHgl3EQfJS1Z/content/tmp_files/2301.11737v1.pdf.txt @@ -0,0 +1,741 @@ +Modeling human road crossing decisions as reward +maximization with visual perception limitations +Yueyang Wang1⋆, Aravinda Ramakrishnan Srinivasan1, Jussi P.P. Jokinen2, Antti Oulasvirta3, Gustav Markkula1 +1 University of Leeds, UK 2 University of Jyv¨askyl¨a, Finland 3 Aalto University, Finland +Abstract—Understanding the interaction between different +road users is critical for road safety and automated vehicles (AVs). +Existing mathematical models on this topic have been proposed +based mostly on either cognitive or machine learning (ML) +approaches. However, current cognitive models are incapable +of simulating road user trajectories in general scenarios, and +ML models lack a focus on the mechanisms generating the +behavior and take a high-level perspective which can cause +failures to capture important human-like behaviors. Here, we +develop a model of human pedestrian crossing decisions based on +computational rationality, an approach using deep reinforcement +learning (RL) to learn boundedly optimal behavior policies given +human constraints, in our case a model of the limited human +visual system. We show that the proposed combined cognitive- +RL model captures human-like patterns of gap acceptance and +crossing initiation time. Interestingly, our model’s decisions are +sensitive to not only the time gap, but also the speed of the +approaching vehicle, something which has been described as +a “bias” in human gap acceptance behavior. However, our +results suggest that this is instead a rational adaption to human +perceptual limitations. Moreover, we demonstrate an approach to +accounting for individual differences in computational rationality +models, by conditioning the RL policy on the parameters of the +human constraints. Our results demonstrate the feasibility of +generating more human-like road user behavior by combining +RL with cognitive models. +Index Terms—Human behavior, computational rationality, +noisy perception, reinforcement learning +I. INTRODUCTION +The interaction between road users is defined as “a situ- +ation where the behavior of at least two road users can be +interpreted as being influenced by the possibility that they +are both intending to occupy the same region of space at the +same time in the near future” [1]. Interdependence between +vehicles and pedestrians makes interactions between road users +instrumental for road safety and automated vehicles(AVs), +which pushes the research into road user interaction. +Pedestrian is the most vulnerable group among all road +users [2], and their behavior is difficult to predict. How drivers +may behave is limited by the machinery of a vehicle and +traffic rules, whereas pedestrians have more freedom, and are +limited only by traffic rules. To better understand pedestrian +behavior, some detailed metrics related to crossing behavior +were investigated [3]–[5]. For example, gap acceptance, where +the gap is defined as the time or spatial distance between +∗ Corresponding author: mn20yw2@leeds.ac.uk +This project has received funding from UK Engineering and Physical +Sciences Research Council under fellowship COMMOTIONS - Computa- +tional Models of Traffic Interactions for Testing of Automated Vehicles - +EP/S005056/1. +the pedestrian and approaching vehicle, is an important met- +ric for understanding the crossing decision [3]. Lobjois and +Cavallo [4] found that speed-dependent gap acceptance was +shown in different age groups i.e., the gap acceptance rate +was higher when the approaching vehicle was faster in a +given time gap. Petzoldt [5] investigated the relationship +between gap acceptance and time to arrival (TTA) estimation. +They speculated that speed-dependent crossing decisions were +caused by the biased TTA estimation. Another important factor +affecting the safety of crossing is cross initiation time (CIT). +Tian et al. [3] observed that CIT was greater when the vehicle +was driven at a higher speed for any given initial TTA, which +led to unsafe behavior. +A descriptive study of road user behavior is insufficient +for AVs to understand and predict other road users’ actions; +therefore, mathematical models of road user behavior are +required for AVs [6]. In recent years, many mechanistic models +have been proposed to generate and understand pedestrian +behavior. For example, rule-based models, such as the social +force model, were successful in traffic flow simulation [7], +[8]. However, they are limited in capturing the details of +road user interactions. To generate more explainable and +accurate road user interactive behavior, cognitive models, such +as the evidence accumulation model, were utilized to model +pedestrian crossing decisions [9], [10]. +With increased computing power and more available data, +machine learning (ML) models have gained increasing at- +tention for road user behavior prediction. Long Short-Term +Memory (LSTM), a variant of Recurrent Neural Networks +(RNN), was used in pedestrian trajectory prediction [11]. To +better predict the interactive behavior between pedestrians, +Alahi et al proposed a ‘Social LSTM’, with a social layer +into the LSTM algorithm, and the model outperformed state- +of-the-art methods [12]. +Many efforts have been made to understand and simulate +the microscopic behavior of road users. ML models can +reproduce accurate trajectories across a diverse range of sce- +narios, and cognitive models can provide the interpretability +of the interactive behavior and the underlying mechanism. +However, both streams of methods have some limitations. In +conventional cognitive models, the modeler should define how +the specific task is completed and the model should be updated +if the environment and task change. This makes it difficult to +simulate road users’ trajectories in general scenarios. Whereas +ML models focus on the minimization of high-level error +metrics, rather than the mechanisms generating the behavior or +arXiv:2301.11737v1 [cs.LG] 27 Jan 2023 + +whether the aspects of behavior that are important to humans +are being captured. Sometimes the model with high accuracy +does not necessarily generate realistic human behavior [13]. +Computational rationality, as a general approach of model- +ing human behavior, has shown promising properties in mod- +eling human-computer interaction (HCI) [14]. This framework +is based on the idea that human behaviors are generated +by cognitive mechanisms that are adapted to the structure +both of the environment and the mind and brain itself [15]. +In this paper, we developed a model of human pedestrian +crossing decisions based on computational rationality, using +deep reinforcement learning (RL) to adopt optimal behavior +policies given human-like constraints. We show that when +we constrain the agent by a simple model of human visual +perception, it reproduces human gap acceptance behavior +qualitatively, including the speed-dependencies, which have +not been previously considered as rational behavior. We also +demonstrate an approach to using computational rationality +to model individual differences, by conditioning RL on the +parameters of human constraints. +II. METHOD +A. Dataset +The dataset used for validation was collected in the previous +experiment reported by Giles et al. [10]. Fig. 1 shows a +birds-eye view of the experiment. A brief summary of the +experimental setup is provided below. 20 participants were +recruited for the experiment. In the experiment, they wore +an HTC Vive Virtual Reality (VR) headset and experienced +the virtual crossing task. The VR environment consisted of a +straight two-lane road with a total width of 5.85 m, with a +zebra crossing at the participant’s location. +In terms of the experimental procedure, participants stood +in front of the zebra crossing. When participants were ready +to start the trial, they turned their head to the right to trigger +the scene. A car at the predefined initial position d0 would +approach the participant at different speeds v0. The experiment +included a mix of scenarios; in this paper, we will only +consider the scenarios where the speed of the approaching +vehicle was constant. The detail of these scenarios is shown +in TABLE I, where also the initial time to arrival (TTA) +τ0 = d0/v0 is listed. Participants pressed the button on the +HTC Vive’s controller when they felt safe to cross. Upon +this button press, CIT was recorded, and the location of the +participant in the virtual environment moved across the zebra +crossing at the speed of 1.31 m/s. This button-press approach +was chosen in favor of physically crossing the road, to reduce +the impact of variability in motor constraints on the crossing +decision. Each participant experienced 6 different constant- +speed trials. Therefore, 120 data trials were used for the +validation of the model. +B. Model +This research aims to model the pedestrian crossing de- +cision under the assumption that human behaves rationally +within limits. Therefore, two models were compared, as shown +Fig. 1. Birds-eye view of the experiment. +in Fig. 2. One is the ideal observer crossing model, in which +the agent has perfect information about the environment. +Another is the model considering the visual limits. In this +model, the agent perceives the environment subject to noise, +but Bayes-optimal perception. +1) Noisy perception: We assume the agent has a noisy +perception of the state of vehicles and perfect knowledge about +their own state. +a) Noisy visual input: The observation obtained by the +agent is according to the principle of the human visual system, +i.e., the sensory input received by our human visual system +is noisy [16]. It is important to consider the nature of this +noise; here we are building on models which assume that +visual noise is introduced at the level of the human retina, +as angular noise [17]. In the current model, we assume that +the agent observes the position of the other agent along its +line of travel by observing the angle below the horizon of +the other agent [18], [19], with a constant Gaussian noise +of standard deviation σv. In practice, this means that the +pedestrian observes the position of the vehicle with a distance- +dependent noise of standard deviation σx(k) = fv[x(k)], +where x(k) is the true world state, and: +fv[x(k)] = |dl| +� +1 − +h +d · tan(arctan h +d + σv) +� +, +where dl is the longitudinal distance between the pedestrian +agent and the crossing point, d is the distance between the +agent and the approaching vehicle, h is the eye height over +the ground of the ego agent, which is set to 1.6 m for all +pedestrian agents for simplicity, and σv could vary between +pedestrians. +b) Kalman filter: There is psychophysical evidence that +human perception system works like a Bayesian optimizer, +and Bayesian methods have been successful in modeling +perception and sensorimotor control [17], [20]. Therefore, +we used a Kalman Filter as a model of the human visual +perception to percept the environment [19]. In our model, +we initialized the Kalman filter with a noisy position of the +vehicle, and a noisy velocity centered at the true velocity with +a standard deviation of all velocity values. At each step, the +TABLE I +VEHICLE APPROACH SCENARIOS +v0 m/s +d0 (m) +τ0 (s) +6.94 +15.90 +2.29 +13.89 +31.81 +2.29 +6.94 +31.81 +4.58 +13.89 +63.61 +4.58 +6.94 +47.71 +6.87 +13.89 +95.42 +6.87 + +Speed +Vehicle(a) +(b) +Fig. 2. +Comparison of models. (a) Ideal observer model, where the agent +had full information about the environment. (b) Model with visual limitations. +Two variants were developed. One was the model without noise magnitude +parameter. Another was the model with noise magnitude parameter, as shown +in the orange dashed box. +Kalman filter received the noisy position about the other agent, +and the output, i.e., the filtered position and velocity of the +vehicle, and the variance of the position and the velocity, was +the input of the RL agent. +2) Reinforcement learning model: In our model, in line +with the theory of computational rationality, we view pedes- +trian behavior as a Partially Observable Markov Decision +Process (POMDP) under bounds posed by perception. RL +algorithm, where the agent interacts with the environment +and learns the optimal strategy by trial and error, can be +used to derive the boundedly optimal policy for this type of +problem [14], [21]. +a) State space S: The time step in the simulation affects +the resolution of the results of the decision time. In our mode, +one time step corresponds to 0.1 seconds, which is suitable for +the dataset we are using. At each time step t, the environment +is in a state st ∈ S. A state contains true information about +the vehicle and the agent, i.e., the position and velocity of the +vehicle and the agent. +b) Action space A: At each time step t, the agent takes +an action at ∈ A. In this paper, in line with the button press in +the experiment, the agent can make the decision to Go or Not +Go. If the Go decision is made, the agent will go straight at +the speed of 1.31 m/s, as in the experiment, resulting either +in a successful crossing or in a collision, and the scenario will +finish. +c) Reward R: In the experiment where the datasets were +collected, the participant’s task was to cross the road as soon as +they felt safe to do so, either before or after the car had passed +them [22]. Therefore, in our model we want the agent to cross +the road in as short a time as possible without a collision. At +each time step t, the agent will receive a negative reward of +0.5×simulationsteps, which helps the agent to cross the road +faster. The agent will be given a reward of 200 when crossing +the road without collision, and a reward of −200 if a collision +happens. The form of this reward function was chosen based +on some initial testing. As the focus of this study is on the +potential effect of noisy perception on the crossing decision, +we kept the reward function simple; future work can further +refine it to better capture human preferences. +d) Observation space O: The agent receives observation +ot ∈ O at each time step. In the ideal observer model, the +agent gets the complete information about the environment. In +the model with visual limits, the agent only observes partial +information about the state of the environment. At each time +step, the agent receives the processed estimates of the position +and velocity and uncertainty about the position and velocity of +the other agent from the Kalman Filter and the exact position +and velocity of the ego agent. +e) Transition function T: The transition function defines +how the current state st changes to the next state st+1 taking +action at. In our model, if Not Go action is chosen, the vehicle +will move according to the kinematic equations with the given +speed, and the position of the agent will not change. Once +Go action is chosen, whether the collision happens will be +calculated, and the corresponding reward will be given to the +agent. Then, the simulation finishes. +f) Deep Q-Networks: +Deep-Q Network (DQN) is a +method using the neural network to learn the optimal policy to +maximize the state-action function (Q function), Q(s, a), the +expected rewards for an action taken in a given state [23]. +DQN is suitable for the problem with a continuous state +space and a discrete action space. For the extensibility of the +model to more complex situations, we utilized an enhanced +version of DQN, Double DQN (DDQN). The Double DQN +(DDQN) structure, which decouples the update of the neural +network parameter for action selection and evaluation, can +avoid the overestimation of the action value [23]. Furthermore, +the dueling network was used, in which the Q-function is +decoupled to a value function and a state-dependent action +advantage A(s, a) function. Compared with the single-stream +DQN, the Duelling DQN shows better performance especially +when different actions lead to a similar value because of the +consideration of the state value in the Q value [24]. We trained +the agent through a two-layer fully connected network, with +512 and 256 nodes. The learning rate and discount factor are +0.001 and 0.99 respectively. To explore the optimal policy, an +ϵ - greedy algorithm was used for exploration: At each time +step t, a random action is chosen with probability ϵ, and the +action with maximum Q value is chosen with probability 1−ϵ. +We decreased ϵ by 10−4 in each learning step. The minimum +value was set to 0.001. +C. Training and fitting +As we don’t know the correct value for σv, and additionally +each participant in the experiment may have an individual σv, +we trained the model for different σv ranging from 0−1 with +a step size of 0.002. We trained the model in two different +ways. First, we trained a separate model for each σv. With this +approach, for any given σv, we can get a boundedly optimal + +External environment +AgentExternal environment +Bayesian +perceptual +filtering +Agent. +Noisepolicy from the model trained with that σv. This approach +is somewhat inconvenient and will not scale well to more +complete models of human constraints, with more parameters +than just one. Therefore, as an alternative approach, we instead +trained just a single model, across all of the different σv, and +in this case we also provided σv as an input to the model, as +illustrated by the dashed line in Fig. 2. In other words, we +conditioned the RL on σv. With this approach, for any given +σv, we get a boundedly optimal policy from this single model, +by also giving it σv as an input. +With the 1.31 m/s walking speed used in the experiment, it +is in fact possible to cross before the vehicle without collision +even for the lowest TTA of 2.29 s. To avoid our agents learning +this trivial policy of always just immediately crossing, in the +model training we also included a lower TTA of 1 s, in which +the agent did not have enough time to cross safely before the +vehicle. +We used three criteria to identify model convergence and +stop training:(1) the collision rate is less than or equal to 0.01, +which means one collision at most happened in the last 100 +episodes; (2) ϵ has reached the lowest value of the epsilon- +greedy algorithm; (3) the difference in average reward between +the last 100 episodes and the last 200 to 100 episodes is less +than 1 to make sure that the average reward does not improve +much and becomes stable and converged. +We also identified the σv that fitted the experimental data +best for each participant by likelihood maximization. We esti- +mated the probability density function (PDF) of CIT predicted +for each model by kernel density estimation, separately for +each of the six scenarios (Table 1). This allowed us to calculate +the model likelihood of each σv for each participant, by +multiplying the model PDF values at the participant’s observed +button press times. Finally, we combined all the PDFs from the +different participants, with tuned σv. We applied this method +to both model types, i.e., separate models for each σv, and a +single model for all σv. In addition to this per-participant fit, +we also selected one σv from the models with separate σv, +which best fitted the entire dataset. +Abbreviations will be introduced here for referring to dif- +ferent model variants; e.g., EXP: experimental data; IO: ideal +observer model; LMD: multiples model with separate visual +limits parameter and one σv fitted across the entire dataset; +LMP: multiple models with separate visual limits parameter +and different σv fitted per participant; LSP: one model with +different visual limits parameters and different σv fitted per +participant. +III. RESULTS +A. Experimental results +First, we reanalyzed the experimental data reported by +Pekkanen et al. [22]. As shown in the first panel on the left +in Fig. 3, this experiment replicated the general finding that +the gap acceptance rate strongly depended on initial TTA, +i.e., more pedestrians crossed before the car when there was +a larger time gap. In addition, we can observe the speed- +dependent gap acceptance rate, i.e., for a given initial TTA, +more pedestrians accepted the gap if the speed was higher [3], +[4]. This effect was strongest at the initial TTA of 4.6 s. +Regarding CIT, which is shown in the first row in Fig. 4, +there was some spread in CIT, both when crossing before and +after the car. +B. Behavior of the model +The results of the model without visual limits (IO) are +shown in the second panel on the left in Fig. 3 and the second +row in Fig. 4. In the model without visual limits, the agent had +full information about the environment, and crossed the road +without collision in the shortest time. Therefore, as shown in +the second row of Fig. 4, the agent always started to cross at +the first time step when it was safe. The second panel in Fig. 3 +shows that the agent always accepted the gap in 2.3 s, 4.6 s, +and 6.9 s initial TTA conditions. +The three graphs on the right in Fig. 3, and the last three +rows in Fig. 4 show the results of the models with visual +limits. In models with visual limits, the agent received the +processed information of the Kalman filter instead of the exact +information about the environment. +As shown in Fig. 3, the models with visual limits captured +the pattern of the gap acceptance rate observed in the exper- +iment. The TTA-dependent gap acceptance rate, i.e., the gap +acceptance rate increased with the initial TTA, was predicted +by all three model types. Unexpectedly, the difference within +the same TTA group was also captured by the model. The +agent was more likely to accept the gap when the vehicle speed +was higher in the given initial TTA. This pattern was shown +by all model types. Overall, the model has a slightly greater +tendency to accept gaps than the human participants. This may +be what is causing the model to show a speed dependency +at 2.3 s TTA (because the model sometimes crosses there, +whereas the humans almost never did), but not at 6.9 s TTA +Fig. 3. Gap acceptance rate by human participants and different models. See the main text for explanations of the model abbreviations. + +EXP +10 +LMD +LMP +LSP +1.0 +0.6 +13.9 m/s +6.9 m/s +0.0 - +1.0 +2.3 +4.6 +1.0 +2.3 +4.6 +1.0 +2.3 +4.6 +1.0 +2.3 +4.6 +6.9 +1.0 +2.3 +4.6 +6.9 +6.9 +6.9 +6.9 +Initial TTA (s) +Initial TTA (s) +Initial TTA (s) +Initial TTA (s) +Initial TTA (s)Fig. 4. Cumulative probability for CIT. The four columns show different initial TTA conditions. See the main text for explanations of the model abbreviations. +(Note: CDF curves are extended to the right after after the cumulative probability reaches y = 1.) +(because the model has already reached full gap acceptance at +this TTA, which the humans had not). +1) Variability within and between individuals: From Fig. 4, +there was some spread in CIT also for model LMD – i.e., the +model showed some within-individual variability – due to the +trial-to-trial variability in visual noise. That model LMP, with +different σv for different participants, came closer to capture +the human variability, suggesting that some of the variability +in the human data is from between-individual differences. +To test this, we did the Akaike Information Criterion (AIC) +analysis of LMD, LMP, and LSP. AIC, which considers both +the log-likelihood and the cost of more parameters, allows +us to compare the performance of different models, and the +preferred model is the one with the minimum AIC value [25]. +The AIC values are 113, 57, and 79 for the LMD, LMP and +LSP respectively. Therefore, LMP, the model with different +individual σv, outperformed other model variants. This sug- +gested that some of the variability in human CIT is due to +between-individual variability in sensory noise. +Fig. 5. Distributions of initial estimated TTA from the output of the visual +perception model, across the different scenarios in the experiment. +2) Speed-dependent gap acceptance: To investigate the +reason why the model showed speed-dependence in its gap +acceptance rate, we calculated the estimated TTA through +the velocity and position estimated by Kalman filter at the +first time step. Fig. 5 shows the distribution, the mean value, +which is shown in the grey vertical line, and the 5th and +95th percentiles of the estimated TTA, which is shown in the +shading area. As shown in Fig. 5, for the same time gap, +the distribution for the estimated TTA was more dispersed at +lower speed conditions. In other words, at low speeds, there +was greater uncertainty about the estimated TTA, which could +be the reason why it is better, from a reward maximization +perspective, to be more careful about crossing in these sit- +uations. This perspective aligns with the findings of Chen +et al. [26], who showed that apparently biased behavior in +a more abstract choice task might also be explained as a +consequence of optimal sequential decisions under uncertainty. +Interestingly, Tian et al. [3] showed that in the road-crossing +context, humans may in practice be achieving this strategy by +making use of relatively simple visual cues. +IV. DISCUSSIONS AND CONCLUSION +We developed a model of human pedestrian crossing de- +cisions based on computational rationality, using deep RL to +adopt optimal behavior policies given human-like constraints. +We show that when we constrain the agent by a simple +model of human visual perception, it reproduces human gap +acceptance behavior qualitatively. Furthermore, the model also +predicts the speed-dependencies that are typically observed in +human gap acceptance. These have previously been considered +as evidence of biases in human pedestrian decision-making, +but our results demonstrate that this type of speed-dependence +is a rational adaption to noisy visual perception. When com- + +Initial TTA: 2.3 s +Initial TTA: 4.6 s +Initial TTA: 6.9 s +1.0 +Prob +6.9 m/s +13.9 m/s +0.0 +Initial TTA: 1 s +1.0 +Prob +6.9 m/s +13.9 m/s +0.0 +1.0 - +Prob +LMD +6.9 m/s +13.9 m/s +0.0 +1.0 +Prob +LMP +6.9 m/s +13.9 m/s +0.0 +1.0 +Prob +6.9 m/s +13.9 m/s +0.0 +2 +2 +2 +6 +8 +10 0 +4 +6 +8 +10 0 +2 +4 +4 +6 +8 +100 +4 +6 +8 +10 +Time (s) +Time (s) +Time (s) +Time (s)Velocity = 6.94 m/s Gap = 2.3 s +Velocity = 6.94 m/s Gap = 4.6 s +Velocity = 6.94 m/s Gap = 6.9 s +0.20 +0.15 +0.10 +0.05 +0.00 +Velocity = 13.89 m/s Gap = 2.3 s +Velocity = 13.89 m/s Gap = 4.6 s +Velocity = 13.89 'm/s Gap ='6.9 s +0.3 +0.2 +0.1 +0.0 +0.0 +2.5 +5.0 +7.5 +10.0 +12.5 +15.00 +5 +10 +15 +20 0 +5 +10 +15 +20 +25 +30 +Estimated TTA (s) +Estimated TTA (s) +Estimated TTA (s)paring the results of LMP and LSP, it shows that these two +approaches learned similar policies, but not identical policies. +We believe that the approach of conditioning RL on constraint +parameters is a promising approach for considering individual +differences. As an early attempt in using computational ratio- +nality in modeling road user interaction, we see the feasibility +to use computational rationality to model road user behavior. +The developed model could be served as the agent model in +the test environment of AVs. +There is ample scope for further improvements to our +model. For example, our agent is more likely to show faster +responses than humans. One reason is that the human percep- +tual filtering might be slower than the Kalman filter we used, +which can process the information without delay. Another +reason is that we have not considered the non-decision time +in the model. We can also observe more variability in human +CITs than in model CITs. This could be due to both between- +and within-individual variability of unmodeled sources. In +addition, our reward function is simple and so far we have not +tuned it. This is enough for our purposes, to show qualitative +patterns of human road-crossing. However, this limitation is +probably the reason why the model has a greater tendency +for gap acceptance than humans. In future work, following a +similar approach to what we did with σv here, we could tune +also for example the time-loss penalty in the reward function. +Furthermore, in the current work, the agent is interacting +with a constant speed approaching vehicle. However, in the +real world, the pedestrian will interact with vehicles with +various kinematic states, which also affect the crossing be- +havior. For example, [22] suggests that human pedestrians are +also perceiving and interpreting vehicle deceleration. A major +advantage of the approach we have taken here is that deep +RL is scalable to much more complex traffic scenarios., far +beyond what is possible with conventional cognitive models. +We therefore conclude that computational rationality overall +holds great promise for applied modeling of human-like road +user interaction behavior. +REFERENCES +[1] G. Markkula, R. Madigan, D. Nathanael, E. Portouli, Y. M. Lee, A. Diet- +rich, J. Billington, A. Schieben, and N. Merat, “Defining interactions: A +conceptual framework for understanding interactive behaviour in human +and automated road traffic,” Theoretical Issues in Ergonomics Science, +vol. 21, no. 6, pp. 728–752, 2020. +[2] “Reported +road +casualties +great +britain, +annual +report: +2021 +- +gov.uk,” +https://www.gov.uk/government/statistics/ +reported-road-casualties-great-britain-annual-report-2021, +accessed: +2023-01-24. +[3] K. Tian, G. Markkula, C. Wei, Y. M. Lee, R. Madigan, N. Merat, and +R. Romano, “Explaining unsafe pedestrian road crossing behaviours +using a psychophysics-based gap acceptance model,” Safety Science, +vol. 154, p. 105837, 2022. +[4] R. Lobjois and V. Cavallo, “Age-related differences in street-crossing +decisions: The effects of vehicle speed and time constraints on gap +selection in an estimation task,” Accident analysis & prevention, vol. 39, +no. 5, pp. 934–943, 2007. +[5] T. Petzoldt, “On the relationship between pedestrian gap acceptance and +time to arrival estimates,” Accident Analysis & Prevention, vol. 72, pp. +127–133, 2014. +[6] F. Camara, N. Bellotto, S. Cosar, F. Weber, D. Nathanael, M. Althoff, +J. Wu, J. Ruenz, A. Dietrich, G. Markkula et al., “Pedestrian models +for autonomous driving part ii: high-level models of human behavior,” +IEEE Transactions on Intelligent Transportation Systems, vol. 22, no. 9, +pp. 5453–5472, 2020. +[7] D. Helbing and P. Molnar, “Social force model for pedestrian dynamics,” +Physical review E, vol. 51, no. 5, p. 4282, 1995. +[8] D. Yang, ¨U. ¨Ozg¨uner, and K. Redmill, “A social force based pedestrian +motion model considering multi-pedestrian interaction with a vehicle,” +ACM Transactions on Spatial Algorithms and Systems (TSAS), vol. 6, +no. 2, pp. 1–27, 2020. +[9] G. Markkula, R. Romano, R. Madigan, C. W. Fox, O. T. Giles, and +N. Merat, “Models of human decision-making as tools for estimating +and optimizing impacts of vehicle automation,” Transportation research +record, vol. 2672, no. 37, pp. 153–163, 2018. +[10] O. Giles, G. Markkula, J. Pekkanen, N. Yokota, N. Matsunaga, N. Merat, +and T. Daimon, “At the zebra crossing: Modelling complex decision +processes with variable-drift diffusion models,” in Proceedings of the +41st annual meeting of the cognitive science society. Cognitive Science +Society, 2019, pp. 366–372. +[11] Y. Wang, L. Jiang, M.-H. Yang, L.-J. Li, M. Long, and L. Fei- +Fei, “Eidetic 3d lstm: A model for video prediction and beyond,” in +International conference on learning representations, 2018. +[12] A. Alahi, K. Goel, V. Ramanathan, A. Robicquet, L. Fei-Fei, and +S. Savarese, “Social lstm: Human trajectory prediction in crowded +spaces,” in Proceedings of the IEEE conference on computer vision and +pattern recognition, 2016, pp. 961–971. +[13] A. R. Srinivasan, Y.-S. Lin, M. Antonello, A. Knittel, M. Hasan, +M. Hawasly, J. Redford, S. Ramamoorthy, M. Leonetti, J. Billington +et al., “Beyond RMSE: Do machine-learned models of road user inter- +action produce human-like behavior?” arXiv preprint arXiv:2206.11110, +2022. +[14] A. Oulasvirta, J. P. Jokinen, and A. Howes, “Computational rationality +as a theory of interaction,” in CHI Conference on Human Factors in +Computing Systems, 2022, pp. 1–14. +[15] R. L. Lewis, A. Howes, and S. Singh, “Computational rationality: Link- +ing mechanism and behavior through bounded utility maximization,” +Topics in cognitive science, vol. 6, no. 2, pp. 279–311, 2014. +[16] A. A. Faisal, L. P. Selen, and D. M. Wolpert, “Noise in the nervous +system,” Nature reviews neuroscience, vol. 9, no. 4, pp. 292–303, 2008. +[17] O.-S. Kwon, D. Tadin, and D. C. Knill, “Unifying account of visual +motion and position perception,” Proceedings of the National Academy +of Sciences, vol. 112, no. 26, pp. 8142–8147, 2015. +[18] T. L. Ooi, B. Wu, and Z. J. He, “Distance determined by the angular +declination below the horizon,” Nature, vol. 414, no. 6860, pp. 197–200, +2001. +[19] G. Markkula, Y.-S. Lin, A. R. Srinivasan, J. Billington, M. Leonetti, +A. H. Kalantari, Y. Yang, Y. M. Lee, R. Madigan, and N. Merat, “Ex- +plaining human interactions on the road requires large-scale integration +of psychological theory,” 2022. +[20] D. C. Knill and A. Pouget, “The bayesian brain: the role of uncertainty +in neural coding and computation,” TRENDS in Neurosciences, vol. 27, +no. 12, pp. 712–719, 2004. +[21] V. Franc¸ois-Lavet, P. Henderson, R. Islam, M. G. Bellemare, J. Pineau +et al., “An introduction to deep reinforcement learning,” Foundations +and Trends® in Machine Learning, vol. 11, no. 3-4, pp. 219–354, 2018. +[22] J. Pekkanen, O. T. Giles, Y. M. Lee, R. Madigan, T. Daimon, N. Merat, +and G. Markkula, “Variable-drift diffusion models of pedestrian road- +crossing decisions,” Computational Brain & Behavior, vol. 5, no. 1, pp. +60–80, 2022. +[23] H. Van Hasselt, A. Guez, and D. Silver, “Deep reinforcement learning +with double q-learning,” in Proceedings of the AAAI conference on +artificial intelligence, vol. 30, no. 1, 2016. +[24] Z. Wang, T. Schaul, M. Hessel, H. Hasselt, M. Lanctot, and N. Freitas, +“Dueling network architectures for deep reinforcement learning,” in +International conference on machine learning. PMLR, 2016, pp. 1995– +2003. +[25] P. A. Murtaugh, “In defense of p values,” Ecology, vol. 95, no. 3, pp. +611–617, 2014. +[26] H. Chen, H. J. Chang, and A. Howes, “Apparently irrational choice +as optimal sequential decision making,” in Proceedings of the AAAI +Conference on Artificial Intelligence, vol. 35, no. 1, 2021, pp. 792–800. + diff --git a/r9FKT4oBgHgl3EQfJS1Z/content/tmp_files/load_file.txt b/r9FKT4oBgHgl3EQfJS1Z/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..67cf93c391ed90fb609a6067db3968c3d256384a --- /dev/null +++ b/r9FKT4oBgHgl3EQfJS1Z/content/tmp_files/load_file.txt @@ -0,0 +1,622 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf,len=621 +page_content='Modeling human road crossing decisions as reward maximization with visual perception limitations Yueyang Wang1⋆, Aravinda Ramakrishnan Srinivasan1, Jussi P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Jokinen2, Antti Oulasvirta3, Gustav Markkula1 1 University of Leeds, UK 2 University of Jyv¨askyl¨a, Finland 3 Aalto University, Finland Abstract—Understanding the interaction between different road users is critical for road safety and automated vehicles (AVs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Existing mathematical models on this topic have been proposed based mostly on either cognitive or machine learning (ML) approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' However, current cognitive models are incapable of simulating road user trajectories in general scenarios, and ML models lack a focus on the mechanisms generating the behavior and take a high-level perspective which can cause failures to capture important human-like behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Here, we develop a model of human pedestrian crossing decisions based on computational rationality, an approach using deep reinforcement learning (RL) to learn boundedly optimal behavior policies given human constraints, in our case a model of the limited human visual system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' We show that the proposed combined cognitive- RL model captures human-like patterns of gap acceptance and crossing initiation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Interestingly, our model’s decisions are sensitive to not only the time gap, but also the speed of the approaching vehicle, something which has been described as a “bias” in human gap acceptance behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' However, our results suggest that this is instead a rational adaption to human perceptual limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Moreover, we demonstrate an approach to accounting for individual differences in computational rationality models, by conditioning the RL policy on the parameters of the human constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Our results demonstrate the feasibility of generating more human-like road user behavior by combining RL with cognitive models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Index Terms—Human behavior, computational rationality, noisy perception, reinforcement learning I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' INTRODUCTION The interaction between road users is defined as “a situ- ation where the behavior of at least two road users can be interpreted as being influenced by the possibility that they are both intending to occupy the same region of space at the same time in the near future” [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Interdependence between vehicles and pedestrians makes interactions between road users instrumental for road safety and automated vehicles(AVs), which pushes the research into road user interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Pedestrian is the most vulnerable group among all road users [2], and their behavior is difficult to predict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' How drivers may behave is limited by the machinery of a vehicle and traffic rules, whereas pedestrians have more freedom, and are limited only by traffic rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' To better understand pedestrian behavior, some detailed metrics related to crossing behavior were investigated [3]–[5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' For example, gap acceptance, where the gap is defined as the time or spatial distance between ∗ Corresponding author: mn20yw2@leeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='uk This project has received funding from UK Engineering and Physical Sciences Research Council under fellowship COMMOTIONS - Computa- tional Models of Traffic Interactions for Testing of Automated Vehicles - EP/S005056/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' the pedestrian and approaching vehicle, is an important met- ric for understanding the crossing decision [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Lobjois and Cavallo [4] found that speed-dependent gap acceptance was shown in different age groups i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=', the gap acceptance rate was higher when the approaching vehicle was faster in a given time gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Petzoldt [5] investigated the relationship between gap acceptance and time to arrival (TTA) estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' They speculated that speed-dependent crossing decisions were caused by the biased TTA estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Another important factor affecting the safety of crossing is cross initiation time (CIT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Tian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' [3] observed that CIT was greater when the vehicle was driven at a higher speed for any given initial TTA, which led to unsafe behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' A descriptive study of road user behavior is insufficient for AVs to understand and predict other road users’ actions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' therefore, mathematical models of road user behavior are required for AVs [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' In recent years, many mechanistic models have been proposed to generate and understand pedestrian behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' For example, rule-based models, such as the social force model, were successful in traffic flow simulation [7], [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' However, they are limited in capturing the details of road user interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' To generate more explainable and accurate road user interactive behavior, cognitive models, such as the evidence accumulation model, were utilized to model pedestrian crossing decisions [9], [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' With increased computing power and more available data, machine learning (ML) models have gained increasing at- tention for road user behavior prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Long Short-Term Memory (LSTM), a variant of Recurrent Neural Networks (RNN), was used in pedestrian trajectory prediction [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' To better predict the interactive behavior between pedestrians, Alahi et al proposed a ‘Social LSTM’, with a social layer into the LSTM algorithm, and the model outperformed state- of-the-art methods [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Many efforts have been made to understand and simulate the microscopic behavior of road users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' ML models can reproduce accurate trajectories across a diverse range of sce- narios, and cognitive models can provide the interpretability of the interactive behavior and the underlying mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' However, both streams of methods have some limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' In conventional cognitive models, the modeler should define how the specific task is completed and the model should be updated if the environment and task change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' This makes it difficult to simulate road users’ trajectories in general scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Whereas ML models focus on the minimization of high-level error metrics, rather than the mechanisms generating the behavior or arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='11737v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='LG] 27 Jan 2023 whether the aspects of behavior that are important to humans are being captured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Sometimes the model with high accuracy does not necessarily generate realistic human behavior [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Computational rationality, as a general approach of model- ing human behavior, has shown promising properties in mod- eling human-computer interaction (HCI) [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' This framework is based on the idea that human behaviors are generated by cognitive mechanisms that are adapted to the structure both of the environment and the mind and brain itself [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' In this paper, we developed a model of human pedestrian crossing decisions based on computational rationality, using deep reinforcement learning (RL) to adopt optimal behavior policies given human-like constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' We show that when we constrain the agent by a simple model of human visual perception, it reproduces human gap acceptance behavior qualitatively, including the speed-dependencies, which have not been previously considered as rational behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' We also demonstrate an approach to using computational rationality to model individual differences, by conditioning RL on the parameters of human constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' METHOD A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Dataset The dataset used for validation was collected in the previous experiment reported by Giles et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' 1 shows a birds-eye view of the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' A brief summary of the experimental setup is provided below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' 20 participants were recruited for the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' In the experiment, they wore an HTC Vive Virtual Reality (VR) headset and experienced the virtual crossing task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' The VR environment consisted of a straight two-lane road with a total width of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='85 m, with a zebra crossing at the participant’s location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' In terms of the experimental procedure, participants stood in front of the zebra crossing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' When participants were ready to start the trial, they turned their head to the right to trigger the scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' A car at the predefined initial position d0 would approach the participant at different speeds v0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' The experiment included a mix of scenarios;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' in this paper, we will only consider the scenarios where the speed of the approaching vehicle was constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' The detail of these scenarios is shown in TABLE I, where also the initial time to arrival (TTA) τ0 = d0/v0 is listed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Participants pressed the button on the HTC Vive’s controller when they felt safe to cross.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Upon this button press, CIT was recorded, and the location of the participant in the virtual environment moved across the zebra crossing at the speed of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='31 m/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' This button-press approach was chosen in favor of physically crossing the road, to reduce the impact of variability in motor constraints on the crossing decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Each participant experienced 6 different constant- speed trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Therefore, 120 data trials were used for the validation of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Model This research aims to model the pedestrian crossing de- cision under the assumption that human behaves rationally within limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Therefore, two models were compared, as shown Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Birds-eye view of the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' One is the ideal observer crossing model, in which the agent has perfect information about the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Another is the model considering the visual limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' In this model, the agent perceives the environment subject to noise, but Bayes-optimal perception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' 1) Noisy perception: We assume the agent has a noisy perception of the state of vehicles and perfect knowledge about their own state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' a) Noisy visual input: The observation obtained by the agent is according to the principle of the human visual system, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=', the sensory input received by our human visual system is noisy [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' It is important to consider the nature of this noise;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' here we are building on models which assume that visual noise is introduced at the level of the human retina, as angular noise [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' In the current model, we assume that the agent observes the position of the other agent along its line of travel by observing the angle below the horizon of the other agent [18], [19], with a constant Gaussian noise of standard deviation σv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' In practice, this means that the pedestrian observes the position of the vehicle with a distance- dependent noise of standard deviation σx(k) = fv[x(k)], where x(k) is the true world state, and: fv[x(k)] = |dl| � 1 − h d · tan(arctan h d + σv) � , where dl is the longitudinal distance between the pedestrian agent and the crossing point, d is the distance between the agent and the approaching vehicle, h is the eye height over the ground of the ego agent, which is set to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='6 m for all pedestrian agents for simplicity, and σv could vary between pedestrians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' b) Kalman filter: There is psychophysical evidence that human perception system works like a Bayesian optimizer, and Bayesian methods have been successful in modeling perception and sensorimotor control [17], [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Therefore, we used a Kalman Filter as a model of the human visual perception to percept the environment [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' In our model, we initialized the Kalman filter with a noisy position of the vehicle, and a noisy velocity centered at the true velocity with a standard deviation of all velocity values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' At each step, the TABLE I VEHICLE APPROACH SCENARIOS v0 m/s d0 (m) τ0 (s) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='94 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='90 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='29 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='89 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='81 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='29 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='94 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='81 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='58 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='89 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='61 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='58 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='94 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='71 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='87 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='89 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='42 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='87 Speed Vehicle(a) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Comparison of models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' (a) Ideal observer model, where the agent had full information about the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' (b) Model with visual limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Two variants were developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' One was the model without noise magnitude parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Another was the model with noise magnitude parameter, as shown in the orange dashed box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Kalman filter received the noisy position about the other agent, and the output, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=', the filtered position and velocity of the vehicle, and the variance of the position and the velocity, was the input of the RL agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' 2) Reinforcement learning model: In our model, in line with the theory of computational rationality, we view pedes- trian behavior as a Partially Observable Markov Decision Process (POMDP) under bounds posed by perception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' RL algorithm, where the agent interacts with the environment and learns the optimal strategy by trial and error, can be used to derive the boundedly optimal policy for this type of problem [14], [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' a) State space S: The time step in the simulation affects the resolution of the results of the decision time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' In our mode, one time step corresponds to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='1 seconds, which is suitable for the dataset we are using.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' At each time step t, the environment is in a state st ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' A state contains true information about the vehicle and the agent, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=', the position and velocity of the vehicle and the agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' b) Action space A: At each time step t, the agent takes an action at ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' In this paper, in line with the button press in the experiment, the agent can make the decision to Go or Not Go.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' If the Go decision is made, the agent will go straight at the speed of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='31 m/s, as in the experiment, resulting either in a successful crossing or in a collision, and the scenario will finish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' c) Reward R: In the experiment where the datasets were collected, the participant’s task was to cross the road as soon as they felt safe to do so, either before or after the car had passed them [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Therefore, in our model we want the agent to cross the road in as short a time as possible without a collision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' At each time step t, the agent will receive a negative reward of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='5×simulationsteps, which helps the agent to cross the road faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' The agent will be given a reward of 200 when crossing the road without collision, and a reward of −200 if a collision happens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' The form of this reward function was chosen based on some initial testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' As the focus of this study is on the potential effect of noisy perception on the crossing decision, we kept the reward function simple;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' future work can further refine it to better capture human preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' d) Observation space O: The agent receives observation ot ∈ O at each time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' In the ideal observer model, the agent gets the complete information about the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' In the model with visual limits, the agent only observes partial information about the state of the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' At each time step, the agent receives the processed estimates of the position and velocity and uncertainty about the position and velocity of the other agent from the Kalman Filter and the exact position and velocity of the ego agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' e) Transition function T: The transition function defines how the current state st changes to the next state st+1 taking action at.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' In our model, if Not Go action is chosen, the vehicle will move according to the kinematic equations with the given speed, and the position of the agent will not change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Once Go action is chosen, whether the collision happens will be calculated, and the corresponding reward will be given to the agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Then, the simulation finishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' f) Deep Q-Networks: Deep-Q Network (DQN) is a method using the neural network to learn the optimal policy to maximize the state-action function (Q function), Q(s, a), the expected rewards for an action taken in a given state [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' DQN is suitable for the problem with a continuous state space and a discrete action space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' For the extensibility of the model to more complex situations, we utilized an enhanced version of DQN, Double DQN (DDQN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' The Double DQN (DDQN) structure, which decouples the update of the neural network parameter for action selection and evaluation, can avoid the overestimation of the action value [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Furthermore, the dueling network was used, in which the Q-function is decoupled to a value function and a state-dependent action advantage A(s, a) function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Compared with the single-stream DQN, the Duelling DQN shows better performance especially when different actions lead to a similar value because of the consideration of the state value in the Q value [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' We trained the agent through a two-layer fully connected network, with 512 and 256 nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' The learning rate and discount factor are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='001 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='99 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' To explore the optimal policy, an ϵ - greedy algorithm was used for exploration: At each time step t, a random action is chosen with probability ϵ, and the action with maximum Q value is chosen with probability 1−ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' We decreased ϵ by 10−4 in each learning step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' The minimum value was set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Training and fitting As we don’t know the correct value for σv, and additionally each participant in the experiment may have an individual σv, we trained the model for different σv ranging from 0−1 with a step size of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' We trained the model in two different ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' First, we trained a separate model for each σv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' With this approach, for any given σv, we can get a boundedly optimal External environment AgentExternal environment Bayesian perceptual filtering Agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Noisepolicy from the model trained with that σv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' This approach is somewhat inconvenient and will not scale well to more complete models of human constraints, with more parameters than just one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Therefore, as an alternative approach, we instead trained just a single model, across all of the different σv, and in this case we also provided σv as an input to the model, as illustrated by the dashed line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' In other words, we conditioned the RL on σv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' With this approach, for any given σv, we get a boundedly optimal policy from this single model, by also giving it σv as an input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' With the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='31 m/s walking speed used in the experiment, it is in fact possible to cross before the vehicle without collision even for the lowest TTA of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='29 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' To avoid our agents learning this trivial policy of always just immediately crossing, in the model training we also included a lower TTA of 1 s, in which the agent did not have enough time to cross safely before the vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' We used three criteria to identify model convergence and stop training:(1) the collision rate is less than or equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='01, which means one collision at most happened in the last 100 episodes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' (2) ϵ has reached the lowest value of the epsilon- greedy algorithm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' (3) the difference in average reward between the last 100 episodes and the last 200 to 100 episodes is less than 1 to make sure that the average reward does not improve much and becomes stable and converged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' We also identified the σv that fitted the experimental data best for each participant by likelihood maximization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' We esti- mated the probability density function (PDF) of CIT predicted for each model by kernel density estimation, separately for each of the six scenarios (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' This allowed us to calculate the model likelihood of each σv for each participant, by multiplying the model PDF values at the participant’s observed button press times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Finally, we combined all the PDFs from the different participants, with tuned σv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' We applied this method to both model types, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=', separate models for each σv, and a single model for all σv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' In addition to this per-participant fit, we also selected one σv from the models with separate σv, which best fitted the entire dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Abbreviations will be introduced here for referring to dif- ferent model variants;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=', EXP: experimental data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' IO: ideal observer model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' LMD: multiples model with separate visual limits parameter and one σv fitted across the entire dataset;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' LMP: multiple models with separate visual limits parameter and different σv fitted per participant;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' LSP: one model with different visual limits parameters and different σv fitted per participant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' RESULTS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Experimental results First, we reanalyzed the experimental data reported by Pekkanen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' As shown in the first panel on the left in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' 3, this experiment replicated the general finding that the gap acceptance rate strongly depended on initial TTA, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=', more pedestrians crossed before the car when there was a larger time gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' In addition, we can observe the speed- dependent gap acceptance rate, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=', for a given initial TTA, more pedestrians accepted the gap if the speed was higher [3], [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' This effect was strongest at the initial TTA of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='6 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Regarding CIT, which is shown in the first row in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' 4, there was some spread in CIT, both when crossing before and after the car.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Behavior of the model The results of the model without visual limits (IO) are shown in the second panel on the left in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' 3 and the second row in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' In the model without visual limits, the agent had full information about the environment, and crossed the road without collision in the shortest time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Therefore, as shown in the second row of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' 4, the agent always started to cross at the first time step when it was safe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' The second panel in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' 3 shows that the agent always accepted the gap in 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='3 s, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='6 s, and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='9 s initial TTA conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' The three graphs on the right in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' 3, and the last three rows in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' 4 show the results of the models with visual limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' In models with visual limits, the agent received the processed information of the Kalman filter instead of the exact information about the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' 3, the models with visual limits captured the pattern of the gap acceptance rate observed in the exper- iment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' The TTA-dependent gap acceptance rate, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=', the gap acceptance rate increased with the initial TTA, was predicted by all three model types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Unexpectedly, the difference within the same TTA group was also captured by the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' The agent was more likely to accept the gap when the vehicle speed was higher in the given initial TTA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' This pattern was shown by all model types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Overall, the model has a slightly greater tendency to accept gaps than the human participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' This may be what is causing the model to show a speed dependency at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='3 s TTA (because the model sometimes crosses there, whereas the humans almost never did), but not at 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='9 s TTA Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Gap acceptance rate by human participants and different models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' See the main text for explanations of the model abbreviations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' EXP 10 LMD LMP LSP 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='6 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='9 m/s 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='9 m/s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='0 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='6 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='6 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='9 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='9 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='9 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='9 Initial TTA (s) Initial TTA (s) Initial TTA (s) Initial TTA (s) Initial TTA (s)Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Cumulative probability for CIT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' The four columns show different initial TTA conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' See the main text for explanations of the model abbreviations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' (Note: CDF curves are extended to the right after after the cumulative probability reaches y = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=') (because the model has already reached full gap acceptance at this TTA, which the humans had not).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' 1) Variability within and between individuals: From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' 4, there was some spread in CIT also for model LMD – i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=', the model showed some within-individual variability – due to the trial-to-trial variability in visual noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' That model LMP, with different σv for different participants, came closer to capture the human variability, suggesting that some of the variability in the human data is from between-individual differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' To test this, we did the Akaike Information Criterion (AIC) analysis of LMD, LMP, and LSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' AIC, which considers both the log-likelihood and the cost of more parameters, allows us to compare the performance of different models, and the preferred model is the one with the minimum AIC value [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' The AIC values are 113, 57, and 79 for the LMD, LMP and LSP respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Therefore, LMP, the model with different individual σv, outperformed other model variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' This sug- gested that some of the variability in human CIT is due to between-individual variability in sensory noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Distributions of initial estimated TTA from the output of the visual perception model, across the different scenarios in the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' 2) Speed-dependent gap acceptance: To investigate the reason why the model showed speed-dependence in its gap acceptance rate, we calculated the estimated TTA through the velocity and position estimated by Kalman filter at the first time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' 5 shows the distribution, the mean value, which is shown in the grey vertical line, and the 5th and 95th percentiles of the estimated TTA, which is shown in the shading area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' 5, for the same time gap, the distribution for the estimated TTA was more dispersed at lower speed conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' In other words, at low speeds, there was greater uncertainty about the estimated TTA, which could be the reason why it is better, from a reward maximization perspective, to be more careful about crossing in these sit- uations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' This perspective aligns with the findings of Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' [26], who showed that apparently biased behavior in a more abstract choice task might also be explained as a consequence of optimal sequential decisions under uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Interestingly, Tian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' [3] showed that in the road-crossing context, humans may in practice be achieving this strategy by making use of relatively simple visual cues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' DISCUSSIONS AND CONCLUSION We developed a model of human pedestrian crossing de- cisions based on computational rationality, using deep RL to adopt optimal behavior policies given human-like constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' We show that when we constrain the agent by a simple model of human visual perception, it reproduces human gap acceptance behavior qualitatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Furthermore, the model also predicts the speed-dependencies that are typically observed in human gap acceptance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' These have previously been considered as evidence of biases in human pedestrian decision-making, but our results demonstrate that this type of speed-dependence is a rational adaption to noisy visual perception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' When com- Initial TTA: 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='3 s Initial TTA: 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='6 s Initial TTA: 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='9 s 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='0 Prob 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='9 m/s 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='9 m/s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='0 Initial TTA: 1 s 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='0 Prob 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='9 m/s 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='9 m/s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='0 - Prob LMD 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='9 m/s 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='9 m/s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='0 Prob LMP 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='9 m/s 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='9 m/s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='0 Prob 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='9 m/s 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='9 m/s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='0 2 2 2 6 8 10 0 4 6 8 10 0 2 4 4 6 8 100 4 6 8 10 Time (s) Time (s) Time (s) Time (s)Velocity = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='94 m/s Gap = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='3 s Velocity = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='94 m/s Gap = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='6 s Velocity = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='94 m/s Gap = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='9 s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='00 Velocity = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='89 m/s Gap = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='3 s Velocity = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='89 m/s Gap = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='6 s Velocity = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content="89 'm/s Gap ='6." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='9 s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='00 5 10 15 20 0 5 10 15 20 25 30 Estimated TTA (s) Estimated TTA (s) Estimated TTA (s)paring the results of LMP and LSP, it shows that these two approaches learned similar policies, but not identical policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' We believe that the approach of conditioning RL on constraint parameters is a promising approach for considering individual differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' As an early attempt in using computational ratio- nality in modeling road user interaction, we see the feasibility to use computational rationality to model road user behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' The developed model could be served as the agent model in the test environment of AVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' There is ample scope for further improvements to our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' For example, our agent is more likely to show faster responses than humans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' One reason is that the human percep- tual filtering might be slower than the Kalman filter we used, which can process the information without delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Another reason is that we have not considered the non-decision time in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' We can also observe more variability in human CITs than in model CITs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' This could be due to both between- and within-individual variability of unmodeled sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' In addition, our reward function is simple and so far we have not tuned it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' This is enough for our purposes, to show qualitative patterns of human road-crossing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' However, this limitation is probably the reason why the model has a greater tendency for gap acceptance than humans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' In future work, following a similar approach to what we did with σv here, we could tune also for example the time-loss penalty in the reward function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Furthermore, in the current work, the agent is interacting with a constant speed approaching vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' However, in the real world, the pedestrian will interact with vehicles with various kinematic states, which also affect the crossing be- havior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' For example, [22] suggests that human pedestrians are also perceiving and interpreting vehicle deceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' A major advantage of the approach we have taken here is that deep RL is scalable to much more complex traffic scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=', far beyond what is possible with conventional cognitive models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' We therefore conclude that computational rationality overall holds great promise for applied modeling of human-like road user interaction behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' REFERENCES [1] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Markkula, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Madigan, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Nathanael, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Portouli, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Lee, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Diet- rich, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Billington, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Schieben, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Merat, “Defining interactions: A conceptual framework for understanding interactive behaviour in human and automated road traffic,” Theoretical Issues in Ergonomics Science, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' 21, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' 6, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' 728–752, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' [2] “Reported road casualties great britain, annual report: 2021 gov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='uk,” https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='gov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='uk/government/statistics/ reported-road-casualties-great-britain-annual-report-2021, accessed: 2023-01-24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' [3] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Tian, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Markkula, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Wei, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Lee, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Madigan, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Merat, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Romano, “Explaining unsafe pedestrian road crossing behaviours using a psychophysics-based gap acceptance model,” Safety Science, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' 154, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' 105837, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' [4] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Lobjois and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Cavallo, “Age-related differences in street-crossing decisions: The effects of vehicle speed and time constraints on gap selection in an estimation task,” Accident analysis & prevention, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' 39, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' 5, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' 934–943, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' [5] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Petzoldt, “On the relationship between pedestrian gap acceptance and time to arrival estimates,” Accident Analysis & Prevention, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' 72, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' 127–133, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' [6] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Camara, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Bellotto, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Cosar, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Weber, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Nathanael, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Althoff, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Wu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Ruenz, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Dietrich, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Markkula et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=', “Pedestrian models for autonomous driving part ii: high-level models of human behavior,” IEEE Transactions on Intelligent Transportation Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' 22, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' 9, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' 5453–5472, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' [7] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Helbing and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Molnar, “Social force model for pedestrian dynamics,” Physical review E, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' 51, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' 5, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' 4282, 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' [8] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Yang, ¨U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' ¨Ozg¨uner, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Redmill, “A social force based pedestrian motion model considering multi-pedestrian interaction with a vehicle,” ACM Transactions on Spatial Algorithms and Systems (TSAS), vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' 6, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' 1–27, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' [9] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Markkula, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Romano, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Madigan, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Fox, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Giles, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Merat, “Models of human decision-making as tools for estimating and optimizing impacts of vehicle automation,” Transportation research record, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' 2672, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' 37, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' 153–163, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' [10] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Giles, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Markkula, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Pekkanen, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Yokota, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Matsunaga, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Merat, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Daimon, “At the zebra crossing: Modelling complex decision processes with variable-drift diffusion models,” in Proceedings of the 41st annual meeting of the cognitive science society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Cognitive Science Society, 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' 366–372.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' [11] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Wang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Jiang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Yang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Li, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Long, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Fei- Fei, “Eidetic 3d lstm: A model for video prediction and beyond,” in International conference on learning representations, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' [12] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Alahi, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Goel, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Ramanathan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Robicquet, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Fei-Fei, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Savarese, “Social lstm: Human trajectory prediction in crowded spaces,” in Proceedings of the IEEE conference on computer vision and pattern recognition, 2016, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' 961–971.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' [13] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Srinivasan, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Lin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Antonello, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Knittel, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Hasan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Hawasly, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Redford, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Ramamoorthy, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Leonetti, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Billington et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=', “Beyond RMSE: Do machine-learned models of road user inter- action produce human-like behavior?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' arXiv preprint arXiv:2206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='11110, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' [14] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Oulasvirta, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Jokinen, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Howes, “Computational rationality as a theory of interaction,” in CHI Conference on Human Factors in Computing Systems, 2022, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' 1–14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' [15] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Lewis, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Howes, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Singh, “Computational rationality: Link- ing mechanism and behavior through bounded utility maximization,” Topics in cognitive science, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' 6, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' 279–311, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' [16] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Faisal, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Selen, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Wolpert, “Noise in the nervous system,” Nature reviews neuroscience, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' 9, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' 292–303, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' [17] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Kwon, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Tadin, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Knill, “Unifying account of visual motion and position perception,” Proceedings of the National Academy of Sciences, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' 112, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' 26, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' 8142–8147, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' [18] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Ooi, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Wu, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' He, “Distance determined by the angular declination below the horizon,” Nature, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' 414, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' 6860, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' 197–200, 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' [19] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Markkula, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Lin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Srinivasan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Billington, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Leonetti, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Kalantari, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Yang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Lee, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Madigan, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Merat, “Ex- plaining human interactions on the road requires large-scale integration of psychological theory,” 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' [20] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Knill and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Pouget, “The bayesian brain: the role of uncertainty in neural coding and computation,” TRENDS in Neurosciences, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' 27, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' 12, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' 712–719, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' [21] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Franc¸ois-Lavet, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Henderson, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Islam, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Bellemare, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Pineau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=', “An introduction to deep reinforcement learning,” Foundations and Trends® in Machine Learning, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' 11, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' 3-4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' 219–354, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' [22] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Pekkanen, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Giles, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Lee, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Madigan, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Daimon, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Merat, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Markkula, “Variable-drift diffusion models of pedestrian road- crossing decisions,” Computational Brain & Behavior, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' 5, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' 60–80, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' [23] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Van Hasselt, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Guez, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Silver, “Deep reinforcement learning with double q-learning,” in Proceedings of the AAAI conference on artificial intelligence, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' 30, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' 1, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' [24] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Wang, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Schaul, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Hessel, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Hasselt, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Lanctot, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Freitas, “Dueling network architectures for deep reinforcement learning,” in International conference on machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' PMLR, 2016, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' 1995– 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' [25] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Murtaugh, “In defense of p values,” Ecology, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' 95, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' 611–617, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' [26] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Chen, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Chang, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' Howes, “Apparently irrational choice as optimal sequential decision making,” in Proceedings of the AAAI Conference on Artificial Intelligence, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' 35, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' 1, 2021, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} +page_content=' 792–800.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9FKT4oBgHgl3EQfJS1Z/content/2301.11737v1.pdf'} diff --git a/stAzT4oBgHgl3EQfcvyC/vector_store/index.faiss b/stAzT4oBgHgl3EQfcvyC/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..a4219067c0951b0b1bc0293597fb6d49367c6bae --- /dev/null +++ b/stAzT4oBgHgl3EQfcvyC/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:c140c0eeaaaa14e2b7e86be47fa77a6fdb36fdd3c85205de28abaec817909998 +size 2883629 diff --git a/t9E3T4oBgHgl3EQf9wvz/content/2301.04820v1.pdf b/t9E3T4oBgHgl3EQf9wvz/content/2301.04820v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..a66627094a0d8ca0130aeb7f6f2197edf906f84f --- /dev/null +++ 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sha256:498dfab3f2e681d8a3223e8f8a633fa919da72ecf4e3a53ae79c5a0d7c583f33 +size 12976173 diff --git a/v9E4T4oBgHgl3EQfXQwv/content/tmp_files/2301.05039v1.pdf.txt b/v9E4T4oBgHgl3EQfXQwv/content/tmp_files/2301.05039v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..a0e883bc0339930608903916a6f4bab636a7bd6a --- /dev/null +++ b/v9E4T4oBgHgl3EQfXQwv/content/tmp_files/2301.05039v1.pdf.txt @@ -0,0 +1,2689 @@ +Dynamic Mode Decomposition of High Reynolds +Number Supersonic Jet Flows +Sami Yamouni∗ +Instituto Tecnol´ogico de Aeron´autica, 12228-900 S˜ao Jos´e dos Campos, SP, Brazil +Carlos Junqueira-Junior† , Jo˜ao Luiz F. Azevedo ‡ +Instituto de Aeron´autica e Espa¸co, 12228-903 S˜ao Jos´e dos Campos, SP, Brazil +William R. Wolf § +Universidade Estadual de Campinas, 13083-970 Campinas, SP, Brazil +Abstract +Current design constraints have encouraged the studies of aeroacoustic fields around +compressible jet flows. The present work addresses the numerical study of unsteady turbu- +lent jet flows as a preparation for future aeroacoustic analyses of main engine rocket plumes. +An in-house large eddy simulation tool is used in order to reproduce high fidelity results +of compressible jet flows. The large eddy simulation formulation is written using a second +order numerical scheme for a finite difference spatial discretization. Numerical simulations +of perfectly expanded jets are performed and the results are compared to the literature. +Dynamic mode decompositions (DMD) of the jet flow, using large size three-dimensional +snapshots, are performed. Three variables are analyzed, namely, the velocity magnitude, +the vorticity magnitude and the divergence of velocity. In particular, two frequencies are +identified and they are linked to flow structures observed in experiments performed by +other authors in the literature. The spatial shapes of the corresponding dynamic modes +are also discussed. +I. +Introduction +One of the main design issues related to launch vehicles lies on noise emission originated by the complex +interaction between the high-temperature/high-velocity exhaustion gases and the atmospheric air. These +emissions, which have high noise levels, can damage the launching structure or even be reflected upon +the vehicle structure itself and the equipment onboard at the top of the vehicles. Moreover, the resulting +pressure fluctuations can damage the solid structure of different parts of the launcher or the onboard scientific +equipment by vibrational acoustic stress. Therefore, it is strongly recommended to consider the load resulted +from acoustic sources over large launching vehicles during take-off and also during the transonic flight. +Moreover, one cannot neglect the energy dissipation effect generated by the acoustic waves even if the +vehicle is far from the ground. Theoretically, all chemical energy should be converted to kinetic energy. +However, in reality, the noise generation consumes part of the chemical energy. +∗Postdoctoral Reasearch Fellow, Graduate Program on Computer Sciences and Electrical Engineering, Departamento de +Ciˆencia e Tecnologia Aeroespacial, DCTA/ITA; E-mail: sami.yamouni@gmail.com. +†Postdoctoral Research Fellow, Graduate Program on Computer Sciences and Electrical Engineering, Departamento de +Ciˆencia e Tecnologia Aeroespacial, DCTA/ITA; E-mail: junior.hmg@gmail.com. +‡Senior Research Engineer, Aerodynamics Division, Departamento de Ciˆencia e Tecnologia Aeroespacial, DCTA/IAE/ALA; +E-mail: joaoluiz.azevedo@gmail.com. Fellow AIAA. +§Assistant Professor, Faculty of Mechanical Engineering; E-mail: wolf@fem.unicamp.br. Member AIAA. +1 of 26 +American Institute of Aeronautics and Astronautics +arXiv:2301.05039v1 [physics.flu-dyn] 12 Jan 2023 + +The acoustics design constraints have encouraged the studies of aeroacoustic fields around compressible +jet flows. Instituto de Aeronautica e Espa¸co (IAE), in Brazil, is interested in this flow configuration for +rocket design applications. Unsteady property fields of the flow are necessary for the aerocoustic study. +Therefore, the present work addresses the numerical study of unsteady turbulent compressible jet flows as a +preparatory step for such aeroacoustic applications in the future. Large eddy simulations (LES) are used in +order to reproduce high fidelity results of the unsteady compressible jet flows. +One issue with the results of such large data sets resulting from LES calculations is precisely the large +amount of data that has to be handled. Hence, with the objective of simplifying a complex flow into a +low-dimensional representation containing the dominant dynamic structures, the use of different techniques +has been proposed. Among these, Proper Orthogonal Decomposition (POD)1–3 and Dynamic Mode Decom- +position (DMD)4 are the more commonly used techniques in the fluid dynamics community. POD selects the +modes depending on their energy content. However, this criterion is not necessarily the most appropriate, +since the energy is not always the key parameter in order to identify the flow structures4 of interest. In +contrast with POD, the modes computed from the DMD approach define characteristic frequencies of the +flow. Hence, DMD is the chosen method for the intended application of the present work. It has already +been applied to various flow configurations, such as cavity flows,4,5 shock wave–turbulent boundary layer +interaction,6 boundary layer flows,7,8 cylinder flows,9,10 combustion chamber flows,11,12 wake behind a flex- +ible membrane4 or jet flows.13–18 Several variations of the DMD algorithm have also been proposed. One +can cite the optimal mode decomposition,19 the sparsity-promoting DMD,20 the extended DMD,21,22 or +the streaming DMD23. Lately, an unbiased noise-robust method has been proposed by Hemati et al.24 to +overcome the adverse influence of measurement errors. This method can be combined with all the previously +listed DMD algorithms. +Therefore, in this context, the main objective of the present work is to apply the DMD algorithm to the +numerical data extracted from large eddy simulations of perfectly-expanded supersonic jet flows at M = 1.4. +Due to the large dimension of this problem, the authors use the streaming version23 of the total-least squares +DMD formulation proposed by Hemati et al.24 The DMD results are compared to numerical and experimental +data available in the literature. +II. +Navier-Stokes Equations +The numerical strategy used in the present study is based on the compressible Navier-Stokes equations +formulated as +∂ρ +∂t + +∂ +∂xj +(ρuj) = 0 , +(1) +∂ +∂t (ρui) + +∂ +∂xj +(ρuiuj) + ∂p +∂xi +− ∂τij +∂xj += 0 , +(2) +∂e +∂t + +∂ +∂xj +[(e + p) uj − τijui + qj] = 0 , +(3) +in which t and xi are independent variables representing time and spatial coordinates of a Cartesian coor- +dinate system x, respectively. The components of the velocity vector u are written as ui, and i = 1, 2, 3. +Density, pressure and total energy per mass unit are denoted by ρ, p and e, respectively. The heat flux +vector, qj, is given by +qj = κ ∂T +∂xj +, +(4) +where T is the static temperature and κ is the thermal conductivity coefficient, which can by expressed by +κ = µCp +Pr +. +(5) +The thermal conductivity coefficient is a function of the specific heat at constant pressure, Cp, of the Prandtl +number, Pr, which is equal to 0.72 for air, and of the dynamic viscosity coefficient, µ. The latter can be +calculated using Sutherland’s law, +µ (T) = µ∞ +� T +T∞ +� 3 +2 T0 + S1 +T + S1 +, +with S1 = 110.4 K . +(6) +2 of 26 +American Institute of Aeronautics and Astronautics + +According to the Stokes hypothesis, the shear-stress tensor, τij, for a Newtonian fluid can be written as +τij = 2µ +� +Sij − 1 +3δijSkk +� +, +(7) +in which the components of rate-of-strain tensor, Sij, are given by +Sij = 1 +2 +� ∂ui +∂xj ++ ∂uj +∂xi +� +. +(8) +In order to close the system of equations the density, the static pressure and the static temperature are +correlated by the equation of state given by +p = ρRT , +(9) +where R is the gas constant, written as +R = Cp − Cv , +(10) +and Cv is the specif heat at constant volume. The total energy per mass unity is given by: +e = +p +γ − 1 + 1 +2ρuiui , +(11) +in which γ is the ratio of specific heats, written as γ = Cp/Cv. +III. +Large Eddy Simulation Filtering +The large eddy simulation is based on the principle of scale separation, which is addressed as a filtering +procedure in a mathematical formalism. A modified version of the the System I filtering approach25 is used +in present work. The original formulation neglects the double correlation term and it is written as +∂ρ +∂t + +∂ +∂xj +(ρ �uj) = 0 , +∂ +∂t (ρ �ui) + +∂ +∂xj +(ρ �ui �uj) + ∂p +∂xi +− ∂ˇτij +∂xj += − ∂ +∂xj +[σij − (τij − ˇτij)] , +∂ˇe +∂t + +∂ +∂xj +[(ˇe + p) �uj] − ∂ˇτij �ui +∂xj ++ ∂ˇqj +∂xj += −B1 − B2 − B3 + B4 + B5 + B6 − B7 . +(12) +The (˜·) notation is used to represent a Frave averaged property. The SGS stress tensor components are +written as σij. The filtering procedure originates two new terms, ˇτij and ˇqj. These new terms are given by +ˇτij = 2µ +� +˜Sij − 1 +3δij ˜Skk +� +, +(13) +where +˜Sij = 1 +2 +� ∂˜ui +∂xj ++ ∂˜uj +∂xi +� +, +(14) +and +ˇqj = ˇκ ∂ ˜T +∂xj +, +(15) +in which +ˇκ = κ +� +˜T +� += +˜µ +� +˜T +� +Cp +Pr +. +(16) +3 of 26 +American Institute of Aeronautics and Astronautics + +The SGS terms of the energy equation, Bi, are given by +B1 = +1 +(γ − 1) +∂ +∂xj +(puj − p �uj) = ∂CvQj +∂xj +, +(17) +B2 = p∂uk +∂xk +− p∂� +uk +∂xk += Πdil , +(18) +B3 = +∂ +∂xj +(σkj� +uk) , +(19) +B4 = σkj +∂ +∂xj +� +uk , +(20) +B5 = τkj +∂ +∂xj +uk − τij +∂ +∂xj +� +uk = ϵ , +(21) +B6 = +∂ +∂xj +(τij �ui − ˇτij �ui) = ∂D +∂xj +, +(22) +B7 = +∂ +∂xj +(qj − ˇqj) . +(23) +The work of Vreman et al.26 +and Vreman25 classify the influence of each term of System I and System +II formulations on a 2-D temporal shear layer flow. The classification, including large, medium, small and +negligible effects, is based on the L2 norm of different terms of the filtered equations. One order of magnitude +separates the norm of each class of terms. Garnier et al.27 compile the analogy as presented in Tab. 1. +Table 1. +Classification of System I terms +Large +convective NS +Medium +diffusive NS, B1, B2 and B3 +Small +B4 and B5 +Negligible +∂ +∂xj (τij − ˇ +τij), B6 and B7 +In practice, the authors of the System I analogy neglect the non-linear terms occuring in the viscous +terms and in the heat fluxes.27 Moreover, some of the terms from the original System I set of equations, Eq. +(12), such as B4 and B5, cannot be written in conservative form. Only the terms with large and medium +influence are considered in the present work. The SGS stress tensor components are written using the SGS +viscosity,28 +σij = −2µsgs +� +ˇSij − 1 +3 +ˇSkk +� ++ 1 +3δijσkk . +(24) +The most important terms of the filtered energy equation are modeled based on the work of Eidson29 and +Vreman.25 They are given by +B1 + B2 = − ∂ +∂xj +� +κsgs +∂ ˜T +∂xj +� +, +(25) +where +κsgs = µsgsCp +Prsgs +. +(26) +Using Eqs. (24), (25), and (26), one can write a simplified version of the System I formulation as +∂ρ +∂t + +∂ +∂xj +(ρ �uj) = 0 , +∂ +∂t (ρ �ui) + +∂ +∂xj +(ρ �ui �uj) + ∂p +∂xi +− ∂τ mod +ij +∂xj ++ 1 +3 +∂ +∂xj +(δijσii) = 0 , +∂ˇe +∂t + +∂ +∂xj +[(ˇe + p) �uj] − +∂ +∂xj +� +τ mod +ij +�ui +� ++ 1 +3 +∂ +∂xj +[(δijσii) �ui] + ∂qmod +j +∂xj += 0 , +(27) +4 of 26 +American Institute of Aeronautics and Astronautics + +where, τ mod +ij +and qmod +j +, include the viscous and the subgrid terms. They can be written as +τ mod +ij += 2 (µ + µsgs) +� +Sij − 1 +3δijSkk +� +(28) +and +qmod +j += (κ + κsgs) ∂T +∂xj +. +(29) +Previous work have shown that the subgrid scale terms are too small when compared to the truncation +error of the second order numerical scheme used in the current research.30–32 Therefore, an implicit LES is +performed in which all subgrid scales terms, [·]sgs, introduced in Eqs. (28) and (29) are neglected. +IV. +Transformation of Coordinates +The formulation used in the current work is written in the a general curviliar coordinate system in order +to facilitate the implementation and add more generality for the CFD tool. The modified System I set of +equations, Eq. (27) can be written in a strong conservative form for a 3-D Cartesian coordinate system as +∂Q +∂t + ∂E +∂x + ∂F +∂y + ∂G +∂z = 0 , +(30) +where Q stands for the filtered conservative properties vector given by +Q = [ρ +ρ˜u +ρ˜v +ρ ˜w +ˇe]T +. +(31) +The flux vectors which represent both the inviscid and viscous fluxes, E, F and G are written as +E = +� +� +� +� +� +� +� +� +� +� +� +� +� +ρ˜u +ρ˜u2 + p − τ mod +xx ++ 1 +3σxx +ρ˜u˜v − τ mod +xy +ρ˜u ˜w − τ mod +xz +� +ˇe + p − τ mod +xx ++ 1 +3σxx +� +˜u − τ mod +xy +˜v − τ mod +xz +˜w + qmod +x +� +� +� +� +� +� +� +� +� +� +� +� +� +, +(32) +F = +� +� +� +� +� +� +� +� +� +� +� +� +� +ρ˜v +ρ˜u˜v − τ mod +xy +ρ˜v2 + p − τ mod +yy ++ 1 +3σyy +ρ˜v ˜w − τ mod +yz +� +ˇe + p − τ mod +yy ++ 1 +3σyy +� +˜v − τ mod +xy +˜u − τ mod +yz +˜w + qmod +y +� +� +� +� +� +� +� +� +� +� +� +� +� +, +(33) +G = +� +� +� +� +� +� +� +� +� +� +� +� +� +ρ ˜w +ρ˜u ˜w − τ mod +xz +ρ˜v ˜w − τ mod +yz +ρ ˜w2 + p − τ mod +zz ++ 1 +3σzz +� +ˇe + p − τ mod +zz ++ 1 +3σzz +� +˜w − τ mod +xz +˜u − τ mod +yz +˜v + qmod +z +� +� +� +� +� +� +� +� +� +� +� +� +� +, +(34) +in which, u, v and w are the velocity components in the Cartesian coordinates, x, y and z respectively. +In the present work the chosen general coordinate transformation is given by +T += +t , +ξ += +ξ (x, y, z, t) , +η += +η (x, y, z, t) , +(35) +ζ += +ζ (x, y, z, t) . +5 of 26 +American Institute of Aeronautics and Astronautics + +Throughout the present work, ξ is the axial jet flow direction, η is the radial direction and ζ is the azimuthal +direction. The derivatives in the general curvilinear coordinate system are calculated as a function of the +derivatives the Cartesian coordinate system by the chain rule. Therefore, one can write +� +� +� +� +� +� +� +� +� +� +� +∂ +∂T +∂ +∂ξ +∂ +∂η +∂ +∂ζ +� +� +� +� +� +� +� +� +� +� +� += +� +���� +1 +xT +yT +zT +0 +xξ +yξ +zξ +0 +xη +yη +zη +0 +xζ +yζ +zζ +� +���� +� +� +� +� +� +� +� +� +� +∂ +∂t +∂ +∂x +∂ +∂y +∂ +∂z +� +� +� +� +� +� +� +� +� +. +(36) +The Jacobian of the transformation, J, is calculated as the inverse of the determinant of the matrix in the +chain rule presented in Eq. (36). Therefore, for the 3-D coordinate transformation, the Jacobian can be +written as +J = (xξyηzζ + xηyζzξ + xζyξzη − xξyζzη − xηyξzζ − xζyηzξ)−1 . +(37) +The metric terms are given by +ξx = J (yηzζ − yζzη) , +ξy = J (zηxζ − zζxη) , +ξz = J (xηyζ − xζyη) , +ηx = J (yηzξ − yξzη) , +ηy = J (zηxξ − zξxη) , +ηz = J (xηyξ − xξyη) , +(38) +ζx = J (yξzη − yηzξ) , +ζy = J (zξxη − zηxξ) , +ζz = J (xξyη − xηyξ) , +ξt = −xT ξx − yT ξy − zT ξz , +ηt = −xT ηx − yT ηy − zT ηz , +ζt = −xT ζx − yT ζy − zT ζz . +One can rewrite Eq. (30), in a conservative form, for the general curvilinear coordinate system as +∂ ˆQ +∂T + ∂ ˆE +∂ξ + ∂ ˆF +∂η + ∂ ˆG +∂ζ = 0 , +(39) +where +ˆQ = J−1Q = J−1 [ρ +ρ˜u +ρ˜v +ρ ˜w +ˇe]T +, +(40) +and the new flux vectors are given by +ˆE = J−1 � +ξtQ + ξxE + ξyF + ξzG +� +, +ˆF = J−1 � +ηtQ + ηxE + ηyF + ηzG +� +, +(41) +ˆG = J−1 � +ζtQ + ζxE + ζyF + ζzG +� +. +Finally, the flux vectors are split in inviscid and viscous part in order to simplify the implementation. +Therefore, Eq. (39) can be rewritten as +∂ ˆQ +∂T + ∂ ˆEe +∂ξ + ∂ ˆFe +∂η + ∂ ˆGe +∂ζ += ∂ ˆEv +∂ξ + ∂ ˆFv +∂η + ∂ ˆGv +∂ζ , +(42) +where the inviscid flux vectors, ˆEe, ˆFe and ˆGe, are given by +ˆEe = J−1 +� +� +� +� +� +� +� +� +� +� +� +� +� +ρU +ρ˜uU + pξx +ρ˜vU + pξy +ρ ˜wU + pξz +(ˇe + p) U − pξt +� +� +� +� +� +� +� +� +� +� +� +� +� +, +(43) +ˆFe = J−1 +� +� +� +� +� +� +� +� +� +� +� +� +� +ρV +ρ˜uV + pηx +ρ˜vV + pηy +ρ ˜wV + pηz +(ˇe + p) V − pηt +� +� +� +� +� +� +� +� +� +� +� +� +� +, +(44) +6 of 26 +American Institute of Aeronautics and Astronautics + +ˆGe = J−1 +� +� +� +� +� +� +� +� +� +� +� +� +� +ρW +ρ˜uW + pζx +ρ˜vW + pζy +ρ ˜wW + pζz +(ˇe + p) W − pζt +� +� +� +� +� +� +� +� +� +� +� +� +� +, +(45) +in which the contravariant velocity components, U, V and W, are calculated as +U = ξt + ξxu + ξyv + ξzw , +V = ηt + ηxu + ηyv + ηzw , +(46) +W = ζt + ζxu + ζyv + ζzw . +The viscous flux vectors, ˆEv, ˆFv and ˆGv, are written as +ˆEv = J−1 +� +� +� +� +� +� +� +� +� +� +� +� +� +0 +ξx +� +τ mod +xx +− 1 +3σxx +� ++ ξyτ mod +xy ++ ξzτ mod +xz +ξxτ mod +xy ++ ξy +� +τ mod +yy +− 1 +3σyy +� ++ ξzτ mod +yz +ξxτ mod +xz ++ ξyτ mod +yz ++ ξz +� +τ mod +zz +− 1 +3σzz +� +ξxβx + ξyβy + ξzβz +� +� +� +� +� +� +� +� +� +� +� +� +� +, +(47) +ˆFv = J−1 +� +� +� +� +� +� +� +� +� +� +� +� +� +0 +ηx +� +τ mod +xx +− 1 +3σxx +� ++ ηyτ mod +xy ++ ηzτ mod +xz +ηxτ mod +xy ++ ηy +� +τ mod +yy +− 1 +3σyy +� ++ ηzτ mod +yz +ηxτ mod +xz ++ ηyτ mod +yz ++ ηz +� +τ mod +zz +− 1 +3σzz +� +ηxβx + ηyβy + ηzβz +� +� +� +� +� +� +� +� +� +� +� +� +� +, +(48) +ˆGv = J−1 +� +� +� +� +� +� +� +� +� +� +� +� +� +0 +ζx +� +τ mod +xx +− 1 +3σxx +� ++ ζyτ mod +xy ++ ζzτ mod +xz +ζxτ mod +xy ++ ζy +� +τ mod +yy +− 1 +3σyy +� ++ ζzτ mod +yz +ζxτ mod +xz ++ ζyτ mod +yz ++ ζz +� +τ mod +zz +− 1 +3σzz +� +ζxβx + ζyβy + ζzβz +� +� +� +� +� +� +� +� +� +� +� +� +� +, +(49) +where βx, βy and βz are defined as +βx = +� +τ mod +xx +− 1 +3σxx +� +˜u + τ mod +xy +˜v + τ mod +xz +˜w − qmod +x +, +βy = τ mod +xy +˜u + +� +τ mod +yy +− 1 +3σyy +� +˜v + τ mod +yz +˜w − qmod +y +, +(50) +βz = τ mod +xz +˜u + τ mod +yz +˜v + +� +τ mod +zz +− 1 +3σzz +� +˜w − qmod +z +. +V. +Dimensionless LES Formulation +A convenient nondimensionalisation is necessary in order to achieve a consistent implementation of the +governing equations of motion. Dimensionless formulation yelds to a more general numerical tool. There is +no need to change the formulation for each configuration intended to be simulated. Moreover, dimensionless +formulation scales all the necessary properties to the same order of magnitude which is a computational +advantage.33 Dimensionless variables are presented in the present section in order perform the nondimen- +sionalisation of Eq. (42). +The dimensionless time, T , is written as function of the freestream speed of sound and of a reference +lenght, D, +T = T a∞ +l +. +(51) +7 of 26 +American Institute of Aeronautics and Astronautics + +In the current work, D represents the jet entrance diameter. This reference lengh is aldo applied to write +the dimensionless length, +l = l +D . +(52) +The dimensionless velocity components are obtained using the freestream speed of sound +vel = +v +a∞ +vel = u, v, w . +(53) +Dimensionless pressure and energy are calculated as +p = +p +ρ∞a2∞ +, +(54) +e = +e +ρ∞a2∞ +. +(55) +Dimensionless density, ρ, temperature, T and viscosity, µ, are calculated using freestream properties +ρ = +ρ +ρ∞ +. +(56) +One can use the dimensionless properties described above in order to write the dimensionless form of the +LES equations as +∂ ˆQ +∂T + ∂ ˆEe +∂ξ + ∂ ˆF e +∂η + ∂ ˆGe +∂ζ += Mj +Re +� +∂ ˆEv +∂ξ + ∂ ˆF v +∂η + ∂ ˆGv +∂ζ +� +, +(57) +where the underlined terms are calculated using non dimensional properties. The jet Mach and Reynolds +numbers are based on the mean jet inlet velocity, Uj, the freestream speed of sound, a∞, density, ρ∞, +viscosity, µ∞ and the reference length, D, +Mj = Uj +a∞ +, +Re = ρ∞UjD +µ∞ +. +(58) +VI. +Numerical Formulation +The governing equations previously described are discretized in a structured finite difference context for +general curvilinear coordinate system.33 The numerical flux is calculated through a central difference scheme +with the explicit addition of the anisotropic artificial dissipation of Turkel and Vatsa.34 The time integration +is performed by an explicit, 2nd-order, 5-stage Runge-Kutta scheme.35,36 Conserved properties and artificial +dissipation terms are properly treated near boundaries in order to assure the physical correctness of the +numerical formulation. +VI.A. +Spatial Discretization +For the sake of simplicity, the formulation discussed in the present section is no longer written using bars, +underbars, etc. However, the reader should notice that the equations are dimensionless and filtered. The LES +equations, presented in Eq. (57), are discretized in space in a finite difference fashion and, then, rewritten as +�∂Q +∂T +� +i,j,k += −RHSi,j,k , +(59) +where RHS is the right hand side of the equation and it is written as function of the numerical flux vectors +at the interfaces between grid points, +RHSi,j,k += +1 +∆ξ +� +Ee(i+ 1 +2 ,j,k) − Ee(i− 1 +2 ,j,k) − Ev(i+ 1 +2 ,j,k) + Ev(i− 1 +2 ,j,k) +� +1 +∆η +� +Fe(i,j+ 1 +2 ,k) − Fe(i,j− 1 +2 ,k) − Fv(i,j+ 1 +2 ,k) + Fv(i,j− 1 +2 ,k) +� +(60) +1 +∆ζ +� +Ge(i,j,k+ 1 +2 ) − Ge(i,j,k− 1 +2 ) − Gv(i,j,k+ 1 +2 ) + Gv(i,j,k− 1 +2 ) +� +. +8 of 26 +American Institute of Aeronautics and Astronautics + +For the general curvilinear coordinate case ∆ξ = ∆η = ∆ζ = 1. The anisotropic artificial dissipation method +of Turkel and Vatsa34 is implemented through the modification of the inviscid flux vectors, Ee, Fe and Ge. +The numerical scheme is nonlinear and allows the selection between artificial dissipation terms of second +and fourth differences, which is very important for capturing discontinuities in the flow. The numerical +fluxes are calculated at interfaces in order to reduce the size of the calculation cell and, therefore, facilitate +the implementation of second derivatives since the the concept of numerical fluxes vectors is used for flux +differencing. Only internal interfaces receive the corresponding artificial dissipation terms, and differences +of the viscous flux vectors use two neighboring points of the interface. +The inviscid flux vectors, with the addition of the artificial dissipation contribution, can be written as +Ee(i± 1 +2 ,j,k) = 1 +2 +� +Ee(i,j,k) + Ee(i±1,j,k) +� +− J−1d(i± 1 +2 ,j,k) , +Fe(i,j± 1 +2 ,k) = 1 +2 +� +Fe(i,j,k) + Fe(i,j±1,k) +� +− J−1d(i,j± 1 +2 ,k) , +(61) +Ge(i,j,k± 1 +2 ) = 1 +2 +� +Ge(i,j,k) + Ge(i,j,k±1) +� +− J−1d(i,j,k± 1 +2 ) , +in which the d(i±1,j,k),d(i,j±1,k) and d(i,j,k±1) terms are the Turkel and Vatsa34 artificial dissipation terms +in the i, j, and k directions respectively. The scaling of the artificial dissipation operator in each coordinate +direction is weighted by its own spectral radius of the corresponding flux Jacobian matrix, which gives the +non-isotropic characteristics of the method.33 The artificial dissipation contribution in the ξ direction is +given by +d(i+ 1 +2 ,j,k) += +λ(i+ 1 +2 ,j,k) +� +ϵ(2) +(i+ 1 +2 ,j,k) +� +W(i+1,j,k) − W(i,j,k) +� +(62) +ϵ(4) +(i+ 1 +2 ,j,k) +� +W(i+2,j,k) − 3W(i+1,j,k) + 3W(i,j,k) − W(i−1,j,k) +� +] , +in which +ϵ(2) +(i+ 1 +2 ,j,k) += +k(2)max +� +νd +(i+1,j,k), νd +(i,j,k) +� +, +(63) +ϵ(4) +(i+ 1 +2 ,j,k) += +max +� +0, k(4) − ϵ(2) +(i+ 1 +2 ,j,k) +� +. +(64) +The original article34 recomends using k(2) = 0.25 and k(4) = 0.016 for the dissipation artificial constants. +The pressure gradient sensor, νd +(i,j,k), for the ξ direction is written as +νd +(i,j,k) = |p(i+1,j,k) − 2p(i,j,k) + p(i−1,j,k)| +p(i+1,j,k) − 2p(i,j,k) + p(i−1,j,k) +. +(65) +The W vector from Eq. (62) is calculated as a function of the conserved variable vector, ˆQ, written in Eq. +(40). The formulation intends to keep the total enthalpy constant in a final converged steady solution, which +is the correct result for the Navier-Stokes equations with Re → ∞. This approach is also valid for the +viscous formulation because the dissipation terms are added to the inviscid flux terms, in which they are +really necessary to avoid nonlinear instabilities of the numerical formulation. The W vector is given by +W = ˆQ + [0 0 0 0 p]T . +(66) +The spectral radius-based scaling factor, λ, for the i − th direction is written +λ(i+ 1 +2 ,j,k) = 1 +2 +�� +λξ +� +(i,j,k) + +� +λξ +� +(i+1,j,k) +� +, +(67) +where +λξ(i,j,k) = λξ +� +1 + +�λη +λξ +�0.5 ++ +�λζ +λξ +�0.5� +. +(68) +The spectral radii, λξ, λη and λζ are given by +λξ += +|U| + a +� +ξ2x + η2y + ζ2z , +λξ += +|V | + a +� +ξ2x + η2y + ζ2z , +(69) +λξ += +|W| + a +� +ξ2x + η2y + ζ2z , +9 of 26 +American Institute of Aeronautics and Astronautics + +in which, U, V and W are the contravariant velocity components in the ξ, η and ζ, previously written in +Eq. (47), and a is the local speed of sound, which can be written as +a = +�γp +ρ . +(70) +The calculation of artificial dissipation terms for the other coordinate directions are completely similar and, +therefore, they are not discussed in the present work. +VI.B. +Time Marching Method +The time marching method used in the present work is a 2nd-order, 5-step Runge-Kutta scheme based on +the work of Jameson and co-workers.35,36 The time integration can be written as +Q(0) +(i,jk,) += +Q(n) +(i,jk,) , +Q(l) +(i,jk,) += +Q(0) +(i,jk,)− +αl∆t(i,j,k)RHS(l−1) +(i,j,k) +l = 1, 2 · · · 5, +Q(n+1) +(i,jk,) += +Q(5) +(i,jk,) , +(71) +in which ∆t is the time step and n and n + 1 indicate the property values at the current and at the next +time step, respectively. The literature35,36 recommends +α1 = 1 +4 , +α2 = 1 +6 , +α3 = 3 +8 , +α4 = 1 +2 , +α5 = 1 , +(72) +in order to improve the numerical stability of the time integration. The present scheme is theoretically stable +for CFL ≤ 2 +√ +2, under a linear analysis.33 +VII. +Boundary Conditions +The present section presents all boundary conditions used for the turbulent compressible jet flow simu- +lation such as inlet, outlet, centerline and far field boundary conditions. Moreover, the numerical treatment +of the centerline singularity and the implementation of the periodic boundary in the azimuthal direction are +also discussed in the end of the section. +VII.A. +Far Field Boundary +Riemann invariants37 are used to implement far field boundary conditions. They are derived from the cha- +racteristic relations for the Euler equations. At the interface of the outer boundary, the following expressions +apply +R− = R− +∞ += +qn∞ − +2 +γ − 1a∞ , +(73) +R+ = R+ +e += +qne − +2 +γ − 1ae , +(74) +where ∞ and e indexes stand for the property in the freestream and in the internal region, respectively. qn +is the velocity component normal to the outer surface, defined as +qn = u · ⃗n , +(75) +and ⃗n is the unit outward normal vector +⃗n = +1 +� +η2x + η2y + η2z +[ηx ηy ηz]T . +(76) +Equation (75) assumes that the η direction is pointing from the jet to the external boundary. Solving for qn +and a, one can obtain +qnf = R+ + R− +2 +, +af = γ − 1 +4 +(R+ − R−) . +(77) +10 of 26 +American Institute of Aeronautics and Astronautics + +The index f is linked to the property at the boundary surface and will be used to update the solution at this +boundary. For a subsonic exit boundary, 0 < qne/ae < 1, the velocity components are derived from internal +properties as +uf += +ue + (qnf − qne)ηx , +vf += +ve + (qnf − qne)ηy , +(78) +wf += +we + (qnf − qne)ηz . +Density and pressure properties are obtained by extrapolating the entropy from the adjacent grid node, +ρf = +� +ργ +ea2 +f +γpe +� +1 +γ−1 +, +pf = +ρfa2 +f +γ +. +For a subsonic entrance, −1 < qne/ae < 0, properties are obtained similarly from the freestream variables as +uf += +u∞ + (qnf − qn∞)ηx , +vf += +v∞ + (qnf − qn∞)ηy , +(79) +wf += +w∞ + (qnf − qn∞)ηz , +ρf = +� +ργ +∞a2 +f +γp∞ +� +1 +γ−1 +. +(80) +For a supersonic exit boundary, qne/ae > 1, the properties are extrapolated from the interior of the domain +as +ρf += +ρe , +uf += +ue , +vf += +ve , +(81) +wf += +we , +ef += +ee , +and for a supersonic entrance, qne/ae < −1, the properties are extrapolated from the freestream variables as +ρf += +ρ∞ , +uf += +u∞ , +vf += +v∞ , +(82) +wf += +w∞ , +ef += +e∞ . +VII.B. +Entrance Boundary +For a jet-like configuration, the entrance boundary is divided in two areas: the jet and the area above it. +The jet entrance boundary condition is implemented through the use of the 1-D characteristic relations for +the 3-D Euler equations for a flat velocity profile. The set of properties then determined is computed from +within and from outside the computational domain. For the subsonic entrance, the v and w components of +the velocity are extrapolated by a zero-order extrapolation from inside the computational domain and the +angle of flow entrance is assumed fixed. The rest of the properties are obtained as a function of the jet Mach +number, which is a known variable. +(u)1,j,k += +uj , +(v)1,j,k += +(v)2,j,k , +(83) +(w)1,j,k += +(w)2,j,k . +11 of 26 +American Institute of Aeronautics and Astronautics + +The dimensionless total temperature and total pressure are defined with the isentropic relations: +Tt = 1 + 1 +2(γ − 1)M 2 +∞ +and +Pt = 1 +γ (Tt)γ/(γ−1) . +(84) +The dimensionless static temperature and pressure are deduced from Eq. (84), resulting in +(T)1,j,k = +Tt +1 + 1 +2(γ − 1)(u2 + v2 + w2)1,j,k +and +(p)1,j,k = 1 +γ (T)γ/(γ−1) +1,j,k +. +(85) +For the supersonic case, all conserved variables receive jet property values. +The far field boundary conditions are implemented outside of the jet area in order to correctly propagate +information comming from the inner domain of the flow to the outter region of the simulation. However, +in the present case, ξ, instead of η, as presented in the previous subsection, is the normal direction used to +define the Riemann invariants. +VII.C. +Exit Boundary Condition +At the exit plane, the same reasoning of the jet entrance boundary is applied. This time, for a subsonic exit, +the pressure is obtained from the outside and all other variables are extrapolated from the interior of the +computational domain by a zero-order extrapolation. The conserved variables are obtained as +(ρ)IMAX,j,k += +(p)IMAX,j,k +(γ − 1)(e)IMAX−1,j,k +, +(86) +(⃗u)IMAX,j,k += +(⃗u)IMAX−1,j,k, +(87) +(ei)IMAX,j,k += +(ρ)IMAX,j,k +� +(e)IMAX−1,j,k + 1 +2(⃗u)IMAX,j,k · (⃗u)IMAX,j,k +� +, +(88) +in which IMAX stands for the last point of the mesh in the axial direction. For the supersonic exit, all +properties are extrapolated from the interior domain. +VII.D. +Centerline Boundary Condition +The centerline boundary is a singularity of the coordinate transformation, and, hence, an adequate treatment +of this boundary must be provided. The conserved properties are extrapolated from the ajacent longitudinal +plane and are averaged in the azimuthal direction in order to define the updated properties at the centerline +of the jet. +The fourth-difference terms of the artificial dissipation scheme, used in the present work, are carefully +treated in order to avoid the five-point difference stencils at the centerline singularity. If one considers the +flux balance at one grid point near the centerline boundary in a certain coordinate direction, let wj denote +a component of the W vector from Eq. (66) and dj denote the corresponding artificial dissipation term at +the mesh point j. In the present example, (∆w)j+ 1 +2 stands for the difference between the solution at the +interface for the points j+1 and j. The fouth-difference of the dissipative fluxes from Eq. (62) can be written +as +dj+ 1 +2 = (∆w)j+ 3 +2 − 2 (∆w)j+ 1 +2 + (∆w)j− 1 +2 . +(89) +Considering the centerline and the point j = 1, as presented in Fig. 1, the calculation of d1+ 1 +2 demands +the (∆w) 1 +2 term, which is unknown since it is outside the computation domain. +In the present work a +extrapolation is performed and given by +(∆w) 1 +2 = − (∆w)1+ 1 +2 . +(90) +This extrapolation modifies the calculation of d1+ 1 +2 that can be written as +dj+ 1 +2 = (∆w)j+ 3 +2 − 3 (∆w)j+ 1 +2 . +(91) +The approach is plausible since the centerline region is smooth and it does not have high gradients of +properties. +12 of 26 +American Institute of Aeronautics and Astronautics + +Figure 1. +Boundary point distribution in the calculation of dissipation operator at the centerline.33 +VII.E. +Periodic Boundary Condition +A periodic condition is implemented between the first (K = 1) and the last point in the azimutal direction +(K = KMAX) in order to close the 3-D computational domain. There are no boundaries in this direction, +since all the points are inside the domain. The first and the last points, in the azimuthal direction, are +superposed in order to facilitate the boundary condition implementation which is given by +(ρ)i,j,KMAX += +(ρ)i,j,1 , +(u)i,j,KMAX += +(u)i,j,1 , +(v)i,j,KMAX += +(v)i,j,1 , +(92) +(w)i,j,KMAX += +(w)i,j,1 , +(e)i,j,KMAX += +(e)i,j,1 . +VIII. +Dynamic Mode Decomposition +VIII.A. +Theoretical Framework +The DMD method provides a spatio-temporal decomposition of the flow into a set of dynamic modes that +are derived from time-resolved snapshots. For example, a generic flow variable, xDMD(x, y, z, t), where x, y, +z and t stand for spatial coordinates and time, respectively, can be represented by +xDMD(x, y, z, t) = +m−1 +� +i=1 +ai exp(λit) φi(x, y, z) . +(93) +Here, ai and λi are the amplitude and the frequency of the spatial mode φi. The underlying mathematics is +closely related to the idea of the Arnoldi algorithm.4 This flow variable, extracted from the simulation, can +be represented in the form of a snapshot sequence X = [x(t1) x(t2) · · · x(tm)] ∈ Rn×m, where x(ti) ∈ Rn +is the i-th snapshot, m denotes the number of snapshots and n, the spatial dimension of each time snapshot. +Each snapshot, x(ti), contains a set of variables depending on the user’s choice. +The present study is +designed to collect data regularly separated in time by ∆t even though recent techniques allow irregularly +spaced sampling in time of the data.38 The authors assume that there exists a linear operator A ∈ Rn×n +connecting two consecutive snapshot giving +xi+1 = A xi +for i = 1, · · · , m − 1. +(94) +The A operator is an approximation of the Koopman operator,39 whose eigen-elements can approximate +the underlying dynamics of the flow, even if such dynamics is nonlinear. The objective of the DMD is the +determination of these characteristics. The selection of the eigen-elements of A is a matter of importance +since the accuracy of the results, as well as the computational costs, both depend significantly on the method +of choice. The strategy used in the present work is a combination of the total least-squares DMD, described +13 of 26 +American Institute of Aeronautics and Astronautics + +j=4 +j=3+1/2 +j=3 +j=2+1/2 +j=2 +j=1+1/2 +j=1 +12 +grid node +X +interfacein Ref. 24, and the streaming DMD algorithm presented in Hemati et al.23 The former technique provides a +noise-aware DMD technique while the latter allows the assimilation “on-the-fly” of new incoming snapshots +and it can even theoretically include an infinite number, m, of snapshots. Hemati et al.40 ran successfully +this combined technique to analyze the dynamics of the flow separation over a flat plate. In practice, the A +DMD operator of Eq. (94) can be defined as A = YX+ using the previously defined snapshot matrix, X, +and its time-shifted version Y = [x(t1 + ∆t) x(t2 + ∆t) · · · x(tm + ∆t)]. In this relation, X+ stands for the +Moore-Penrose pseudoinverse of X. The solution of the problem in the present form is prohibitively expensive +in terms of CPU and memory costs. The streaming DMD approach suggests a solution to reformulate A in +order to be able to handle large dimension problems. First, the augmented snapshot matrix, Z = [X Y]T , is +built.24 After substitution, a low-dimensional version of A, ˜ +A, can be obtained under the form +˜ +A = QT +x +� +0 +I +� +Qz Gz QT +z +� +0 +I +� +QxG+ +x +∈ Rr×r , +(95) +where r is the rank of X, Qx and Qz are obtained from the QR-decomposition of X and Z, respectively. +Hence, one could write that X = QxRx and Z = QzRz. Therefore, one can also write that Gz = RzRT +z and +Gx = RxGzRT +z . This procedure allows an incremental update of new available snapshots, without storing all +of them in memory. Moreover, in this expression, the total number snapshot, m, does not appear anymore. +During the streaming DMD process, a POD compression is included allowing the user to choose the rank +of the DMD operator, r. The DMD modes and frequencies are given by the eigenvectors and eigenvalues of +˜ +A, such that φi is the i-th eigenvector with the associated eigenvalue, µi. Hence, the associated growth rate +and frequency of the i-th DMD mode are given by +σi = log(|µi|) +∆t +and +ωi = arg(µi) +∆t +. +(96) +Finally, λi = log(µi)/∆t. Another interesting aspect of the DMD, is that knowing the first snapshot and +the eigenvalues of the DMD operator, one can predict the temporal behavior of the mode. Indeed, using a +discretized version of Eq. (93) expressed at any time instant k = 1, · · · , m − 1, +xk = +m−1 +� +i=1 +θi(k) φi, +(97) +where θi are the temporal coefficients of the eigenvectors φi. It comes directly, using Eq. (94), that +xk+1 += +A xk += +m−1 +� +i=1 +θi(k)A φi += +m−1 +� +i=1 +θi(k)µiφi += +Ak x1 += +m−1 +� +i=1 +θi(1)µk +i φi, +(98) +Using the work of Ref. 41, the matrix of the initial coefficient can be calculated using the relation +θ(1) = φ+x1. +(99) +VIII.B. +Choice of the Parameters +Two main parameters are considered in the DMD framework initially introduced by Schmid.4 +The first +one is ∆t, the constant time-step between two consecutive snapshots, while the second one is m, the total +number of snapshots. Both of them require a good knowledge of the physical phenomenon under study. +According to Schmid,4 the sample rate must be sufficiently high, about three times the Nyquist cutoff, to +capture correctly the dynamics of an oscillatory flow. The idea is, then, to tune the sampling frequency +based on the phenomenon the user wants to study. However, following Chen et al.,42 using a high sample +rate, the snapshots are likely to be correlated in time. This is a problem since the method impose the use +of a linear independent dataset to work properly. Finally, a high number, m, of snapshots could also affect +the linear independency of the snapshots. In the algorithm used in the present work, a Gram-Schmidt step +is included in the process to address this problem.10,23 +14 of 26 +American Institute of Aeronautics and Astronautics + +IX. +Case of a High Reynolds Number Supersonic Jet Flow +The present section is devoted to the study of a supersonic perfectly expanded jet flow. The geometry and +flow configurations of interest are presented, followed by large eddy simulation results, which are compared +to analytical, numerical and experimental data from the literature.43,44 The LES results provide a database +for three DMD studies using the velocity magnitude, the vorticity, represented by the Q criterion, and the +divergence of the flow velocity. +IX.A. +Geometry and Mesh Configurations +Figure 2 illustrates a three-dimensional view of the representative domain for the jet flow simulations. The +geometry resembles a frustrum of a cone with the jet entering the computational domain through the small +base at x = 0, and leaving the domain at the large base at x = 30D. +The radii of the entrance and +exit plans are approximately 8D and 9D, respectively. The authors have chosen not to include the nozzle +geometry in the computational domain. Hence, the jet entrance is located at x = 0, for |r|/D ≤ 0.5, where +|r| = +� +y2 + z2 is the distance from the centerline in the radial direction and D is the incoming jet diameter. +The computational domain is created in two steps. First, a 2-D region is generated. In the sequence, this +region is rotated around the horizontal direction, x, indicated by the discontinuous blue line in Fig. 2, in +order to generate a fully 3-D geometry. The rotation approach generates a singularity at the centerline of +the domain. The treatment of this region is discussed in the boundary conditions section. +The commercial mesh generator ANSYS® ICEM CFD45 is used for the creation of the 2-D domain for +an azimuthal plane. The zones of this geometry are created based on results from simulations of previous +work31 in order to refine the mesh in the shear layer region of the flow until x = 10D, after the end of the +potential core. The mesh is, then, coarsened towards the outer regions of the domain in order to dissipate +properties of the flow far from the jet. Such mesh refinement approach can avoid reflection of information +into the domain. The radial and longitudinal dimensions of the smallest distance between mesh points of +the computational grid are given by (∆r)min = 0.002 and (∆x)min = 0.0126, respectively. This minimal +spacing occurs at the lipline of the jet and at the entrance of the computational domain. These dimensions +are based on a reference grid of Mendez et al.43,46 The resulting computational grid is composed by 537 +points in the axial direction, 442 points in the radial direction and 360 points in the azimuthal direction, +yielding approximately 85 million grid points. For further details about the mesh generation, the reader is +referred to the work of Junqueira-Junior.32 +Figure 2. +3-D view of two XY slices of the grid, located above and below the centerline highlighted by a discontinuous +blue line. The red arrow indicates the jet entrance inside the domain. +15 of 26 +American Institute of Aeronautics and Astronautics + +07 +-5 +5 +10 +15 +x +5 +20 +25 +5 +30IX.B. +Flow Configuration +The flow is characterized by an unheated perfectly expanded jet with a Mach number of 1.4 at the domain +entrance. Therefore, the pressure ratio, PR = Pj/P∞, and the temperature ratio, TR = Tj/T∞, between +the jet exit and the ambient freestream are equal to one, PR = 1 and TR = 1. The time step used in the +simulation is constant and equal to 2.0 × 10−4 in dimensionless form. The Reynolds number of the jet is +Re = 1.57 × 106, based on the jet entrance diameter. This flow configuration is chosen due to the absence of +strong shocks waves. Strong discontinuities must be carefully treated using numerical approaches, and the +authors did not want to deal with those issues at the present time. Moreover, numerical and experimental +data for a perfectly expanded jet flow configuration, such as the one used in the present work, are available +in the literature such as the work of Mendez et al.43,46 and the work of Bridges and Wernet.44 +Properties of flow at the inlet and at the far field regions have to be provided to the code in order to +impose the boundary conditions. Density, ρ, temperature, T, velocity, U, Reynolds number, Re, and specific +heat at constant volume, Cv, are provided in the dimensionless form to the simulation. These dimensionless +properties are given by +ρj = 1.00 , +ρ∞ = 1.00 , +Tj = 1.00 , +T∞ = 1.00 , +(100) +Uj = 1.4 , +U∞ = 0.00 , +Rej = 1.57 × 106 , +Cv = 1.786 , +where the j subscript stands for property at the jet entrance and the ∞ subscript stands for property at the +far field region. +IX.C. +Data Extraction Procedure +For the present study, data are extracted after a preliminary simulation is run in order to achieve a statistically +steady state condition for the jet flow. This initial preliminary simulation lasts 96 dimensionless time units. +For the current jet exit Mach number of Mj = 1.4, this simulation time represents approximately 3 flow- +through times (FTT). One flow-through time is the time for a particle to cross the entire domain from the +jet entrance to the domain exit. After the flow initialization process, the simulations are restarted and run +for another period of time in which data of the flow are extracted and recorded at a fixed frequency. +Table 2. +Data extraction characteristics +Simulation +∆t c∞/D +No. Extractions +Grid Size +Total Time +FTT +LES statistics +0.06 +4096 +500 × 425 (2-D) +245.76 +≈ 8 +DMD +0.12 +256 +473 × 412 × 180 (3-D) +30.72 +≈ 1 +The temporal characteristics of the data extraction are displayed in Tab. 2 for both the LES statistics and +the DMD computations. These data processing methods are very different one from each other, especially +because of the grid dimension. In the present work, the LES statistics are computed only along 2-D surfaces, +whereas DMD calculations use three-dimensional snapshots as input. The snapshots extracted during the +DMD process have more than 35 million points and they are stored in the PLOT3D formata, adapted for +structured meshes. The memory size of one snapshot, used for the DMD calculations, is about 1.5 Gb. On +the other hand, the time-dependent LES surfaces are all included in one single CGNS file of 40 Gb. Finally, +the total simulation time, necessary for obtaining the LES statistics, is higher than that used in Ref. 47 +for the same purpose, and this can be considered as a fairly large time sample for an LES calculation. As +indicated in Tab. 2, a total of 8 flow-through times have been used in order to obtain the LES statistics. +IX.D. +Large Eddy Simulation Results +In this subsection, 2-D distributions of properties and profiles are collected from the compressible LES +simulation and compared with numerical and experimental results from the literature.43,44,46 A longitudinal +ahttps://www.grc.nasa.gov/www/wind/valid/plot3d.html +16 of 26 +American Institute of Aeronautics and Astronautics + +(a) Time averaged axial velocity component. +(b) RMS value of the fluctuating part of the axial velocity component. +(c) Turbulent kinetic energy. +Figure 3. +Contour plots of longitudinal planes of statistically converged jet properties. +The white line defines the +potential core of the jet, where u = 0.95Uj. +plane view of the statistically-converged time-averaged distributions of three flow properties, namely, axial +velocity component, ⟨U⟩, RMS value of the fluctuating part of the axial velocity component, urms, and +turbulent kinetic energy, k, are presented in Fig. 3. The statistical properties of the LES results are calculated +using much more snapshots and with a more refined time increment than the numerical reference data.43,46 +Each variable displays a fairly smooth flow field, confirming the good statistical convergence of the results. +Moreover, the contours of ⟨U⟩, urms and k display a classical shape, with urms spreading along with the jet +shear layer and with high values of k at the beginning of the mixing layer. The white solid line defines the +jet potential core region, U 95% +j +, which is a characteristic parameter of jet flows. The potential core length, +δ95% +j +, is defined as the distance from the jet entrance and along the centerline until the jet velocity reaches +95% of the velocity of the jet at the inlet. +In line with previous work, reported in Ref. 31, the current simulation aims to reduce the error with +respect to the experimental data in Ref.44 by refining the grid in the jet potential core. Table 3 presents the +size of the potential cores for the current simulation, compared to the numerical results in Refs. 31, 43 and +46. The table presents the relative error compared to the experimental data.44 The present LES calculations +are performed on the same grid geometry used in Ref. 31, but with more points inside the potential core. +As one can see in the table, the error has been reduced from 26% to 22%. The grid used in the present work +needs to be further refined in order to overcome the dissipative characteristics of 2nd-order scheme used and, +17 of 26 +American Institute of Aeronautics and Astronautics + +1 +1.4 +0 +-1 +-0.1 +2 +2 +4 +6 +8 +10 +12 +14 +x0.3 +1 +0 +-1 +2 +2 +4 +6 +8 +10 +12 +14 +x0.09 +1 +0 +-1 +D +2 +2 +4 +6 +8 +10 +12 +14 +xhence, keep reducing the magnitude of error when compared to experimental data. +Table 3. +Potential core length comparisons. +Simulation +δ95% +j +Relative error +Current work +7.05 +22% +Junqueira-Junior et al.31 +6.84 +26% +Mendez et al.43,46 +8.35 +8% +The evolution of the averaged axial component of velocity and the evolution of the RMS value of the +fluctuating part of the axial component of velocity along the centerline and the lipline are illustrated in Figs. +4 and 5, respectively. The solid line stands for the results of the present case, the open square symbols +represent the LES results of Mendez et al.,43,46 while the triangular symbols stand for the experimental data +of Bridges and Wernet.44 The lipline is the surface defined over r = 0.5D, which represents the boundary +of the jet at the entrance of the domain. The comparison of profiles indicates that distributions of ⟨U⟩/Uj +along the centerline correlates well with the references until x = 7.0D, where the grid has good resolution. +The time averaged axial component of velocity start to correlate poorly with the reference when the mesh +spacing increases, x > 7.0D, due to the mesh coarsening in the streamwise direction. The mesh coarsening +is used in order to add artificial dissipation towards the exit of the domain, since the numerical framework +does not have a sponge zone implemented. The time averaged axial component of velocity calculated along +the lipline correlates well with the references until x ≈ 6.0D. The magnitude of ⟨U⟩/Uj along the lipline is +understimated for x > 6.0D. +(a) Centerline +(b) Lipline +Figure 4. Averaged axial component of velocity along the centerline and lipline. The solid line stands for the results of +the present case, the open square symbols represent the numerical references43, 46 and the triangular symbols are the +experimental data.44 +The distribution of urms/Uj calculated along the centerline fits the numerical and experimental reference +distributions of the same property for x < 4.0D. However, it presents an overestimated distribution of +urms/Uj when compared with both numerical and experimental data at other positions along the centerline. +The numerical reference has also calculated an overestimated distribution of urms/Ujalong the centerline +when compared to the experimental reference at x > 5.0D. The distribution of urms/Uj along the lipline +calculated by the current work and by the numerical reference present similar behavior. Nonetheless, the +distributions are overestimated when compared to the experimental data. +IX.E. +Dynamic Mode Decomposition Results +The streaming version23 of the total-least-squares DMD algorithm24 on volumetric data extracted during the +large eddy simulations described in Sect. IX is computed in the present work. Considering that the DMD +calculation is performed in serial mode, the computer memory is the limiting factor to compute the DMD +18 of 26 +American Institute of Aeronautics and Astronautics + +口 +F +D +口 +4 +口 +口 +口 +44 +口 +口 +0 +口 +4 +.0.8 +口 +4 +口 +U>/U +口 +口 +口 +口 +V +口 + ++ +口 +0.6 +口 +口 +0.4 +1 +0 +5 +10 +15 +20 +x0.8 +0.7 +J>/U, +口 +口 +口± +0.6 +口 +占 +*± ± +口 +口 +0.5 +口 +口 +口 +0.4 +、 +5 +10 +15 +20 +x(a) Centerline +(b) Lipline +Figure 5. +RMS value of the fluctuating part of the axial component of velocity along the centerline and lipline. The +solid line stands for the results of the present case, the open square symbols represent the numerical references43, 46 +and the triangular symbols are the experimental data.44 +modes. The DMD calculation is run on a single processor with 128 GB of RAM. According to Hemati +et al.,23 the computational cost of the algorithm to calculate the DMD eigen-elements is O(nr2), where +n and r are the snapshot dimension and the maximum rank of the DMD operator, respectively. For the +latter parameter, the streaming version of the DMD algorithm includes a compression step allowing to set +it arbitrarily. Then, the choice of these parameters is a compromise between spatial and spectral resolution. +The jet entrance, the potential core and the near field of the jet are included in the computational domain +in order to prioritize the spatial aspects of the flow. +Therefore, the results should include the aerodynamic structures as well as the generated acoustic waves. +However, the original snapshots have been under-sampled in spatial resolution in order to handle manageable +snapshots. The dimensions of the snapshots are specified in Tab. 2, counting approximatively 35 million +grid points. Finally, considering 256 snapshots without subtracting the mean, r has been set equal to 50, +which was the higher affordable number of retained modes in relation to the available computer memory. +In the present case, three variables were extracted from the LES calculations. Hence, three different DMD +reconstruction procedures were performed, using snapshots of the velocity magnitude, the vorticity, based +on the Q criterion, and the divergence of the velocity. In the following subsections, results are discussed +regarding their spectral content (Sect. IX.E.1) and spatial shape (Sect. IX.E.2). +IX.E.1. +Spectral Analysis +Figure 6 displays three different ways of representing the DMD spectrum obtained after the DMD compu- +tation using snapshots of the velocity magnitude. Figure 6(a) presents the 50 eigenvalues of A DMD linear +operator. The symbols are colored by the initial amplitude of the DMD modes, ∥θi(1)∥, which are defined +in Eq. (99). The choice of this parameter to differentiate the dynamic modes comes from Eqs. (93) and (98). +The initial amplitude of the DMD modes has also been taken into account by Sayadi et al.7 All dynamic +modes which are located inside the unit circle are stable. The only one DMD mode located on the unit circle +is a steady mode which, in general, retrieves the mean characteristics of the flow.42 The stable dynamic +modes are unsteady and have a complex conjugate, symmetric with respect to the Im(µi) = 0 axis. In +Fig. 6 (b), the growth rate of each mode, σi, is plotted versus the frequency, ωi. A mode is stable if σi is +negative, which is in agreement with the discussion considering Fig. 6(a). Finally, Fig. 6 (c) presents the +most amplified DMD mode as a function of the Strouhal number. Four dynamic modes displaying a high +amplitude have been selected. In Figs. 6(b) and (c), it appears that the stability of the mode is not linked +with its initial amplitude. The DMD Mode 5 is more stable than the DMD Mode 7 (σ5 < σ7). However, +∥θ5(1)∥ is larger than ∥θ7(1)∥. Therefore, one can state that the dynamic mode 5 is initially more amplified +than the dynamic mode 7, but it decays more quickly as the simulation advances in time. +Figures 7 and 8 show two different sets of spectra obtained from the DMD computations using snapshots +19 of 26 +American Institute of Aeronautics and Astronautics + +0.15 +口 +口 +口 +0.1 +口 +口 +口 +口 +口 +口 +口 +口 +口 +口 +口 +口 +口 +0.05 +口 +口 +口 +口 +口 +口 +口口 +5 +10 +15 +20 +x口 +0.15 +口 +0.05 +0 +0 +5 +10 +15 +20 +x(a) Eigenvalues of A, µ +(b) Eigenvalues of the DMD modes, λ +(c) Initial amplitude of the DMD modes, +∥θi(1)∥ +Figure 6. +Spectra from DMD computation using snapshots of velocity magnitude. +In (a) and (b), the symbols are +colored by the mode amplitude, ∥θi(1)∥. +of the vorticity, based on the Q-criterion, and the divergence of the velocity, respectively. Once again, all +dynamic modes are stable, but the one representing the mean flow is neutrally stable. More DMD modes +have been highlighted by a number in order to identify them in each spectrum. +(a) Eigenvalues of A, µ +(b) Eigenvalues of the DMD modes, λ +(c) Initial amplitude of the DMD modes, +∥θi(1)∥ +Figure 7. +Spectra from DMD computation using snapshots of vorticity. In (a) and (b), the symbols are colored by the +mode amplitude, ∥θi(1)∥. +One can observe in Figs. 6(c), 7(c) and 8(c) that every spectra contain a dynamic mode at St ≈ 0.25 +and at St ≈ 0.48. The clustering around specific frequencies for different DMD analyses denotes important +dynamic activity at these frequencies. The DMD modes associated to each frequency are shown in Tab. 4. +These characteristic frequencies coincide with the experimental far field pressure peaks observed by Bridges +et al.44 In the next subsection, the spatial shapes of these dynamic modes, given in Tab. 4, are discussed in +more detail. +Table 4. +Characteristic frequencies and associated DMD modes +St +Velocity magnitude +Vorticity +Divergence of velocity +0.25 +3 +3 +7 +0.48 +7 +7 +5 +20 of 26 +American Institute of Aeronautics and Astronautics + +0 +3 +-0.5 +5 +b +-20 +0 +20 +wi103 +101 +100 +0 +0.5 +1.5 +2 +2.5 +3 +1S19 +15 +6 +0.5 +7 +5 +3 +(ri)wl +0 +-0.5 +-1 +-0.5 +0 +0.5 +Re(μ)9 +-0.5 +15 +5 +b +19 +-20 +0 +20 +wi105 +II (L)@ I +3 +9 +15 +0 +0.5 +1.5 +2 +2.5 +3 +1S0.5 +0 +7 +5 +3 +(ri)wl +0 +-0.5 +-1 +-0.5 +0 +0.5 +Re(μ;)(a) Eigenvalues of A, µ +(b) Eigenvalues of the DMD modes, λ +(c) Initial amplitude of the DMD modes, +∥θi(1)∥ +Figure 8. +Spectra from DMD computation using snapshots of divergence of velocity. In (a) and (b), the symbols are +colored by the mode amplitude, ∥θi(1)∥. +IX.E.2. +Spatial Modes Analysis +The averaged axial velocity component of the steady DMD mode is shown in Fig. 9, using the same color +coding for the contours as the LES mean flow illustrated in Fig. 3. One can notice a white gap around +the centerline of the flow. The gap is created because the radial coordinate of the snapshot grid starts at +the 20th point in the radial direction, in order to reduce the computational cost of the DMD computation. +The DMD mode has been reconstructed by multiplying the mode shape by its initial amplitude ∥θ0(1)∥. A +fairly good agreement between the DMD calculation and the large eddy simulation is found regarding the +potential core length as well as the contour levels, even considering that the sample rate and the number of +snapshots are quite different in the DMD calculation when compared to the LES statistics calculation. +Figure 9. +Slice of the three dimensional steady DMD mode for the velocity magnitude. +The white line defines the +potential core limits. Contours are the same as in Fig. 3. +As mentioned in the previous subsection, the experimental far field pressure spectrum of Bridges et al.44 +displays two peaks at St ≈ 0.25 and St ≈ 0.48. +Modes at the same frequencies are observable in the +three DMD analyses performed in the present study. Figure 10 displays the DMD modes associated to the +first frequency, St ≈ 0.25, while Fig. 11 shows the DMD modes associated to St ≈ 0.48. In both figures, +isosurfaces and 2-D cut planes of velocity magnitude, vorticity (Q criterion) and divergence of velocity +21 of 26 +American Institute of Aeronautics and Astronautics + +20 +0.5 +(ri)wl +0 +-0.5 +1 +-0.5 +0 +0.5 +Re(μ;)6 +5 +-0.5 +b +20 +3 +-20 +0 +20 +wi103 +II (L)@ I +20 +5 +7 +102 +3 +0 +0.5 +1 +1.5 +2 +2.5 +3 +1S3 +2 +1 +0 +-1 +-2 +-3 +0 +2 +4 +6 +8 +10 +Xare presented. Considering the high Reynolds number of the present work, and the rapid transition from +laminar flow at the jet inlet to a turbulent jet mixing layer, it is possible to observe coherent behavior in +the jet dynamics. Moreover, the three variables, for which the DMD computations were performed, bring +different information about the flow dynamics. While the vorticity modes seem to enlighten the mixing layer +dynamics, the velocity magnitude as well as the divergence of velocity seem to highlight the aeracoustic +dynamics. +(a) DMD mode 3 – Velocity magnitude +(b) DMD mode 3 – Velocity magnitude +(c) DMD mode 3 – Vorticity (Q criterion) +(d) DMD mode 3 – Vorticity (Q criterion) +(e) DMD mode 7 – Divergence of velocity +(f) DMD mode 7 – Divergence of velocity +Figure 10. +Visualization of the DMD modes found at St ≈ 0.25. (a) and (b) display the real part of Mode 3 extracted +from the DMD analysis using snapshots of velocity magnitude, (c) and (d) display the real part of Mode 3 extracted +from the DMD analysis using snapshots of vorticity (Q criterion), and (e) and (f) display the real part of Mode 7 +extracted from the DMD analysis using snapshots of divergence of velocity. +The left and right columns show 3-D +isosurfaces and 2-D cut-plane visualizations of the modes, respectively (positive in red and negative in blue). +Figures 10 and 11 indicate that, until x ≈ 1, small coherent vortical structures are growing in the +jet mixing layer, generating small acoustic waves. Further downstream, the flow has already transitioned +and large acoustic waves are generated and are propagated in the downstream direction. As expected, the +wavelength of the large acoustic waves depends on the DMD mode frequency. One can see, when comparing, +for instance, Fig. 10(a) with Fig. 11(a), or Fig. 10(e) with Fig. 11(e), that the wavelength of the coherent +22 of 26 +American Institute of Aeronautics and Astronautics + +0 +2 +3 +0 +2 +4 +6 +8 +10 +X3 +2 +2 +3 +0 +2 +4 +6 +8 +10 +X0 +1 +2 +3 +0 +2 +4 +6 +8 +10 +X3 +2 +1 +2 +3 +0 +2 +4 +6 +8 +10 +X3 +-1 +2 +3 +0 +2 +4 +6 +8 +10 +X3 +2 +1 +2 +3 +0 +2 +4 +6 +8 +10 +X(a) DMD mode 7 – Velocity magnitude +(b) DMD mode 7 – Velocity magnitude +(c) DMD mode 7 – Vorticity (Q criterion) +(d) DMD mode 7 – Vorticity (Q criterion) +(e) DMD mode 5 – Divergence of velocity +(f) DMD mode 5 – Divergence of velocity +Figure 11. +Visualization of the DMD modes found at St ≈ 0.48. (a) and (b) display the real part of Mode 7 extracted +from the DMD analysis using snapshots of velocity magnitude, (c) and (d) display the real part of Mode 7 extracted +from the DMD analysis using snapshots of vorticity (Q criterion), and (e) and (f) display the real part of Mode 5 +extracted from the DMD analysis using snapshots of divergence of velocity. +The left and right columns show 3-D +isosurfaces and 2-D cut-plane visualizations of the modes, respectively (positive in red and negative in blue). +structures is divided by two when the frequency is doubled. Moreover, it is easy to verify, for instance, in +Fig. 11(f), the relation between the wavelength of the large acoustic waves and the actual frequency of the +DMD mode, ωi. +Another interesting aspect is the presence of small vortices in the inner mixing layer, at the interface +with the potential core. These structures are visible in Figs. 10(d) and 11(d). Unfortunately, due to the +absence of grid points along the centerline itself in the grid used to extract the data for in the present DMD +calculations, the influence of these small vortices at the end of the potential core is not accessible in the +present case. Future work should consider a snapshot grid covering all the inner part of the jet. Finally, one +can see in Fig. 11(c) that the vortex filaments in the mixing layer seem to suffer a three-dimensional helicoidal +distortion around the jet mixing layer. The work of Violato and Scarano,48 who performed experiments for a +low Reynolds free water jet, using time-resolved tomographic particle image velocimetry (TR-TOMO PIV), +23 of 26 +American Institute of Aeronautics and Astronautics + +2 +3 +0 +2 +4 +6 +8 +10 +X3 +2 +1 +2 +3 +0 +2 +4 +6 +8 +10 +X3 +0 +1 +-2 +3 +0 +2 +4 +6 +8 +10 +X3 +1 +2 +3 +0 +2 +4 +6 +8 +10 +X3 +-1 +-2 +3 +0 +2 +4 +6 +8 +10 +X3 +2 +2 +3 +0 +2 +4 +6 +8 +10 +Xcan certainly help in the understanding of this type of fundamental aspect in the current jet dynamics. +X. +Concluding Remarks +The present work is concerned with the study of the aerodynamics of a perfectly expanded supersonic +jet flow. It is expected that the flow data and the reduced order model here generated could be used in the +future for performing aeroacoustic studies of jet flows. An implicit large eddy simulation (LES) formulation +for compressible flows, based on the System I set of equations, is used. A streaming version of the total-least- +squares DMD algorithm is chosen to run concurrently with the LES simulation and provide an additional +form of studying the more relevant aspects of the jet dynamics. +LES of a high Reynolds perfectly expanded supersonic jet flow configuration is performed on a compu- +tational mesh with 85 million grid points. Statistical data are extracted from the simulations and present +good agreement with the numerical and experimental reference work, at least near the jet inlet region where +the mesh is well refined. However, this is not the case when the jet moves away from the domain entrance. +As a result, the potential core length calculated by the present LES is underestimated. Such behavior could +be expected since the low order numerical scheme of the numerical solver presently used would probably +require quite extensive mesh refinements. The work also presents three DMD analyses, which have been +performed by extracting large three-dimensional snapshots from the LES results. These DMD computations +concerned the velocity magnitude, the vorticity, based on the Q criterion, and the divergence of the velocity. +Two frequencies are identified for which all DMD calculations identify a dynamic mode with relevant flow +structures. These frequencies agree with those of relevant dynamics identified in previous experimental work +available in the literature. The analysis of all the dynamic modes brought new insights on the jet dynamics +regarding the vortical structures and the acoustic wave patterns. +At the time of this writing, the LES solver is being adapted in order to include parallel I/O features. This +capability will open new opportunities in term of additional grid resolution that would allow a reduction in +the difference between the results calculated by the authors and other data, computational or experimental, +available in the literature. Moreover, the DMD algorithm here implemented should also be parallelized in +order to allow handling larger snapshots and, hence, the extraction of more information from the flow, espe- +cially at the centerline of the jet and further downstream of the jet entrance. Hopefully, these modifications +will allow sufficient mesh refinement, both for the LES calculations and for DMD analyses, that the present +tool will be useful for studies of the jet aeroacoustics. +Acknowledgments +The authors gratefully acknowledge the partial support for this research provided by Conselho Nacional +de Desenvolvimento Cient´ıfico e Tecnol´ogico, CNPq, under the Research Grants No. 309985/2013-7, No. +400844/2014-1, No. 443839/2014-0 and No. 150450/2016-8. The authors are also indebted to the partial +financial support received from Funda¸c˜ao de Amparo `a Pesquisa do Estado de S˜ao Paulo, FAPESP, under +the Research Grants No. 2013/07375-0 and No. 2013/21535-0. +References +1Lumley, J. L., Stochastic Tools in Turbulence, Academic Press, New York, 1970. +2Sirovich, L., “Turbulence and the Dynamics of Coherent Structures. Part I: Coherent Structures,” Quarterly of Applied +Mathematics, Vol. 45, No. 3, 1987, pp. 561–571. +3Berkooz, G., Holmes, P., and Lumley, J. L., “The Proper Orthogonal Decomposition in the Analysis of Turbulent Flows,” +Annual Review of Fluid Mechanics, Vol. 25, No. 1, 1993, pp. 539–575. +4Schmid, P. J., “Dynamic Mode Decomposition of Numerical and Experimental Data,” Journal of Fluid Mechanics, +Vol. 656, 2010, pp. 5–28. +5Seena, A. and Sung, H. J., “Dynamic Mode Decomposition of Turbulent Cavity Flows for Self-Sustained Oscillations,” +International Journal of Heat and Fluid Flow, Vol. 32, No. 6, 2011, pp. 1098–1110. +6Grilli, M., Schmid, P. J., Hickel, S., and Adams, N. A., “Analysis of Unsteady Behaviour in Shockwave Turbulent +Boundary Layer Interaction,” Journal of Fluid Mechanics, Vol. 700, 2012, pp. 16–28. +7Sayadi, T., Schmid, P., Nichols, J. W., and Moin, P., “Dynamic Mode Decomposition of Controlled H- and K-Type +Transitions,” Tech. rep., Annual Research Briefs, 2013. +8Tang, Z. and Jiang, N., “Dynamic Mode Decomposition of Hairpin Vortices Generated by a Hemisphere Protuberance,” +Science China Physics, Mechanics and Astronomy, Vol. 55, No. 1, 2012, pp. 118–124. +24 of 26 +American Institute of Aeronautics and Astronautics + +9Bagheri, S., “Koopman-Mode Decomposition of the Cylinder Wake,” Journal of Fluid Mechanics, Vol. 726, 2013, pp. 596– +623. +10Tissot, G., Cordier, L., Benard, N., and Noack, B. R., “Dynamic Mode Decomposition of PIV Measurements for Cylinder +Wake Flow in Turbulent Regime,” Proceedings of Turbulence and Shear Flow Phenomena Conferences, Eight International +Symposium on Turbulence and Shear Flow Phenomena, Begel House Inc., 2013. +11Jourdain, G., Eriksson, L.-E., Kim, S. H., and Sohn, C. H., “Application of Dynamic Mode Decomposition to Acoustic- +Modes Identification and Damping in a 3-Dimensional Chamber with Baffled Injectors,” Journal of Sound and Vibration, +Vol. 332, No. 18, 2013, pp. 4308–4323. +12Abou-Taouk, A., Sadasivuni, S., L¨orstad, D., Ghenadie, B., and Eriksson, L.-E., “CFD Analysis and Application of +Dynamic Mode Decomposition for Resonant-Mode Identification and Damping in an SGT-100 DLE Combustion System,” +Proceedings of the 7th European Combustion Meeting, 2015. +13L´arusson, R., Hafsteinsson, H. E., Andersson, N., and Eriksson, L.-E., “Investigation of Supersonic Jet Flow Using Modal +Decomposition,” 20th AIAA/CEAS Aeroacoustics Conference, 2014. +14Schmid, P. J., “Application of the Dynamic Mode Decomposition to Experimental Data,” Experiments in Fluids, Vol. 50, +No. 4, 2011, pp. 1123–1130. +15Schmid, P. J., Li, L., Juniper, M., and Pust, O., “Applications of the Dynamic Mode Decomposition,” Theoretical and +Computational Fluid Dynamics, Vol. 25, No. 1-4, 2011, pp. 249–259. +16Schmid, P. J., Violato, D., and Scarano, F., “Decomposition of Time-Resolved Tomographic PIV,” Experiments in Fluids, +Vol. 52, No. 6, 2012, pp. 1567–1579. +17Stegeman, P., Soria, J., and Ooi, A., “Dynamic Mode Decomposition of Near Nozzle Instabilities in Large-Eddy Simula- +tions of Under-Expanded Circular Jets.” Australasian Fluid Mechanics Conference, RMIT University, 08–11 December 2014, +pp. 1–4. +18Wan, Z.-H., Zhou, L., Wang, B.-F., and Sun, D.-J., “Dynamic Mode Decomposition of Forced Spatially Developed +Transitional Jets,” European Journal of Mechanics-B/Fluids, Vol. 51, 2015, pp. 16–26. +19Wynn, A., Pearson, D., Ganapathisubramani, B., and Goulart, P., “Optimal Mode Decomposition for Unsteady Flows,” +Journal of Fluid Mechanics, Vol. 733, 2013, pp. 473–503. +20Jovanovi´c, M. R., Schmid, P. J., and Nichols, J. W., “Sparsity-Promoting Dynamic Mode Decomposition,” Physics of +Fluids, Vol. 26, No. 2, 2014, pp. 024103. +21Williams, M. O., Kevrekidis, I. G., and Rowley, C. W., “A Data–Driven Approximation of the Koopman Operator: +Extending Dynamic Mode Decomposition,” Journal of Nonlinear Science, Vol. 25, No. 6, 2015, pp. 1307–1346. +22Williams, M., Rowley, C. W., and Kevrekidis, I. G., “A Kernel-Based Approach to Data-Driven Koopman Spectral +Analysis,” AIMS Journals, arxiv.org/pdf/1411.2260v4.pdf , 2015. +23Hemati, M. S., Williams, M. O., and Rowley, C. W., “Dynamic Mode Decomposition for Large and Streaming Datasets,” +Physics of Fluids, Vol. 26, No. 11, 2014, pp. 111701. +24Hemati, M. S. and Rowley, C. W., “De-Biasing the Dynamic Mode Decomposition for Applied Koopman Spectral +Analysis,” arxiv.org/pdf/1502.03854v2.pdf , 2015. +25Vreman, A. W., Direct and Large-Eddy Simulation of the Compressible Turbulent Mixing Layer, Ph.D. thesis, Universiteit +Twente, 1995. +26Vremant, B., Geurts, B., and Kuerten, H., “A Priori Tests of Large Eddy Simulation of the Compressible Plane Mixing +Layer,” Journal of Engineering Mathematics, Vol. 29, 1995, pp. 299–327. +27Garnier, E., Adams, N., and Sagaut, P., Large Eddy Simulation for Compressible Flows, Springer, 2009. +28Sagaut, P., Large Eddy Simulation for Incompressible Flows, Springer, 2002. +29Eidson, T. M., “Numerical Simulation of the Turbulent Rayleigh–B´enard Problem Using Subgrid Modelling,” Journal of +Fluid Mechanics, Vol. 158, 1985, pp. 245–268. +30Junqueira-Junior, C., Yamouni, S., Azevedo, J. L. F., and Wolf, W. R., “Large Eddy Simulations of Supersonic Jet Flows +for Aeroacoustic Applications,” AIAA Paper No. 2015-3306, Proceedings of the 33rd AIAA Applied Aerodynamics Conference, +Dallas, TX, June 2015. +31Junqueira-Junior, C., Yamouni, S., Azevedo, J. L. F., and Wolf, W. R., “Influence of Different Subgrid Scale Models +in LES of Supersonic Jet Flows,” AIAA Paper No. 2016-4093, Proceedings of the 46th AIAA Fluid Dynamics Conference, +Washington, DC, June 2016. +32Junqueira-Junior, C., Development of a Parallel Solver for Large Eddy Simulation of Supersonic Jet Flows, Ph.D. thesis, +Instituto Tecnol´ogico de Aeron´autica, S˜ao Jos´e dos Campos, SP, Brasil, 2016. +33Bigarella, E. D. V., Three-Dimensional Turbulent Flow Over Aerospace Configurations, M.Sc. Thesis, Instituto Tec- +nol´ogico de Aeron´autica, S˜ao Jos´e dos Campos, SP, Brasil, 2002. +34Turkel, E. and Vatsa, V. N., “Effect of Artificial Viscosity on Three-Dimensional Flow Solutions,” AIAA Journal, Vol. 32, +No. 1, 1994, pp. 39–45. +35Jameson, A. and Mavriplis, D., “Finite Volume Solution of the Two-Dimensional Euler Equations on a Regular Triangular +Mesh,” AIAA Journal, Vol. 24, No. 4, Apr. 1986, pp. 611–618. +36Jameson, A., Schmidt, W., and Turkel, E., “Numerical Solutions of the Euler Equations by Finite Volume Methods +Using Runge-Kutta Time-Stepping Schemes,” AIAA Paper 81–1259, Proceedings of the AIAA 14th Fluid and Plasma Dynamic +Conference, Palo Alto, Californa, USA, June 1981. +37Long, L. N., Khan, M., and Sharp, H. T., “A Massively Parallel Three-Dimensional Euler/Navier-Stokes Method,” AIAA +Journal, Vol. 29, No. 5, 1991, pp. 657–666. +38Tu, J. H., Rowley, C. W., Kutz, J. N., and Shang, J. K., “Spectral Analysis of Fluid Flows Using Sub-Nyquist-Rate PIV +Data,” Experiments in Fluids, Vol. 55, No. 9, 2014, pp. 1–13. +39Rowley, C. W., Mezi´c, I., Bagheri, S., Schlatter, P., and Henningson, D. S., “Spectral Analysis of Nonlinear Flows,” +Journal of Fluid Mechanics, Vol. 641, 2009, pp. 115–127. +25 of 26 +American Institute of Aeronautics and Astronautics + +40Hemati, M. S., Deem, E. A., Williams, M. O., Rowley, C. W., and Cattafesta, L. N., “Improving Separation Control with +Noise-Robust Variants of Dynamic Mode Decomposition,” 54th AIAA Aerospace Sciences Meeting, 2016. +41Kutz, J. N., Fu, X., and Brunton, S. L., “Multiresolution Dynamic Mode Decomposition,” SIAM Journal on Applied +Dynamical Systems, Vol. 15, No. 2, 2016, pp. 713–735. +42Chen, K. K., Tu, J. H., and Rowley, C. W., “Variants of Dynamic Mode Decomposition: Boundary Condition, Koopman, +and Fourier Analyses,” Journal of Nonlinear Science, Vol. 22, No. 6, 2012, pp. 887–915. +43Mendez, S., Shoeybi, M., Sharma, A., Ham, F. E., Lele, S. K., and Moin, P., “Large-Eddy Simulations of Perfectly- +Expanded Supersonic Jets: Quality Assessment and Validation,” AIAA Paper No. 2010-0271, Proceedings of the 48th AIAA +Aerospace Sciences Meeting Including The New Horizons Forum and Aerospace Exposition, Orlando, FL, January 2010. +44Bridges, J. and Wernet, M. P., “Turbulence Associated with Broadband Shock Noise in Hot Jets,” AIAA Paper No. 2008- +2834, Proceedings of the 14th AIAA/CEAS Aeroacoustics Conference and 29th AIAA Aeroacoustics Conference, Vancouver, +BC, Canada, May 2008. +45ANSYS, “http://www.ansys.com/,” . +46Mendez, S., Shoeybi, M., Sharma, A., Ham, F. E., Lele, S. K., and Moin, P., “Large-Eddy Simulations of Perfectly- +Expanded Supersonic Jets Using an Unstructured Solver,” AIAA Journal, Vol. 50, No. 5, May 2012, pp. 1103–1118. +47Bres, G. A., Nichols, J. W., Lele, S. K., and Ham, F. E., “Towards Best Practices for Jet Noise Predictions with +Unstructured Large Eddy Simulations,” AIAA Paper No. 2012-2965, Proceedings of the 42nd AIAA Fluid Dynamics Conference +and Exhibit, New Orleans, Louisiana, June 2012. +48Violato, D. and Scarano, F., “Three-Dimensional Vortex Analysis and Aeroacoustic Source Characterization of Jet Core +Breakdown,” Physics of Fluids, Vol. 25, No. 1, 2013, pp. 015112. +26 of 26 +American Institute of Aeronautics and Astronautics + diff --git a/v9E4T4oBgHgl3EQfXQwv/content/tmp_files/load_file.txt b/v9E4T4oBgHgl3EQfXQwv/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..1e46de49bc90dc92d0f439721cfc80403ee91a03 --- /dev/null +++ b/v9E4T4oBgHgl3EQfXQwv/content/tmp_files/load_file.txt @@ -0,0 +1,1172 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf,len=1171 +page_content='Dynamic Mode Decomposition of High Reynolds Number Supersonic Jet Flows Sami Yamouni∗ Instituto Tecnol´ogico de Aeron´autica, 12228-900 S˜ao Jos´e dos Campos, SP, Brazil Carlos Junqueira-Junior† , Jo˜ao Luiz F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Azevedo ‡ Instituto de Aeron´autica e Espa¸co, 12228-903 S˜ao Jos´e dos Campos, SP, Brazil William R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Wolf § Universidade Estadual de Campinas, 13083-970 Campinas, SP, Brazil Abstract Current design constraints have encouraged the studies of aeroacoustic fields around compressible jet flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' The present work addresses the numerical study of unsteady turbu- lent jet flows as a preparation for future aeroacoustic analyses of main engine rocket plumes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' An in-house large eddy simulation tool is used in order to reproduce high fidelity results of compressible jet flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' The large eddy simulation formulation is written using a second order numerical scheme for a finite difference spatial discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Numerical simulations of perfectly expanded jets are performed and the results are compared to the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Dynamic mode decompositions (DMD) of the jet flow, using large size three-dimensional snapshots, are performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Three variables are analyzed, namely, the velocity magnitude, the vorticity magnitude and the divergence of velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' In particular, two frequencies are identified and they are linked to flow structures observed in experiments performed by other authors in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' The spatial shapes of the corresponding dynamic modes are also discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Introduction One of the main design issues related to launch vehicles lies on noise emission originated by the complex interaction between the high-temperature/high-velocity exhaustion gases and the atmospheric air.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' These emissions, which have high noise levels, can damage the launching structure or even be reflected upon the vehicle structure itself and the equipment onboard at the top of the vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Moreover, the resulting pressure fluctuations can damage the solid structure of different parts of the launcher or the onboard scientific equipment by vibrational acoustic stress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Therefore, it is strongly recommended to consider the load resulted from acoustic sources over large launching vehicles during take-off and also during the transonic flight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Moreover, one cannot neglect the energy dissipation effect generated by the acoustic waves even if the vehicle is far from the ground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Theoretically, all chemical energy should be converted to kinetic energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' However, in reality, the noise generation consumes part of the chemical energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' ∗Postdoctoral Reasearch Fellow, Graduate Program on Computer Sciences and Electrical Engineering, Departamento de Ciˆencia e Tecnologia Aeroespacial, DCTA/ITA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' E-mail: sami.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='yamouni@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' †Postdoctoral Research Fellow, Graduate Program on Computer Sciences and Electrical Engineering, Departamento de Ciˆencia e Tecnologia Aeroespacial, DCTA/ITA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' E-mail: junior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='hmg@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' ‡Senior Research Engineer, Aerodynamics Division, Departamento de Ciˆencia e Tecnologia Aeroespacial, DCTA/IAE/ALA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' E-mail: joaoluiz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='azevedo@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Fellow AIAA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' §Assistant Professor, Faculty of Mechanical Engineering;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' E-mail: wolf@fem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='unicamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='br.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Member AIAA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 1 of 26 American Institute of Aeronautics and Astronautics arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='05039v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='flu-dyn] 12 Jan 2023 The acoustics design constraints have encouraged the studies of aeroacoustic fields around compressible jet flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Instituto de Aeronautica e Espa¸co (IAE), in Brazil, is interested in this flow configuration for rocket design applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Unsteady property fields of the flow are necessary for the aerocoustic study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Therefore, the present work addresses the numerical study of unsteady turbulent compressible jet flows as a preparatory step for such aeroacoustic applications in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Large eddy simulations (LES) are used in order to reproduce high fidelity results of the unsteady compressible jet flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' One issue with the results of such large data sets resulting from LES calculations is precisely the large amount of data that has to be handled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Hence, with the objective of simplifying a complex flow into a low-dimensional representation containing the dominant dynamic structures, the use of different techniques has been proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Among these, Proper Orthogonal Decomposition (POD)1–3 and Dynamic Mode Decom- position (DMD)4 are the more commonly used techniques in the fluid dynamics community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' POD selects the modes depending on their energy content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' However, this criterion is not necessarily the most appropriate, since the energy is not always the key parameter in order to identify the flow structures4 of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' In contrast with POD, the modes computed from the DMD approach define characteristic frequencies of the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Hence, DMD is the chosen method for the intended application of the present work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' It has already been applied to various flow configurations, such as cavity flows,4,5 shock wave–turbulent boundary layer interaction,6 boundary layer flows,7,8 cylinder flows,9,10 combustion chamber flows,11,12 wake behind a flex- ible membrane4 or jet flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='13–18 Several variations of the DMD algorithm have also been proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' One can cite the optimal mode decomposition,19 the sparsity-promoting DMD,20 the extended DMD,21,22 or the streaming DMD23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Lately, an unbiased noise-robust method has been proposed by Hemati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='24 to overcome the adverse influence of measurement errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' This method can be combined with all the previously listed DMD algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Therefore, in this context, the main objective of the present work is to apply the DMD algorithm to the numerical data extracted from large eddy simulations of perfectly-expanded supersonic jet flows at M = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Due to the large dimension of this problem, the authors use the streaming version23 of the total-least squares DMD formulation proposed by Hemati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='24 The DMD results are compared to numerical and experimental data available in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Navier-Stokes Equations The numerical strategy used in the present study is based on the compressible Navier-Stokes equations formulated as ∂ρ ∂t + ∂ ∂xj (ρuj) = 0 , (1) ∂ ∂t (ρui) + ∂ ∂xj (ρuiuj) + ∂p ∂xi − ∂τij ∂xj = 0 , (2) ∂e ∂t + ∂ ∂xj [(e + p) uj − τijui + qj] = 0 , (3) in which t and xi are independent variables representing time and spatial coordinates of a Cartesian coor- dinate system x, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' The components of the velocity vector u are written as ui, and i = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Density, pressure and total energy per mass unit are denoted by ρ, p and e, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' The heat flux vector, qj, is given by qj = κ ∂T ∂xj , (4) where T is the static temperature and κ is the thermal conductivity coefficient, which can by expressed by κ = µCp Pr .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' (5) The thermal conductivity coefficient is a function of the specific heat at constant pressure, Cp, of the Prandtl number, Pr, which is equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='72 for air, and of the dynamic viscosity coefficient, µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' The latter can be calculated using Sutherland’s law, µ (T) = µ∞ � T T∞ � 3 2 T0 + S1 T + S1 , with S1 = 110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='4 K .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' (6) 2 of 26 American Institute of Aeronautics and Astronautics According to the Stokes hypothesis, the shear-stress tensor, τij, for a Newtonian fluid can be written as τij = 2µ � Sij − 1 3δijSkk � , (7) in which the components of rate-of-strain tensor, Sij, are given by Sij = 1 2 � ∂ui ∂xj + ∂uj ∂xi � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' (8) In order to close the system of equations the density, the static pressure and the static temperature are correlated by the equation of state given by p = ρRT , (9) where R is the gas constant, written as R = Cp − Cv , (10) and Cv is the specif heat at constant volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' The total energy per mass unity is given by: e = p γ − 1 + 1 2ρuiui , (11) in which γ is the ratio of specific heats, written as γ = Cp/Cv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Large Eddy Simulation Filtering The large eddy simulation is based on the principle of scale separation, which is addressed as a filtering procedure in a mathematical formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' A modified version of the the System I filtering approach25 is used in present work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' The original formulation neglects the double correlation term and it is written as ∂ρ ∂t + ∂ ∂xj (ρ �uj) = 0 , ∂ ∂t (ρ �ui) + ∂ ∂xj (ρ �ui �uj) + ∂p ∂xi − ∂ˇτij ∂xj = − ∂ ∂xj [σij − (τij − ˇτij)] , ∂ˇe ∂t + ∂ ∂xj [(ˇe + p) �uj] − ∂ˇτij �ui ∂xj + ∂ˇqj ∂xj = −B1 − B2 − B3 + B4 + B5 + B6 − B7 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' (12) The (˜·) notation is used to represent a Frave averaged property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' The SGS stress tensor components are written as σij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' The filtering procedure originates two new terms, ˇτij and ˇqj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' These new terms are given by ˇτij = 2µ � ˜Sij − 1 3δij ˜Skk � , (13) where ˜Sij = 1 2 � ∂˜ui ∂xj + ∂˜uj ∂xi � , (14) and ˇqj = ˇκ ∂ ˜T ∂xj , (15) in which ˇκ = κ � ˜T � = ˜µ � ˜T � Cp Pr .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' (16) 3 of 26 American Institute of Aeronautics and Astronautics The SGS terms of the energy equation, Bi, are given by B1 = 1 (γ − 1) ∂ ∂xj (puj − p �uj) = ∂CvQj ∂xj , (17) B2 = p∂uk ∂xk − p∂� uk ∂xk = Πdil , (18) B3 = ∂ ∂xj (σkj� uk) , (19) B4 = σkj ∂ ∂xj � uk , (20) B5 = τkj ∂ ∂xj uk − τij ∂ ∂xj � uk = ϵ , (21) B6 = ∂ ∂xj (τij �ui − ˇτij �ui) = ∂D ∂xj , (22) B7 = ∂ ∂xj (qj − ˇqj) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' (23) The work of Vreman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='26 and Vreman25 classify the influence of each term of System I and System II formulations on a 2-D temporal shear layer flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' The classification, including large, medium, small and negligible effects, is based on the L2 norm of different terms of the filtered equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' One order of magnitude separates the norm of each class of terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Garnier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='27 compile the analogy as presented in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Classification of System I terms Large convective NS Medium diffusive NS, B1, B2 and B3 Small B4 and B5 Negligible ∂ ∂xj (τij − ˇ τij), B6 and B7 In practice, the authors of the System I analogy neglect the non-linear terms occuring in the viscous terms and in the heat fluxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='27 Moreover, some of the terms from the original System I set of equations, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' (12), such as B4 and B5, cannot be written in conservative form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Only the terms with large and medium influence are considered in the present work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' The SGS stress tensor components are written using the SGS viscosity,28 σij = −2µsgs � ˇSij − 1 3 ˇSkk � + 1 3δijσkk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' (24) The most important terms of the filtered energy equation are modeled based on the work of Eidson29 and Vreman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='25 They are given by B1 + B2 = − ∂ ∂xj � κsgs ∂ ˜T ∂xj � , (25) where κsgs = µsgsCp Prsgs .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' (26) Using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' (24), (25), and (26), one can write a simplified version of the System I formulation as ∂ρ ∂t + ∂ ∂xj (ρ �uj) = 0 , ∂ ∂t (ρ �ui) + ∂ ∂xj (ρ �ui �uj) + ∂p ∂xi − ∂τ mod ij ∂xj + 1 3 ∂ ∂xj (δijσii) = 0 , ∂ˇe ∂t + ∂ ∂xj [(ˇe + p) �uj] − ∂ ∂xj � τ mod ij �ui � + 1 3 ∂ ∂xj [(δijσii) �ui] + ∂qmod j ∂xj = 0 , (27) 4 of 26 American Institute of Aeronautics and Astronautics where, τ mod ij and qmod j , include the viscous and the subgrid terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' They can be written as τ mod ij = 2 (µ + µsgs) � Sij − 1 3δijSkk � (28) and qmod j = (κ + κsgs) ∂T ∂xj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' (29) Previous work have shown that the subgrid scale terms are too small when compared to the truncation error of the second order numerical scheme used in the current research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='30–32 Therefore, an implicit LES is performed in which all subgrid scales terms, [·]sgs, introduced in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' (28) and (29) are neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Transformation of Coordinates The formulation used in the current work is written in the a general curviliar coordinate system in order to facilitate the implementation and add more generality for the CFD tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' The modified System I set of equations, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' (27) can be written in a strong conservative form for a 3-D Cartesian coordinate system as ∂Q ∂t + ∂E ∂x + ∂F ∂y + ∂G ∂z = 0 , (30) where Q stands for the filtered conservative properties vector given by Q = [ρ ρ˜u ρ˜v ρ ˜w ˇe]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' (31) The flux vectors which represent both the inviscid and viscous fluxes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' E,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' F and G are written as E = � � � � � � � � � � � � � ρ˜u ρ˜u2 + p − τ mod xx + 1 3σxx ρ˜u˜v − τ mod xy ρ˜u ˜w − τ mod xz � ˇe + p − τ mod xx + 1 3σxx � ˜u − τ mod xy ˜v − τ mod xz ˜w + qmod x � � � � � � � � � � � � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' (32) F = � � � � � � � � � � � � � ρ˜v ρ˜u˜v − τ mod xy ρ˜v2 + p − τ mod yy + 1 3σyy ρ˜v ˜w − τ mod yz � ˇe + p − τ mod yy + 1 3σyy � ˜v − τ mod xy ˜u − τ mod yz ˜w + qmod y � � � � � � � � � � � � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' (33) G = � � � � � � � � � � � � � ρ ˜w ρ˜u ˜w − τ mod xz ρ˜v ˜w − τ mod yz ρ ˜w2 + p − τ mod zz + 1 3σzz � ˇe + p − τ mod zz + 1 3σzz � ˜w − τ mod xz ˜u − τ mod yz ˜v + qmod z � � � � � � � � � � � � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' (34) in which,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' v and w are the velocity components in the Cartesian coordinates,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' y and z respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' In the present work the chosen general coordinate transformation is given by T = t , ξ = ξ (x, y, z, t) , η = η (x, y, z, t) , (35) ζ = ζ (x, y, z, t) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 5 of 26 American Institute of Aeronautics and Astronautics Throughout the present work, ξ is the axial jet flow direction, η is the radial direction and ζ is the azimuthal direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' The derivatives in the general curvilinear coordinate system are calculated as a function of the derivatives the Cartesian coordinate system by the chain rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Therefore, one can write � � � � � � � � � � � ∂ ∂T ∂ ∂ξ ∂ ∂η ∂ ∂ζ � � � � � � � � � � � = � ���� 1 xT yT zT 0 xξ yξ zξ 0 xη yη zη 0 xζ yζ zζ � ���� � � � � � � � � � ∂ ∂t ∂ ∂x ∂ ∂y ∂ ∂z � � � � � � � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' (36) The Jacobian of the transformation, J, is calculated as the inverse of the determinant of the matrix in the chain rule presented in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' (36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Therefore, for the 3-D coordinate transformation, the Jacobian can be written as J = (xξyηzζ + xηyζzξ + xζyξzη − xξyζzη − xηyξzζ − xζyηzξ)−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' (37) The metric terms are given by ξx = J (yηzζ − yζzη) , ξy = J (zηxζ − zζxη) , ξz = J (xηyζ − xζyη) , ηx = J (yηzξ − yξzη) , ηy = J (zηxξ − zξxη) , ηz = J (xηyξ − xξyη) , (38) ζx = J (yξzη − yηzξ) , ζy = J (zξxη − zηxξ) , ζz = J (xξyη − xηyξ) , ξt = −xT ξx − yT ξy − zT ξz , ηt = −xT ηx − yT ηy − zT ηz , ζt = −xT ζx − yT ζy − zT ζz .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' One can rewrite Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' (30), in a conservative form, for the general curvilinear coordinate system as ∂ ˆQ ∂T + ∂ ˆE ∂ξ + ∂ ˆF ∂η + ∂ ˆG ∂ζ = 0 , (39) where ˆQ = J−1Q = J−1 [ρ ρ˜u ρ˜v ρ ˜w ˇe]T , (40) and the new flux vectors are given by ˆE = J−1 � ξtQ + ξxE + ξyF + ξzG � , ˆF = J−1 � ηtQ + ηxE + ηyF + ηzG � , (41) ˆG = J−1 � ζtQ + ζxE + ζyF + ζzG � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Finally, the flux vectors are split in inviscid and viscous part in order to simplify the implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Therefore, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' (39) can be rewritten as ∂ ˆQ ∂T + ∂ ˆEe ∂ξ + ∂ ˆFe ∂η + ∂ ˆGe ∂ζ = ∂ ˆEv ∂ξ + ∂ ˆFv ∂η + ∂ ˆGv ∂ζ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' (42) where the inviscid flux vectors,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' ˆEe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' ˆFe and ˆGe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' are given by ˆEe = J−1 � � � � � � � � � � � � � ρU ρ˜uU + pξx ρ˜vU + pξy ρ ˜wU + pξz (ˇe + p) U − pξt � � � � � � � � � � � � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' (43) ˆFe = J−1 � � � � � � � � � � � � � ρV ρ˜uV + pηx ρ˜vV + pηy ρ ˜wV + pηz (ˇe + p) V − pηt � � � � � � � � � � � � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' (44) 6 of 26 American Institute of Aeronautics and Astronautics ˆGe = J−1 � � � � � � � � � � � � � ρW ρ˜uW + pζx ρ˜vW + pζy ρ ˜wW + pζz (ˇe + p) W − pζt � � � � � � � � � � � � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' (45) in which the contravariant velocity components,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' U,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' V and W,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' are calculated as U = ξt + ξxu + ξyv + ξzw ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' V = ηt + ηxu + ηyv + ηzw ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' (46) W = ζt + ζxu + ζyv + ζzw .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' The viscous flux vectors,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' ˆEv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' ˆFv and ˆGv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' are written as ˆEv = J−1 � � � � � � � � � � � � � 0 ξx � τ mod xx − 1 3σxx � + ξyτ mod xy + ξzτ mod xz ξxτ mod xy + ξy � τ mod yy − 1 3σyy � + ξzτ mod yz ξxτ mod xz + ξyτ mod yz + ξz � τ mod zz − 1 3σzz � ξxβx + ξyβy + ξzβz � � � � � � � � � � � � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' (47) ˆFv = J−1 � � � � � � � � � � � � � 0 ηx � τ mod xx − 1 3σxx � + ηyτ mod xy + ηzτ mod xz ηxτ mod xy + ηy � τ mod yy − 1 3σyy � + ηzτ mod yz ηxτ mod xz + ηyτ mod yz + ηz � τ mod zz − 1 3σzz � ηxβx + ηyβy + ηzβz � � � � � � � � � � � � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' (48) ˆGv = J−1 � � � � � � � � � � � � � 0 ζx � τ mod xx − 1 3σxx � + ζyτ mod xy + ζzτ mod xz ζxτ mod xy + ζy � τ mod yy − 1 3σyy � + ζzτ mod yz ζxτ mod xz + ζyτ mod yz + ζz � τ mod zz − 1 3σzz � ζxβx + ζyβy + ζzβz � � � � � � � � � � � � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' (49) where βx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' βy and βz are defined as βx = � τ mod xx − 1 3σxx � ˜u + τ mod xy ˜v + τ mod xz ˜w − qmod x ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' βy = τ mod xy ˜u + � τ mod yy − 1 3σyy � ˜v + τ mod yz ˜w − qmod y ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' (50) βz = τ mod xz ˜u + τ mod yz ˜v + � τ mod zz − 1 3σzz � ˜w − qmod z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Dimensionless LES Formulation A convenient nondimensionalisation is necessary in order to achieve a consistent implementation of the governing equations of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Dimensionless formulation yelds to a more general numerical tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' There is no need to change the formulation for each configuration intended to be simulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Moreover, dimensionless formulation scales all the necessary properties to the same order of magnitude which is a computational advantage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='33 Dimensionless variables are presented in the present section in order perform the nondimen- sionalisation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' (42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' The dimensionless time, T , is written as function of the freestream speed of sound and of a reference lenght, D, T = T a∞ l .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' (51) 7 of 26 American Institute of Aeronautics and Astronautics In the current work, D represents the jet entrance diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' This reference lengh is aldo applied to write the dimensionless length, l = l D .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' (52) The dimensionless velocity components are obtained using the freestream speed of sound vel = v a∞ vel = u, v, w .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' (53) Dimensionless pressure and energy are calculated as p = p ρ∞a2∞ , (54) e = e ρ∞a2∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' (55) Dimensionless density, ρ, temperature, T and viscosity, µ, are calculated using freestream properties ρ = ρ ρ∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' (56) One can use the dimensionless properties described above in order to write the dimensionless form of the LES equations as ∂ ˆQ ∂T + ∂ ˆEe ∂ξ + ∂ ˆF e ∂η + ∂ ˆGe ∂ζ = Mj Re � ∂ ˆEv ∂ξ + ∂ ˆF v ∂η + ∂ ˆGv ∂ζ � , (57) where the underlined terms are calculated using non dimensional properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' The jet Mach and Reynolds numbers are based on the mean jet inlet velocity, Uj, the freestream speed of sound, a∞, density, ρ∞, viscosity, µ∞ and the reference length, D, Mj = Uj a∞ , Re = ρ∞UjD µ∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' (58) VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Numerical Formulation The governing equations previously described are discretized in a structured finite difference context for general curvilinear coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='33 The numerical flux is calculated through a central difference scheme with the explicit addition of the anisotropic artificial dissipation of Turkel and Vatsa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='34 The time integration is performed by an explicit, 2nd-order, 5-stage Runge-Kutta scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='35,36 Conserved properties and artificial dissipation terms are properly treated near boundaries in order to assure the physical correctness of the numerical formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Spatial Discretization For the sake of simplicity, the formulation discussed in the present section is no longer written using bars, underbars, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' However, the reader should notice that the equations are dimensionless and filtered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' The LES equations, presented in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' (57),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' are discretized in space in a finite difference fashion and,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' rewritten as �∂Q ∂T � i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='k = −RHSi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='k ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' (59) where RHS is the right hand side of the equation and it is written as function of the numerical flux vectors at the interfaces between grid points,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' RHSi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='k = 1 ∆ξ � Ee(i+ 1 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='k) − Ee(i− 1 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='k) − Ev(i+ 1 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='k) + Ev(i− 1 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='k) � 1 ∆η � Fe(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='j+ 1 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='k) − Fe(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='j− 1 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='k) − Fv(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='j+ 1 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='k) + Fv(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='j− 1 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='k) � (60) 1 ∆ζ � Ge(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='k+ 1 2 ) − Ge(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='k− 1 2 ) − Gv(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='k+ 1 2 ) + Gv(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='k− 1 2 ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 8 of 26 American Institute of Aeronautics and Astronautics For the general curvilinear coordinate case ∆ξ = ∆η = ∆ζ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' The anisotropic artificial dissipation method of Turkel and Vatsa34 is implemented through the modification of the inviscid flux vectors, Ee, Fe and Ge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' The numerical scheme is nonlinear and allows the selection between artificial dissipation terms of second and fourth differences, which is very important for capturing discontinuities in the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' The numerical fluxes are calculated at interfaces in order to reduce the size of the calculation cell and, therefore, facilitate the implementation of second derivatives since the the concept of numerical fluxes vectors is used for flux differencing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Only internal interfaces receive the corresponding artificial dissipation terms, and differences of the viscous flux vectors use two neighboring points of the interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' The inviscid flux vectors,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' with the addition of the artificial dissipation contribution,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' can be written as Ee(i± 1 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='k) = 1 2 � Ee(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='k) + Ee(i±1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='k) � − J−1d(i± 1 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='k) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Fe(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='j± 1 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='k) = 1 2 � Fe(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='k) + Fe(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='j±1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='k) � − J−1d(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='j± 1 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='k) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' (61) Ge(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='k± 1 2 ) = 1 2 � Ge(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='k) + Ge(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='k±1) � − J−1d(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='k± 1 2 ) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' in which the d(i±1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='k),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='d(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='j±1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='k) and d(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='k±1) terms are the Turkel and Vatsa34 artificial dissipation terms in the i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' and k directions respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' The scaling of the artificial dissipation operator in each coordinate direction is weighted by its own spectral radius of the corresponding flux Jacobian matrix, which gives the non-isotropic characteristics of the method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='33 The artificial dissipation contribution in the ξ direction is given by d(i+ 1 2 ,j,k) = λ(i+ 1 2 ,j,k) � ϵ(2) (i+ 1 2 ,j,k) � W(i+1,j,k) − W(i,j,k) � (62) ϵ(4) (i+ 1 2 ,j,k) � W(i+2,j,k) − 3W(i+1,j,k) + 3W(i,j,k) − W(i−1,j,k) � ] , in which ϵ(2) (i+ 1 2 ,j,k) = k(2)max � νd (i+1,j,k), νd (i,j,k) � , (63) ϵ(4) (i+ 1 2 ,j,k) = max � 0, k(4) − ϵ(2) (i+ 1 2 ,j,k) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' (64) The original article34 recomends using k(2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='25 and k(4) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='016 for the dissipation artificial constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' The pressure gradient sensor, νd (i,j,k), for the ξ direction is written as νd (i,j,k) = |p(i+1,j,k) − 2p(i,j,k) + p(i−1,j,k)| p(i+1,j,k) − 2p(i,j,k) + p(i−1,j,k) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' (65) The W vector from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' (62) is calculated as a function of the conserved variable vector, ˆQ, written in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' (40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' The formulation intends to keep the total enthalpy constant in a final converged steady solution, which is the correct result for the Navier-Stokes equations with Re → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' This approach is also valid for the viscous formulation because the dissipation terms are added to the inviscid flux terms, in which they are really necessary to avoid nonlinear instabilities of the numerical formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' The W vector is given by W = ˆQ + [0 0 0 0 p]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' (66) The spectral radius-based scaling factor, λ, for the i − th direction is written λ(i+ 1 2 ,j,k) = 1 2 �� λξ � (i,j,k) + � λξ � (i+1,j,k) � , (67) where λξ(i,j,k) = λξ � 1 + �λη λξ �0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='5 + �λζ λξ �0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='5� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' (68) The spectral radii, λξ, λη and λζ are given by λξ = |U| + a � ξ2x + η2y + ζ2z , λξ = |V | + a � ξ2x + η2y + ζ2z , (69) λξ = |W| + a � ξ2x + η2y + ζ2z , 9 of 26 American Institute of Aeronautics and Astronautics in which, U, V and W are the contravariant velocity components in the ξ, η and ζ, previously written in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' (47), and a is the local speed of sound, which can be written as a = �γp ρ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' (70) The calculation of artificial dissipation terms for the other coordinate directions are completely similar and, therefore, they are not discussed in the present work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Time Marching Method The time marching method used in the present work is a 2nd-order, 5-step Runge-Kutta scheme based on the work of Jameson and co-workers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='35,36 The time integration can be written as Q(0) (i,jk,) = Q(n) (i,jk,) , Q(l) (i,jk,) = Q(0) (i,jk,)− αl∆t(i,j,k)RHS(l−1) (i,j,k) l = 1, 2 · · · 5, Q(n+1) (i,jk,) = Q(5) (i,jk,) , (71) in which ∆t is the time step and n and n + 1 indicate the property values at the current and at the next time step, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' The literature35,36 recommends α1 = 1 4 , α2 = 1 6 , α3 = 3 8 , α4 = 1 2 , α5 = 1 , (72) in order to improve the numerical stability of the time integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' The present scheme is theoretically stable for CFL ≤ 2 √ 2, under a linear analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='33 VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Boundary Conditions The present section presents all boundary conditions used for the turbulent compressible jet flow simu- lation such as inlet, outlet, centerline and far field boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Moreover, the numerical treatment of the centerline singularity and the implementation of the periodic boundary in the azimuthal direction are also discussed in the end of the section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Far Field Boundary Riemann invariants37 are used to implement far field boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' They are derived from the cha- racteristic relations for the Euler equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' At the interface of the outer boundary, the following expressions apply R− = R− ∞ = qn∞ − 2 γ − 1a∞ , (73) R+ = R+ e = qne − 2 γ − 1ae , (74) where ∞ and e indexes stand for the property in the freestream and in the internal region, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' qn is the velocity component normal to the outer surface, defined as qn = u · ⃗n , (75) and ⃗n is the unit outward normal vector ⃗n = 1 � η2x + η2y + η2z [ηx ηy ηz]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' (76) Equation (75) assumes that the η direction is pointing from the jet to the external boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Solving for qn and a, one can obtain qnf = R+ + R− 2 , af = γ − 1 4 (R+ − R−) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' (77) 10 of 26 American Institute of Aeronautics and Astronautics The index f is linked to the property at the boundary surface and will be used to update the solution at this boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' For a subsonic exit boundary, 0 < qne/ae < 1, the velocity components are derived from internal properties as uf = ue + (qnf − qne)ηx , vf = ve + (qnf − qne)ηy , (78) wf = we + (qnf − qne)ηz .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Density and pressure properties are obtained by extrapolating the entropy from the adjacent grid node, ρf = � ργ ea2 f γpe � 1 γ−1 , pf = ρfa2 f γ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' For a subsonic entrance, −1 < qne/ae < 0, properties are obtained similarly from the freestream variables as uf = u∞ + (qnf − qn∞)ηx , vf = v∞ + (qnf − qn∞)ηy , (79) wf = w∞ + (qnf − qn∞)ηz , ρf = � ργ ∞a2 f γp∞ � 1 γ−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' (80) For a supersonic exit boundary, qne/ae > 1, the properties are extrapolated from the interior of the domain as ρf = ρe , uf = ue , vf = ve , (81) wf = we , ef = ee , and for a supersonic entrance, qne/ae < −1, the properties are extrapolated from the freestream variables as ρf = ρ∞ , uf = u∞ , vf = v∞ , (82) wf = w∞ , ef = e∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Entrance Boundary For a jet-like configuration, the entrance boundary is divided in two areas: the jet and the area above it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' The jet entrance boundary condition is implemented through the use of the 1-D characteristic relations for the 3-D Euler equations for a flat velocity profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' The set of properties then determined is computed from within and from outside the computational domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' For the subsonic entrance, the v and w components of the velocity are extrapolated by a zero-order extrapolation from inside the computational domain and the angle of flow entrance is assumed fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' The rest of the properties are obtained as a function of the jet Mach number, which is a known variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' (u)1,j,k = uj , (v)1,j,k = (v)2,j,k , (83) (w)1,j,k = (w)2,j,k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 11 of 26 American Institute of Aeronautics and Astronautics The dimensionless total temperature and total pressure are defined with the isentropic relations: Tt = 1 + 1 2(γ − 1)M 2 ∞ and Pt = 1 γ (Tt)γ/(γ−1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' (84) The dimensionless static temperature and pressure are deduced from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' (84), resulting in (T)1,j,k = Tt 1 + 1 2(γ − 1)(u2 + v2 + w2)1,j,k and (p)1,j,k = 1 γ (T)γ/(γ−1) 1,j,k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' (85) For the supersonic case, all conserved variables receive jet property values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' The far field boundary conditions are implemented outside of the jet area in order to correctly propagate information comming from the inner domain of the flow to the outter region of the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' However, in the present case, ξ, instead of η, as presented in the previous subsection, is the normal direction used to define the Riemann invariants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Exit Boundary Condition At the exit plane, the same reasoning of the jet entrance boundary is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' This time, for a subsonic exit, the pressure is obtained from the outside and all other variables are extrapolated from the interior of the computational domain by a zero-order extrapolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' The conserved variables are obtained as (ρ)IMAX,j,k = (p)IMAX,j,k (γ − 1)(e)IMAX−1,j,k , (86) (⃗u)IMAX,j,k = (⃗u)IMAX−1,j,k, (87) (ei)IMAX,j,k = (ρ)IMAX,j,k � (e)IMAX−1,j,k + 1 2(⃗u)IMAX,j,k · (⃗u)IMAX,j,k � , (88) in which IMAX stands for the last point of the mesh in the axial direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' For the supersonic exit, all properties are extrapolated from the interior domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Centerline Boundary Condition The centerline boundary is a singularity of the coordinate transformation, and, hence, an adequate treatment of this boundary must be provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' The conserved properties are extrapolated from the ajacent longitudinal plane and are averaged in the azimuthal direction in order to define the updated properties at the centerline of the jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' The fourth-difference terms of the artificial dissipation scheme, used in the present work, are carefully treated in order to avoid the five-point difference stencils at the centerline singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' If one considers the flux balance at one grid point near the centerline boundary in a certain coordinate direction, let wj denote a component of the W vector from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' (66) and dj denote the corresponding artificial dissipation term at the mesh point j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' In the present example, (∆w)j+ 1 2 stands for the difference between the solution at the interface for the points j+1 and j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' The fouth-difference of the dissipative fluxes from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' (62) can be written as dj+ 1 2 = (∆w)j+ 3 2 − 2 (∆w)j+ 1 2 + (∆w)j− 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' (89) Considering the centerline and the point j = 1, as presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 1, the calculation of d1+ 1 2 demands the (∆w) 1 2 term, which is unknown since it is outside the computation domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' In the present work a extrapolation is performed and given by (∆w) 1 2 = − (∆w)1+ 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' (90) This extrapolation modifies the calculation of d1+ 1 2 that can be written as dj+ 1 2 = (∆w)j+ 3 2 − 3 (∆w)j+ 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' (91) The approach is plausible since the centerline region is smooth and it does not have high gradients of properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 12 of 26 American Institute of Aeronautics and Astronautics Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Boundary point distribution in the calculation of dissipation operator at the centerline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='33 VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Periodic Boundary Condition A periodic condition is implemented between the first (K = 1) and the last point in the azimutal direction (K = KMAX) in order to close the 3-D computational domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' There are no boundaries in this direction, since all the points are inside the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' The first and the last points, in the azimuthal direction, are superposed in order to facilitate the boundary condition implementation which is given by (ρ)i,j,KMAX = (ρ)i,j,1 , (u)i,j,KMAX = (u)i,j,1 , (v)i,j,KMAX = (v)i,j,1 , (92) (w)i,j,KMAX = (w)i,j,1 , (e)i,j,KMAX = (e)i,j,1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Dynamic Mode Decomposition VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Theoretical Framework The DMD method provides a spatio-temporal decomposition of the flow into a set of dynamic modes that are derived from time-resolved snapshots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' For example, a generic flow variable, xDMD(x, y, z, t), where x, y, z and t stand for spatial coordinates and time, respectively, can be represented by xDMD(x, y, z, t) = m−1 � i=1 ai exp(λit) φi(x, y, z) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' (93) Here, ai and λi are the amplitude and the frequency of the spatial mode φi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' The underlying mathematics is closely related to the idea of the Arnoldi algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='4 This flow variable, extracted from the simulation, can be represented in the form of a snapshot sequence X = [x(t1) x(t2) · · · x(tm)] ∈ Rn×m, where x(ti) ∈ Rn is the i-th snapshot, m denotes the number of snapshots and n, the spatial dimension of each time snapshot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Each snapshot, x(ti), contains a set of variables depending on the user’s choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' The present study is designed to collect data regularly separated in time by ∆t even though recent techniques allow irregularly spaced sampling in time of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='38 The authors assume that there exists a linear operator A ∈ Rn×n connecting two consecutive snapshot giving xi+1 = A xi for i = 1, · · · , m − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' (94) The A operator is an approximation of the Koopman operator,39 whose eigen-elements can approximate the underlying dynamics of the flow, even if such dynamics is nonlinear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' The objective of the DMD is the determination of these characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' The selection of the eigen-elements of A is a matter of importance since the accuracy of the results, as well as the computational costs, both depend significantly on the method of choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' The strategy used in the present work is a combination of the total least-squares DMD, described 13 of 26 American Institute of Aeronautics and Astronautics j=4 j=3+1/2 j=3 j=2+1/2 j=2 j=1+1/2 j=1 12 grid node X interfacein Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 24, and the streaming DMD algorithm presented in Hemati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='23 The former technique provides a noise-aware DMD technique while the latter allows the assimilation “on-the-fly” of new incoming snapshots and it can even theoretically include an infinite number, m, of snapshots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Hemati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='40 ran successfully this combined technique to analyze the dynamics of the flow separation over a flat plate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' In practice, the A DMD operator of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' (94) can be defined as A = YX+ using the previously defined snapshot matrix, X, and its time-shifted version Y = [x(t1 + ∆t) x(t2 + ∆t) · · · x(tm + ∆t)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' In this relation, X+ stands for the Moore-Penrose pseudoinverse of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' The solution of the problem in the present form is prohibitively expensive in terms of CPU and memory costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' The streaming DMD approach suggests a solution to reformulate A in order to be able to handle large dimension problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' First, the augmented snapshot matrix, Z = [X Y]T , is built.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='24 After substitution, a low-dimensional version of A, ˜ A, can be obtained under the form ˜ A = QT x � 0 I � Qz Gz QT z � 0 I � QxG+ x ∈ Rr×r , (95) where r is the rank of X, Qx and Qz are obtained from the QR-decomposition of X and Z, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Hence, one could write that X = QxRx and Z = QzRz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Therefore, one can also write that Gz = RzRT z and Gx = RxGzRT z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' This procedure allows an incremental update of new available snapshots, without storing all of them in memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Moreover, in this expression, the total number snapshot, m, does not appear anymore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' During the streaming DMD process, a POD compression is included allowing the user to choose the rank of the DMD operator, r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' The DMD modes and frequencies are given by the eigenvectors and eigenvalues of ˜ A, such that φi is the i-th eigenvector with the associated eigenvalue, µi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Hence, the associated growth rate and frequency of the i-th DMD mode are given by σi = log(|µi|) ∆t and ωi = arg(µi) ∆t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' (96) Finally, λi = log(µi)/∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Another interesting aspect of the DMD, is that knowing the first snapshot and the eigenvalues of the DMD operator, one can predict the temporal behavior of the mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Indeed, using a discretized version of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' (93) expressed at any time instant k = 1, · · · , m − 1, xk = m−1 � i=1 θi(k) φi, (97) where θi are the temporal coefficients of the eigenvectors φi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' It comes directly, using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' (94), that xk+1 = A xk = m−1 � i=1 θi(k)A φi = m−1 � i=1 θi(k)µiφi = Ak x1 = m−1 � i=1 θi(1)µk i φi, (98) Using the work of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 41, the matrix of the initial coefficient can be calculated using the relation θ(1) = φ+x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' (99) VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Choice of the Parameters Two main parameters are considered in the DMD framework initially introduced by Schmid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='4 The first one is ∆t, the constant time-step between two consecutive snapshots, while the second one is m, the total number of snapshots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Both of them require a good knowledge of the physical phenomenon under study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' According to Schmid,4 the sample rate must be sufficiently high, about three times the Nyquist cutoff, to capture correctly the dynamics of an oscillatory flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' The idea is, then, to tune the sampling frequency based on the phenomenon the user wants to study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' However, following Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=',42 using a high sample rate, the snapshots are likely to be correlated in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' This is a problem since the method impose the use of a linear independent dataset to work properly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Finally, a high number, m, of snapshots could also affect the linear independency of the snapshots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' In the algorithm used in the present work, a Gram-Schmidt step is included in the process to address this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='10,23 14 of 26 American Institute of Aeronautics and Astronautics IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Case of a High Reynolds Number Supersonic Jet Flow The present section is devoted to the study of a supersonic perfectly expanded jet flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' The geometry and flow configurations of interest are presented, followed by large eddy simulation results, which are compared to analytical, numerical and experimental data from the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='43,44 The LES results provide a database for three DMD studies using the velocity magnitude, the vorticity, represented by the Q criterion, and the divergence of the flow velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Geometry and Mesh Configurations Figure 2 illustrates a three-dimensional view of the representative domain for the jet flow simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' The geometry resembles a frustrum of a cone with the jet entering the computational domain through the small base at x = 0, and leaving the domain at the large base at x = 30D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' The radii of the entrance and exit plans are approximately 8D and 9D, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' The authors have chosen not to include the nozzle geometry in the computational domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Hence, the jet entrance is located at x = 0, for |r|/D ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='5, where |r| = � y2 + z2 is the distance from the centerline in the radial direction and D is the incoming jet diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' The computational domain is created in two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' First, a 2-D region is generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' In the sequence, this region is rotated around the horizontal direction, x, indicated by the discontinuous blue line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 2, in order to generate a fully 3-D geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' The rotation approach generates a singularity at the centerline of the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' The treatment of this region is discussed in the boundary conditions section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' The commercial mesh generator ANSYS® ICEM CFD45 is used for the creation of the 2-D domain for an azimuthal plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' The zones of this geometry are created based on results from simulations of previous work31 in order to refine the mesh in the shear layer region of the flow until x = 10D, after the end of the potential core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' The mesh is, then, coarsened towards the outer regions of the domain in order to dissipate properties of the flow far from the jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Such mesh refinement approach can avoid reflection of information into the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' The radial and longitudinal dimensions of the smallest distance between mesh points of the computational grid are given by (∆r)min = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='002 and (∆x)min = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='0126, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' This minimal spacing occurs at the lipline of the jet and at the entrance of the computational domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' These dimensions are based on a reference grid of Mendez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='43,46 The resulting computational grid is composed by 537 points in the axial direction, 442 points in the radial direction and 360 points in the azimuthal direction, yielding approximately 85 million grid points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' For further details about the mesh generation, the reader is referred to the work of Junqueira-Junior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='32 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 3-D view of two XY slices of the grid, located above and below the centerline highlighted by a discontinuous blue line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' The red arrow indicates the jet entrance inside the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 15 of 26 American Institute of Aeronautics and Astronautics 07 5 5 10 15 x 5 20 25 5 30IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Flow Configuration The flow is characterized by an unheated perfectly expanded jet with a Mach number of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='4 at the domain entrance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Therefore, the pressure ratio, PR = Pj/P∞, and the temperature ratio, TR = Tj/T∞, between the jet exit and the ambient freestream are equal to one, PR = 1 and TR = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' The time step used in the simulation is constant and equal to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='0 × 10−4 in dimensionless form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' The Reynolds number of the jet is Re = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='57 × 106, based on the jet entrance diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' This flow configuration is chosen due to the absence of strong shocks waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Strong discontinuities must be carefully treated using numerical approaches, and the authors did not want to deal with those issues at the present time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Moreover, numerical and experimental data for a perfectly expanded jet flow configuration, such as the one used in the present work, are available in the literature such as the work of Mendez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='43,46 and the work of Bridges and Wernet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='44 Properties of flow at the inlet and at the far field regions have to be provided to the code in order to impose the boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Density, ρ, temperature, T, velocity, U, Reynolds number, Re, and specific heat at constant volume, Cv, are provided in the dimensionless form to the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' These dimensionless properties are given by ρj = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='00 , ρ∞ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='00 , Tj = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='00 , T∞ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='00 , (100) Uj = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='4 , U∞ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='00 , Rej = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='57 × 106 , Cv = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='786 , where the j subscript stands for property at the jet entrance and the ∞ subscript stands for property at the far field region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Data Extraction Procedure For the present study, data are extracted after a preliminary simulation is run in order to achieve a statistically steady state condition for the jet flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' This initial preliminary simulation lasts 96 dimensionless time units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' For the current jet exit Mach number of Mj = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='4, this simulation time represents approximately 3 flow- through times (FTT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' One flow-through time is the time for a particle to cross the entire domain from the jet entrance to the domain exit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' After the flow initialization process, the simulations are restarted and run for another period of time in which data of the flow are extracted and recorded at a fixed frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Data extraction characteristics Simulation ∆t c∞/D No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Extractions Grid Size Total Time FTT LES statistics 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='06 4096 500 × 425 (2-D) 245.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='76 ≈ 8 DMD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='12 256 473 × 412 × 180 (3-D) 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='72 ≈ 1 The temporal characteristics of the data extraction are displayed in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 2 for both the LES statistics and the DMD computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' These data processing methods are very different one from each other, especially because of the grid dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' In the present work, the LES statistics are computed only along 2-D surfaces, whereas DMD calculations use three-dimensional snapshots as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' The snapshots extracted during the DMD process have more than 35 million points and they are stored in the PLOT3D formata, adapted for structured meshes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' The memory size of one snapshot, used for the DMD calculations, is about 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='5 Gb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' On the other hand, the time-dependent LES surfaces are all included in one single CGNS file of 40 Gb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Finally, the total simulation time, necessary for obtaining the LES statistics, is higher than that used in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 47 for the same purpose, and this can be considered as a fairly large time sample for an LES calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' As indicated in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 2, a total of 8 flow-through times have been used in order to obtain the LES statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Large Eddy Simulation Results In this subsection, 2-D distributions of properties and profiles are collected from the compressible LES simulation and compared with numerical and experimental results from the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='43,44,46 A longitudinal ahttps://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='grc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='nasa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='gov/www/wind/valid/plot3d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='html 16 of 26 American Institute of Aeronautics and Astronautics (a) Time averaged axial velocity component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' (b) RMS value of the fluctuating part of the axial velocity component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' (c) Turbulent kinetic energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Contour plots of longitudinal planes of statistically converged jet properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' The white line defines the potential core of the jet, where u = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='95Uj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' plane view of the statistically-converged time-averaged distributions of three flow properties, namely, axial velocity component, ⟨U⟩, RMS value of the fluctuating part of the axial velocity component, urms, and turbulent kinetic energy, k, are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' The statistical properties of the LES results are calculated using much more snapshots and with a more refined time increment than the numerical reference data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='43,46 Each variable displays a fairly smooth flow field, confirming the good statistical convergence of the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Moreover, the contours of ⟨U⟩, urms and k display a classical shape, with urms spreading along with the jet shear layer and with high values of k at the beginning of the mixing layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' The white solid line defines the jet potential core region, U 95% j , which is a characteristic parameter of jet flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' The potential core length, δ95% j , is defined as the distance from the jet entrance and along the centerline until the jet velocity reaches 95% of the velocity of the jet at the inlet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' In line with previous work, reported in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 31, the current simulation aims to reduce the error with respect to the experimental data in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='44 by refining the grid in the jet potential core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Table 3 presents the size of the potential cores for the current simulation, compared to the numerical results in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 31, 43 and 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' The table presents the relative error compared to the experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='44 The present LES calculations are performed on the same grid geometry used in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 31, but with more points inside the potential core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' As one can see in the table, the error has been reduced from 26% to 22%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' The grid used in the present work needs to be further refined in order to overcome the dissipative characteristics of 2nd-order scheme used and, 17 of 26 American Institute of Aeronautics and Astronautics 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='4 0 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='1 2 2 4 6 8 10 12 14 x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='3 1 0 1 2 2 4 6 8 10 12 14 x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='09 1 0 1 D 2 2 4 6 8 10 12 14 xhence, keep reducing the magnitude of error when compared to experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Potential core length comparisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Simulation δ95% j Relative error Current work 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='05 22% Junqueira-Junior et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='31 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='84 26% Mendez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='43,46 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='35 8% The evolution of the averaged axial component of velocity and the evolution of the RMS value of the fluctuating part of the axial component of velocity along the centerline and the lipline are illustrated in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 4 and 5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' The solid line stands for the results of the present case, the open square symbols represent the LES results of Mendez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=',43,46 while the triangular symbols stand for the experimental data of Bridges and Wernet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='44 The lipline is the surface defined over r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='5D, which represents the boundary of the jet at the entrance of the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' The comparison of profiles indicates that distributions of ⟨U⟩/Uj along the centerline correlates well with the references until x = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='0D, where the grid has good resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' The time averaged axial component of velocity start to correlate poorly with the reference when the mesh spacing increases, x > 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='0D, due to the mesh coarsening in the streamwise direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' The mesh coarsening is used in order to add artificial dissipation towards the exit of the domain, since the numerical framework does not have a sponge zone implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' The time averaged axial component of velocity calculated along the lipline correlates well with the references until x ≈ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='0D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' The magnitude of ⟨U⟩/Uj along the lipline is understimated for x > 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='0D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' (a) Centerline (b) Lipline Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Averaged axial component of velocity along the centerline and lipline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' The solid line stands for the results of the present case, the open square symbols represent the numerical references43, 46 and the triangular symbols are the experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='44 The distribution of urms/Uj calculated along the centerline fits the numerical and experimental reference distributions of the same property for x < 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='0D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' However, it presents an overestimated distribution of urms/Uj when compared with both numerical and experimental data at other positions along the centerline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' The numerical reference has also calculated an overestimated distribution of urms/Ujalong the centerline when compared to the experimental reference at x > 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='0D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' The distribution of urms/Uj along the lipline calculated by the current work and by the numerical reference present similar behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Nonetheless, the distributions are overestimated when compared to the experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Dynamic Mode Decomposition Results The streaming version23 of the total-least-squares DMD algorithm24 on volumetric data extracted during the large eddy simulations described in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' IX is computed in the present work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Considering that the DMD calculation is performed in serial mode, the computer memory is the limiting factor to compute the DMD 18 of 26 American Institute of Aeronautics and Astronautics 口 F D 口 4 口 口 口 44 口 口 0 口 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='8 口 4 口 U>/U 口 口 口 口 V 口 ++ 口 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='6 口 口 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='4 1 0 5 10 15 20 x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='7 J>/U, 口 口 口± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='6 口 占 ± ± 口 口 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='5 口 口 口 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='4 、 5 10 15 20 x(a) Centerline (b) Lipline Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' RMS value of the fluctuating part of the axial component of velocity along the centerline and lipline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' The solid line stands for the results of the present case, the open square symbols represent the numerical references43, 46 and the triangular symbols are the experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='44 modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' The DMD calculation is run on a single processor with 128 GB of RAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' According to Hemati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=',23 the computational cost of the algorithm to calculate the DMD eigen-elements is O(nr2), where n and r are the snapshot dimension and the maximum rank of the DMD operator, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' For the latter parameter, the streaming version of the DMD algorithm includes a compression step allowing to set it arbitrarily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Then, the choice of these parameters is a compromise between spatial and spectral resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' The jet entrance, the potential core and the near field of the jet are included in the computational domain in order to prioritize the spatial aspects of the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Therefore, the results should include the aerodynamic structures as well as the generated acoustic waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' However, the original snapshots have been under-sampled in spatial resolution in order to handle manageable snapshots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' The dimensions of the snapshots are specified in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 2, counting approximatively 35 million grid points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Finally, considering 256 snapshots without subtracting the mean, r has been set equal to 50, which was the higher affordable number of retained modes in relation to the available computer memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' In the present case, three variables were extracted from the LES calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Hence, three different DMD reconstruction procedures were performed, using snapshots of the velocity magnitude, the vorticity, based on the Q criterion, and the divergence of the velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' In the following subsections, results are discussed regarding their spectral content (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='1) and spatial shape (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Spectral Analysis Figure 6 displays three different ways of representing the DMD spectrum obtained after the DMD compu- tation using snapshots of the velocity magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Figure 6(a) presents the 50 eigenvalues of A DMD linear operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' The symbols are colored by the initial amplitude of the DMD modes, ∥θi(1)∥, which are defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' (99).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' The choice of this parameter to differentiate the dynamic modes comes from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' (93) and (98).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' The initial amplitude of the DMD modes has also been taken into account by Sayadi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='7 All dynamic modes which are located inside the unit circle are stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' The only one DMD mode located on the unit circle is a steady mode which, in general, retrieves the mean characteristics of the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='42 The stable dynamic modes are unsteady and have a complex conjugate, symmetric with respect to the Im(µi) = 0 axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 6 (b), the growth rate of each mode, σi, is plotted versus the frequency, ωi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' A mode is stable if σi is negative, which is in agreement with the discussion considering Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 6(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Finally, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 6 (c) presents the most amplified DMD mode as a function of the Strouhal number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Four dynamic modes displaying a high amplitude have been selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' In Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 6(b) and (c), it appears that the stability of the mode is not linked with its initial amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' The DMD Mode 5 is more stable than the DMD Mode 7 (σ5 < σ7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' However, ∥θ5(1)∥ is larger than ∥θ7(1)∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Therefore, one can state that the dynamic mode 5 is initially more amplified than the dynamic mode 7, but it decays more quickly as the simulation advances in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Figures 7 and 8 show two different sets of spectra obtained from the DMD computations using snapshots 19 of 26 American Institute of Aeronautics and Astronautics 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='15 口 口 口 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='1 口 口 口 口 口 口 口 口 口 口 口 口 口 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='05 口 口 口 口 口 口 口口 5 10 15 20 x口 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='15 口 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='05 0 0 5 10 15 20 x(a) Eigenvalues of A, µ (b) Eigenvalues of the DMD modes, λ (c) Initial amplitude of the DMD modes, ∥θi(1)∥ Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Spectra from DMD computation using snapshots of velocity magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' In (a) and (b), the symbols are colored by the mode amplitude, ∥θi(1)∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' of the vorticity, based on the Q-criterion, and the divergence of the velocity, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Once again, all dynamic modes are stable, but the one representing the mean flow is neutrally stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' More DMD modes have been highlighted by a number in order to identify them in each spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' (a) Eigenvalues of A, µ (b) Eigenvalues of the DMD modes, λ (c) Initial amplitude of the DMD modes, ∥θi(1)∥ Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Spectra from DMD computation using snapshots of vorticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' In (a) and (b), the symbols are colored by the mode amplitude, ∥θi(1)∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' One can observe in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 6(c), 7(c) and 8(c) that every spectra contain a dynamic mode at St ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='25 and at St ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' The clustering around specific frequencies for different DMD analyses denotes important dynamic activity at these frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' The DMD modes associated to each frequency are shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' These characteristic frequencies coincide with the experimental far field pressure peaks observed by Bridges et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='44 In the next subsection, the spatial shapes of these dynamic modes, given in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 4, are discussed in more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Characteristic frequencies and associated DMD modes St Velocity magnitude Vorticity Divergence of velocity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='25 3 3 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='48 7 7 5 20 of 26 American Institute of Aeronautics and Astronautics 0 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='5 5 b 20 0 20 wi103 101 100 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='5 3 1S19 15 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='5 7 5 3 (ri)wl 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='5 Re(μ)9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='5 15 5 b 19 20 0 20 wi105 II (L)@ I 3 9 15 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='5 3 1S0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='5 0 7 5 3 (ri)wl 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='5 Re(μ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=')(a) Eigenvalues of A, µ (b) Eigenvalues of the DMD modes, λ (c) Initial amplitude of the DMD modes, ∥θi(1)∥ Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Spectra from DMD computation using snapshots of divergence of velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' In (a) and (b), the symbols are colored by the mode amplitude, ∥θi(1)∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Spatial Modes Analysis The averaged axial velocity component of the steady DMD mode is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 9, using the same color coding for the contours as the LES mean flow illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' One can notice a white gap around the centerline of the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' The gap is created because the radial coordinate of the snapshot grid starts at the 20th point in the radial direction, in order to reduce the computational cost of the DMD computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' The DMD mode has been reconstructed by multiplying the mode shape by its initial amplitude ∥θ0(1)∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' A fairly good agreement between the DMD calculation and the large eddy simulation is found regarding the potential core length as well as the contour levels, even considering that the sample rate and the number of snapshots are quite different in the DMD calculation when compared to the LES statistics calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Slice of the three dimensional steady DMD mode for the velocity magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' The white line defines the potential core limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Contours are the same as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' As mentioned in the previous subsection, the experimental far field pressure spectrum of Bridges et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='44 displays two peaks at St ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='25 and St ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Modes at the same frequencies are observable in the three DMD analyses performed in the present study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Figure 10 displays the DMD modes associated to the first frequency, St ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='25, while Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 11 shows the DMD modes associated to St ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' In both figures, isosurfaces and 2-D cut planes of velocity magnitude, vorticity (Q criterion) and divergence of velocity 21 of 26 American Institute of Aeronautics and Astronautics 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='5 (ri)wl 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='5 Re(μ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=')6 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='5 b 20 3 20 0 20 wi103 II (L)@ I 20 5 7 102 3 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='5 3 1S3 2 1 0 1 2 3 0 2 4 6 8 10 Xare presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Considering the high Reynolds number of the present work, and the rapid transition from laminar flow at the jet inlet to a turbulent jet mixing layer, it is possible to observe coherent behavior in the jet dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Moreover, the three variables, for which the DMD computations were performed, bring different information about the flow dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' While the vorticity modes seem to enlighten the mixing layer dynamics, the velocity magnitude as well as the divergence of velocity seem to highlight the aeracoustic dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' (a) DMD mode 3 – Velocity magnitude (b) DMD mode 3 – Velocity magnitude (c) DMD mode 3 – Vorticity (Q criterion) (d) DMD mode 3 – Vorticity (Q criterion) (e) DMD mode 7 – Divergence of velocity (f) DMD mode 7 – Divergence of velocity Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Visualization of the DMD modes found at St ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' (a) and (b) display the real part of Mode 3 extracted from the DMD analysis using snapshots of velocity magnitude, (c) and (d) display the real part of Mode 3 extracted from the DMD analysis using snapshots of vorticity (Q criterion), and (e) and (f) display the real part of Mode 7 extracted from the DMD analysis using snapshots of divergence of velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' The left and right columns show 3-D isosurfaces and 2-D cut-plane visualizations of the modes, respectively (positive in red and negative in blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Figures 10 and 11 indicate that, until x ≈ 1, small coherent vortical structures are growing in the jet mixing layer, generating small acoustic waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Further downstream, the flow has already transitioned and large acoustic waves are generated and are propagated in the downstream direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' As expected, the wavelength of the large acoustic waves depends on the DMD mode frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' One can see, when comparing, for instance, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 10(a) with Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 11(a), or Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 10(e) with Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 11(e),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' that the wavelength of the coherent ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='22 of 26 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='American Institute of Aeronautics and Astronautics ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='2 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='X(a) DMD mode 7 – Velocity magnitude ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='(b) DMD mode 7 – Velocity magnitude ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='(c) DMD mode 7 – Vorticity (Q criterion) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='(d) DMD mode 7 – Vorticity (Q criterion) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='(e) DMD mode 5 – Divergence of velocity ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='(f) DMD mode 5 – Divergence of velocity ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Visualization of the DMD modes found at St ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' (a) and (b) display the real part of Mode 7 extracted from the DMD analysis using snapshots of velocity magnitude, (c) and (d) display the real part of Mode 7 extracted from the DMD analysis using snapshots of vorticity (Q criterion), and (e) and (f) display the real part of Mode 5 extracted from the DMD analysis using snapshots of divergence of velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' The left and right columns show 3-D isosurfaces and 2-D cut-plane visualizations of the modes, respectively (positive in red and negative in blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' structures is divided by two when the frequency is doubled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Moreover, it is easy to verify, for instance, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 11(f), the relation between the wavelength of the large acoustic waves and the actual frequency of the DMD mode, ωi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Another interesting aspect is the presence of small vortices in the inner mixing layer, at the interface with the potential core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' These structures are visible in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 10(d) and 11(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Unfortunately, due to the absence of grid points along the centerline itself in the grid used to extract the data for in the present DMD calculations, the influence of these small vortices at the end of the potential core is not accessible in the present case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Future work should consider a snapshot grid covering all the inner part of the jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Finally, one can see in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 11(c) that the vortex filaments in the mixing layer seem to suffer a three-dimensional helicoidal distortion around the jet mixing layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' The work of Violato and Scarano,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='48 who performed experiments for a low Reynolds free water jet,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' using time-resolved tomographic particle image velocimetry (TR-TOMO PIV),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 23 of 26 American Institute of Aeronautics and Astronautics 2 3 0 2 4 6 8 10 X3 2 1 2 3 0 2 4 6 8 10 X3 0 1 2 3 0 2 4 6 8 10 X3 1 2 3 0 2 4 6 8 10 X3 1 2 3 0 2 4 6 8 10 X3 2 2 3 0 2 4 6 8 10 Xcan certainly help in the understanding of this type of fundamental aspect in the current jet dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Concluding Remarks The present work is concerned with the study of the aerodynamics of a perfectly expanded supersonic jet flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' It is expected that the flow data and the reduced order model here generated could be used in the future for performing aeroacoustic studies of jet flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' An implicit large eddy simulation (LES) formulation for compressible flows, based on the System I set of equations, is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' A streaming version of the total-least- squares DMD algorithm is chosen to run concurrently with the LES simulation and provide an additional form of studying the more relevant aspects of the jet dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' LES of a high Reynolds perfectly expanded supersonic jet flow configuration is performed on a compu- tational mesh with 85 million grid points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Statistical data are extracted from the simulations and present good agreement with the numerical and experimental reference work, at least near the jet inlet region where the mesh is well refined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' However, this is not the case when the jet moves away from the domain entrance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' As a result, the potential core length calculated by the present LES is underestimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Such behavior could be expected since the low order numerical scheme of the numerical solver presently used would probably require quite extensive mesh refinements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' The work also presents three DMD analyses, which have been performed by extracting large three-dimensional snapshots from the LES results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' These DMD computations concerned the velocity magnitude, the vorticity, based on the Q criterion, and the divergence of the velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Two frequencies are identified for which all DMD calculations identify a dynamic mode with relevant flow structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' These frequencies agree with those of relevant dynamics identified in previous experimental work available in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' The analysis of all the dynamic modes brought new insights on the jet dynamics regarding the vortical structures and the acoustic wave patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' At the time of this writing, the LES solver is being adapted in order to include parallel I/O features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' This capability will open new opportunities in term of additional grid resolution that would allow a reduction in the difference between the results calculated by the authors and other data, computational or experimental, available in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Moreover, the DMD algorithm here implemented should also be parallelized in order to allow handling larger snapshots and, hence, the extraction of more information from the flow, espe- cially at the centerline of the jet and further downstream of the jet entrance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Hopefully, these modifications will allow sufficient mesh refinement, both for the LES calculations and for DMD analyses, that the present tool will be useful for studies of the jet aeroacoustics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Acknowledgments The authors gratefully acknowledge the partial support for this research provided by Conselho Nacional de Desenvolvimento Cient´ıfico e Tecnol´ogico, CNPq, under the Research Grants No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 309985/2013-7, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 400844/2014-1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 443839/2014-0 and No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 150450/2016-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' The authors are also indebted to the partial financial support received from Funda¸c˜ao de Amparo `a Pesquisa do Estado de S˜ao Paulo, FAPESP, under the Research Grants No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 2013/07375-0 and No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 2013/21535-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' References 1Lumley, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', Stochastic Tools in Turbulence, Academic Press, New York, 1970.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 2Sirovich, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', “Turbulence and the Dynamics of Coherent Structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Part I: Coherent Structures,” Quarterly of Applied Mathematics, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 45, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 3, 1987, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 561–571.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 3Berkooz, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', Holmes, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', and Lumley, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', “The Proper Orthogonal Decomposition in the Analysis of Turbulent Flows,” Annual Review of Fluid Mechanics, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 25, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 1, 1993, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 539–575.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 4Schmid, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', “Dynamic Mode Decomposition of Numerical and Experimental Data,” Journal of Fluid Mechanics, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 656, 2010, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 5–28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 5Seena, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' and Sung, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', “Dynamic Mode Decomposition of Turbulent Cavity Flows for Self-Sustained Oscillations,” International Journal of Heat and Fluid Flow, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 32, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 6, 2011, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 1098–1110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 6Grilli, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', Schmid, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', Hickel, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', and Adams, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', “Analysis of Unsteady Behaviour in Shockwave Turbulent Boundary Layer Interaction,” Journal of Fluid Mechanics, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 700, 2012, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 16–28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 7Sayadi, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', Schmid, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', Nichols, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', and Moin, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', “Dynamic Mode Decomposition of Controlled H- and K-Type Transitions,” Tech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', Annual Research Briefs, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 8Tang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' and Jiang, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', “Dynamic Mode Decomposition of Hairpin Vortices Generated by a Hemisphere Protuberance,” Science China Physics, Mechanics and Astronomy, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 55, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 1, 2012, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 118–124.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 24 of 26 American Institute of Aeronautics and Astronautics 9Bagheri, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', “Koopman-Mode Decomposition of the Cylinder Wake,” Journal of Fluid Mechanics, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 726, 2013, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 596– 623.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 10Tissot, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', Cordier, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', Benard, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', and Noack, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', “Dynamic Mode Decomposition of PIV Measurements for Cylinder Wake Flow in Turbulent Regime,” Proceedings of Turbulence and Shear Flow Phenomena Conferences, Eight International Symposium on Turbulence and Shear Flow Phenomena, Begel House Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 11Jourdain, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', Eriksson, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='-E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', Kim, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', and Sohn, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', “Application of Dynamic Mode Decomposition to Acoustic- Modes Identification and Damping in a 3-Dimensional Chamber with Baffled Injectors,” Journal of Sound and Vibration, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 332, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 18, 2013, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 4308–4323.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 12Abou-Taouk, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', Sadasivuni, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', L¨orstad, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', Ghenadie, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', and Eriksson, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='-E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', “CFD Analysis and Application of Dynamic Mode Decomposition for Resonant-Mode Identification and Damping in an SGT-100 DLE Combustion System,” Proceedings of the 7th European Combustion Meeting, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 13L´arusson, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', Hafsteinsson, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', Andersson, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', and Eriksson, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='-E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', “Investigation of Supersonic Jet Flow Using Modal Decomposition,” 20th AIAA/CEAS Aeroacoustics Conference, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 14Schmid, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', “Application of the Dynamic Mode Decomposition to Experimental Data,” Experiments in Fluids, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 50, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 4, 2011, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 1123–1130.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 15Schmid, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', Li, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', Juniper, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', and Pust, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', “Applications of the Dynamic Mode Decomposition,” Theoretical and Computational Fluid Dynamics, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 25, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 1-4, 2011, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 249–259.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 16Schmid, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', Violato, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', and Scarano, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', “Decomposition of Time-Resolved Tomographic PIV,” Experiments in Fluids, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 52, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 6, 2012, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 1567–1579.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 17Stegeman, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', Soria, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', and Ooi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', “Dynamic Mode Decomposition of Near Nozzle Instabilities in Large-Eddy Simula- tions of Under-Expanded Circular Jets.” Australasian Fluid Mechanics Conference, RMIT University, 08–11 December 2014, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 1–4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 18Wan, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', Zhou, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', Wang, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', and Sun, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', “Dynamic Mode Decomposition of Forced Spatially Developed Transitional Jets,” European Journal of Mechanics-B/Fluids, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 51, 2015, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 16–26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 19Wynn, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', Pearson, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', Ganapathisubramani, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', and Goulart, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', “Optimal Mode Decomposition for Unsteady Flows,” Journal of Fluid Mechanics, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 733, 2013, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 473–503.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 20Jovanovi´c, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', Schmid, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', and Nichols, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', “Sparsity-Promoting Dynamic Mode Decomposition,” Physics of Fluids, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 26, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 2, 2014, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 024103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 21Williams, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', Kevrekidis, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', and Rowley, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', “A Data–Driven Approximation of the Koopman Operator: Extending Dynamic Mode Decomposition,” Journal of Nonlinear Science, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 25, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 6, 2015, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 1307–1346.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 22Williams, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', Rowley, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', and Kevrekidis, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', “A Kernel-Based Approach to Data-Driven Koopman Spectral Analysis,” AIMS Journals, arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='org/pdf/1411.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='2260v4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='pdf , 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 23Hemati, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', Williams, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', and Rowley, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', “Dynamic Mode Decomposition for Large and Streaming Datasets,” Physics of Fluids, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 26, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 11, 2014, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 111701.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 24Hemati, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' and Rowley, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', “De-Biasing the Dynamic Mode Decomposition for Applied Koopman Spectral Analysis,” arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='org/pdf/1502.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='03854v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='pdf , 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 25Vreman, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', Direct and Large-Eddy Simulation of the Compressible Turbulent Mixing Layer, Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' thesis, Universiteit Twente, 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 26Vremant, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', Geurts, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', and Kuerten, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', “A Priori Tests of Large Eddy Simulation of the Compressible Plane Mixing Layer,” Journal of Engineering Mathematics, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 29, 1995, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 299–327.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 27Garnier, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', Adams, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', and Sagaut, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', Large Eddy Simulation for Compressible Flows, Springer, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 28Sagaut, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', Large Eddy Simulation for Incompressible Flows, Springer, 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 29Eidson, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', “Numerical Simulation of the Turbulent Rayleigh–B´enard Problem Using Subgrid Modelling,” Journal of Fluid Mechanics, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 158, 1985, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 245–268.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 30Junqueira-Junior, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', Yamouni, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', Azevedo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', and Wolf, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', “Large Eddy Simulations of Supersonic Jet Flows for Aeroacoustic Applications,” AIAA Paper No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 2015-3306, Proceedings of the 33rd AIAA Applied Aerodynamics Conference, Dallas, TX, June 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 31Junqueira-Junior, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', Yamouni, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', Azevedo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', and Wolf, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', “Influence of Different Subgrid Scale Models in LES of Supersonic Jet Flows,” AIAA Paper No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 2016-4093, Proceedings of the 46th AIAA Fluid Dynamics Conference, Washington, DC, June 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 32Junqueira-Junior, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', Development of a Parallel Solver for Large Eddy Simulation of Supersonic Jet Flows, Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' thesis, Instituto Tecnol´ogico de Aeron´autica, S˜ao Jos´e dos Campos, SP, Brasil, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 33Bigarella, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', Three-Dimensional Turbulent Flow Over Aerospace Configurations, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='Sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' Thesis, Instituto Tec- nol´ogico de Aeron´autica, S˜ao Jos´e dos Campos, SP, Brasil, 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 34Turkel, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' and Vatsa, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', “Effect of Artificial Viscosity on Three-Dimensional Flow Solutions,” AIAA Journal, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 32, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 1, 1994, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 39–45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 35Jameson, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' and Mavriplis, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', “Finite Volume Solution of the Two-Dimensional Euler Equations on a Regular Triangular Mesh,” AIAA Journal, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 24, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 4, Apr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 1986, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 611–618.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 36Jameson, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', Schmidt, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', and Turkel, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', “Numerical Solutions of the Euler Equations by Finite Volume Methods Using Runge-Kutta Time-Stepping Schemes,” AIAA Paper 81–1259, Proceedings of the AIAA 14th Fluid and Plasma Dynamic Conference, Palo Alto, Californa, USA, June 1981.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 37Long, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', Khan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', and Sharp, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', “A Massively Parallel Three-Dimensional Euler/Navier-Stokes Method,” AIAA Journal, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 29, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 5, 1991, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 657–666.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 38Tu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', Rowley, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', Kutz, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', and Shang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', “Spectral Analysis of Fluid Flows Using Sub-Nyquist-Rate PIV Data,” Experiments in Fluids, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 55, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 9, 2014, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 1–13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 39Rowley, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', Mezi´c, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', Bagheri, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', Schlatter, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', and Henningson, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', “Spectral Analysis of Nonlinear Flows,” Journal of Fluid Mechanics, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 641, 2009, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 115–127.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 25 of 26 American Institute of Aeronautics and Astronautics 40Hemati, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', Deem, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', Williams, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', Rowley, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', and Cattafesta, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', “Improving Separation Control with Noise-Robust Variants of Dynamic Mode Decomposition,” 54th AIAA Aerospace Sciences Meeting, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 41Kutz, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', Fu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', and Brunton, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', “Multiresolution Dynamic Mode Decomposition,” SIAM Journal on Applied Dynamical Systems, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 15, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 2, 2016, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 713–735.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 42Chen, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', Tu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', and Rowley, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', “Variants of Dynamic Mode Decomposition: Boundary Condition, Koopman, and Fourier Analyses,” Journal of Nonlinear Science, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 22, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 6, 2012, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 887–915.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 43Mendez, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', Shoeybi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', Sharma, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', Ham, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', Lele, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', and Moin, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', “Large-Eddy Simulations of Perfectly- Expanded Supersonic Jets: Quality Assessment and Validation,” AIAA Paper No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 2010-0271, Proceedings of the 48th AIAA Aerospace Sciences Meeting Including The New Horizons Forum and Aerospace Exposition, Orlando, FL, January 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 44Bridges, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' and Wernet, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', “Turbulence Associated with Broadband Shock Noise in Hot Jets,” AIAA Paper No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 2008- 2834, Proceedings of the 14th AIAA/CEAS Aeroacoustics Conference and 29th AIAA Aeroacoustics Conference, Vancouver, BC, Canada, May 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 45ANSYS, “http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='ansys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content='com/,” .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 46Mendez, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', Shoeybi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', Sharma, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', Ham, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', Lele, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', and Moin, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', “Large-Eddy Simulations of Perfectly- Expanded Supersonic Jets Using an Unstructured Solver,” AIAA Journal, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 50, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 5, May 2012, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 1103–1118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 47Bres, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', Nichols, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', Lele, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', and Ham, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', “Towards Best Practices for Jet Noise Predictions with Unstructured Large Eddy Simulations,” AIAA Paper No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 2012-2965, Proceedings of the 42nd AIAA Fluid Dynamics Conference and Exhibit, New Orleans, Louisiana, June 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 48Violato, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' and Scarano, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=', “Three-Dimensional Vortex Analysis and Aeroacoustic Source Characterization of Jet Core Breakdown,” Physics of Fluids, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 25, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 1, 2013, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 015112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} +page_content=' 26 of 26 American Institute of Aeronautics and Astronautics' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E4T4oBgHgl3EQfXQwv/content/2301.05039v1.pdf'} diff --git a/vNE0T4oBgHgl3EQfbwA4/content/2301.02352v1.pdf b/vNE0T4oBgHgl3EQfbwA4/content/2301.02352v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..a18ac08e9250686891183405b2c349365f24bbd1 --- /dev/null +++ b/vNE0T4oBgHgl3EQfbwA4/content/2301.02352v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:a25ff2c6e8e1d7654dd38bbb4c4dd3ff44dfac38a771aa86af3a8252a6fc5a83 +size 1955542 diff --git a/vNE0T4oBgHgl3EQfbwA4/vector_store/index.faiss b/vNE0T4oBgHgl3EQfbwA4/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..a9b0eb253ec37a11422e73ed9f7504cb93d73a14 --- /dev/null +++ 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pp. 1–35 +doi:10.1093/biostatistics/Manuscript˙cohort2 +A joint Bayesian hierarchical model for +estimating SARS-CoV-2 diagnostic and +subgenomic RNA viral dynamics and +seroconversion +Tracy Q. Dong1∗ and Elizabeth R. Brown1,2 +1Vaccine and Infectious Disease Division, Fred Hutchinson Cancer Center +2Department of Biostatistics, University of Washington +qdong@fredhutch.org +Summary +Understanding the viral dynamics and immunizing antibodies of the severe acute respiratory +syndrome coronavirus 2 (SARS-CoV-2) is crucial for devising better therapeutic and prevention +strategies for COVID-19. Here, we present a Bayesian hierarchical model that jointly estimates the +diagnostic RNA viral load reflecting genomic materials of SARS-CoV-2, the subgenomic RNAs +(sgRNA) viral load reflecting active viral replication, and the rate and timing of seroconversion +reflecting presence of antibodies. Our proposed method accounts for the dynamical relationship +and correlation structure between the two types of viral load, allows for borrowing of information +between viral load and antibody data, and identifies potential correlates of viral load character- +istics and propensity for seroconversion. We demonstrate the features of the joint model through +application to the COVID-19 PEP study and conduct a cross-validation exercise to illustrate the +∗To whom correspondence should be addressed. +© The Author 2023. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oup.com +arXiv:2301.03714v1 [stat.AP] 9 Jan 2023 + +2 +T.Q. Dong and E.R. Brown +model’s ability to impute the sgRNA viral trajectories for people who only had diagnostic viral +load data. +Key words: Joint models; Bayesian hierarchical models; SARS-CoV-2; Correlates; Viral load; Serocon- +version. +1. Introduction +The severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) has spread rapidly around +the world, resulting in more than 6 million deaths worldwide to date (World Health Organization, +2022). Current knowledge regarding SARS-CoV-2 viral dynamics is built on diagnostic viral +load detected via real-time reverse transcription polymerase chain reaction (RT-PCR)(Cevik and +others, 2021; Gastine and others, 2021; Mellis and others, 2022; Pajon and others, 2022), the +most sensitive and widely used technique for the diagnosis of coronavirus disease 2019 (COVID- +19). However, the diagnostic RT-PCR tests give limited insight into infectivity because the results +may reflect residual genomic material rather than replicating virus (Singanayagam and others, +2020). Viral culture and quantification by plaque assays provide a gold standard for assessing +infectivity but are not readily available at large scale for a diagnostic assay. Therefore, detection +of subgenomic RNAs (sgRNA) in clinical samples has been suggested as an additional diagnostic +tool to track infectious virus because sgRNA transcripts indicate active viral replication (Dagotto +and others, 2021; Bravo and others, 2022). Characterization of both SARS-CoV-2 diagnostic and +sgRNA viral dynamics is crucial for understanding the pathogenesis of the virus, identifying +patients most at risk , and optimizing the period of self-isolation to reduce onward transmission +while reducing unnecessary loss of productivity. +Developing a model for SARS-CoV-2 viral load is of particular interest because viral load +is an important outcome of COVID-19 treatment, vaccine, and non-vaccine prevention studies + +Joint Bayesian model for SARS-CoV-2 viral load and seroconversion +3 +but viral load trajectories are rarely fully observed (?Gottlieb and others, 2021; Goldberg and +others, 2021; Chaccour and others, 2021; Portal-Celhay and others, 2022; Levine-Tiefenbrun and +others, 2022; Eyre and others, 2022). One commonly used approach is compartmental models +based on ordinary differential equations (ODE) (Goyal and others, 2020; N´eant and others, 2021; +Ke and others, 2021; Challenger and others, 2022). While the ODE-based models are useful for +capturing cell-level within-host viral kinetics and simulating potential effects of antiviral therapies +under various assumptions, the model parameters cannot be easily translated into key viral +trajectory characteristics such as peak viral load and shedding duration. Therefore, Bayesian +mixed-effects models are a popular alternative for estimating population- and individual-level +viral load trajectories (Kissler and others, 2021a,b; Stankiewicz Karita and others, 2021; Jones +and others, 2021; Singanayagam and others, 2021; Deming and others, 2022). This approach +has the advantage of being able to directly model viral load features and incorporate covariates +and correlation structures to explore potential associations between viral dynamics and patient +characteristics. So far, all existing models estimate the diagnostic and sgRNA viral load separately +without considering any dependency between the two. +In addition to viral dynamics, serology characteristics of SARS-CoV-2 infection, especially +the rate and timing of seroconversion defined by the development of neutralizing antibodies, are +also of interest. Not all persons recovering from SARS-CoV-2 infection develop antibodies (Oved +and others, 2020; Pathela and others, 2021; Wellinghausen and others, 2020). Understanding the +natural immunity acquired after SARS-CoV-2 infection and the correlates of seroconversion is +crucial for studying disease transmission, vaccine effectiveness, and population-wide seropreva- +lence and immunity. Since most serological studies were conducted among hospitalized people +who exhibited moderate to severe COVID-19 symptoms and were tracked in the later course of +the infection (Lou and others, 2020; Yadav and others, 2021; Masi´a and others, 2021), little is +known regarding seroconversion rate and time in outpatients settings and among persons with + +4 +T.Q. Dong and E.R. Brown +very mild or asymptomatic infection (Centers for Disease Control and Prevention, 2021). +In this paper, we develop a Bayesian hierarchical model that jointly analyzes SARS-CoV-2 +diagnostic and sgRNA viral load data and interval-censored seroconversion data to estimate viral +trajectories, their association with patient characteristics, and the correlates, rate, and time of +seroconversion. Our proposed model has several advantages compared to the existing models: +first, it takes into account the dynamical relationship between the diagnostic PCR and sgRNA +viral trajectories and can be used to impute sgRNA viral trajectories for patients who only had +diagnostic viral load data. In addition, by jointly modeling viral load and antibody measurements, +our model allows the borrowing of information across different data types to better estimate key +virological and serological parameters. Our model also uses a pair of indicator variables for the +observed and true viral shedding status to allow direct modeling of false positive and false negative +cases. Finally, our model explicitly accounts for the increased uncertainty in RT-PCR viral load +measurements below the test’s limit of quantification (LoQ). +In Section 2, we introduce a motivating example of the COVID-19 post-exposure prophylaxis +(PEP) study (Barnabas and others, 2021). Section 3 describes the model formulation and infer- +ence procedure. In Section 4, we apply our methods to analyze data collected in the COVID-19 +PEP study and summarize the results. We end the paper with a discussion in Section 5. +2. A motivating example: the COVID-19 PEP Study +The COVID-19 PEP Study was a double-blinded, household-randomized controlled trial compar- +ing hydroxychloroquine to placebo-like control for prevention of SARS-CoV-2 infection (Barnabas +and others, 2021). The study was remotely conducted between March and August 2020 before +COVID-19 vaccines were available and enrolled 829 initially asymptomatic household contacts +and healthcare workers recently exposed (<96 hours) to persons with laboratory-confirmed SARS- +CoV-2 infection from 41 U.S. states. The study found no differences between hydroxychloroquine + +Joint Bayesian model for SARS-CoV-2 viral load and seroconversion +5 +and placebo-like control arms in SARS-CoV-2 acquisition. +Participants provided demographic information via enrollment surveys. They also completed +daily symptom questionnaires and self-collected daily mid-turbinate swabs for 14 consecutive +days. Swabs were tested for SARS-CoV-2 RNA at the University of Washington (UW) Virology +Laboratory using RT-PCR targeting nucleocapsid genes N1 and N2. Samples with detectable +SARS-CoV-2 RNA via RT-PCR were further tested for sgRNA targeting sgE (Bruce and others, +2021). +RT-PCR for both N and sgE were performed to quantify the viral titer via the cycle threshold +(Ct), defined as the number of thermal cycles needed to amplify sampled viral RNA to a detectable +level. Quantification standards were run with both assays to convert Ct value to viral load as +copies/ml during assay validation. Both tests used a Ct cutoff of 40, corresponding to a lower +limit of detection (LoD) of approximately 102.4 copies/ml. In addition, the diagnostic RT-PCR +had a limit of quantification (LoQ), defined as the lowest amount of viral RNA in a sample that +can be quantitatively determined with acceptable precision accuracy (Forootan and others, 2017), +of approximately 104.9 copies/ml, corresponding to a Ct value of 31. Therefore, the conversion +from the diagnostic Ct values to viral concentrations suffers higher uncertainty for nasal swabs +with observed Ct values between the LoD and LoQ. +Dried blood spot (DBS) samples were voluntarily provided by a subset of participants on study +day 1, 14 and 28. The DBS samples were assayed to detect Immunoglobulin G (IgG) antibodies +to SARS-CoV-2 spike protein at the UW Department of Laboratory Medicine & Pathology. For +brevity, we will use “seroconversion” to refer to anti-spike IgG seroconversion for the rest of the +paper. However, the model can be generalized to accommodate other types of seroonversion as +well. More information can be found in the Discussion section. + +6 +T.Q. Dong and E.R. Brown +3. Methods +We assume that the diagnostic and sgRNA viral load trajectories consist of a proliferation phase +with exponential growth in viral RNA concentration, followed by a clearance phase characterized +by exponential decay in viral RNA concentration. On the scale of log10 copies/ml, this roughly +corresponds to a linear increase to a peak followed by a linear decrease. Figure 1 shows a graphical +illustration of the latent viral RNA concentration trends overlay with the observed diagnostic and +sgRNA viral load measurements over time for an individual participant. We describe the details +of the model in the following sections. +3.1 +Modeling the diagnostic RNA viral load +Let i be index for participants and j be index for nasal swabs. For person i, let wa,i represent +time from shedding onset to peak viral load, wb,i represent time from peak to viral clearance, +and vp,i represent the magnitude of peak viral load with respect to LoD (Figure 1). We use the +latent peak as a reference point and let s represent time since the latent peak viral load. As such, +we have s = 0 at the latent peak viral load, s = −wa,i at shedding onset, and s = wb,i at viral +clearance. For the jth swab collected from person i, sij is the time from peak viral load to swab +collection, and the latent diagnostic viral load at sij, denoted by µy(sij), is +µy(sij) = LoD + vp,i + +� vp,i +wa,i +� +× sij × 1 [sij ⩽ 0] + +� +− vp,i +wb,i +� +× sij × 1 [sij > 0] , +where 1 [·] is an indicator function. In particular, +� +vp,i +wa,i +� +and +� +− vp,i +wb,i +� +represent the upward and +downward slopes respectively for the linear change in diagnostic viral trajectory over time. To +model the person-level variability in wa,i, wb,i and vp,i while accounting for potential correlation +among these parameters, we use a multivariate normal distribution for the random effects on the + +Joint Bayesian model for SARS-CoV-2 viral load and seroconversion +7 +log scale: +log ([vp,i, wa,i, wb,i]⊺) ∼ MVN (µlog,i, Σlog) +µlog,i[1] = µlvp + β⊺ +vpXvp,i +µlog,i[2] = µlwa + β⊺ +waXwa,i +µlog,i[3] = µlwb + β⊺ +wbXwb,i +where Xvp,i, Xwa,i and Xwb,i are vectors of person-level covariates for peak viral load, time from +shedding onset to peak, and time from peak to viral clearance, respectively; and βvp, βwa and +βwb are the corresponding vectors of coefficients for these covariates. +We use a pair of indicator variables to model the true positive rate (TPR) and true negative +rate (TNR) for the observed diagnostic viral detection. Let yij denote the observed viral RNA +concentration in log10 copies/ml for swab j of person i, Bij be the fully observed indicator variable +of whether the swab had detectable diagnostic RNA via RT-PCR (i.e., Bij = 1 [yij > LoD]), and +let Sij be the latent indicator variable of whether person i was truly experiencing diagnostic viral +shedding when swab j was collected (i.e., Sij = 1 [−wa,i ⩽ sij ⩽ wb,i]). We specify the following +model: +Bij|Sij ∼ Bernoulli (expit(α0 + α1Sij)) . +In particular, the true positive rate (TPR), defined as the probability of detecting diagnostic +RNA from a nasal swab when a person was experiencing viral shedding, is +TPR = E[Bij = 1|Sij = 1] = expit(α0 + α1); +and the true negative rate (TNR), defined as the probability of not detecting viral RNA from a +nasal swab when a person was clear of viral RNA, is +TNR = E[Bij = 0|Sij = 0] = 1 − expit(α0). + +8 +T.Q. Dong and E.R. Brown +The distribution of a swab’s observed diagnostic viral load depends on whether the test result +is a true/false positive/negative. When the jth swab collected from person i has no detectable +diagnostic RNA, i.e., Bij = 0, the observed viral load equals to the LoD. When a swab does +have detectable viral RNA, i.e., Bij = 1, we further consider two scenarios: if the result is false +positive, i.e., Sij = 0, we use a normal distribution centered around some viral load level above +the LOD with a moderate standard deviation: +yij|Bij = 1, Sij = 0 ∼ Normal +� +LoD+, 0.5 +� +. +If the result is true positive, i.e., Sij = 1, we use a normal distribution centered around the latent +viral trajectory and a standard deviation that depends on LoQ: +yij|Bij = 1, Sij = 1 ∼ Normal (µy(sij), σy (Qij)) , +where +σ2 +y (Qij) = σ2 +yy × (1 + Qij × δQ) . +Our specification the variance σ2 +y aims to account for the increased uncertainty in viral load +measurements for swabs with diagnostic RNA concentration below the LoQ. Specifically, we let +σ2 +yy be the base variance term denoting the unmeasured variability in the observations, and Qij +be an indicator variable of whether the observed viral concentration from swab j lies between +LoD and LoQ (i.e., Qij = 1 [LoD < yij < LoQ]). When a swab has an observed diagnostic RNA +viral load between LoD and LoQ, i.e., Qij = 1, the variance term will be higher than the base +level by a factor of δQ to account for the extra uncertainty in these observations. +Finally, since the latent peak viral load is not observable, we estimate the time from the +observed peak to the latent peak for person i, denoted tp,i. If person i’s observed peak viral load +was recorded within the first two days of follow-up, we will assume the latent peak happens at + +Joint Bayesian model for SARS-CoV-2 viral load and seroconversion +9 +the latest one day after the observed peak: +tp,i ∼ Normal (µtpL, σtpL) truncated [−∞, 1], +where µtpL is negative. If person i’s observed peak viral load was recorded during the last two +days of follow-up, we will assume the latent peak happens at the earliest one day before the +observed peak: +tp,i ∼ Normal (µtpR, σtpR) truncated [−1, ∞], +where µtpR is positive. Otherwise, we assume the latent peak happens around the time of the +observed peak: +tp,i ∼ Normal (0, σtp) . +We can specify σtpL, σtpR, and σtp to reflect our prior belief in how variable the time differences +tp,i are among the participants. +3.2 +Modeling the sgRNA viral load +We model the sgRNA viral load in relation to the diagnostic viral trajectory. For clarity, we +define a parallel set of notations for the sgRNA viral load by adding ′ after the previously defined +notations. We assume that the diagnostic and sgRNA viral shedding starts at the same time. +In addition, we know that sgRNA viral clearance occurs no later than the diagnostic RNA viral +clearance (Figure 1). +Let td,i denote time from the latent peak diagnostic viral load to the latent peak sgRNA viral +load for person i. We use the following truncated normal distribution for t′ +d,i: +t′ +d,i ∼ Normal (µtd, 2) truncated [−wa,i, wb,i] +The time from sgRNA shedding onset to peak viral load for person i, denoted w′ +a,i, can be + +10 +T.Q. Dong and E.R. Brown +calculated as +w′ +a,i = wa,i + t′ +d,i. +We use a truncated normal distribution to model the log-transformed time from sgRNA viral +clearance to diagnostic RNA rival clearance, denoted w′ +d,i: +log +� +w′ +d,i +� +∼ Normal (µlwd, σlwd) truncated +� +0, log +� +wb,i − t′ +d,i +�� +. +The time from peak sgRNA viral load to viral clearance for person i, denoted w′ +b,i, can be +calculated as +w′ +b,i = wa,i + wb,i − w′ +a,i − w′ +d,i. +As for the magnitude of the peak sgRNA viral load v′ +p,i, we parameterize it as the product +of the diagnostic peak viral load and a multiplier qi, i.e., v′ +p,i = qi × vp,i. Since the peak sgRNA +viral load is always lower than the peak diagnostic RNA viral load, we model the multiplier qi +using a beta distribution: +qi ∼ Beta (γ1, γ2) +In the PEP study, only the swabs with detectable diagnostic RNA were tested for sgRNA. +Therefore, we model the TPR and TNR of sgRNA detection conditional on the observed diag- +nostic viral load measurement: +B′ +ij|S′ +ij, Bij ∼ Bernoulli +� +Bij × +� +expit(α′ +0 + α′ +1S′ +ij) +� � +. +Specifically, when a swab was negative for diagnostic RNA, i.e., Bij = 0, it would also be negative +for sgRNA, i.e., B′ +ij = 0. Otherwise, the probability of detecting sgRNA from a nasal swab when +a person was experiencing sgRNA viral shedding is +TPR′ = E[B′ +ij = 1|S′ +ij = 1, Bij = 1] = expit(α′ +0 + α′ +1); + +Joint Bayesian model for SARS-CoV-2 viral load and seroconversion +11 +and the probability of not detecting sgRNA from a nasal swab when a person was clear of sgRNA +is +TNR′ = E[B′ +ij = 0|S′ +ij = 0, Bij = 1] = 1 − expit(α′ +0). +The distribution of the observed sgRNA viral load follows a similar structure as the diagnostic +viral load and depends on whether the sgRNA test result is a true/false positive/negative. When +the jth swab collected from person i has no detectable sgRNA, i.e., B′ +ij = 0, the observed +viral load equals to the LoD. When a swab does have detectable sgRNA and the result is false +positive, we use a normal distribution centered around some viral load level above the LOD with +a moderate standard deviation: +y′ +ij|B′ +ij = 1, S′ +ij = 0 ∼ Normal +� +LoD+, 0.5 +� +. +Finally, if the sgRNA viral load detected is true positive, we use a normal distribution: +y′ +ij|B′ +ij = 1, S′ +ij = 1 ∼ Normal +� +µ′ +y(s′ +ij), σ′ +y +� +, +where +µ′ +y(s′ +ij) = LoD′ + v′ +p,i + +� +v′ +p,i +w′ +a,i +� +× s′ +ij × 1 +� +s′ +ij ⩽ 0 +� ++ +� +− v′ +p,i +w′ +b,i +� +× s′ +ij × 1 +� +s′ +ij > 0 +� +. +3.3 +Modeling seroconversion rate and time +Since not everyone infected with SARS-CoV-2 would eventually develop neutralizing antibodies, +we use an indicator variable Ci to model whether person i would ever seroconvert: +Ci ∼ Bernoulli (expit (βC0 + β⊺ +CXC,i)) , +where XC,i are vectors of potential correlates of seroconversion. If a person developed antibodies +after infection, then the time from viral shedding onset to seroconversion, denoted ws,i, is modeled + +12 +T.Q. Dong and E.R. Brown +as +ws,i|Ci = 1 ∼ Gamma (κ1, κ2) +If a person never seroconverts, ws,i = ∞. +In our motivating example of the COVID-19 PEP study, DBS samples were only collected on +day 1, 14 and/or 28 of the study. Therefore, the servoconversion time is censored. Specifically, if +a participant only had negative IgG samples, his/her seroconversion time is right-censored at the +last negative antibody test; if a participant only had positive IgG samples, his/her seroconversion +time is left-censored at the first positive antibody test; if a participant had both negative and +positive IgG samples, his/her seroconversion time is interval-censored between the last negative +test and the first positive test. Finally, since not all participants had provided DBS samples with +confirmed results, only those who had antibody data available will contribute to the likelihood +of the model. +3.4 +Prior elucidation +Inference for all model parameters can be conducted under a Bayesian framework. The prior +distributions for virological parameters {µlvp, µlwa, µlwb, µlwd} can be specified using normal dis- +tributions with means equal to the estimated diagnostic and sgRNA peak viral load, time from +shedding onset to peak and time from peak to shedding cessation from previous literature. For +variance and covariance parameters +� +σ2 +yy, σ +′2 +y , Σlog +� +, we can use weakly informative priors such +as the Cauchy distribution and inverse wishart distribution. Multivariate normal distributions +with zero means can be used as priors for the coefficients of potential correlates of viral dynamics +and seroconversion {βvp, βwa, βwb, βC}. For the model parameters related to TPR and TNR of +diagnostic and sgRNA detection {α0, α1, α′ +0, α′ +1}, we can use relatively tight normal priors be- +cause we expect TPR and TNR to be high. Similarly, we can specify a moderately informative +beta prior for δQ since we do not expect the extra uncertainty in the observed diagnostic viral load + +Joint Bayesian model for SARS-CoV-2 viral load and seroconversion +13 +below LoQ to be huge. As for the positive-valued parameters {γ1, γ2, κ1, κ2}, we can use gamma +distributions as priors such that the resulting prior distributions for qi and wi have reasonable +values as suggested by previous literature. +3.5 +sgRNA viral load imputation +In a resource-constrained setting, it is common that everyone in a study was tested for diagnostic +RNA but only a subset were tested for sgRNA. In this case, our model can be used to impute the +sgRNA viral trajectories for people who only had diagnostic RNA viral load and covariates data +by treating their sgRNA measurements as missing data. Specifically, we can combine the people +without sgRNA samples with the people who had complete data and fit the model to everyone to +estimate the posterior distributions of +� +v′ +p,i, w′ +a,i, w′ +b,i +� +by borrowing information from the rest +of the parameters. +4. Application to the COVID-19 PEP study +4.1 +Model implementation +We applied our methods to analyze data collected from the 80 COVID-19 PEP study participants +who had at least 2 positive sgRNA samples during the first 14 days of follow-up (Figures S1-2 +of the Supplement). We chose this subset of participants because we are interested in examining +the viral dynamics and seroconversion among SARS-CoV-2-infected people who had sustained +viral shedding. Among them, 33 participants had at least 1 DBS sample with confirmed results, +hence their anti-spike IgG antibody data contributed to the seroconversion part of the model. +For our primary model, the following covariates were included as correlates for peak diagnostic +viral load (Xvp): age, sex at birth, treatment arm (hydroxychloroquine vs. placebo), and presence +of COVID-19 symptoms defined based on the case definition approved by the U.S. Centers for +Disease Control and Prevention (CDC) (Centers for Disease Control and Prevention, 2020). The + +14 +T.Q. Dong and E.R. Brown +peak diagnostic viral load, vp, was included as a correlate for seroconversion probability (XC). +All analysis was implemented in R 4.2.1 (R Core Team, 2022) using JAGS 4.3.0 (Plummer and +others, 2003). More details, including prior specification, Markov Chain Monte Carlo (MCMC) +setup, and model diagnostics, can be found in Section S1 of the Supplement. +4.2 +Model results +On average, the diagnostic viral load reaches a peak of 8.0 (95% credible interval, CI: [7.8, 8.1]) +log10 copies/ml after 3.8 (95% CI: [3.2, 4.7]) days and then clears after 10.5 (95% CI: [10.1, 10.9]) +days. The sgRNA viral load reaches a peak of 6.0 (95% CI: [5.9, 6.2]) log10 copies/ml approxi- +mately 0.6 (95% CI: [0.3, 0.8]) day after the peak diagnostic viral load and then clears after only +5.7 (95% CI: [5.3, 6.1]) days, approximately 4.3 (95% CI: [4.0, 4.6]) days before the diagnostic +viral clearance (Table 1 and Figure 2). The estimated infectious period, as approximated by the +total duration of sgRNA viral shedding, is 10.1 days (95% CI: [9.4, 11.0]). The posterior means +and 95% CIs of individual viral load trajectories are shown in Figures S1-2 in the Supplement. +We estimated the multiplicative factors associated with various patient characteristics for an +increase in peak diagnostic viral load on the log10 copies/ml scale (Figure 3). The posterior means +of the parameters indicated that being older, male, in the hydroxychloroquine arm, and reporting +COVID-19 symptoms were positively associated with peak viral load in our data. However, all the +posterior 95% CIs were wide and overlapped with 1, indicating that there was high uncertainty +associated with these estimates. +To identify specific symptoms that might potentially be associated with peak diagnostic vi- +ral load, we fitted additional models by replacing the COVID-19 symptoms indicator with other +symptom indicators defined based on the Flu-PRO patient-reported outcome instrument (Powers +and others, 2015), including fever, change in taste or smell, and chest/throat/nose/body/gastrointestinal +symptoms. Figure 4 shows the posterior means and 95% CIs of the multiplicative factors associ- + +Joint Bayesian model for SARS-CoV-2 viral load and seroconversion +15 +ated with various symptoms for an increase in peak diagnostic viral load on the log10 copies/ml +scale. Among all symptoms investigated, having body symptoms (chills, myalgia, headache, or +fatigue) was most strongly associated with a higher peak diagnostic viral load (multiplicative fac- +tor = 1.25, 95% CI: [1.07, 1.50]). That is, on the log10 copies/ml scale, the peak diagnostic RNA +viral load was estimated to be 25% (95% CI: [7%, 50%]) higher among SARS-CoV-2 patients +who experienced chills, myalgia, headache, or fatigue than those who did not experience these +symptoms. The rest of the symptoms either had close-to-zero or positive estimated associations, +but all their 95% CIs overlapped with 1. +The overall seroconversion rate was 75.0% (95% CI: [62.5%, 85.0%]). The peak diagnostic viral +load was estimated to be positively associated with the probability of seroconversion (odds ratio += 1.24, 95% CI: [1.00, 1.54], for every 10-fold increase in diagnostic peak viral load). Among those +who seroconverted, the mean time from viral shedding onset to the presence of detectable IgG +antibodies is 14.4 days (95% CI: [12.5, 16.6]) (Table 1 and Figure 2). +To identify any symptoms that might potentially be associated with seroconversion rate, we +fitted additional models by replacing peak diagnostic viral load with symptom indicators. Figure +5 shows the posterior means and 95% CIs of the odds ratio of seroconversion associated with +various symptoms. All symptoms investigated were estimated to be positively associated with +the probability of seroconversion but all posterior 95% CIs overlapped with 1. +The probabilities that a sample collected during viral shedding would be tested positive +for diagnostic and sgRNA (TPR) are 91.8% (95% CI: [90.1%, 93.3%]) and 89.6% (95% CI: +[87.4%, 91.6%]) respectively; the probabilities that a sample collected outside the viral shed- +ding periods would be tested negative for diagnostic and sgRNA (TNR) are 99.5% (95% CI: +[99.4%, 99.6%]) and 99.6% (95% CI: [99.5%, 99.7%]) respectively (Table 1). Note that these val- +ues account for not only the properties inherent to the RT-PCR tests such as sensitivity and +specificity, but also the potential errors that occurred due to misplaced swabs, contaminated + +16 +T.Q. Dong and E.R. Brown +samples, or improper swabbing that were common in outpatient household studies. +Our model results indicated that there were no significant correlations among peak diagnostic +viral load, time from shedding onset to peak, and time from peak to diagnostic viral clearance +(Table S1 in the Supplement). As for the extra uncertainty in diagnostic viral load measurements +below LoQ (104.9 copies/ml, corresponding to a Ct value of 31), our model estimated that the +variance in the observed diagnostic viral load (σ2 +y) was approximately 25% (95% CI: [13%, 38%]) +higher if the observed values were below LoQ as compared to those above LoQ. +4.3 +Cross-validation for sgRNA viral load imputation and seroconversion probability estimation +We conducted an 10-fold cross-validation exercise to examine our model’s ability to impute sgRNA +viral load and seroconversion for people who only have diagnostic viral load and covariate data. +Specifically, we randomly divided the 80 participants into 10 groups of 8 people. For each itera- +tion, we removed the sgRNA viral load data (and antibody data if available) from one group of +participants and fitted the model using the rest of the observed data. In figures S3-4 in the Sup- +plement, we present the observed and imputed sgRNA viral load trajectories and the estimated +probability of seroconversion from all iterations. The posterior means of the sgRNA viral trajec- +tories were very close to the masked sgRNA viral load measurements. As expected, the imputed +sgRNA viral load had wider posterior 95% CIs than those of the diagnostic viral trajectories. +The estimated seroconversion probabilities ranged from 64% to 89%, a reasonable range given +the masked antibody data. +5. Discussion +In this paper, we presented a joint Bayesian hierarchical model that allows for borrowing of +information between viral load and antibody data to make inferences on key characteristics of +diagnostic and sgRNA viral dynamics and seroconversion. Our method can be used to identify + +Joint Bayesian model for SARS-CoV-2 viral load and seroconversion +17 +potential correlates of viral load trajectories and propensity for seroconversion, to estimate the +seroconversion time after a SARS-CoV-2 infection, and to impute sgRNA viral load for people +who only have diagnostic RNA viral load data. By jointly modeling the diagnostic and sgRNA +viral trajectories, our method is able to take into account the dynamical relationship and corre- +lation structure between the two types of viral load. Our model also uses indicator variables to +represent latent and observed viral shedding status separately, accounting for false positive and +false negative viral detection due to not only the RT-PCR test properties but also the potential +sample collection errors in outpatient study settings. As a demonstrating example, we imple- +mented our proposed methods using JAGS in R and analyzed data collected from 80 participants +of the COVID-19 PEP study who had at least 2 positive sgRNA samples during the first 14 days +of follow-up. A cross-validation exercise was also conducted to examine the model’s ability to +impute sgRNA viral load trajectories. +Based on the COVID-19 PEP data, our model-estimated population-level viral load trajecto- +ries were characterized by a rapid rise to reach viral peak and subsequent slower decline, similar +to what has been demonstrated in previous literature (Goyal and others, 2020; N´eant and others, +2021; Ke and others, 2021; Kissler and others, 2021a,b; Stankiewicz Karita and others, 2021; +Jones and others, 2021; Singanayagam and others, 2021; Deming and others, 2022). The model +results on diagnostic and sgRNA viral shedding duration were consistent with previous findings +that SARS-CoV-2 infectivity usually terminates after 10 days (van Kampen and others, 2021; +W¨olfel and others, 2020; Bullard and others, 2020; Arons and others, 2020; The COVID-19 +Investigation Team, 2020; Buder and others, 2022) while SARS-CoV-2 RNA is detectable in res- +piratory samples for an average of 17 days (Cevik and others, 2021). In addition, our analysis +confirms that anti-spike IgG seroconversion is positively associated with viral load (Liu and oth- +ers, 2021; Masi´a and others, 2021). To our knowledge, our model is the first that quantifies the +association between peak SARS-CoV-2 viral load and IgG seroconversion rate and the time from + +18 +T.Q. Dong and E.R. Brown +SARS-CoV-2 infection to IgG seroconversion. +One limitation of our paper is the small sample size of the COVID-19 PEP data especially the +limited number of people with seroconversion data. Repeating the analysis with data from more +participants and more frequent antibody sampling schedule may increase the precision of the +parameter estimates. In addition, we only focused on the people with sustained viral shedding, +which we subset based on the criteria of having at least 2 positive sgRNA samples during the +first 14 days. Extending our method to include people who only had one positive sample is +possible. However, adjustment needs to be made to adequately account for the possible cases of +intermittent viral detection at the very end of viral shedding, clearance of the virus by innate +immunity due to low viral inoculum, or/and limited infections attributed to immune priming by +prior seasonal coronaviruses (Stankiewicz Karita and others, 2021). +In addition to identifying correlates of peak diagnostic viral load, as demonstrated in the +analysis of PEP data, our model can be used to investigate factors associated with other viral +dynamics characteristics as well. For example, we can estimate the potential association between +vaccination and viral shedding duration by using vaccination status as a covariate for time from +viral shedding onset to peak (Xwa,i) and/or time from peak to viral clearance (Xwb,i). Although +we only focused on anti-spike IgG seroconversion in this paper, our model can be generalized +to study other immune responses, such as the development of anti-nucleocapsid IgG and Im- +munoglobulin M (IgM) antibodies, if relevant data is available. Given the flexible structure of the +joint hierarchical model, it is straight forward to extend our methods to model multiple immune +markers simultaneously. +Our model will be a valuable tool for future SARS-CoV-2 vaccine studies that focus on assess- +ing the vaccine effect on transmission (VET)(Kennedy-Shaffer and others, 2021; Follmann and +Fay, 2022). Since detection of sgRNA indicates active viral replication and tracks infectious virus, +our methods can be used to impute sgRNA viral load and better estimate proxies of VET. Our + +Joint Bayesian model for SARS-CoV-2 viral load and seroconversion +19 +model can also be used to better understand the potential impact of interventions that reduce +viral load on seroconversion rate. For example, previous vaccine studies have shown that vacci- +nated people tended to have lower viral load at diagnosis and have lower rate of seroconversion as +compared to the control group (Pajon and others, 2022; Follmann and others, 2022). Models like +ours could help explain how much of the decrease in seroconversion rates is due to vaccination +and how much is due to the vaccination effect on viral load. Finally, our proposed model can +facilitate analysis of household transmission studies by providing estimated viral load trajectories +for each individual in a household to inform the direction of transmission. +6. Software +Software in the form of R code and JAGS code is available online at https://github.com/ +dq0708/joint_vl_sero. +7. Supplementary Material +Supplementary material is available online at http://biostatistics.oxfordjournals.org. +Acknowledgments +Funding for the project was provided by Bill & Melinda Gates Foundation. The authors thank +the Hydroxychloroquine COVID-19 PEP Study Team and participants for providing scientific +insights and data. Special thanks to Dr. Ruanne Barnabas and Dr. Anna Bershteyn for reviewing +the manuscript drafts and providing valuable suggestions. We also thank Dr. Leigh Fisher for +additional advice regarding RT-PCR viral load quantification. +Conflict of Interest: None declared. + +20 +REFERENCES +References +Arons, Melissa M, Hatfield, Kelly M, Reddy, Sujan C, Kimball, Anne, James, Alli- +son, Jacobs, Jesica R, Taylor, Joanne, Spicer, Kevin, Bardossy, Ana C, Oakley, +Lisa P and others. (2020). Presymptomatic SARS-CoV-2 infections and transmission in a +skilled nursing facility. New England journal of medicine 382(22), 2081–2090. +Barnabas, Ruanne V., Brown, Elizabeth R., Bershteyn, Anna, Stankiewicz Karita, +Helen C., Johnston, Christine, Thorpe, Lorna E., Kottkamp, Angelica, Neuzil, +Kathleen M., Laufer, Miriam K., Deming, Meagan, Paasche-Orlow, Michael K., +Kissinger, Patricia J., Luk, Alfred, Paolino, Kristopher, Landovitz, Raphael J., +Hoffman, Risa, Schaafsma, Torin T., Krows, Meighan L., Thomas, Katherine K., +Morrison, Susan, Haugen, Harald S., Kidoguchi, Lara, Wener, Mark, Greninger, +Alexander L., Huang, Meei-Li, Jerome, Keith R., Wald, Anna, Celum, Connie, +Chu, Helen Y. and others. (2021, 3). +Hydroxychloroquine as postexposure prophylaxis +to prevent severe acute respiratory syndrome coronavirus 2 infection. +Annals of Internal +Medicine 174(3), 344–352. +Bravo, Marta Santos, Berengua, Carla, Mar´ın, Pilar, Esteban, Montserrat, Ro- +driguez, Cristina, del Cuerpo, Margarita, Mir´o, Elisenda, Cuesta, Genoveva, +Mosquera, Mar, S´anchez-Palomino, Sonsoles, Vila, Jordi, Rabella, N´uria and +others. (2022). Viral culture confirmed SARS-CoV-2 subgenomic RNA value as a good surro- +gate marker of infectivity. Journal of Clinical Microbiology 60(1), e01609–21. +Bruce, Emily A, Mills, Margaret G, Sampoleo, Reigran, Perchetti, Garrett A, +Huang, Meei-Li, Despres, Hannah W, Schmidt, Madaline M, Roychoudhury, Pavi- +tra, Shirley, David J, Jerome, Keith R and others. (2021). Predicting infectivity: com- + +REFERENCES +21 +paring four PCR-based assays to detect culturable SARS-CoV-2 in clinical samples. EMBO +Molecular Medicine 14, e15290. +Buder, Felix, Bauswein, Markus, Magnus, Clara L, Audebert, Franz, Lang, Hen- +riette, Kundel, Christof, Distler, Karin, Reuschel, Edith, Lubnow, Matthias, +Mueller, Thomas and others. (2022). Contribution of high viral loads, detection of viral +antigen and seroconversion to severe acute respiratory syndrome coronavirus 2 infectivity. The +Journal of infectious diseases 225(2), 190–198. +Bullard, Jared, Dust, Kerry, Funk, Duane, Strong, James E, Alexander, David, +Garnett, Lauren, Boodman, Carl, Bello, Alexander, Hedley, Adam, Schiffman, +Zachary and others. (2020). Predicting infectious severe acute respiratory syndrome coron- +avirus 2 from diagnostic samples. Clinical infectious diseases 71(10), 2663–2666. +Centers for Disease Control and Prevention. (2020). Coronavirus disease 2019 (COVID- +19) 2020 interim case definition, approved August 5, 2020. https://ndc.services.cdc.gov/ +case-definitions/coronavirus-disease-2019-2020-08-05/. +[Online; accessed 2022-11- +02]. +Centers for Disease Control and Prevention. (2021). +Science Brief: SARS-CoV- +2 Infection-induced and Vaccine-induced Immunity. +https://www.cdc.gov/coronavirus/ +2019-ncov/science/science-briefs/vaccine-induced-immunity.html#. +[Online; +ac- +cessed 2022-11-21]. +Cevik, Muge, Tate, Matthew, Lloyd, Ollie, Maraolo, Alberto Enrico, Schafers, +Jenna and Ho, Antonia. (2021). +SARS-CoV-2, SARS-CoV, and MERS-CoV viral load +dynamics, duration of viral shedding, and infectiousness: a systematic review and meta-analysis. +The lancet microbe 2(1), e13–e22. + +22 +REFERENCES +Chaccour, Carlos, Casellas, Aina, Blanco-Di Matteo, Andr´es, Pineda, I˜nigo, +Fernandez-Montero, Alejandro, Ruiz-Castillo, Paula, Richardson, Mary-Ann, +Rodr´ıguez-Mateos, Mariano, Jord´an-Iborra, Carlota, Brew, Joe and others. +(2021). The effect of early treatment with ivermectin on viral load, symptoms and humoral +response in patients with non-severe COVID-19: A pilot, double-blind, placebo-controlled, ran- +domized clinical trial. EClinicalMedicine 32, 100720. +Challenger, +Joseph +D, +Foo, +Cher +Y, +Wu, +Yue, +Yan, +Ada +WC, +Marjaneh, +Mahdi Moradi, Liew, Felicity, Thwaites, Ryan S, Okell, Lucy C and Cunnington, +Aubrey J. (2022). Modelling upper respiratory viral load dynamics of SARS-CoV-2. BMC +medicine 20(1), 1–20. +Dagotto, Gabriel, Mercado, Noe B, Martinez, David R, Hou, Yixuan J, Nkolola, +Joseph P, Carnahan, Robert H, Crowe, James E, Baric, Ralph S and Barouch, +Dan H. (2021). Comparison of subgenomic and total RNA in SARS-CoV-2-challenged rhesus +macaques. Journal of virology 95(8), e02370–20. +Deming, Meagan E., Dong, Tracy Q., Agrawal, Vaidehi, Mills, Margaret G., +Huang, Meei-Li W., Greninger, Alexander L., Jerome, Keith R., Wener, Mark H., +Paasche-Orlow, Michael K., Kissinger, Patricia, Luk, Alfred, Hoffman, Risa M., +Stewart, Jenell, Kottkamp, Angelica C., Bershteyn, Anna, Chu, Helen Y., +Stankiewicz Karita, Helen C., Johnston, Christine M., Wald, Anna, Barnabas, +Ruanne, Brown, Elizabeth and others. (2022). Detection and kinetics of subgenomic SARS- +CoV-2 RNA viral load in longitudinal diagnostic RNA positive samples. The Journal of Infec- +tious Diseases 226(5), 788–796. +Eyre, David W, Taylor, Donald, Purver, Mark, Chapman, David, Fowler, Tom, +Pouwels, Koen B, Walker, A Sarah and Peto, Tim EA. (2022). Effect of Covid-19 vac- + +REFERENCES +23 +cination on transmission of alpha and delta variants. New England Journal of Medicine 386(8), +744–756. +Follmann, +Dean, +Janes, +Holly +E, +Buhule, +Olive +D, +Zhou, +Honghong, +Gi- +rard, Bethany, Marks, Kristen, Kotloff, Karen, Desjardins, Micha¨el, Corey, +Lawrence, Neuzil, Kathleen M and others. (2022). +Antinucleocapsid antibodies after +SARS-CoV-2 infection in the blinded phase of the randomized, placebo-controlled mRNA-1273 +COVID-19 vaccine efficacy clinical trial. Annals of internal medicine 175(9), 1258–1265. +Follmann, Dean A and Fay, Michael P. (2022, 03). Vaccine efficacy at a point in time. +Biostatistics. kxac008. +Forootan, Amin, Sj¨oback, Robert, Bj¨orkman, Jens, Sj¨ogreen, Bj¨orn, Linz, Lucas +and Kubista, Mikael. (2017). Methods to determine limit of detection and limit of quantifi- +cation in quantitative real-time PCR (qPCR). Biomolecular Detection and Quantification 12, +1–6. +Gastine, Silke, Pang, Juanita, Boshier, Florencia AT, Carter, Simon J, Lons- +dale, Dagan O, Cortina-Borja, Mario, Hung, Ivan FN, Breuer, Judy, Kloprogge, +Frank and Standing, Joseph F. (2021). Systematic review and patient-level meta-analysis +of SARS-CoV-2 viral dynamics to model response to antiviral therapies. Clinical Pharmacology +& Therapeutics 110(2), 321–333. +Goldberg, Elad, Zvi, Haim Ben, Sheena, Liron, Sofer, Summer, Krause, Ilan, Sklan, +Ella H and Shlomai, Amir. (2021). A real-life setting evaluation of the effect of remdesivir +on viral load in COVID-19 patients admitted to a large tertiary centre in Israel. +Clinical +Microbiology and Infection 27(6), 917–e1. +Gottlieb, Robert L, Nirula, Ajay, Chen, Peter, Boscia, Joseph, Heller, Barry, +Morris, Jason, Huhn, Gregory, Cardona, Jose, Mocherla, Bharat, Stosor, + +24 +REFERENCES +Valentina and others. (2021). Effect of bamlanivimab as monotherapy or in combination +with etesevimab on viral load in patients with mild to moderate COVID-19: a randomized +clinical trial. Jama 325(7), 632–644. +Goyal, Ashish, Cardozo-Ojeda, E. Fabian and Schiffer, Joshua T. (2020). Potency and +timing of antiviral therapy as determinants of duration of SARS-CoV-2 shedding and intensity +of inflammatory response. Science Advances 6(47), eabc7112. +Jones, Terry C., Biele, Guido, M¨uhlemann, Barbara, Veith, Talitha, Schnei- +der, Julia, Beheim-Schwarzbach, J¨orn, Bleicker, Tobias, Tesch, Julia, Schmidt, +Marie Luisa, Sander, Leif Erik, Kurth, Florian, Menzel, Peter, Schwarzer, +Rolf, Zuchowski, Marta, Hofmann, J¨org, Krumbholz, Andi, Stein, Angela, Edel- +mann, Anke, Corman, Victor Max and others. (2021). Estimating infectiousness through- +out SARS-CoV-2 infection course. Science 373(6551), eabi5273. +Ke, Ruian, Zitzmann, Carolin, Ho, David D, Ribeiro, Ruy M and Perelson, Alan S. +(2021). In vivo kinetics of SARS-CoV-2 infection and its relationship with a person’s infec- +tiousness. Proceedings of the National Academy of Sciences 118(49), e2111477118. +Kennedy-Shaffer, Lee, Kahn, Rebecca and Lipsitch, Marc. (2021). Estimating vaccine +efficacy against transmission via effect on viral load. Epidemiology (Cambridge, Mass.) 32(6), +820. +Kissler, Stephen M, Fauver, Joseph R, Mack, Christina, Olesen, Scott W, Tai, +Caroline, Shiue, Kristin Y, Kalinich, Chaney C, Jednak, Sarah, Ott, Isabel M, +Vogels, Chantal BF and others. (2021a). Viral dynamics of acute SARS-CoV-2 infection +and applications to diagnostic and public health strategies. PLoS biology 19(7), e3001333. +Kissler, Stephen M., Fauver, Joseph R., Mack, Christina, Tai, Caroline G., Bre- +ban, Mallery I., Watkins, Anne E., Samant, Radhika M., Anderson, Deverick J., + +REFERENCES +25 +Metti, Jessica, Khullar, Gaurav, Baits, Rachel, MacKay, Matthew, Salgado, +Daisy, Baker, Tim, Dudley, Joel T., Mason, Christopher E., Ho, David D., +Grubaugh, Nathan D. and others. (2021b). +Viral dynamics of SARS-CoV-2 variants in +vaccinated and unvaccinated Persons. New England Journal of Medicine 385, 2489–2491. +Levine-Tiefenbrun, Matan, Yelin, Idan, Alapi, Hillel, Herzel, Esma, Kuint, Jacob, +Chodick, Gabriel, Gazit, Sivan, Patalon, Tal and Kishony, Roy. (2022). Waning of +SARS-CoV-2 booster viral-load reduction effectiveness. Nature communications 13(1), 1–4. +Liu, Weimin, Russell, Ronnie M, Bibollet-Ruche, Frederic, Skelly, Ashwin N, +Sherrill-Mix, Scott, Freeman, Drew A, Stoltz, Regina, Lindemuth, Emily, Lee, +Fang-Hua, Sterrett, Sarah and others. (2021). +Predictors of nonseroconversion after +SARS-CoV-2 infection. Emerging Infectious Diseases 27(9), 2454. +Lou, Bin, Li, Ting-Dong, Zheng, Shu-Fa, Su, Ying-Ying, Li, Zhi-Yong, Liu, Wei, +Yu, Fei, Ge, Sheng-Xiang, Zou, Qian-Da, Yuan, Quan and others. (2020). Serology +characteristics of SARS-CoV-2 infection after exposure and post-symptom onset. European +Respiratory Journal 56(2), 2000763. +Masi´a, Mar, Telenti, Guillermo, Fern´andez, Marta, Garc´ıa, Jos´e A, Agull´o, +Vanesa, Padilla, Sergio, Garc´ıa-Abell´an, Javier, Guill´en, Luc´ıa, Mascarell, +Paula, Asenjo, Jos´e C and others. (2021). SARS-CoV-2 seroconversion and viral clear- +ance in patients hospitalized with COVID-19: viral load predicts antibody response. Open +forum infectious diseases 8(2), ofab005. +Mellis, Alexandra M, Meece, Jennifer K, Halasa, Natasha B, Chappell, James D, +McLean, Huong Q, Grijalva, Carlos G, Hanson, Kayla E, Zhu, Yuwei, Kim, Ahra, +Deyoe, Jessica, Ivacic, Lynn C, Reed, Carrie, Talbot, H Keipp and others. (2022, 05). + +26 +REFERENCES +SARS-CoV-2 virus dynamics in recently infected people—data from a household transmission +study. The Journal of Infectious Diseases 226(10), 1699–1703. +N´eant, Nad`ege, Lingas, Guillaume, Le Hingrat, Quentin, Ghosn, Jade, Engel- +mann, Ilka, Lepiller, Quentin, Gaymard, Alexandre, Ferr´e, Virginie, Hartard, +C´edric, Plantier, Jean-Christophe, Thibault, Vincent, Marlet, Julien, Montes, +Brigitte, Bouiller, Kevin, Lescure, Franc¸ois-Xavier, Timsit, Jean-Franc¸ois, +Faure, Emmanuel, Poissy, Julien, Chidiac, Christian, Raffi, Franc¸ois, Kim- +moun, Antoine, Etienne, Manuel, Richard, Jean-Christophe, Tattevin, Pierre, +Garot, Denis, Le Moing, Vincent, Bachelet, Delphine, Tardivon, Coralie, Du- +val, Xavier, Yazdanpanah, Yazdan, Mentr´e, France, Laou´enan, C´edric, Visseaux, +Benoit, Guedj, J´er´emie, for the French COVID Cohort Investigators and others. +(2021). Modeling SARS-CoV-2 viral kinetics and association with mortality in hospitalized pa- +tients from the French COVID cohort. Proceedings of the National Academy of Sciences 118(8), +e2017962118. +Oved, Kfir, Olmer, Liraz, Shemer-Avni, Yonat, Wolf, Tamar, Supino-Rosin, Lia, +Prajgrod, George, Shenhar, Yotam, Payorsky, Irina, Cohen, Yuval, Kohn, Yishai +and others. (2020). +Multi-center nationwide comparison of seven serology assays reveals a +SARS-CoV-2 non-responding seronegative subpopulation. EClinicalMedicine 29, 100651. +Pajon, Rolando, Paila, Yamuna D, Girard, Bethany, Dixon, Groves, Kacena, +Katherine, Baden, Lindsey R, El Sahly, Hana M, Essink, Brandon, Mullane, +Kathleen M, Frank, Ian and others. (2022). Initial analysis of viral dynamics and cir- +culating viral variants during the mRNA-1273 Phase 3 COVE trial. Nature Medicine 28(4), +823–830. +Pathela, Preeti, Crawley, Addie, Weiss, Don, Maldin, Beth, Cornell, Jennifer, + +REFERENCES +27 +Purdin, Jeff, Schumacher, Pamela K, Marovich, Stacey, Li, Joyce, Daskalakis, +Demetre and others. (2021). Seroprevalence of severe acute respiratory syndrome coronavirus +2 following the largest initial epidemic wave in the United States: findings from New York City, +13 May to 21 July 2020. The Journal of infectious diseases 224(2), 196–206. +Plummer, Martyn and others. (2003). JAGS: A program for analysis of Bayesian graphical +models using Gibbs sampling. Proceedings of the 3rd international workshop on distributed +statistical computing 124(125.10), 1–10. +Portal-Celhay, Cynthia, Forleo-Neto, Eduardo, Eagan, Will, Musser, Bret J, +Davis, John D, Turner, Kenneth C, Norton, Thomas, Hooper, Andrea T, Hamil- +ton, Jennifer D, Pan, Cynthia and others. (2022). Virologic efficacy of casirivimab and +imdevimab COVID-19 antibody combination in outpatients with SARS-CoV-2 infection: A +phase 2 dose-ranging randomized clinical trial. JAMA network open 5(8), e2225411–e2225411. +Powers, John H, Guerrero, M Lourdes, Leidy, Nancy Kline, Fairchok, Mary P, +Rosenberg, Alice, Hern´andez, Andr´es, Stringer, Sonja, Schofield, Christina, +Rodr´ıguez-Zulueta, Patricia, Kim, Katherine and others. (2015). Development of the +Flu-PRO: a patient-reported outcome (PRO) instrument to evaluate symptoms of influenza. +BMC infectious diseases 16(1), 1–11. +R Core Team. (2022). R: A language and environment for statistical computing. R Foundation +for Statistical Computing, Vienna, Austria. +Singanayagam, Anika, Hakki, Seran, Dunning, Jake, Madon, Kieran J., Crone, +Michael A., Koycheva, Aleksandra, Derqui-Fernandez, Nieves, Barnett, Jack L., +Whitfield, Michael G., Varro, Robert, Charlett, Andre, Kundu, Rhia, Fenn, +Joe, +Cutajar, +Jessica, +Quinn, +Valerie, +Conibear, +Emily, +Barclay, +Wendy, + +28 +REFERENCES +Freemont, Paul S., Taylor, Graham P., Ahmad, Shazaad, Zambon, Maria, Fer- +guson, Neil M., Lalvani, Ajit, Badhan, Anjna, Dustan, Simon, Tejpal, Chitra, +Ketkar, Anjeli V., Narean, Janakan Sam, Hammett, Sarah, McDermott, Eimear, +Pillay, Timesh, Houston, Hamish, Luca, Constanta, Samuel, Jada, Bremang, +Samuel, Evetts, Samuel, Poh, John, Anderson, Charlotte, Jackson, David, Miah, +Shahjahan, Ellis, Joanna and others. (2021, 11). Community transmission and viral load +kinetics of the SARS-CoV-2 delta (B. 1.617. 2) variant in vaccinated and unvaccinated individ- +uals in the UK: a prospective, longitudinal, cohort study. The Lancet Infectious Diseases 22(2), +183–195. +Singanayagam, Anika, Patel, Monika, Charlett, Andre, Bernal, Jamie Lopez, Sal- +iba, Vanessa, Ellis, Joanna, Ladhani, Shamez, Zambon, Maria and Gopal, Robin. +(2020). Duration of infectiousness and correlation with RT-PCR cycle threshold values in cases +of COVID-19, England, January to May 2020. Eurosurveillance 25(32), 2001483. +Stankiewicz Karita, Helen C., Dong, Tracy Q., Johnston, Christine, Neuzil, Kath- +leen M., Paasche-Orlow, Michael K., Kissinger, Patricia J., Bershteyn, Anna, +Thorpe, Lorna E., Deming, Meagan, Kottkamp, Angelica, Laufer, Miriam, Lan- +dovitz, Raphael J., Luk, Alfred, Hoffman, Risa, Roychoudhury, Pavitra, Mag- +aret, Craig A., Grenninger, Alexander L., Huang, Meei-Li, Jerome, Keith R., +Wener, Mark, Celum, Connie, Chu, Helen Y., Baeten, Jared M., Wald, Anna, +Barnabas, Ruanne V., Brown, Elizabeth R. and others. (2021). Trajectory of viral RNA +load among persons with incident SARS-CoV-2 G614 infection (Wuhan strain) in relation to +COVID-19 symptom onset and severity. JAMA Network Open 5(1), e2142796. +The COVID-19 Investigation Team. (2020). Clinical and virologic characteristics of the + +REFERENCES +29 +first 12 patients with coronavirus disease 2019 (COVID-19) in the United States. +Nature +medicine 26(6), 861–868. +van Kampen, Jeroen JA, van de Vijver, David AMC, Fraaij, Pieter LA, Haagmans, +Bart L, Lamers, Mart M, Okba, Nisreen, van den Akker, Johannes PC, Endeman, +Henrik, Gommers, Diederik AMPJ, Cornelissen, Jan J and others. (2021). Duration +and key determinants of infectious virus shedding in hospitalized patients with coronavirus +disease-2019 (COVID-19). Nature communications 12(1), 1–6. +Wellinghausen, Nele, Plonn´e, Dietmar, Voss, Meike, Ivanova, Ralitsa, Frodl, +Reinhard and Deininger, Susanne. (2020). +SARS-CoV-2-IgG response is different in +COVID-19 outpatients and asymptomatic contact persons. Journal of Clinical Virology 130, +104542. +W¨olfel, Roman, Corman, Victor M, Guggemos, Wolfgang, Seilmaier, Michael, +Zange, Sabine, M¨uller, Marcel A, Niemeyer, Daniela, Jones, Terry C, Vollmar, +Patrick, Rothe, Camilla and others. (2020). Virological assessment of hospitalized patients +with COVID-2019. Nature 581(7809), 465–469. +World Health Organization. (2022). Coronavirus disease (COVID-19) pandemic. https: +//www.who.int/emergencies/diseases/novel-coronavirus-2019. [Online; accessed 2022- +10-12]. +Yadav, Arun Kumar, Ghosh, S, Kotwal, Atul, Kaushik, SK, Bobdey, Saurabh, Sahu, +Rajesh, Kapoor, Suraj, Faujdar, DS, Teli, Prabhakar T and Anand, Vivek. (2021). +Seroconversion among COVID-19 patients admitted in a dedicated COVID hospital: a longi- +tudinal prospective study of 1000 patients. medical journal armed forces india 77, S379–S384. + +30 +REFERENCES +Fig. 1. Graphical illustration of the joint model. Along the x-axis is time since the latent peak diagnostic +RNA viral load in day. Along the y-axis is the magnitude of viral load in log10 copies/ml. The limit of +detection (LoD) for both diagnostic and sgRNA RT-PCR tests were approximately 102.4 copies/ml. In +addition, the diagnostic RT-PCR had a limit of quantification (LoQ) of approximately 104.9 copies/ml. +Black dots represent the observed diagnostic RNA viral load from RT-PCR tests, and blue segments +represent the latent diagnostic RNA viral load trajectory. Red dots represent the observed sgRNA viral +load from RT-PCR tests, and maroon segments represent the latent sgRNA viral load trajectory. The +black and red dots were actual data from one participant from the COVID-19 PEP study. wa and w′ +a +represent time from shedding onset to peak for diagnostic and sgRNA viral load respectively. wb and +w′ +b represent time from peak to viral clearance for diagnostic and sgRNA viral load respectively. w′ +d +represents the time from sgRNA viral clearance to diagnostic RNA viral clearance. tp represents time +from the observed peak diagnostic RNA viral load to the latent peak diagnostic RNA viral load, and +t′ +d represents time from the latent peak diagnostic RNA viral load to the latent peak sgRNA viral load. +Finally, vp represents the magnitude of peak diagnostic RNA viral load with respect to LoD, and q is +the multiplicative factor of peak sgRNA viral load relative to peak diagnostic RNA viral load. + +10 +tp +td' +6 +LoQ +vp' =q*vp +4 +Viral +LoD +wa +wb +wa' +wb' +wd' +-10 +-5 +0 +5 +10 +Time since the latent peak diagnositc RNA viral load (day)REFERENCES +31 +Fig. 2. The posterior means and 95% credible intervals (CI) of the population-level diagnostic and sgRNA +viral load trajectories and time of anti-spike IgG seroconversion (among those who seroconverted). +Fig. 3. The posterior means and 95% credible intervals of the multiplicative factors associated with +various patient characteristics for an increase in peak diagnostic viral load on the log10 copies/ml scale. + +Diagnostic RNA viral load +9 +SgRNA viral load +Time of seroconversion +8 +Viral load (log10 copies/mL) +6 +5 +4 +3 +LoD +0 +2 +4 +6 +8 +10 +12 +14 +16 +18 +Day since viral shedding onsetAge (+10 years) +Male +HCQ Arm (vs. Placebo Arm) +COVID-19 Symptomatic (vs. Asymptomatic) +0.8 +1.0 +1.2 +1.4 +Multiplicative factor32 +REFERENCES +Fig. 4. The posterior means and 95% credible intervals of the multiplicative factors associated with +various symptoms for an increase in peak diagnostic viral load on the log10 copies/ml scale. Specifically, +“COVID-19” symptoms were defined based on the case definition approved by the U.S. Centers for Disease +Control and Prevention (Centers for Disease Control and Prevention, 2020); the rest were defined based +on the Flu-PRO patient-reported outcome instrument (Powers and others, 2015): “Chest” symptoms +include cough and shortness of breath; “Taste or Smell” symptoms include olfactory and taste disorders; +“Nose” symptoms include congestion, runny nose, and olfactory disorder; “Body” symptoms include +chills, myalgia, headache, and fatigue; “GI” symptoms include diarrhea, nausea, and vomit; “Throat” +symptoms include sore throat and taste disorder. + +Nose symptoms +Chest symptoms +Gastrointestinal symptoms +Change in taste or smell +Throat symptoms +COVID-19 symptoms +Fever +Body symptoms +0.8 +1.0 +1.2 +1.4 +1.6 +Multiplicative factorREFERENCES +33 +Fig. 5. The posterior means and 95% credible intervals of the odds ratio of seroconversion associated +with various symptoms. Specifically, “COVID-19” symptoms were defined based on the case definition +approved by the U.S. Centers for Disease Control and Prevention (Centers for Disease Control and Pre- +vention, 2020); “Chest” symptoms include cough and shortness of breath; “Taste or Smell” symptoms +include olfactory and taste disorders; “Nose” symptoms include congestion, runny nose, and olfactory dis- +order; “Body” symptoms include chills, myalgia, headache, and fatigue; “GI” symptoms include diarrhea, +nausea, and vomit; “Throat” symptoms include sore throat and taste disorder. + +Gastrointestinal symptoms +Fever +Change in taste or smell +Chest symptoms +COVID-19 symptoms +Body symptoms +Throat symptoms +Nose symptoms +0 +1 +2 +3 +4 +Odds ratio34 +REFERENCES +Table 1. The posterior means and 95% credible intervals (CI) for selected quantities of interest. In +particular, n = 80 is the total number of people with diagnostic RNA viral load data. +Type +Quantity of interest +Estimand +Posterior mean [95% CI] +Diagnostic RNA +Mean time from shedding onset to peak viral load (days) +�n +i=1 ˆwa,i/n +3.8 [3.2 − 4.7] +viral load +Mean time from peak viral load to viral clearance (days) +�n +i=1 ˆwb,i/n +10.5 [10.1 − 10.9] +Mean peak viral load (log10 copies/ml) +�n +i=1 ˆvp,i/n + LoD +8.0 [7.8 − 8.1] +True Positive Rate +expit(ˆα0 + ˆα1) +91.8% [90.1% − 93.3%] +True Negative Rate +1 − expit(ˆα0) +99.5% [99.4% − 99.6%] +sgRNA +Mean time from shedding onset to peak viral load (days) +�n +i=1 ˆw′ +a,i/n +4.4 [3.8 − 5.3] +viral load +Mean time from peak viral load to viral clearance (days) +�n +i=1 ˆw′ +b,i/n +5.7 [5.3 − 6.1] +Mean peak viral load (log10 copies/ml) +�n +i=1 ˆv′ +p,i/n + LoD′ +6.0 [5.9 − 6.2] +True Positive Rate +expit(ˆα′ +0 + ˆα′ +1) +89.6% [87.4% − 91.6%] +True Negative Rate +1 − expit(ˆα′ +0) +99.6% [99.5% − 99.7%] +Anti-spike IgG +The overall seroconversion rate +�n +i=1 ˆCi/n +75.0% [62.5% − 85.0%] +Seroconversion +The odds ratio of seroconversion associated with +exp +� +ˆβC[2] +� +1.24 [1.00 − 1.54] +every 10-fold increase in peak diagnostic viral copies +Mean time from infection to seroconversion (days) +ˆκ1/ ˆκ2 +14.4 [12.5 − 16.6] + +REFERENCES +35 +[Received XXX; revised XXX; accepted for publication XXX ] + diff --git a/wdE2T4oBgHgl3EQfLgZh/content/tmp_files/load_file.txt b/wdE2T4oBgHgl3EQfLgZh/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d99da5e5bb5db12c2c5ae82b868022353fc6b34e --- /dev/null +++ b/wdE2T4oBgHgl3EQfLgZh/content/tmp_files/load_file.txt @@ -0,0 +1,851 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf,len=850 +page_content='Biostatistics (2023), 0, 0, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' 1–35 doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='1093/biostatistics/Manuscript˙cohort2 A joint Bayesian hierarchical model for estimating SARS-CoV-2 diagnostic and subgenomic RNA viral dynamics and seroconversion Tracy Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Dong1∗ and Elizabeth R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Brown1,2 1Vaccine and Infectious Disease Division, Fred Hutchinson Cancer Center 2Department of Biostatistics, University of Washington qdong@fredhutch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='org Summary Understanding the viral dynamics and immunizing antibodies of the severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) is crucial for devising better therapeutic and prevention strategies for COVID-19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Here, we present a Bayesian hierarchical model that jointly estimates the diagnostic RNA viral load reflecting genomic materials of SARS-CoV-2, the subgenomic RNAs (sgRNA) viral load reflecting active viral replication, and the rate and timing of seroconversion reflecting presence of antibodies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Our proposed method accounts for the dynamical relationship and correlation structure between the two types of viral load, allows for borrowing of information between viral load and antibody data, and identifies potential correlates of viral load character- istics and propensity for seroconversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' We demonstrate the features of the joint model through application to the COVID-19 PEP study and conduct a cross-validation exercise to illustrate the ∗To whom correspondence should be addressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' © The Author 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Published by Oxford University Press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' All rights reserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' For permissions, please e-mail: journals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='permissions@oup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='com arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='03714v1 [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='AP] 9 Jan 2023 2 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Dong and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Brown model’s ability to impute the sgRNA viral trajectories for people who only had diagnostic viral load data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Key words: Joint models;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Bayesian hierarchical models;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' SARS-CoV-2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Correlates;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Viral load;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Serocon- version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Introduction The severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) has spread rapidly around the world, resulting in more than 6 million deaths worldwide to date (World Health Organization, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Current knowledge regarding SARS-CoV-2 viral dynamics is built on diagnostic viral load detected via real-time reverse transcription polymerase chain reaction (RT-PCR)(Cevik and others, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Gastine and others, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Mellis and others, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Pajon and others, 2022), the most sensitive and widely used technique for the diagnosis of coronavirus disease 2019 (COVID- 19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' However, the diagnostic RT-PCR tests give limited insight into infectivity because the results may reflect residual genomic material rather than replicating virus (Singanayagam and others, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Viral culture and quantification by plaque assays provide a gold standard for assessing infectivity but are not readily available at large scale for a diagnostic assay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Therefore, detection of subgenomic RNAs (sgRNA) in clinical samples has been suggested as an additional diagnostic tool to track infectious virus because sgRNA transcripts indicate active viral replication (Dagotto and others, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Bravo and others, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Characterization of both SARS-CoV-2 diagnostic and sgRNA viral dynamics is crucial for understanding the pathogenesis of the virus, identifying patients most at risk , and optimizing the period of self-isolation to reduce onward transmission while reducing unnecessary loss of productivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Developing a model for SARS-CoV-2 viral load is of particular interest because viral load is an important outcome of COVID-19 treatment, vaccine, and non-vaccine prevention studies Joint Bayesian model for SARS-CoV-2 viral load and seroconversion 3 but viral load trajectories are rarely fully observed (?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='Gottlieb and others, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Goldberg and others, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Chaccour and others, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Portal-Celhay and others, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Levine-Tiefenbrun and others, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Eyre and others, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' One commonly used approach is compartmental models based on ordinary differential equations (ODE) (Goyal and others, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' N´eant and others, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Ke and others, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Challenger and others, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' While the ODE-based models are useful for capturing cell-level within-host viral kinetics and simulating potential effects of antiviral therapies under various assumptions, the model parameters cannot be easily translated into key viral trajectory characteristics such as peak viral load and shedding duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Therefore, Bayesian mixed-effects models are a popular alternative for estimating population- and individual-level viral load trajectories (Kissler and others, 2021a,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Stankiewicz Karita and others, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Jones and others, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Singanayagam and others, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Deming and others, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' This approach has the advantage of being able to directly model viral load features and incorporate covariates and correlation structures to explore potential associations between viral dynamics and patient characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' So far, all existing models estimate the diagnostic and sgRNA viral load separately without considering any dependency between the two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' In addition to viral dynamics, serology characteristics of SARS-CoV-2 infection, especially the rate and timing of seroconversion defined by the development of neutralizing antibodies, are also of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Not all persons recovering from SARS-CoV-2 infection develop antibodies (Oved and others, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Pathela and others, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Wellinghausen and others, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Understanding the natural immunity acquired after SARS-CoV-2 infection and the correlates of seroconversion is crucial for studying disease transmission, vaccine effectiveness, and population-wide seropreva- lence and immunity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Since most serological studies were conducted among hospitalized people who exhibited moderate to severe COVID-19 symptoms and were tracked in the later course of the infection (Lou and others, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Yadav and others, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Masi´a and others, 2021), little is known regarding seroconversion rate and time in outpatients settings and among persons with 4 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Dong and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Brown very mild or asymptomatic infection (Centers for Disease Control and Prevention, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' In this paper, we develop a Bayesian hierarchical model that jointly analyzes SARS-CoV-2 diagnostic and sgRNA viral load data and interval-censored seroconversion data to estimate viral trajectories, their association with patient characteristics, and the correlates, rate, and time of seroconversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Our proposed model has several advantages compared to the existing models: first, it takes into account the dynamical relationship between the diagnostic PCR and sgRNA viral trajectories and can be used to impute sgRNA viral trajectories for patients who only had diagnostic viral load data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' In addition, by jointly modeling viral load and antibody measurements, our model allows the borrowing of information across different data types to better estimate key virological and serological parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Our model also uses a pair of indicator variables for the observed and true viral shedding status to allow direct modeling of false positive and false negative cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Finally, our model explicitly accounts for the increased uncertainty in RT-PCR viral load measurements below the test’s limit of quantification (LoQ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' In Section 2, we introduce a motivating example of the COVID-19 post-exposure prophylaxis (PEP) study (Barnabas and others, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Section 3 describes the model formulation and infer- ence procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' In Section 4, we apply our methods to analyze data collected in the COVID-19 PEP study and summarize the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' We end the paper with a discussion in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' A motivating example: the COVID-19 PEP Study The COVID-19 PEP Study was a double-blinded, household-randomized controlled trial compar- ing hydroxychloroquine to placebo-like control for prevention of SARS-CoV-2 infection (Barnabas and others, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' The study was remotely conducted between March and August 2020 before COVID-19 vaccines were available and enrolled 829 initially asymptomatic household contacts and healthcare workers recently exposed (<96 hours) to persons with laboratory-confirmed SARS- CoV-2 infection from 41 U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' The study found no differences between hydroxychloroquine Joint Bayesian model for SARS-CoV-2 viral load and seroconversion 5 and placebo-like control arms in SARS-CoV-2 acquisition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Participants provided demographic information via enrollment surveys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' They also completed daily symptom questionnaires and self-collected daily mid-turbinate swabs for 14 consecutive days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Swabs were tested for SARS-CoV-2 RNA at the University of Washington (UW) Virology Laboratory using RT-PCR targeting nucleocapsid genes N1 and N2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Samples with detectable SARS-CoV-2 RNA via RT-PCR were further tested for sgRNA targeting sgE (Bruce and others, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' RT-PCR for both N and sgE were performed to quantify the viral titer via the cycle threshold (Ct), defined as the number of thermal cycles needed to amplify sampled viral RNA to a detectable level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Quantification standards were run with both assays to convert Ct value to viral load as copies/ml during assay validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Both tests used a Ct cutoff of 40, corresponding to a lower limit of detection (LoD) of approximately 102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='4 copies/ml.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' In addition, the diagnostic RT-PCR had a limit of quantification (LoQ), defined as the lowest amount of viral RNA in a sample that can be quantitatively determined with acceptable precision accuracy (Forootan and others, 2017), of approximately 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='9 copies/ml, corresponding to a Ct value of 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Therefore, the conversion from the diagnostic Ct values to viral concentrations suffers higher uncertainty for nasal swabs with observed Ct values between the LoD and LoQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Dried blood spot (DBS) samples were voluntarily provided by a subset of participants on study day 1, 14 and 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' The DBS samples were assayed to detect Immunoglobulin G (IgG) antibodies to SARS-CoV-2 spike protein at the UW Department of Laboratory Medicine & Pathology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' For brevity, we will use “seroconversion” to refer to anti-spike IgG seroconversion for the rest of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' However, the model can be generalized to accommodate other types of seroonversion as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' More information can be found in the Discussion section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' 6 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Dong and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Brown 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Methods We assume that the diagnostic and sgRNA viral load trajectories consist of a proliferation phase with exponential growth in viral RNA concentration, followed by a clearance phase characterized by exponential decay in viral RNA concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' On the scale of log10 copies/ml, this roughly corresponds to a linear increase to a peak followed by a linear decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Figure 1 shows a graphical illustration of the latent viral RNA concentration trends overlay with the observed diagnostic and sgRNA viral load measurements over time for an individual participant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' We describe the details of the model in the following sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='1 Modeling the diagnostic RNA viral load Let i be index for participants and j be index for nasal swabs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' For person i, let wa,i represent time from shedding onset to peak viral load, wb,i represent time from peak to viral clearance, and vp,i represent the magnitude of peak viral load with respect to LoD (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' We use the latent peak as a reference point and let s represent time since the latent peak viral load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' As such, we have s = 0 at the latent peak viral load, s = −wa,i at shedding onset, and s = wb,i at viral clearance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' For the jth swab collected from person i, sij is the time from peak viral load to swab collection, and the latent diagnostic viral load at sij, denoted by µy(sij), is µy(sij) = LoD + vp,i + � vp,i wa,i � × sij × 1 [sij ⩽ 0] + � − vp,i wb,i � × sij × 1 [sij > 0] , where 1 [·] is an indicator function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' In particular, � vp,i wa,i � and � − vp,i wb,i � represent the upward and downward slopes respectively for the linear change in diagnostic viral trajectory over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' To model the person-level variability in wa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' wb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='i and vp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='i while accounting for potential correlation among these parameters,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' we use a multivariate normal distribution for the random effects on the Joint Bayesian model for SARS-CoV-2 viral load and seroconversion 7 log scale: log ([vp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' wa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' wb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='i]⊺) ∼ MVN (µlog,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Σlog) µlog,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='i[1] = µlvp + β⊺ vpXvp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='i µlog,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='i[2] = µlwa + β⊺ waXwa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='i µlog,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='i[3] = µlwb + β⊺ wbXwb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='i where Xvp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Xwa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='i and Xwb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='i are vectors of person-level covariates for peak viral load,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' time from shedding onset to peak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' and time from peak to viral clearance,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' and βvp, βwa and βwb are the corresponding vectors of coefficients for these covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' We use a pair of indicator variables to model the true positive rate (TPR) and true negative rate (TNR) for the observed diagnostic viral detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Let yij denote the observed viral RNA concentration in log10 copies/ml for swab j of person i, Bij be the fully observed indicator variable of whether the swab had detectable diagnostic RNA via RT-PCR (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=', Bij = 1 [yij > LoD]), and let Sij be the latent indicator variable of whether person i was truly experiencing diagnostic viral shedding when swab j was collected (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=', Sij = 1 [−wa,i ⩽ sij ⩽ wb,i]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' We specify the following model: Bij|Sij ∼ Bernoulli (expit(α0 + α1Sij)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' In particular, the true positive rate (TPR), defined as the probability of detecting diagnostic RNA from a nasal swab when a person was experiencing viral shedding, is TPR = E[Bij = 1|Sij = 1] = expit(α0 + α1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' and the true negative rate (TNR), defined as the probability of not detecting viral RNA from a nasal swab when a person was clear of viral RNA, is TNR = E[Bij = 0|Sij = 0] = 1 − expit(α0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' 8 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Dong and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Brown The distribution of a swab’s observed diagnostic viral load depends on whether the test result is a true/false positive/negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' When the jth swab collected from person i has no detectable diagnostic RNA, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=', Bij = 0, the observed viral load equals to the LoD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' When a swab does have detectable viral RNA, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=', Bij = 1, we further consider two scenarios: if the result is false positive, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=', Sij = 0, we use a normal distribution centered around some viral load level above the LOD with a moderate standard deviation: yij|Bij = 1, Sij = 0 ∼ Normal � LoD+, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='5 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' If the result is true positive, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=', Sij = 1, we use a normal distribution centered around the latent viral trajectory and a standard deviation that depends on LoQ: yij|Bij = 1, Sij = 1 ∼ Normal (µy(sij), σy (Qij)) , where σ2 y (Qij) = σ2 yy × (1 + Qij × δQ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Our specification the variance σ2 y aims to account for the increased uncertainty in viral load measurements for swabs with diagnostic RNA concentration below the LoQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Specifically, we let σ2 yy be the base variance term denoting the unmeasured variability in the observations, and Qij be an indicator variable of whether the observed viral concentration from swab j lies between LoD and LoQ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=', Qij = 1 [LoD < yij < LoQ]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' When a swab has an observed diagnostic RNA viral load between LoD and LoQ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=', Qij = 1, the variance term will be higher than the base level by a factor of δQ to account for the extra uncertainty in these observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Finally, since the latent peak viral load is not observable, we estimate the time from the observed peak to the latent peak for person i, denoted tp,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' If person i’s observed peak viral load was recorded within the first two days of follow-up, we will assume the latent peak happens at Joint Bayesian model for SARS-CoV-2 viral load and seroconversion 9 the latest one day after the observed peak: tp,i ∼ Normal (µtpL, σtpL) truncated [−∞, 1], where µtpL is negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' If person i’s observed peak viral load was recorded during the last two days of follow-up, we will assume the latent peak happens at the earliest one day before the observed peak: tp,i ∼ Normal (µtpR, σtpR) truncated [−1, ∞], where µtpR is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Otherwise, we assume the latent peak happens around the time of the observed peak: tp,i ∼ Normal (0, σtp) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' We can specify σtpL, σtpR, and σtp to reflect our prior belief in how variable the time differences tp,i are among the participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='2 Modeling the sgRNA viral load We model the sgRNA viral load in relation to the diagnostic viral trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' For clarity, we define a parallel set of notations for the sgRNA viral load by adding ′ after the previously defined notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' We assume that the diagnostic and sgRNA viral shedding starts at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' In addition, we know that sgRNA viral clearance occurs no later than the diagnostic RNA viral clearance (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Let td,i denote time from the latent peak diagnostic viral load to the latent peak sgRNA viral load for person i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' We use the following truncated normal distribution for t′ d,i: t′ d,i ∼ Normal (µtd, 2) truncated [−wa,i, wb,i] The time from sgRNA shedding onset to peak viral load for person i, denoted w′ a,i, can be 10 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Dong and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Brown calculated as w′ a,i = wa,i + t′ d,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' We use a truncated normal distribution to model the log-transformed time from sgRNA viral clearance to diagnostic RNA rival clearance, denoted w′ d,i: log � w′ d,i � ∼ Normal (µlwd, σlwd) truncated � 0, log � wb,i − t′ d,i �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' The time from peak sgRNA viral load to viral clearance for person i, denoted w′ b,i, can be calculated as w′ b,i = wa,i + wb,i − w′ a,i − w′ d,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' As for the magnitude of the peak sgRNA viral load v′ p,i, we parameterize it as the product of the diagnostic peak viral load and a multiplier qi, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=', v′ p,i = qi × vp,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Since the peak sgRNA viral load is always lower than the peak diagnostic RNA viral load, we model the multiplier qi using a beta distribution: qi ∼ Beta (γ1, γ2) In the PEP study, only the swabs with detectable diagnostic RNA were tested for sgRNA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Therefore, we model the TPR and TNR of sgRNA detection conditional on the observed diag- nostic viral load measurement: B′ ij|S′ ij, Bij ∼ Bernoulli � Bij × � expit(α′ 0 + α′ 1S′ ij) � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Specifically, when a swab was negative for diagnostic RNA, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=', Bij = 0, it would also be negative for sgRNA, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=', B′ ij = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Otherwise, the probability of detecting sgRNA from a nasal swab when a person was experiencing sgRNA viral shedding is TPR′ = E[B′ ij = 1|S′ ij = 1, Bij = 1] = expit(α′ 0 + α′ 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Joint Bayesian model for SARS-CoV-2 viral load and seroconversion 11 and the probability of not detecting sgRNA from a nasal swab when a person was clear of sgRNA is TNR′ = E[B′ ij = 0|S′ ij = 0, Bij = 1] = 1 − expit(α′ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' The distribution of the observed sgRNA viral load follows a similar structure as the diagnostic viral load and depends on whether the sgRNA test result is a true/false positive/negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' When the jth swab collected from person i has no detectable sgRNA, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=', B′ ij = 0, the observed viral load equals to the LoD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' When a swab does have detectable sgRNA and the result is false positive, we use a normal distribution centered around some viral load level above the LOD with a moderate standard deviation: y′ ij|B′ ij = 1, S′ ij = 0 ∼ Normal � LoD+, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='5 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Finally, if the sgRNA viral load detected is true positive, we use a normal distribution: y′ ij|B′ ij = 1, S′ ij = 1 ∼ Normal � µ′ y(s′ ij), σ′ y � , where µ′ y(s′ ij) = LoD′ + v′ p,i + � v′ p,i w′ a,i � × s′ ij × 1 � s′ ij ⩽ 0 � + � − v′ p,i w′ b,i � × s′ ij × 1 � s′ ij > 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='3 Modeling seroconversion rate and time Since not everyone infected with SARS-CoV-2 would eventually develop neutralizing antibodies, we use an indicator variable Ci to model whether person i would ever seroconvert: Ci ∼ Bernoulli (expit (βC0 + β⊺ CXC,i)) , where XC,i are vectors of potential correlates of seroconversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' If a person developed antibodies after infection, then the time from viral shedding onset to seroconversion, denoted ws,i, is modeled 12 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Dong and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Brown as ws,i|Ci = 1 ∼ Gamma (κ1, κ2) If a person never seroconverts, ws,i = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' In our motivating example of the COVID-19 PEP study, DBS samples were only collected on day 1, 14 and/or 28 of the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Therefore, the servoconversion time is censored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Specifically, if a participant only had negative IgG samples, his/her seroconversion time is right-censored at the last negative antibody test;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' if a participant only had positive IgG samples, his/her seroconversion time is left-censored at the first positive antibody test;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' if a participant had both negative and positive IgG samples, his/her seroconversion time is interval-censored between the last negative test and the first positive test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Finally, since not all participants had provided DBS samples with confirmed results, only those who had antibody data available will contribute to the likelihood of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='4 Prior elucidation Inference for all model parameters can be conducted under a Bayesian framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' The prior distributions for virological parameters {µlvp, µlwa, µlwb, µlwd} can be specified using normal dis- tributions with means equal to the estimated diagnostic and sgRNA peak viral load, time from shedding onset to peak and time from peak to shedding cessation from previous literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' For variance and covariance parameters � σ2 yy, σ ′2 y , Σlog � , we can use weakly informative priors such as the Cauchy distribution and inverse wishart distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Multivariate normal distributions with zero means can be used as priors for the coefficients of potential correlates of viral dynamics and seroconversion {βvp, βwa, βwb, βC}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' For the model parameters related to TPR and TNR of diagnostic and sgRNA detection {α0, α1, α′ 0, α′ 1}, we can use relatively tight normal priors be- cause we expect TPR and TNR to be high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Similarly, we can specify a moderately informative beta prior for δQ since we do not expect the extra uncertainty in the observed diagnostic viral load Joint Bayesian model for SARS-CoV-2 viral load and seroconversion 13 below LoQ to be huge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' As for the positive-valued parameters {γ1, γ2, κ1, κ2}, we can use gamma distributions as priors such that the resulting prior distributions for qi and wi have reasonable values as suggested by previous literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='5 sgRNA viral load imputation In a resource-constrained setting, it is common that everyone in a study was tested for diagnostic RNA but only a subset were tested for sgRNA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' In this case, our model can be used to impute the sgRNA viral trajectories for people who only had diagnostic RNA viral load and covariates data by treating their sgRNA measurements as missing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Specifically, we can combine the people without sgRNA samples with the people who had complete data and fit the model to everyone to estimate the posterior distributions of � v′ p,i, w′ a,i, w′ b,i � by borrowing information from the rest of the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Application to the COVID-19 PEP study 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='1 Model implementation We applied our methods to analyze data collected from the 80 COVID-19 PEP study participants who had at least 2 positive sgRNA samples during the first 14 days of follow-up (Figures S1-2 of the Supplement).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' We chose this subset of participants because we are interested in examining the viral dynamics and seroconversion among SARS-CoV-2-infected people who had sustained viral shedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Among them, 33 participants had at least 1 DBS sample with confirmed results, hence their anti-spike IgG antibody data contributed to the seroconversion part of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' For our primary model, the following covariates were included as correlates for peak diagnostic viral load (Xvp): age, sex at birth, treatment arm (hydroxychloroquine vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' placebo), and presence of COVID-19 symptoms defined based on the case definition approved by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Centers for Disease Control and Prevention (CDC) (Centers for Disease Control and Prevention, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' The 14 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Dong and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Brown peak diagnostic viral load, vp, was included as a correlate for seroconversion probability (XC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' All analysis was implemented in R 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='1 (R Core Team, 2022) using JAGS 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='0 (Plummer and others, 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' More details, including prior specification, Markov Chain Monte Carlo (MCMC) setup, and model diagnostics, can be found in Section S1 of the Supplement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='2 Model results On average, the diagnostic viral load reaches a peak of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='0 (95% credible interval, CI: [7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='8, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='1]) log10 copies/ml after 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='8 (95% CI: [3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='2, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='7]) days and then clears after 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='5 (95% CI: [10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='1, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='9]) days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' The sgRNA viral load reaches a peak of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='0 (95% CI: [5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='9, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='2]) log10 copies/ml approxi- mately 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='6 (95% CI: [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='8]) day after the peak diagnostic viral load and then clears after only 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='7 (95% CI: [5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='3, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='1]) days, approximately 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='3 (95% CI: [4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='0, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='6]) days before the diagnostic viral clearance (Table 1 and Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' The estimated infectious period, as approximated by the total duration of sgRNA viral shedding, is 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='1 days (95% CI: [9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='4, 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='0]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' The posterior means and 95% CIs of individual viral load trajectories are shown in Figures S1-2 in the Supplement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' We estimated the multiplicative factors associated with various patient characteristics for an increase in peak diagnostic viral load on the log10 copies/ml scale (Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' The posterior means of the parameters indicated that being older, male, in the hydroxychloroquine arm, and reporting COVID-19 symptoms were positively associated with peak viral load in our data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' However, all the posterior 95% CIs were wide and overlapped with 1, indicating that there was high uncertainty associated with these estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' To identify specific symptoms that might potentially be associated with peak diagnostic vi- ral load, we fitted additional models by replacing the COVID-19 symptoms indicator with other symptom indicators defined based on the Flu-PRO patient-reported outcome instrument (Powers and others, 2015), including fever, change in taste or smell, and chest/throat/nose/body/gastrointestinal symptoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Figure 4 shows the posterior means and 95% CIs of the multiplicative factors associ- Joint Bayesian model for SARS-CoV-2 viral load and seroconversion 15 ated with various symptoms for an increase in peak diagnostic viral load on the log10 copies/ml scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Among all symptoms investigated, having body symptoms (chills, myalgia, headache, or fatigue) was most strongly associated with a higher peak diagnostic viral load (multiplicative fac- tor = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='25, 95% CI: [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='07, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='50]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' That is, on the log10 copies/ml scale, the peak diagnostic RNA viral load was estimated to be 25% (95% CI: [7%, 50%]) higher among SARS-CoV-2 patients who experienced chills, myalgia, headache, or fatigue than those who did not experience these symptoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' The rest of the symptoms either had close-to-zero or positive estimated associations, but all their 95% CIs overlapped with 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' The overall seroconversion rate was 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='0% (95% CI: [62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='5%, 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='0%]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' The peak diagnostic viral load was estimated to be positively associated with the probability of seroconversion (odds ratio = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='24, 95% CI: [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='00, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='54], for every 10-fold increase in diagnostic peak viral load).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Among those who seroconverted, the mean time from viral shedding onset to the presence of detectable IgG antibodies is 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='4 days (95% CI: [12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='5, 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='6]) (Table 1 and Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' To identify any symptoms that might potentially be associated with seroconversion rate, we fitted additional models by replacing peak diagnostic viral load with symptom indicators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Figure 5 shows the posterior means and 95% CIs of the odds ratio of seroconversion associated with various symptoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' All symptoms investigated were estimated to be positively associated with the probability of seroconversion but all posterior 95% CIs overlapped with 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' The probabilities that a sample collected during viral shedding would be tested positive for diagnostic and sgRNA (TPR) are 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='8% (95% CI: [90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='1%, 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='3%]) and 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='6% (95% CI: [87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='4%, 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='6%]) respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' the probabilities that a sample collected outside the viral shed- ding periods would be tested negative for diagnostic and sgRNA (TNR) are 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='5% (95% CI: [99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='4%, 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='6%]) and 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='6% (95% CI: [99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='5%, 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='7%]) respectively (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Note that these val- ues account for not only the properties inherent to the RT-PCR tests such as sensitivity and specificity, but also the potential errors that occurred due to misplaced swabs, contaminated 16 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Dong and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Brown samples, or improper swabbing that were common in outpatient household studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Our model results indicated that there were no significant correlations among peak diagnostic viral load, time from shedding onset to peak, and time from peak to diagnostic viral clearance (Table S1 in the Supplement).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' As for the extra uncertainty in diagnostic viral load measurements below LoQ (104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='9 copies/ml, corresponding to a Ct value of 31), our model estimated that the variance in the observed diagnostic viral load (σ2 y) was approximately 25% (95% CI: [13%, 38%]) higher if the observed values were below LoQ as compared to those above LoQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='3 Cross-validation for sgRNA viral load imputation and seroconversion probability estimation We conducted an 10-fold cross-validation exercise to examine our model’s ability to impute sgRNA viral load and seroconversion for people who only have diagnostic viral load and covariate data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Specifically, we randomly divided the 80 participants into 10 groups of 8 people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' For each itera- tion, we removed the sgRNA viral load data (and antibody data if available) from one group of participants and fitted the model using the rest of the observed data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' In figures S3-4 in the Sup- plement, we present the observed and imputed sgRNA viral load trajectories and the estimated probability of seroconversion from all iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' The posterior means of the sgRNA viral trajec- tories were very close to the masked sgRNA viral load measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' As expected, the imputed sgRNA viral load had wider posterior 95% CIs than those of the diagnostic viral trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' The estimated seroconversion probabilities ranged from 64% to 89%, a reasonable range given the masked antibody data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Discussion In this paper, we presented a joint Bayesian hierarchical model that allows for borrowing of information between viral load and antibody data to make inferences on key characteristics of diagnostic and sgRNA viral dynamics and seroconversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Our method can be used to identify Joint Bayesian model for SARS-CoV-2 viral load and seroconversion 17 potential correlates of viral load trajectories and propensity for seroconversion, to estimate the seroconversion time after a SARS-CoV-2 infection, and to impute sgRNA viral load for people who only have diagnostic RNA viral load data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' By jointly modeling the diagnostic and sgRNA viral trajectories, our method is able to take into account the dynamical relationship and corre- lation structure between the two types of viral load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Our model also uses indicator variables to represent latent and observed viral shedding status separately, accounting for false positive and false negative viral detection due to not only the RT-PCR test properties but also the potential sample collection errors in outpatient study settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' As a demonstrating example, we imple- mented our proposed methods using JAGS in R and analyzed data collected from 80 participants of the COVID-19 PEP study who had at least 2 positive sgRNA samples during the first 14 days of follow-up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' A cross-validation exercise was also conducted to examine the model’s ability to impute sgRNA viral load trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Based on the COVID-19 PEP data, our model-estimated population-level viral load trajecto- ries were characterized by a rapid rise to reach viral peak and subsequent slower decline, similar to what has been demonstrated in previous literature (Goyal and others, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' N´eant and others, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Ke and others, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Kissler and others, 2021a,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Stankiewicz Karita and others, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Jones and others, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Singanayagam and others, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Deming and others, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' The model results on diagnostic and sgRNA viral shedding duration were consistent with previous findings that SARS-CoV-2 infectivity usually terminates after 10 days (van Kampen and others, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' W¨olfel and others, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Bullard and others, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Arons and others, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' The COVID-19 Investigation Team, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Buder and others, 2022) while SARS-CoV-2 RNA is detectable in res- piratory samples for an average of 17 days (Cevik and others, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' In addition, our analysis confirms that anti-spike IgG seroconversion is positively associated with viral load (Liu and oth- ers, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Masi´a and others, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' To our knowledge, our model is the first that quantifies the association between peak SARS-CoV-2 viral load and IgG seroconversion rate and the time from 18 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Dong and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Brown SARS-CoV-2 infection to IgG seroconversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' One limitation of our paper is the small sample size of the COVID-19 PEP data especially the limited number of people with seroconversion data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Repeating the analysis with data from more participants and more frequent antibody sampling schedule may increase the precision of the parameter estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' In addition, we only focused on the people with sustained viral shedding, which we subset based on the criteria of having at least 2 positive sgRNA samples during the first 14 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Extending our method to include people who only had one positive sample is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' However, adjustment needs to be made to adequately account for the possible cases of intermittent viral detection at the very end of viral shedding, clearance of the virus by innate immunity due to low viral inoculum, or/and limited infections attributed to immune priming by prior seasonal coronaviruses (Stankiewicz Karita and others, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' In addition to identifying correlates of peak diagnostic viral load, as demonstrated in the analysis of PEP data, our model can be used to investigate factors associated with other viral dynamics characteristics as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' For example, we can estimate the potential association between vaccination and viral shedding duration by using vaccination status as a covariate for time from viral shedding onset to peak (Xwa,i) and/or time from peak to viral clearance (Xwb,i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Although we only focused on anti-spike IgG seroconversion in this paper, our model can be generalized to study other immune responses, such as the development of anti-nucleocapsid IgG and Im- munoglobulin M (IgM) antibodies, if relevant data is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Given the flexible structure of the joint hierarchical model, it is straight forward to extend our methods to model multiple immune markers simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Our model will be a valuable tool for future SARS-CoV-2 vaccine studies that focus on assess- ing the vaccine effect on transmission (VET)(Kennedy-Shaffer and others, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Follmann and Fay, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Since detection of sgRNA indicates active viral replication and tracks infectious virus, our methods can be used to impute sgRNA viral load and better estimate proxies of VET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Our Joint Bayesian model for SARS-CoV-2 viral load and seroconversion 19 model can also be used to better understand the potential impact of interventions that reduce viral load on seroconversion rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' For example, previous vaccine studies have shown that vacci- nated people tended to have lower viral load at diagnosis and have lower rate of seroconversion as compared to the control group (Pajon and others, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Follmann and others, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Models like ours could help explain how much of the decrease in seroconversion rates is due to vaccination and how much is due to the vaccination effect on viral load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Finally, our proposed model can facilitate analysis of household transmission studies by providing estimated viral load trajectories for each individual in a household to inform the direction of transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Software Software in the form of R code and JAGS code is available online at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='com/ dq0708/joint_vl_sero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Supplementary Material Supplementary material is available online at http://biostatistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='oxfordjournals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Acknowledgments Funding for the project was provided by Bill & Melinda Gates Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' The authors thank the Hydroxychloroquine COVID-19 PEP Study Team and participants for providing scientific insights and data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Special thanks to Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Ruanne Barnabas and Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Anna Bershteyn for reviewing the manuscript drafts and providing valuable suggestions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' We also thank Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Leigh Fisher for additional advice regarding RT-PCR viral load quantification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Conflict of Interest: None declared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' 20 REFERENCES References Arons, Melissa M, Hatfield, Kelly M, Reddy, Sujan C, Kimball, Anne, James, Alli- son, Jacobs, Jesica R, Taylor, Joanne, Spicer, Kevin, Bardossy, Ana C, Oakley, Lisa P and others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Presymptomatic SARS-CoV-2 infections and transmission in a skilled nursing facility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' New England journal of medicine 382(22), 2081–2090.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Barnabas, Ruanne V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=', Brown, Elizabeth R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=', Bershteyn, Anna, Stankiewicz Karita, Helen C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=', Johnston, Christine, Thorpe, Lorna E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=', Kottkamp, Angelica, Neuzil, Kathleen M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=', Laufer, Miriam K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=', Deming, Meagan, Paasche-Orlow, Michael K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=', Kissinger, Patricia J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=', Luk, Alfred, Paolino, Kristopher, Landovitz, Raphael J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=', Hoffman, Risa, Schaafsma, Torin T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=', Krows, Meighan L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=', Thomas, Katherine K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=', Morrison, Susan, Haugen, Harald S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=', Kidoguchi, Lara, Wener, Mark, Greninger, Alexander L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=', Huang, Meei-Li, Jerome, Keith R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=', Wald, Anna, Celum, Connie, Chu, Helen Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' and others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' (2021, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Hydroxychloroquine as postexposure prophylaxis to prevent severe acute respiratory syndrome coronavirus 2 infection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Annals of Internal Medicine 174(3), 344–352.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Bravo, Marta Santos, Berengua, Carla, Mar´ın, Pilar, Esteban, Montserrat, Ro- driguez, Cristina, del Cuerpo, Margarita, Mir´o, Elisenda, Cuesta, Genoveva, Mosquera, Mar, S´anchez-Palomino, Sonsoles, Vila, Jordi, Rabella, N´uria and others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Viral culture confirmed SARS-CoV-2 subgenomic RNA value as a good surro- gate marker of infectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Journal of Clinical Microbiology 60(1), e01609–21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Bruce, Emily A, Mills, Margaret G, Sampoleo, Reigran, Perchetti, Garrett A, Huang, Meei-Li, Despres, Hannah W, Schmidt, Madaline M, Roychoudhury, Pavi- tra, Shirley, David J, Jerome, Keith R and others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Predicting infectivity: com- REFERENCES 21 paring four PCR-based assays to detect culturable SARS-CoV-2 in clinical samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' EMBO Molecular Medicine 14, e15290.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Buder, Felix, Bauswein, Markus, Magnus, Clara L, Audebert, Franz, Lang, Hen- riette, Kundel, Christof, Distler, Karin, Reuschel, Edith, Lubnow, Matthias, Mueller, Thomas and others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Contribution of high viral loads, detection of viral antigen and seroconversion to severe acute respiratory syndrome coronavirus 2 infectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' The Journal of infectious diseases 225(2), 190–198.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Bullard, Jared, Dust, Kerry, Funk, Duane, Strong, James E, Alexander, David, Garnett, Lauren, Boodman, Carl, Bello, Alexander, Hedley, Adam, Schiffman, Zachary and others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Predicting infectious severe acute respiratory syndrome coron- avirus 2 from diagnostic samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Clinical infectious diseases 71(10), 2663–2666.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Centers for Disease Control and Prevention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Coronavirus disease 2019 (COVID- 19) 2020 interim case definition, approved August 5, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' https://ndc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='cdc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='gov/ case-definitions/coronavirus-disease-2019-2020-08-05/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' [Online;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' accessed 2022-11- 02].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Centers for Disease Control and Prevention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Science Brief: SARS-CoV- 2 Infection-induced and Vaccine-induced Immunity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='cdc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='gov/coronavirus/ 2019-ncov/science/science-briefs/vaccine-induced-immunity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='html#.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' [Online;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' ac- cessed 2022-11-21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Cevik, Muge, Tate, Matthew, Lloyd, Ollie, Maraolo, Alberto Enrico, Schafers, Jenna and Ho, Antonia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' SARS-CoV-2, SARS-CoV, and MERS-CoV viral load dynamics, duration of viral shedding, and infectiousness: a systematic review and meta-analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' The lancet microbe 2(1), e13–e22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' 22 REFERENCES Chaccour, Carlos, Casellas, Aina, Blanco-Di Matteo, Andr´es, Pineda, I˜nigo, Fernandez-Montero, Alejandro, Ruiz-Castillo, Paula, Richardson, Mary-Ann, Rodr´ıguez-Mateos, Mariano, Jord´an-Iborra, Carlota, Brew, Joe and others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' The effect of early treatment with ivermectin on viral load, symptoms and humoral response in patients with non-severe COVID-19: A pilot, double-blind, placebo-controlled, ran- domized clinical trial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' EClinicalMedicine 32, 100720.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Challenger, Joseph D, Foo, Cher Y, Wu, Yue, Yan, Ada WC, Marjaneh, Mahdi Moradi, Liew, Felicity, Thwaites, Ryan S, Okell, Lucy C and Cunnington, Aubrey J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Modelling upper respiratory viral load dynamics of SARS-CoV-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' BMC medicine 20(1), 1–20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Dagotto, Gabriel, Mercado, Noe B, Martinez, David R, Hou, Yixuan J, Nkolola, Joseph P, Carnahan, Robert H, Crowe, James E, Baric, Ralph S and Barouch, Dan H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Comparison of subgenomic and total RNA in SARS-CoV-2-challenged rhesus macaques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Journal of virology 95(8), e02370–20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Deming, Meagan E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=', Dong, Tracy Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=', Agrawal, Vaidehi, Mills, Margaret G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=', Huang, Meei-Li W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=', Greninger, Alexander L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=', Jerome, Keith R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=', Wener, Mark H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=', Paasche-Orlow, Michael K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=', Kissinger, Patricia, Luk, Alfred, Hoffman, Risa M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=', Stewart, Jenell, Kottkamp, Angelica C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=', Bershteyn, Anna, Chu, Helen Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=', Stankiewicz Karita, Helen C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=', Johnston, Christine M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=', Wald, Anna, Barnabas, Ruanne, Brown, Elizabeth and others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Detection and kinetics of subgenomic SARS- CoV-2 RNA viral load in longitudinal diagnostic RNA positive samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' The Journal of Infec- tious Diseases 226(5), 788–796.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Eyre, David W, Taylor, Donald, Purver, Mark, Chapman, David, Fowler, Tom, Pouwels, Koen B, Walker, A Sarah and Peto, Tim EA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Effect of Covid-19 vac- REFERENCES 23 cination on transmission of alpha and delta variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' New England Journal of Medicine 386(8), 744–756.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Follmann, Dean, Janes, Holly E, Buhule, Olive D, Zhou, Honghong, Gi- rard, Bethany, Marks, Kristen, Kotloff, Karen, Desjardins, Micha¨el, Corey, Lawrence, Neuzil, Kathleen M and others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Antinucleocapsid antibodies after SARS-CoV-2 infection in the blinded phase of the randomized, placebo-controlled mRNA-1273 COVID-19 vaccine efficacy clinical trial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Annals of internal medicine 175(9), 1258–1265.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Follmann, Dean A and Fay, Michael P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' (2022, 03).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Vaccine efficacy at a point in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Biostatistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' kxac008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Forootan, Amin, Sj¨oback, Robert, Bj¨orkman, Jens, Sj¨ogreen, Bj¨orn, Linz, Lucas and Kubista, Mikael.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Methods to determine limit of detection and limit of quantifi- cation in quantitative real-time PCR (qPCR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Biomolecular Detection and Quantification 12, 1–6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Gastine, Silke, Pang, Juanita, Boshier, Florencia AT, Carter, Simon J, Lons- dale, Dagan O, Cortina-Borja, Mario, Hung, Ivan FN, Breuer, Judy, Kloprogge, Frank and Standing, Joseph F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Systematic review and patient-level meta-analysis of SARS-CoV-2 viral dynamics to model response to antiviral therapies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Clinical Pharmacology & Therapeutics 110(2), 321–333.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Goldberg, Elad, Zvi, Haim Ben, Sheena, Liron, Sofer, Summer, Krause, Ilan, Sklan, Ella H and Shlomai, Amir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' A real-life setting evaluation of the effect of remdesivir on viral load in COVID-19 patients admitted to a large tertiary centre in Israel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Clinical Microbiology and Infection 27(6), 917–e1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Gottlieb, Robert L, Nirula, Ajay, Chen, Peter, Boscia, Joseph, Heller, Barry, Morris, Jason, Huhn, Gregory, Cardona, Jose, Mocherla, Bharat, Stosor, 24 REFERENCES Valentina and others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Effect of bamlanivimab as monotherapy or in combination with etesevimab on viral load in patients with mild to moderate COVID-19: a randomized clinical trial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Jama 325(7), 632–644.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Goyal, Ashish, Cardozo-Ojeda, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Fabian and Schiffer, Joshua T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Potency and timing of antiviral therapy as determinants of duration of SARS-CoV-2 shedding and intensity of inflammatory response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Science Advances 6(47), eabc7112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Jones, Terry C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=', Biele, Guido, M¨uhlemann, Barbara, Veith, Talitha, Schnei- der, Julia, Beheim-Schwarzbach, J¨orn, Bleicker, Tobias, Tesch, Julia, Schmidt, Marie Luisa, Sander, Leif Erik, Kurth, Florian, Menzel, Peter, Schwarzer, Rolf, Zuchowski, Marta, Hofmann, J¨org, Krumbholz, Andi, Stein, Angela, Edel- mann, Anke, Corman, Victor Max and others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Estimating infectiousness through- out SARS-CoV-2 infection course.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Science 373(6551), eabi5273.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Ke, Ruian, Zitzmann, Carolin, Ho, David D, Ribeiro, Ruy M and Perelson, Alan S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' In vivo kinetics of SARS-CoV-2 infection and its relationship with a person’s infec- tiousness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Proceedings of the National Academy of Sciences 118(49), e2111477118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Kennedy-Shaffer, Lee, Kahn, Rebecca and Lipsitch, Marc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Estimating vaccine efficacy against transmission via effect on viral load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Epidemiology (Cambridge, Mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=') 32(6), 820.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Kissler, Stephen M, Fauver, Joseph R, Mack, Christina, Olesen, Scott W, Tai, Caroline, Shiue, Kristin Y, Kalinich, Chaney C, Jednak, Sarah, Ott, Isabel M, Vogels, Chantal BF and others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' (2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Viral dynamics of acute SARS-CoV-2 infection and applications to diagnostic and public health strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' PLoS biology 19(7), e3001333.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Kissler, Stephen M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=', Fauver, Joseph R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=', Mack, Christina, Tai, Caroline G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=', Bre- ban, Mallery I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=', Watkins, Anne E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=', Samant, Radhika M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=', Anderson, Deverick J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=', REFERENCES 25 Metti, Jessica, Khullar, Gaurav, Baits, Rachel, MacKay, Matthew, Salgado, Daisy, Baker, Tim, Dudley, Joel T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=', Mason, Christopher E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=', Ho, David D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=', Grubaugh, Nathan D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' and others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' (2021b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Viral dynamics of SARS-CoV-2 variants in vaccinated and unvaccinated Persons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' New England Journal of Medicine 385, 2489–2491.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Levine-Tiefenbrun, Matan, Yelin, Idan, Alapi, Hillel, Herzel, Esma, Kuint, Jacob, Chodick, Gabriel, Gazit, Sivan, Patalon, Tal and Kishony, Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Waning of SARS-CoV-2 booster viral-load reduction effectiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Nature communications 13(1), 1–4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Liu, Weimin, Russell, Ronnie M, Bibollet-Ruche, Frederic, Skelly, Ashwin N, Sherrill-Mix, Scott, Freeman, Drew A, Stoltz, Regina, Lindemuth, Emily, Lee, Fang-Hua, Sterrett, Sarah and others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Predictors of nonseroconversion after SARS-CoV-2 infection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Emerging Infectious Diseases 27(9), 2454.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Lou, Bin, Li, Ting-Dong, Zheng, Shu-Fa, Su, Ying-Ying, Li, Zhi-Yong, Liu, Wei, Yu, Fei, Ge, Sheng-Xiang, Zou, Qian-Da, Yuan, Quan and others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Serology characteristics of SARS-CoV-2 infection after exposure and post-symptom onset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' European Respiratory Journal 56(2), 2000763.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Masi´a, Mar, Telenti, Guillermo, Fern´andez, Marta, Garc´ıa, Jos´e A, Agull´o, Vanesa, Padilla, Sergio, Garc´ıa-Abell´an, Javier, Guill´en, Luc´ıa, Mascarell, Paula, Asenjo, Jos´e C and others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' SARS-CoV-2 seroconversion and viral clear- ance in patients hospitalized with COVID-19: viral load predicts antibody response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Open forum infectious diseases 8(2), ofab005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Mellis, Alexandra M, Meece, Jennifer K, Halasa, Natasha B, Chappell, James D, McLean, Huong Q, Grijalva, Carlos G, Hanson, Kayla E, Zhu, Yuwei, Kim, Ahra, Deyoe, Jessica, Ivacic, Lynn C, Reed, Carrie, Talbot, H Keipp and others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' (2022, 05).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' 26 REFERENCES SARS-CoV-2 virus dynamics in recently infected people—data from a household transmission study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' The Journal of Infectious Diseases 226(10), 1699–1703.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' N´eant,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Nad`ege,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Lingas,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Guillaume,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Le Hingrat,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Quentin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Ghosn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Jade,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Engel- mann,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Ilka,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Lepiller,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Quentin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Gaymard,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Alexandre,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Ferr´e,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Virginie,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Hartard,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' C´edric,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Plantier,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Jean-Christophe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Thibault,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Vincent,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Marlet,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Julien,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Jean-Franc¸ois,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Faure,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Emmanuel,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Poissy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Julien,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Chidiac,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Christian,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Raffi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Franc¸ois,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Kim- moun,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Antoine,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Etienne,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Manuel,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Richard,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Jean-Christophe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Tattevin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Pierre,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Garot,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Denis,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Le Moing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Vincent,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Bachelet,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Delphine,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Tardivon,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Coralie,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Du- val,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Xavier,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Yazdanpanah,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Yazdan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Mentr´e,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' France,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Laou´enan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' C´edric,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Visseaux,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Benoit,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Guedj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' J´er´emie,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' for the French COVID Cohort Investigators and others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Modeling SARS-CoV-2 viral kinetics and association with mortality in hospitalized pa- tients from the French COVID cohort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Proceedings of the National Academy of Sciences 118(8), e2017962118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Oved, Kfir, Olmer, Liraz, Shemer-Avni, Yonat, Wolf, Tamar, Supino-Rosin, Lia, Prajgrod, George, Shenhar, Yotam, Payorsky, Irina, Cohen, Yuval, Kohn, Yishai and others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Multi-center nationwide comparison of seven serology assays reveals a SARS-CoV-2 non-responding seronegative subpopulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' EClinicalMedicine 29, 100651.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Pajon, Rolando, Paila, Yamuna D, Girard, Bethany, Dixon, Groves, Kacena, Katherine, Baden, Lindsey R, El Sahly, Hana M, Essink, Brandon, Mullane, Kathleen M, Frank, Ian and others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Initial analysis of viral dynamics and cir- culating viral variants during the mRNA-1273 Phase 3 COVE trial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Nature Medicine 28(4), 823–830.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Pathela, Preeti, Crawley, Addie, Weiss, Don, Maldin, Beth, Cornell, Jennifer, REFERENCES 27 Purdin, Jeff, Schumacher, Pamela K, Marovich, Stacey, Li, Joyce, Daskalakis, Demetre and others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Seroprevalence of severe acute respiratory syndrome coronavirus 2 following the largest initial epidemic wave in the United States: findings from New York City, 13 May to 21 July 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' The Journal of infectious diseases 224(2), 196–206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Plummer, Martyn and others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' JAGS: A program for analysis of Bayesian graphical models using Gibbs sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Proceedings of the 3rd international workshop on distributed statistical computing 124(125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='10), 1–10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Portal-Celhay, Cynthia, Forleo-Neto, Eduardo, Eagan, Will, Musser, Bret J, Davis, John D, Turner, Kenneth C, Norton, Thomas, Hooper, Andrea T, Hamil- ton, Jennifer D, Pan, Cynthia and others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Virologic efficacy of casirivimab and imdevimab COVID-19 antibody combination in outpatients with SARS-CoV-2 infection: A phase 2 dose-ranging randomized clinical trial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' JAMA network open 5(8), e2225411–e2225411.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Powers, John H, Guerrero, M Lourdes, Leidy, Nancy Kline, Fairchok, Mary P, Rosenberg, Alice, Hern´andez, Andr´es, Stringer, Sonja, Schofield, Christina, Rodr´ıguez-Zulueta, Patricia, Kim, Katherine and others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Development of the Flu-PRO: a patient-reported outcome (PRO) instrument to evaluate symptoms of influenza.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' BMC infectious diseases 16(1), 1–11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' R Core Team.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' R: A language and environment for statistical computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' R Foundation for Statistical Computing, Vienna, Austria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Singanayagam, Anika, Hakki, Seran, Dunning, Jake, Madon, Kieran J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=', Crone, Michael A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=', Koycheva, Aleksandra, Derqui-Fernandez, Nieves, Barnett, Jack L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=', Whitfield, Michael G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=', Varro, Robert, Charlett, Andre, Kundu, Rhia, Fenn, Joe, Cutajar, Jessica, Quinn, Valerie, Conibear, Emily, Barclay, Wendy, 28 REFERENCES Freemont, Paul S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=', Taylor, Graham P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=', Ahmad, Shazaad, Zambon, Maria, Fer- guson, Neil M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=', Lalvani, Ajit, Badhan, Anjna, Dustan, Simon, Tejpal, Chitra, Ketkar, Anjeli V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=', Narean, Janakan Sam, Hammett, Sarah, McDermott, Eimear, Pillay, Timesh, Houston, Hamish, Luca, Constanta, Samuel, Jada, Bremang, Samuel, Evetts, Samuel, Poh, John, Anderson, Charlotte, Jackson, David, Miah, Shahjahan, Ellis, Joanna and others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' (2021, 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Community transmission and viral load kinetics of the SARS-CoV-2 delta (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='617.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' 2) variant in vaccinated and unvaccinated individ- uals in the UK: a prospective, longitudinal, cohort study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' The Lancet Infectious Diseases 22(2), 183–195.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Singanayagam, Anika, Patel, Monika, Charlett, Andre, Bernal, Jamie Lopez, Sal- iba, Vanessa, Ellis, Joanna, Ladhani, Shamez, Zambon, Maria and Gopal, Robin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Duration of infectiousness and correlation with RT-PCR cycle threshold values in cases of COVID-19, England, January to May 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Eurosurveillance 25(32), 2001483.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Stankiewicz Karita, Helen C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=', Dong, Tracy Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=', Johnston, Christine, Neuzil, Kath- leen M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=', Paasche-Orlow, Michael K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=', Kissinger, Patricia J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=', Bershteyn, Anna, Thorpe, Lorna E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=', Deming, Meagan, Kottkamp, Angelica, Laufer, Miriam, Lan- dovitz, Raphael J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=', Luk, Alfred, Hoffman, Risa, Roychoudhury, Pavitra, Mag- aret, Craig A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=', Grenninger, Alexander L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=', Huang, Meei-Li, Jerome, Keith R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=', Wener, Mark, Celum, Connie, Chu, Helen Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=', Baeten, Jared M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=', Wald, Anna, Barnabas, Ruanne V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=', Brown, Elizabeth R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' and others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Trajectory of viral RNA load among persons with incident SARS-CoV-2 G614 infection (Wuhan strain) in relation to COVID-19 symptom onset and severity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' JAMA Network Open 5(1), e2142796.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' The COVID-19 Investigation Team.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Clinical and virologic characteristics of the REFERENCES 29 first 12 patients with coronavirus disease 2019 (COVID-19) in the United States.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Nature medicine 26(6), 861–868.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' van Kampen, Jeroen JA, van de Vijver, David AMC, Fraaij, Pieter LA, Haagmans, Bart L, Lamers, Mart M, Okba, Nisreen, van den Akker, Johannes PC, Endeman, Henrik, Gommers, Diederik AMPJ, Cornelissen, Jan J and others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Duration and key determinants of infectious virus shedding in hospitalized patients with coronavirus disease-2019 (COVID-19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Nature communications 12(1), 1–6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Wellinghausen, Nele, Plonn´e, Dietmar, Voss, Meike, Ivanova, Ralitsa, Frodl, Reinhard and Deininger, Susanne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' SARS-CoV-2-IgG response is different in COVID-19 outpatients and asymptomatic contact persons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Journal of Clinical Virology 130, 104542.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' W¨olfel, Roman, Corman, Victor M, Guggemos, Wolfgang, Seilmaier, Michael, Zange, Sabine, M¨uller, Marcel A, Niemeyer, Daniela, Jones, Terry C, Vollmar, Patrick, Rothe, Camilla and others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Virological assessment of hospitalized patients with COVID-2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Nature 581(7809), 465–469.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' World Health Organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Coronavirus disease (COVID-19) pandemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' https: //www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='who.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='int/emergencies/diseases/novel-coronavirus-2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' [Online;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' accessed 2022- 10-12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Yadav, Arun Kumar, Ghosh, S, Kotwal, Atul, Kaushik, SK, Bobdey, Saurabh, Sahu, Rajesh, Kapoor, Suraj, Faujdar, DS, Teli, Prabhakar T and Anand, Vivek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Seroconversion among COVID-19 patients admitted in a dedicated COVID hospital: a longi- tudinal prospective study of 1000 patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' medical journal armed forces india 77, S379–S384.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' 30 REFERENCES Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Graphical illustration of the joint model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Along the x-axis is time since the latent peak diagnostic RNA viral load in day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Along the y-axis is the magnitude of viral load in log10 copies/ml.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' The limit of detection (LoD) for both diagnostic and sgRNA RT-PCR tests were approximately 102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='4 copies/ml.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' In addition, the diagnostic RT-PCR had a limit of quantification (LoQ) of approximately 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='9 copies/ml.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Black dots represent the observed diagnostic RNA viral load from RT-PCR tests, and blue segments represent the latent diagnostic RNA viral load trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Red dots represent the observed sgRNA viral load from RT-PCR tests, and maroon segments represent the latent sgRNA viral load trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' The black and red dots were actual data from one participant from the COVID-19 PEP study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' wa and w′ a represent time from shedding onset to peak for diagnostic and sgRNA viral load respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' wb and w′ b represent time from peak to viral clearance for diagnostic and sgRNA viral load respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' w′ d represents the time from sgRNA viral clearance to diagnostic RNA viral clearance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' tp represents time from the observed peak diagnostic RNA viral load to the latent peak diagnostic RNA viral load, and t′ d represents time from the latent peak diagnostic RNA viral load to the latent peak sgRNA viral load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Finally, vp represents the magnitude of peak diagnostic RNA viral load with respect to LoD, and q is the multiplicative factor of peak sgRNA viral load relative to peak diagnostic RNA viral load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=" 10 tp td' 6 LoQ vp' =q*vp 4 Viral LoD wa wb wa' wb' wd' 10 5 0 5 10 Time since the latent peak diagnositc RNA viral load (day)REFERENCES 31 Fig." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' The posterior means and 95% credible intervals (CI) of the population-level diagnostic and sgRNA viral load trajectories and time of anti-spike IgG seroconversion (among those who seroconverted).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' The posterior means and 95% credible intervals of the multiplicative factors associated with various patient characteristics for an increase in peak diagnostic viral load on the log10 copies/ml scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Diagnostic RNA viral load 9 SgRNA viral load Time of seroconversion 8 Viral load (log10 copies/mL) 6 5 4 3 LoD 0 2 4 6 8 10 12 14 16 18 Day since viral shedding onsetAge (+10 years) Male HCQ Arm (vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Placebo Arm) COVID-19 Symptomatic (vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Asymptomatic) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='4 Multiplicative factor32 REFERENCES Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' The posterior means and 95% credible intervals of the multiplicative factors associated with various symptoms for an increase in peak diagnostic viral load on the log10 copies/ml scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Specifically, “COVID-19” symptoms were defined based on the case definition approved by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Centers for Disease Control and Prevention (Centers for Disease Control and Prevention, 2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' the rest were defined based on the Flu-PRO patient-reported outcome instrument (Powers and others, 2015): “Chest” symptoms include cough and shortness of breath;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' “Taste or Smell” symptoms include olfactory and taste disorders;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' “Nose” symptoms include congestion, runny nose, and olfactory disorder;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' “Body” symptoms include chills, myalgia, headache, and fatigue;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' “GI” symptoms include diarrhea, nausea, and vomit;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' “Throat” symptoms include sore throat and taste disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Nose symptoms Chest symptoms Gastrointestinal symptoms Change in taste or smell Throat symptoms COVID-19 symptoms Fever Body symptoms 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='6 Multiplicative factorREFERENCES 33 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' The posterior means and 95% credible intervals of the odds ratio of seroconversion associated with various symptoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Specifically, “COVID-19” symptoms were defined based on the case definition approved by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Centers for Disease Control and Prevention (Centers for Disease Control and Pre- vention, 2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' “Chest” symptoms include cough and shortness of breath;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' “Taste or Smell” symptoms include olfactory and taste disorders;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' “Nose” symptoms include congestion, runny nose, and olfactory dis- order;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' “Body” symptoms include chills, myalgia, headache, and fatigue;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' “GI” symptoms include diarrhea, nausea, and vomit;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' “Throat” symptoms include sore throat and taste disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Gastrointestinal symptoms Fever Change in taste or smell Chest symptoms COVID-19 symptoms Body symptoms Throat symptoms Nose symptoms 0 1 2 3 4 Odds ratio34 REFERENCES Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' The posterior means and 95% credible intervals (CI) for selected quantities of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' In particular, n = 80 is the total number of people with diagnostic RNA viral load data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' Type Quantity of interest Estimand Posterior mean [95% CI] Diagnostic RNA Mean time from shedding onset to peak viral load (days) �n i=1 ˆwa,i/n 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='8 [3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='2 − 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='7] viral load Mean time from peak viral load to viral clearance (days) �n i=1 ˆwb,i/n 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='5 [10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='1 − 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='9] Mean peak viral load (log10 copies/ml) �n i=1 ˆvp,i/n + LoD 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='0 [7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='8 − 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='1] True Positive Rate expit(ˆα0 + ˆα1) 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='8% [90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='1% − 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='3%] True Negative Rate 1 − expit(ˆα0) 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='5% [99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='4% − 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='6%] sgRNA Mean time from shedding onset to peak viral load (days) �n i=1 ˆw′ a,i/n 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='4 [3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='8 − 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='3] viral load Mean time from peak viral load to viral clearance (days) �n i=1 ˆw′ b,i/n 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='7 [5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='3 − 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='1] Mean peak viral load (log10 copies/ml) �n i=1 ˆv′ p,i/n + LoD′ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='0 [5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='9 − 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='2] True Positive Rate expit(ˆα′ 0 + ˆα′ 1) 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='6% [87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='4% − 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='6%] True Negative Rate 1 − expit(ˆα′ 0) 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='6% [99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='5% − 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='7%] Anti-spike IgG The overall seroconversion rate �n i=1 ˆCi/n 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='0% [62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='5% − 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='0%] Seroconversion The odds ratio of seroconversion associated with exp � ˆβC[2] � 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='24 [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='00 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='54] every 10-fold increase in peak diagnostic viral copies Mean time from infection to seroconversion (days) ˆκ1/ ˆκ2 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='4 [12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='5 − 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content='6] REFERENCES 35 [Received XXX;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' revised XXX;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} +page_content=' accepted for publication XXX ]' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE2T4oBgHgl3EQfLgZh/content/2301.03714v1.pdf'} diff --git a/wtE2T4oBgHgl3EQfgwev/vector_store/index.faiss b/wtE2T4oBgHgl3EQfgwev/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..5462dc059bafa4660a04ab6e7c3b3282dd82f320 --- /dev/null +++ b/wtE2T4oBgHgl3EQfgwev/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:aedd2a7107386e3d396637e72eb51bd44fb376fdec473d6f2456e73e002e0c20 +size 19267629 diff --git a/wtFPT4oBgHgl3EQfPjQO/content/tmp_files/2301.13038v1.pdf.txt b/wtFPT4oBgHgl3EQfPjQO/content/tmp_files/2301.13038v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..6711fb7ee46160cd5a929cdf35de0ecf911e2d5e --- /dev/null +++ b/wtFPT4oBgHgl3EQfPjQO/content/tmp_files/2301.13038v1.pdf.txt @@ -0,0 +1,1409 @@ +MNRAS 000, 1–13 (2023) +Preprint 31 January 2023 +Compiled using MNRAS LATEX style file v3.0 +First Light And Reionisation Epoch Simulations (FLARES) XI: +[O iii] emitting galaxies at 5 < 𝑧 < 10 +Stephen M. Wilkins1,2★, Christopher C. Lovell3, Aswin P. Vijayan4,5, Dimitrios Irodotou6, +Nathan J. Adams7, William J. Roper1, Joseph Caruana8,2, Jorryt Matthee9, Louise T. C. Seeyave1, +Christopher J. Conselice7, Pablo G. Pérez-González10, Jack C. Turner1, James M. S. Donnellan1 +1Astronomy Centre, University of Sussex, Falmer, Brighton BN1 9QH, UK +2Institute of Space Sciences and Astronomy, University of Malta, Msida MSD 2080, Malta +3Institute of Cosmology and Gravitation, University of Portsmouth, Burnaby Road, Portsmouth, PO1 3FX, UK +4Cosmic Dawn Center (DAWN) +5DTU-Space, Technical University of Denmark, Elektrovej 327, DK-2800 Kgs. Lyngby, Denmark +6Department of Physics, University of Helsinki, Gustaf Hällströmin katu 2, FI-00014, Helsinki, Finland +7Jodrell Bank Centre for Astrophysics, University of Manchester, Oxford Road, Manchester, UK +8Department of Physics, Faculty of Science, University of Malta, Msida MSD 2080, Malta +9 Department of Physics, ETH Zürich, Wolfgang-Pauli-Strasse 27, 8093 Zürich, Switzerland +10 Centro de Astrobiología (CAB), CSIC-INTA, Ctra. de Ajalvir km 4, Torrejón de Ardoz, E-28850, Madrid, Spain +Accepted XXX. Received YYY; in original form ZZZ +ABSTRACT +JWST has now made it possible to probe the rest-frame optical line emission of high-redshift galaxies extending to 𝑧 ≈ 9, and +potentially beyond. To aid in the interpretation of these emerging constraints, in this work we explore predictions for [O iii] +emission in high-redshift galaxies using the First Light and Reionisation Epoch Simulations (Flares). We produce predictions +for the [O iii] luminosity function, its correlation with the UV luminosity, and the distribution of equivalent widths (EWs). +We also explore how the [O iii] EW correlates with physical properties including specific star formation rate, metallicity, and +dust attenuation. Our predictions are largely consistent with recent observational constraints on the luminosity function, average +equivalent widths, and line ratios. However, they fail to reproduce the observed tail of high-EW sources and the number density +of extreme line emitters. Possibilities to explain these discrepancies include an additional source of ionising photons and/or +greater stochasticity in star formation in the model or photometric scatter and/or bias in the observations. With JWST now +rapidly building larger samples and a wider range of emission lines the answer to this remaining discrepancy should be available +imminently. +Key words: methods: numerical – galaxies: formation – galaxies: evolution – galaxies: high-redshift – galaxies: extinction – +infrared: galaxies +1 INTRODUCTION +With the successful commissioning of JWST, the detailed study of +rest-frame optical emission of distant, high-redshift (𝑧 > 3) galaxies +has now opened up. Already, dozens of arcmin2 of deep > 2 µm +imaging have been obtained (e.g. Bagley et al. 2022). This has al- +lowed us to identify samples of 𝑧 > 10 galaxies for the first time (e.g. +Adams et al. 2023; Atek et al. 2023; Castellano et al. 2022; Donnan +et al. 2022; Finkelstein et al. 2022; Naidu et al. 2022), probe the rest- +frame optical spectral energy distributions at 𝑧 > 6 (e.g. Adams et al. +2023), and study optical morphologies to high-redshift (e.g. Ferreira +et al. 2022b,a; Kartaltepe et al. 2022). At the same time, the first +spectroscopic constraints have emerged making use of both JWST’s +multi-object (MOS) and wide field slitless spectroscopic (WFSS) +modes to study distant, high-redshift galaxies (e.g. Kashino et al. +★ E-mail: s.wilkins@sussex.ac.uk +2022; Matthee et al. 2022; Sun et al. 2022; Tacchella et al. 2022; +Trump et al. 2022; Trussler et al. 2022; Curti et al. 2023; Katz et al. +2023). With many more spectroscopic observations underway, large +samples will soon emerge, providing new insights into the physical +properties of distant galaxies as well as accurately constraining their +redshifts. +The primary target of spectroscopic studies of distant galaxies is +the various rest-frame optical emission lines. These not only permit +an unambiguous determination of the redshift, but carry a wealth of +information about the source of ionising photons and the composi- +tion and properties of the interstellar medium (ISM) in these distant +galaxies. +While a range of optical lines are now potentially accessible, in this +work we focus on the [O iii]𝜆𝜆4960, 5008Å(hereafter [O iii]) doublet. +In this work we consider both the combined line luminosity and EW +denoted by [O iii] and individual lines denoted by e.g. [O iii]𝜆5008. +The combined [O iii] line flux is, at least for 𝑍 ≈ 0.001 − 0.01, sim- +© 2023 The Authors +arXiv:2301.13038v1 [astro-ph.GA] 30 Jan 2023 + +2 +Stephen M. Wilkins et al. +ilar to H𝛼 while also being accessible at higher-redshift, 𝑧 ≈ 9 (c.f. +𝑧 ≈ 6.6 for H𝛼) with JWST’s near-infrared instruments alone. For typ- +ical metallicities this line is correlated with the ionising photon pro- +duction, though is also sensitive to extreme metallicities (𝑍 < 0.001, +𝑍 > 0.01) and the conditions of the ISM. Crucially, observational +constraints have already emerged for statistically useful samples of +galaxies (e.g. Sun et al. 2022; Matthee et al. 2022). +To fully realise the constraining potential of these observations, in +this work we make predictions for the [O iii] properties of galaxies at +𝑧 = 5−10 using the First Light And Reionisation Epoch Simulations +(Flares Lovell et al. 2021). Flares is a suite of hydrodynamical sim- +ulations employing the Eagle physics model (Schaye et al. 2015a; +Crain et al. 2015a), but with a strategy designed to efficiently ex- +tend the range of masses and luminosities simulated relative to the +original Eagle reference simulation. Flares has also been used to +study the evolution of galaxy sizes (Roper et al. 2022, 2023), colours +(Wilkins et al. 2022b), star formation and metal enrichment histories +(Wilkins et al. 2022a), and the emergence of passive galaxies (Lovell +et al. 2022). This work builds on earlier efforts to model the nebular +line emission in large samples of simulated high-redshift galaxies +(e.g Wilkins et al. 2013, 2020; Vijayan et al. 2021). In this work +we explore predictions for a range of properties including the [O iii] +luminosity function, the correlation of 𝐿[O iii] with the UV luminos- +ity, the equivalent width distribution, the correlation with physical +properties, and the impact of dust attenuation. We also compare these +predictions with recent observational constraints. +This paper is structured as follows: in Section 2 we explore the key +physics driving [O iii] emission in star forming galaxies, including +the dependence on the star formation and metal enrichment history +(§2.1), geometry (§2.1.2), dust (§2.1.3), and the assumed initial mass +function and stellar population synthesis model (§2.2). In Section +3 we describe the Flares project, including details of our spectral +energy distribution modelling procedure (§3.1). Then in Section 4 we +present our predictions for the [O iii] properties of galaxies in Flares. +In this section we also explore the impact of dust (§4.5) and the +correlation of [O iii] emission with key physical properties (§4.6). In +Section 5 we then compare our predictions with recent observations +from Hubble and JWST. We then summarise our findings in this work +and present our conclusion in Section §6. +While this article is focused on the prominent [O iii] doublet we +also make predictions for several other strong rest-frame optical emis- +sion lines including [O ii]372.6,372.9 nm, [Ne iii]386.9,396.7 nm, +H𝛽, and H𝛼. These are presented in the Appendix. +2 THEORETICAL BACKGROUND +Ionising photons produced by massive stars, active galactic nuclei +(AGN), or other phenomena (e.g. shocks) lead to the formation of +H ii regions. Within these regions various physical processes result in +the formation of nebular line and continuum emission. The strength +of individual emission lines are sensitive to the shape and normalisa- +tion of the ionising spectrum, alongside the properties of the ionised +region, including its geometry, composition, and other physical prop- +erties. +Amongst the most prominent and useful lines is the [O iii] dou- +blet. In this section we explore how the luminosities and equivalent +widths of the [O iii] doublet are affected by the star formation history, +metallicity, and geometry. To do this we employ smooth parametric +star formation histories, a single metallicity shared by both the stel- +lar population and surrounding gas, and a simple screen model for +reprocessing by dust and gas (i.e. stellar populations are all equally +6.0 +6.5 +7.0 +7.5 +8.0 +8.5 +9.0 +( +/ +) +1e-05 +0.0001 +0.001 +0.002 +0.003 +0.004 +0.006 +0.008 +0.01 +0.014 +0.02 +0.03 +0.04 +43 +44 +45 +46 +47 +( +/ +) +Figure 1. The specific ionising photon luminosity of a simple stellar popula- +tion as a function of age and metallicity assuming BPASS v2.2.1 and Chabrier +(2003) IMF with 𝑚up = 300 M⊙. +affected by dust). Initially, we assume the following fiducial param- +eters and model choices: to model the stellar emission we use the +v2.2.1 of the Binary Population And Spectral Synthesis (BPASS) +stellar population synthesis (SPS) model and a Chabrier (2003) ini- +tial mass function (IMF), while to model the nebular emission we +use version 17.03 of the cloudy photo-ionisation code (Ferland et al. +2017) and assume a reference ionising parameter (𝑈ref) of 0.01, a +solar abundance pattern, and no escape of ionising photons. Most of +these assumptions are explored in this section. +2.1 Star formation and metal enrichment history +In a stellar population the production of Lyman-continuum (LyC) +photons is dominated by hot, massive, and short lived stars. Conse- +quently, the LyC luminosity (�𝑛LyC) drops precipitously as the stellar +population ages. This is demonstrated in Figure 1, where we show +the specific LyC luminosity as a function of age and metallicity. +This reveals that the LyC luminosity drops by a factor of ≈ 104 as a +population ages from 𝑡 = 1 → 100 Myr. As lower-metallicity stars +can attain higher temperatures, the LyC luminosity is lower at high +metallicity, at least at young ages (< 10 Myr). For older stellar pop- +ulations this trend reverses as the most massive stars evolve off the +main sequence faster. +The consequence of this strong dependency on age and metallicity +is that any emission line is strongly sensitive to the star formation +and metal enrichment history. In Figure 2 we explore how the spe- +cific (i.e. per unit stellar mass formed) [O iii] luminosity, equivalent +width (EW), and ratio to the H𝛽 luminosity are affected by the star +formation history. Here we present results assuming four simple star +formation histories: an instantaneous burst, an exponentially declin- +ing (𝜏 = −100 Myr), constant, and exponentially increasing star +formation (𝜏 = 100 Myr). Unsurprisingly in each case the specific +luminosity and EW drops, at least initially. With no replenishment of +massive stars in the instantaneous model the luminosity and EW drop +by four orders of magnitude over 𝑡 = 1 → 100 Myr. For the other +models both drop more slowly, with the luminosity (EW) dropping +by ≈ 10× (≈ 4×) over 𝑡 = 1 → 100 Myr. At this point the ex- +ponentially increasing model plateaus as the rapidly increasing star +formation rate balances the accumulation of longer lived lower mass +stars. Conversely in the exponentially declining model the luminosity +MNRAS 000, 1–13 (2023) + +FLARES XI: O iii emitters +3 +24 +26 +28 +30 +32 +34 +( +[OIII]/ +) += += += +( = +) +( = +) +101 +102 +103 +[OIII]/Å +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +( +/ +) +10 +1 +100 +101 +[OIII]/ +H +Figure 2. The specific [O iii] luminosity, [O iii] equivalent width, and of +[O iii]/H𝛽 luminosity ratio as a function of the duration of star formation +assuming four different star formation histories: an instantaneous burst (dotted +line), exponentially decreasing star formation (dot-dashed line), constant star +formation (solid line), and exponentially increasing star formation (dashed +line), and three metallicities: 𝑍 = 10−4, 10−3, 10−2. +and EW rapidly drop, eventually joining an instantaneous burst. In +the constant model the specific luminosity and EW drop at a near +constant fractional rate. For scenarios with continuing star formation +the [O iii]/H𝛽 luminosity ratio is largely constant since the shape of +the ionising spectrum remains roughly constant. Conversely, for an +instantaneous burst the [O iii]/H𝛽 ratio drops as the population ages +due to the changing shape of the ionising spectrum. +Figure 2 also presents predictions for each star formation history +scenario for three different metallicities: 𝑍 = 10−4, 10−3, and 10−2. +Metallicity has an impact through both changing the ionising contin- +uum normalisation and shape, and changing the composition of the +nebular region itself. To show this more clearly, in Figure 3 we show +the luminosity, EW, and [O iii]/H𝛽 ratio as a function of metallicity +assuming 10 Myr constant star formation. In the top (luminosity) +panel we also show the LyC luminosity. While the LyC luminosity +33.5 +34.0 +34.5 +35.0 +( +[OIII]/ +) +101 +102 +103 +[OIII]/Å +5.0 +4.5 +4.0 +3.5 +3.0 +2.5 +2.0 +1.5 +10 +1 +100 +101 +[OIII]/ +H +45.5 +46.0 +46.5 +47.0 +( +/ +) +Figure 3. The specific [O iii] luminosity, [O iii] equivalent width, and of +[O iii]/H𝛽 luminosity ratio as a function of metallicity assuming 10 Myr +constant star formation. Also shown on the top panel is the sensitivity of the +LyC luminosity. +increases to lower metallicity, the [O iii] luminosity (and thus EW) +reaches a peak at 𝑍 ∼ 0.005, dropping rapidly on either side. At low- +metallicity the decrease is driven by dropping abundance of Oxygen, +while the drop to high-metallicities is due to both the falling LyC +luminosity and changing shape of the ionising spectrum. +2.1.1 Composition +In our modelling so far we have assumed that the composition of the +nebular gas follows a Solar composition. However, at high-redshift, +galaxies will be increasingly enhanced with 𝛼-elements due to the +shorter timescale for their production (Steidel et al. 2016). To explore +the impact of 𝛼-enhancement in Figure 4 we show the specific [O iii] +luminosity, [O iii] equivalent width, and [O iii]/H𝛽 luminosity ratio +as a function of metallicity for different 𝛼-enhancements assuming +10 Myr constant star formation. Since the underlying stellar models +assume Solar composition, this modelling is not self-consistent; nev- +MNRAS 000, 1–13 (2023) + +4 +Stephen M. Wilkins et al. +33.5 +34.0 +34.5 +35.0 +( +[OIII]/ +) +101 +102 +103 +[OIII]/Å +[ / +] = +. +[ / +] = . +[ / +] = . +[ / +] = . +[ / +] = . +[ / +] = . +[ / +] = . +5.0 +4.5 +4.0 +3.5 +3.0 +2.5 +2.0 +1.5 +10 +1 +100 +101 +[OIII]/ +H +Figure 4. The same as Figure 3 but showing different levels of 𝛼-enhancement +in the nebular gas. +ertheless, it allows us to explore the potential impact. This reveals +that boosting the enhancement to [𝛼/Fe]= 0.6 boosts [O iii] lumi- +nosities and EWs by ≈ 0.1 dex at 𝑍 < 0.003. At 𝑍 > 0.005 the effect +becomes small and leads to a suppression at super-solar metallicities. +2.1.2 Geometry +In our cloudy modelling the geometry of the Hii region is encapsu- +lated by the LyC escape fraction and the ionisation parameter (𝑈). 𝑈 +encodes the the number of LyC photons per atom. +In the modelling thus far we have assumed a fixed reference ionisa- +tion parameter of 𝑈ref = 0.01 referenced at 𝑡 = 1 Myr and 𝑍 = 0.01. +What this means is that the actual ionisation parameter assumed by +our cloudy implementation varies according to the metallicity and +age of the stellar population such that, +𝑈 = 𝑈ref +� 𝑄 +𝑄ref +�1/3 +(1) +where 𝑈 and 𝑄 are the ionisation parameter and ionising photon +33.5 +34.0 +34.5 +35.0 +( +[OIII]/ +) +101 +102 +103 +[OIII]/Å += +. += +. += +. += +. += +. += +. += +. += +. += . +5.0 +4.5 +4.0 +3.5 +3.0 +2.5 +2.0 +1.5 +10 +1 +100 +101 +[OIII]/ +H +Figure 5. As Figure 3 but showing the impact of varying the reference +ionisation parameter 𝑈ref. +luminosity of the stellar population and 𝑈ref and 𝑄ref are that for the +reference population. +To see how the ionisation parameter affects our predictions, in Fig- +ure 5 we show the specific luminosity, EW, and [O iii]/H𝛽 line ratio +as a function of metallicity but assuming several different reference +ionisation parameters, log10 𝑈ref = −4 → 0. This reveals the com- +plex sensitivity of [O iii] to 𝑈. The [O iii] mass-weighted luminosity +peaks assuming log10 𝑈ref ≈ −2 (i.e. our default value) at 𝑍 = 0.003. +The location of the peak is sensitive to the choice of 𝑈 shifting to +lower metallicity with higher 𝑈. This shift is sufficient such that at +very low metallicity (𝑍 < 0.0001) the [O iii] emission is highest for +higher values of 𝑈. +2.1.3 Dust +Like any other optical photons, the [O iii] line emission is susceptible +to dust attenuation. For a simple screen dust geometry, EWs will be +unaffected by dust since both the continuum and line emission would +MNRAS 000, 1–13 (2023) + +FLARES XI: O iii emitters +5 +be attenuated by the same amount. In reality, however, dust and stars +have a more complex geometry, likely leaving EWs sensitive to dust +attenuation. For example, a model in which young stars and their +associated H ii regions are attenuated more strongly than older stars +would naturally result in line emission suffering higher attenuation +than the underlying continuum, reducing the equivalent width. On +the other hand, if star formation is preferentially taking place on the +outskirts of galaxies, where the attenuation is lower, then the attenua- +tion of the overall continuum could be higher, boosting EWs relative +to their intrinsic values. This is an important consideration, since +Flares includes both a birth cloud component and a wider attenu- +ation for each star particle determined by the distribution of metals +along the line-of-sight. In the context of the Flares predictions this +is explored in 4.5. +2.2 Initial Mass Function and Stellar Population Synthesis +Model +Finally, any predicted observational quantity is also going to be af- +fected by the choice of Initial Mass Function (IMF) and Stellar Pop- +ulation Synthesis (SPS) model. In Figure 6 we show the predicted +luminosity and EW as a function of metallicity for three different SPS +models, including BPASS (our default), the Flexible Stellar Popula- +tion Synthesis (FSPS Conroy et al. 2009) code, and the Bruzual & +Charlot (2003) (BC03) models. Due the inclusion of binary inter- +actions the BPASS model generally yields higher LyC luminosities, +resulting in higher line luminosities and EWs (Stanway et al. 2016). +At 𝑍 < 0.001 FSPS yields slightly higher luminosities, but EWs that +are ≈ 50% higher than BPASS due to fainter continuum emission. +Since the IMF controls the relative proportions of stars and line +emission is driven by the most massive stars the luminosity, and +potentially EWs, will be sensitive to the shape of the IMF. In Figure +7 we show the predicted specific line luminosities and EWs as a +function of the high-mass slope of the IMF (𝛼3) using the BPASS +and FSPS models which include this flexibility. In both FSPS and +BPASS the parameter 𝛼3 describes the slope of the IMF at > 1 M⊙ +(c.f. Salpeter: 𝛼 = 2.3). We do this for three metallicities but in each +case assume 10 Myr constant star formation. Unsurprisingly, this +reveals that line luminosities are strongly impacted by the choice of +𝛼3 with the line luminosity increasing by ≈ 0.5 dex for changes to +the slope of 0.35. However, EWs are only subtly affected since the +optical continuum emission is also enhanced. +3 FIRST LIGHT AND REIONISATION EPOCH +SIMULATIONS +In this study we make use of the First Light And Reionisation Epoch +Simulations (Flares). Flares is introduced in Lovell et al. (2021) +and Vijayan et al. (2021) and we refer the reader to those papers +and references therein for a detailed introduction. In brief, Flares +is a suite of hydrodynamical re-simulations. The core1 Flares suite +adopts the AGNdT9 variant of the Eagle simulation project (Schaye +et al. 2015b; Crain et al. 2015b) with identical resolution to the Eagle +reference run. The core suite consists of 40 14/ℎ cMpc radius re- +simulations of regions selected from a large (3.2 Gpc)3 dark matter +only simulation. The selected regions span a large range in over- +density (at 𝑧 ≈ 4.7): 𝛿 + 1 ≈ −1 → 1, with over-representation of +1 In addition to the core runs Flares includes a range of simulations explor- +ing changes to the physics model. +33.5 +34.0 +34.5 +35.0 +( +[OIII]/ +) +101 +102 +103 +[OIII]/Å +. . +. +5 +4 +3 +2 +10 +1 +100 +101 +[OIII]/ +H +45.5 +46.0 +46.5 +47.0 +( +/ +) +Figure 6. As Figure 3 but for two additional population synthesis models: +FSPS and BC03. +the extremes of the density distribution. This yields a large range of +galaxy and halo masses across a wide range of environments. This +Flares strategy allows us, by appropriately weighting each galaxy, +to predict galaxy distribution functions and scaling relations across +a wider range of masses and luminosities than possible in a similarly +sized periodic volume. +3.1 Spectral Energy Distribution modelling +The spectral energy distribution (SED) modelling of galaxies in +Flares is described in depth in Vijayan et al. (2021). In short, we +associate every star particle2 in the simulation with a pure stellar +SED based on its mass, age, and metallicity using v2.2.1 of the Bi- +nary Population And Spectral Synthesis (Stanway & Eldridge 2018, +BPASS) stellar population synthesis (SPS) library, and assume a +(Chabrier 2003) initial mass function (IMF). +2 The initial star particle mass is ≈ 2 × 106 M⊙. +MNRAS 000, 1–13 (2023) + +6 +Stephen M. Wilkins et al. +33.5 +34.0 +34.5 +35.0 +35.5 +( +[OIII]/ +) += += += +. +. . += +. . += +101 +102 +103 +[OIII]/Å +1.6 +1.8 +2.0 +2.2 +2.4 +2.6 +2.8 +3.0 +10 +1 +100 +101 +[OIII]/ +H +Figure 7. The dependence of the line luminosity and EW on the high-mass +slope of the IMF 𝛼3 assuming BPASS and FSPS. +In the top panel of Figure 8 we present the specific (i.e. per unit +stellar mass) LyC photon production rate predicted for galaxies in +Flares. This quantity simply depends on the star formation and +metal enrichment histories of galaxies, since it is independent of +reprocessing by dust and gas. This reveals a clear downward trend +at high-masses (𝑀★ > 109.5 M⊙) but remains flatter at lower mass. +This is primarily due to the strong evolution of stellar metallicities +across this mass-range (see Wilkins et al. 2022a) combined with +the strong dependence of LyC photon production on metallicity (see +§2.1). Figure 8 also reveals significant redshift evolution, with the +specific production rate declining by ≈ 0.6 dex from 𝑧 = 10 → 5. +Since the mass-metallicity relationship evolves weakly with redshift +over this range, this decrease reflects the changing star formation +histories of galaxies, in particular the increase in average ages. +3.1.1 Nebular emission modelling +Once we have assigned a stellar SED we then associate each star +particle with an ionisation bounded H ii region using version 17.03 +of the cloudy photo-ionisation code (Ferland et al. 2017). Specifi- +cally, we use the pure stellar spectrum as the incident radiation field, +assume the metallicity of the nebula is identical to the star particle, +a solar abundance pattern, a covering fraction of 1 (corresponding +to a LyC escape fraction of ≈ 0 for an ionisation bound nebula) and +a metallicity and age dependent ionisation parameter referenced at +𝑡 = 1 Myr and 𝑍 = 0.01 of 𝑈 = 0.01. At other ages and metallicities +the assumed ionisation parameter is scaled from this reference value +using the ratio of the ionising luminosities of the two populations. +A consequence of this is that nebular SEDs are formed from a wide +range of ionisation parameters, with the maximum approximately +that of the reference value. The bottom panel of Figure 8 shows the +effective (ionising photon luminosity weighted) ionisation parame- +ter of galaxies in Flares. Unsurprisingly, these are clustered around +our reference ionisation parameter, though show some modest de- +crease to higher masses, reflecting the shift to higher metallicities. +The assumption of a solar abundance pattern was a decision made +to align with the BPASS SPS library. However, galaxies at high- +redshift are observed (e.g Cullen et al. 2021) and predicted (e.g +Wilkins et al. 2022a) to be strongly enhanced with 𝛼-elements, in- +cluding Oxygen. Wilkins et al. (2022a) found typical enhancements +of [𝛼/Fe]= 0.6 − 0.8 at 5 < 𝑧 < 10. Our modelling in §2.1.1 revealed +that this level of 𝛼-enhancement can boost the [O iii] luminosity and +EW fluxes by ≈ 0.1 dex. In a future iteration of Flares we plan +to address this self-consistently by using the newest version (v2.3) +of the BPASS models which include 𝛼-enhancement in the stellar +atmosphere modelling. +3.1.2 Dust attenuation +As described in Vijayan et al. (2021) in Flares we implement a +two component dust attenuation model. First, we associate young +stellar populations (with age less than 10 Myr, following Charlot & +Fall (2000) that birth clouds disperse along these timescales) with +a metallicity dependent dusty birth cloud. Secondly, for each star +particle (and associated H ii region) we apply attenuation due to dust +in the intervening inter-stellar medium. This is determined by cal- +culating the line-of-sight surface density of metals along the spatial +𝑧-axis, for each star particle, and converting this to an optical depth. +For both components we assume a simple 𝜆−1 dependence of the +attenuation. In Vijayan et al. in-prep we explore some of the wider +features of this dust model. +4 PREDICTIONS +We now explore predictions for the [O iii] properties of galaxies in +Flares, including the [O iii] luminosity function, correlation with +UV luminosity, and equivalent width distribution. We then explore +how our predictions are impacted by dust attenuation and how the +[O iii] EW correlates with other physical properties. +4.1 [O iii] luminosity function +We begin by exploring the shape and redshift evolution of the [O iii] +luminosity function (LF), shown in Figure 9. The shape of the [O iii] +LF broadly follows that of UV LF showing a clear drop at high- +luminosities, consistent with an exponential like drop-off, and at +MNRAS 000, 1–13 (2023) + +FLARES XI: O iii emitters +7 +Figure 8. The evolution of the specific LyC photon production rate ( �𝑁LyC/s−1 M−1 +⊙ ) [top] and effective ionisation parameter 𝑈eff [bottom] from 𝑧 = 5 → 10 +predicted by Flares. The effective ionisation parameter in this context is the combination of all star particles weighted by their LyC luminosity. The thick grey +line denotes the 𝑧 = 5 relation in both figures. +fainter luminosities a power-law behaviour (Vijayan et al. 2021). The +UV LF itself tracks the evolution of galaxy stellar mass function +(GSMF, see Vijayan et al. 2021) but with a steeper drop-off due +to the effect of dust attenuation. Figure 9 also shows the intrinsic +[O iii] luminosity function revealing that, like the far-UV luminosity +function (see Vijayan et al. 2021), the density of the brightest galaxies +is suppressed by dust attenuation. The impact of dust attenuation is +explored in more detail below in §4.5. Figure 9 also includes a +comparison with observations, this is discussed below in Section 5. +4.2 [O iii]–UV luminosity relation +The evolution of the [O iii] luminosity function across this redshift +interval largely tracks that of the rest-frame far-UV luminosity func- +tion. To show this more clearly, in Figure 10 we show the relationship +between the [O iii] luminosity and the far-UV luminosity, again from +𝑧 = 5 → 10. Firstly, this reveals a flat relationship, i.e. 𝐿[O iii] tracks +𝐿FUV. Since the most luminous galaxies in Flares have signifi- +cantly higher metallicity (see Wilkins et al. 2022a) and that [O iii] +luminosities drop precipitously with metallicity (see Figure 3, §2.1) +this is perhaps surprising and we might naively expect to see a drop +in 𝐿[O iii]/𝐿FUV with 𝐿FUV. However, in Flares [O iii] emission if +generally less susceptible to dust than the UV compensating for the +metallicity driven drop in [O iii] emission. +4.3 [O iii] equivalent width distribution +Next, in Figure 11, we show predictions for the evolution of the +relationship between the rest-frame [O iii] equivalent width and far- +UV luminosity. This reveals a predominantly flat relationship at all +redshifts, with a slight decline at the brightest luminosities. This +decline is driven by the higher metallicity combined with slightly +older ages in these galaxies (see Wilkins et al. 2022a). +4.4 [O iii]𝜆5008-H𝛽 ratio +An additional useful, observationally accessible, diagnostic is the +ratio of the [O iii]𝜆5008 to H𝛽 line luminosities. Predictions from +Flares for this ratio are shown in Figure 12. Unlike the EW, this +ratio is not particularly sensitive to the star formation history, at least +for actively star forming galaxies. The ratio is however sensitive to +extreme metallicities (𝑍 < 0.001, 𝑍 > 0.01,) where it rapidly drops, +as well as the ionisation parameter 𝑈. As the majority of Flares +galaxies span the range 𝑍 = 0.001 − 0.01 and we assume a single +reference ionisation parameter (and thus have a narrow range of +effective ionisation parameters) it is not surprising that the Flares +predictions are tightly clustered around ≈ 5. +4.5 Impact of reprocessing by dust +To further explore the impact of dust on our predictions, in Figure +13 we show the ratio of the intrinsic to attenuated line luminosities +(top panel) and equivalent widths (bottom) as a function of the dust- +attenuated far-UV luminosity. +As already hinted at in our comparison of the attenuated and +intrinsic luminosity functions dust attenuation increases with both the +far-UV and [O iii] luminosity. Compared to the intrinsic luminosity +(not shown) the attenuation continues to increase with increasing +luminosity. However, compared to the dust-attenuated (observed) +luminosity the attenuation flattens since the most heavily obscured +galaxies have lower observed luminosities. In the context of Flares +dust attenuation is due to both a birth cloud component, linked to +the metallicity of each star particle, and an ISM component linked +to the intervening surface density of metals. The more massive (and +intrinsically bright) a galaxy generally the higher the stellar (see +Lovell et al. 2022) and gas-phase metallicity. More massive galaxies +also have larger gas reservoirs and thus metal (and dust) surface +densities. +The effect of dust attenuation on equivalent widths is more +complex. Our faintest galaxies show mild suppression peaking at +𝐿FUV ≈ 1029 erg/s−1/Hz−1 where the EW is reduced by 0.1 dex. +At brighter far-UV luminosities the impact of dust on the EW dust +declines and eventually leads to a small enhancement of EWs in the +most UV luminous galaxies. The interpretation here is that in fainter +galaxies, young [O iii] generating stellar populations undergo slightly +higher dust attenuation than the wider continuum generating popu- +MNRAS 000, 1–13 (2023) + +1 M'l +z=10 +z=9 +z=8 +z=7 +z=6 +z=5 +30.0 +log1o(LFuv/erg s-1 Hz-1) +46 +29.5 +29.0 +45 +28.5 +9 +10 +11 +9 +10 +11 +9 +10 +11 +9 +10 +11 +9 +10 +11 +9 +10 +11 +log10(M+/Mo)z=10 +z=9 +z=8 +z=7 +z=6 +z=5 +30.0 +log1o(LFuv/erg s-1 Hz-1) +-2.0 +log10(Ueff) +29.5 +-2.2 +29.0 +28.5 +-2.4 +9 +10 +11 +9 +10 +11 +9 +10 +11 +9 +10 +11 +9 +10 +11 +9 +10 +11 +log10(M+/Mo)8 +Stephen M. Wilkins et al. +7 +6 +5 +4 +3 +2 += +( +) +( = ) += += +42.0 +42.5 +43.0 +43.5 +7 +6 +5 +4 +3 +2 += +42.0 +42.5 +43.0 +43.5 += +. ( +) +. ( +) +42.0 +42.5 +43.0 +43.5 += +( +[OIII]5007/ +) +[ / +] +Figure 9. The evolution of the [O iii]500.8nm luminosity function from 𝑧 = 5 → 10 predicted by Flares. The dark thin line shows the observed (dust-attenuated) +luminosity function while the thicker fainter line shows the intrinsic LF. The dashed line is the 𝑧 = 5 LF to highlight the evolution of the luminosity function. +Observational constraints on the LF from Sun et al. (2022) and Matthee et al. (2022) are also shown at 𝑧 ≈ 6. +lation. In the context of the Flares model this is expected due to the +addition of a birth cloud dust component. In the most luminous galax- +ies, which also roughly corresponds to the most attenuated systems, +this additional birth cloud is sub-dominant to dust in the wider ISM. +In these galaxies, the enhancement of EWs relative to the intrinsic +values is explained by the fact that the [O iii] producing stellar popu- +lations are preferentially found on the outskirts of galaxies compared +to the continuum generating populations. +4.6 Correlation with Physical Properties +We next explore, in Figure 14, how the [O iii] equivalent width cor- +relates with key physical properties, including the specific star for- +mation rate, total stellar metallicity, the stellar metallicity of young +stellar populations, and the ionising photon production efficiency +𝜉ion. Here, the star formation rate is defined as the mass of stars +that have formed in the last 10 Myr, the stellar metallicity is the mass +weighted stellar metallicity, and, in common with most observational +studies, 𝜉ion as the ratio of the Lyman continuum (ionising) photon +production rate (�𝑛LyC) to the rest-frame observed UV luminosity. +This reveals a correlation, albeit relatively weak (𝑟 = 0.44), with +specific star formation rate, such that galaxies with the highest [O iii] +EW generally have higher specific star formation rates. This is ex- +pected since [O iii] emission is driven by young stars while the optical +continuum includes a contribution from older stellar populations. +The relationship between the [O iii] EW and stellar metallicity +is more complex, with two clear branches. Because of the steep +shape of the galaxy stellar mass function, and the existence of a +tight mass–metallicity relation, the vast majority of our galaxies have +𝑍★ = 0.001. These faint, low-mass galaxies exhibit a range of EWs +spanning ≈ 100 − 2000 with the scatter driven by several effects, +including the star formation history, dust, but crucially the metallicity +of the [O iii] population, not just the overall metallicity. The second, +upper, branch corresponds to more massive, luminous galaxies with +higher-metallicities. The metallicities of the [O iii] producing stel- +lar populations in these galaxies falls beyond the peak in the [O iii] +luminosity–metallicity relation (see Figure 3). Since this branch cor- +responds to the most massive and luminous systems it is also strongly +impacted by dust attenuation, which has the effect of increasing the +scatter in the EW but, as found previously, not significantly reducing +it. +Finally, in the bottom panel of Figure 14, we show the relationship +between the EW and the ionising photon production efficiency. This +shows a clear correlation (𝑟 = 0.68) such that the most extreme +emitters have the largest production efficiencies. This is of course +not a surprise considering [O iii] line emission is, at least in Flares, +driven by the production of ionising photons by massive stars. +5 COMPARISON WITH OBSERVATIONAL CONSTRAINTS +We now turn our attention to a comparison with recent observational +constraints from Hubble, Spitzer, and JWST, including De Barros +et al. (2019), Endsley et al. (2022), Matthee et al. (2022), and Sun +et al. (2022). The former two studies infer the [O iii] + H𝛽 properties +MNRAS 000, 1–13 (2023) + +FLARES XI: O iii emitters +9 +3.0 +2.5 +2.0 +1.5 +1.0 +0.5 +( +[OIII]/ +) += += += +28.5 +29.0 +29.5 +30.0 +3.0 +2.5 +2.0 +1.5 +1.0 +0.5 +( +[OIII]/ +) += +28.5 +29.0 +29.5 +30.0 += +. ( +) +28.5 +29.0 +29.5 +30.0 += +( +/ +) +Figure 10. The relationship between the [O iii] luminosity and the far-UV luminosity, expressed as a ratio, predicted by Flares. The outlined black line show +the median [O iii] luminosity while the grey line shows the median at 𝑧 = 5. The two shaded regions show the central 68% and 95% ranges. The dashed line +shows the relationship obtained by Matthee et al. (2022). Note: the FUV luminosity included in the line ratio is expressed in units of erg/s not erg/s/Hz. +from broadband photometry, while the latter two use spectroscopic +observations recently obtained by JWST. +5.1 Photometric Constraints +The predicted strength of the combination of the [O iii] and H𝛽 +emission is sufficient to significantly boost broadband fluxes by up- +to ≈ 0.3 dex (e.g. Wilkins et al. 2013, 2020). With one band en- +compassing the line emission and a second probing the (strong line +emission free) continuum, it is then possible to infer the luminosi- +ties and equivalent widths of these combined lines. In particular, +combining Hubble and Spitzer/IRAC observations (e.g. Smit et al. +2014; Roberts-Borsani et al. 2016; De Barros et al. 2019; Endsley +et al. 2021) or, more recently, JWST (e.g. Endsley et al. 2022), it is +possible to constrain the combined [O iii] and H𝛽 emission. +In Figure 15 we compare the combined [O iii] + H𝛽 EWs predicted +by Flares with photometric constraints from De Barros et al. (2019) +and Endsley et al. (2022) at 𝑧 = 6 − 9. In both cases we find good +agreement between the predicted and observed median EWs. This +supports the findings of Wilkins et al. (2022b) where we made a direct +comparison between the broadband Hubble and Spitzer colours of +observed galaxies and galaxies predicted by Flares, finding good +agreement. However, while we broadly reproduce the median EW, +we fail to reproduce the high-EW tail observed in both De Barros +et al. (2019) and Endsley et al. (2022). In the context of [O iii] driven +by stellar populations there is little remaining model flexibility, since +our assumptions tend to maximise the possible EWs. For example, +assuming an alternative reference ionisation parameter 𝑈 would not +significantly increase EWs since our adopted value (0.01) maximises +the EW, as demonstrated in Figure 5 (§2.1.2). This suggests the +explanation lies either with the observations themselves, or there is +a significant additional source of ionising photons present in high- +redshift galaxies. While Flares predicts AGN are present in these +galaxies, their contribution to the LyC luminosity is, on average, +relatively small (see Kuusisto et al., in-prep). +5.2 Spectroscopic constraints +As noted in the introduction, a handful of spectroscopic constraints on +[O iii] are now available at 𝑧 > 6. At present these samples are small, +but will rapidly grow thanks to large spectroscopic programmes such +as JADES and CEERS. +Sun et al. (2022) presented observations of four serendipitously +discovered emission line galaxies at 𝑧 = 6.11−6.35 in JWST NIRCam +wide field slitless spectroscopy (WFSS) comissioning data. These +galaxies exhibit [O iii]5008Å EWs ranging from 100 − 1200 with a +median [O iii] + H𝛽 EW 546 ± 77Å, consistent with our predictions +of ≈ 500 − 600 at the same redshift and luminosity range. Sun +et al. (2022) also present constraints on the [O iii]5008Å luminosity +function, and these are presented in Figure 9. These are somewhat +higher than the predictions from Flares, but the sample size is small +and consequently the statistical uncertainties are very large. Since +the observations are based on a single NIRCam pointing there is also +the possibility of significant field-to-field variation (see Thomas et +al. - in-prep). +More recently, Matthee et al. (2022) presented spectroscopic con- +MNRAS 000, 1–13 (2023) + +10 +Stephen M. Wilkins et al. +101 +102 +103 +([OIII])/Å += += += +28.5 +29.0 +29.5 +30.0 +101 +102 +103 +([OIII])/Å += +28.5 +29.0 +29.5 +30.0 += +28.5 +29.0 +29.5 +30.0 += +( +/ +) +Figure 11. The same as Figure 10 but showing the rest-frame equivalent width of [O iii]. +straints from the JWST NIRCam WFSS Emission-line galaxies and +Intergalactic Gas in the Epoch of Reionization (EIGER, Kashino +et al. 2022) survey. Matthee et al. (2022) present both pure spectro- +scopic constraints alongside spectro-photometric constraints, com- +bining line fluxes from the WFSS with photometry from the NIRCam +imaging. +The combined [O iii] + H𝛽 equivalent widths of the Matthee et al. +(2022) observations are shown in Figure 15. Here we choose to +compare with the spectro-photometric constraints since these allow +a better estimation of the continuum flux than the pure spectroscopic +observations alone. We find good agreement between the predicted +and observed median of the EW distribution. However, like with the +photometric constraints we find an excess in the number of high-EW +sources. +Matthee et al. (2022) also constrain the relationship between +the far-UV and [O iii] luminosity, shown in Figure 10 and the +[O iii]5008Å luminosity function, shown in Figure 9. The normal- +isation of the 𝐿FUV-𝐿[O iii] relationship matches that predicted by +Flares, though the slope is somewhat steeper, resulting in fainter +galaxies having higher ratios than predicted by Flares. At faint +(𝐿[O iii] < 1043 erg/s) luminosities the Flares predictions pro- +vide an excellent match to EIGER luminosity function. However, at +brighter luminosities the EIGER constraints tend to be higher than +predicted by Flares, suggesting an excess of bright [O iii] emitting +galaxies. One possible explanation here is cosmic variance. Indeed, +the EIGER field contains an over-density at 𝑧 ≈ 6.77 to which three +of four sources with 𝑀FUV < −22 belong. While the EIGER con- +straints lie above our predictions, which include dust attenuation, +they are however comparable to our intrinsic predictions. Another +possibility is then that we have over-predicted the amount of dust +attenuation in these systems. +Spectroscopic observations also have the advantage that they can +separate the contributions of [O iii] and H𝛽, allowing us to measure +line ratios. Matthee et al. (2022) measure [O iii]5008/H𝛽 (R3) line +ration in their sample, finding an average value of 6.3. This is slightly +larger than our predicted typical values (4 − 5) possibly suggesting +the need for higher ionisation parameters (see §2.1.2). +6 CONCLUSIONS +In this work we have explored the [O iii] properties of the galaxy +population in the First Light And Reionisation Epoch Simulations +(Flares). The Flares strategy enables us to predict the properties +of galaxies over a wide range of masses and luminosities at high- +redshift. +Our main conclusions are: +• The [O iii] luminosity function (LF) predicted by Flares de- +clines sharply with redshift with the density of sources dropping by +≈ 1 dex from 𝑧 = 5 to 10. The bright-end of the LF is strongly +impacted by dust with a suppression of 1 dex at 𝐿 ∼ 1043.5 erg/s. +While the faint end of the LF is well matched to recent spectroscopic +constraints (Sun et al. 2022; Matthee et al. 2022) we predict fewer +very bright sources than Matthee et al. (2022) with one possible +explanation being comsic variance. +• We predict a flat, un-evolving, relationship between the far-UV +and [O iii] line luminosities. While the intrinsic ratio falls to higher +MNRAS 000, 1–13 (2023) + +FLARES XI: O iii emitters +11 +1 +2 +3 +4 +6 +10 +[OIII]5007/ += += += +28.5 +29.0 +29.5 +30.0 +1 +2 +3 +4 +6 +10 +[OIII]5007/ += +28.5 +29.0 +29.5 +30.0 += +. ( +) +28.5 +29.0 +29.5 +30.0 += +( +/ +) +Figure 12. The same as Figure 10 but showing the ratio of the [O iii]𝜆5008 to H𝛽 line luminosities. The point shows the observational constraints of Matthee +et al. (2022) for their full sample of 117 [O iii] emitters. +luminosity due to the effect of increasing metallicity this is moderated +by the growing impact of dust. +• We predict a median [O iii] rest-frame equivalent width (EW) +of ≈ 500Å at 𝑧 = 5. This declines slightly with far-UV luminosity +and stellar mass and increases to higher-redshift. At low UV lumi- +nosities dust-attenuated EWs are slightly smaller than the intrinsic +values. However, at higher luminosities (𝑀 < −21), dust-attenuated +EWs are slightly higher than the intrinsic values indicating that the +continuum emission is more heavily attenuated than the line. The +interpretation here is that the young [O iii] producing stellar popu- +lation are preferentially found on the outskirts of galaxies compared +to the continuum producing population. We find that the [O iii] EW +correlates weakly with specific star formation rate but more strongly +with ionising photon production efficiency. The relationship with +metallicity is more complex with two clear branches. +• Our median EWs are consistent with both recent photometric +(De Barros et al. 2019; Endsley et al. 2022) and spectroscopic (Sun +et al. 2022; Matthee et al. 2022) constraints. However, we fail to +predict the tail of galaxies with extremely high (> 2000Å) EWs +found by these studies possibly suggesting an additional source of +ionising photons in these systems. +• We predict [O iii]5008Å/H𝛽 ratios of ≈ 4 − 5, slightly smaller +than those found by Matthee et al. (2022). However, our ratios are +strongly affected by our assumed reference ionisation parameter. +Spectroscopic constraints of galaxies in the distant Universe will +imminently be transformed by surveys such as JADES and CEERS. +Together with other cycle 1/2 observations these will vastly increase +the sample size and dynamic range of spectroscopic observations of +[O iii] and other lines yielding new constraints on physical models in +this epoch of the Universe’s history. +ACKNOWLEDGEMENTS +We thank the Eagle team for their efforts in developing the Eagle +simulation code. We wish to thank Scott Kay and Adrian Jenkins +for their invaluable help getting up and running with the Eagle +resimulation code. +This work used the DiRAC@Durham facility managed by the +Institute for Computational Cosmology on behalf of the STFC +DiRAC HPC Facility (www.dirac.ac.uk). The equipment was funded +by BEIS capital funding via STFC capital grants ST/K00042X/1, +ST/P002293/1, ST/R002371/1 and ST/S002502/1, Durham Univer- +sity and STFC operations grant ST/R000832/1. DiRAC is part of the +National e-Infrastructure. We also wish to acknowledge the following +open source software packages used in the analysis: Scipy (Virtanen +et al. 2020), Astropy (Robitaille et al. 2013), Matplotlib (Hunter +2007) and WebPlotDigitizer (Rohatgi 2020). +APV acknowledges support from the Carlsberg Foundation (grant +no CF20-0534). PAT acknowledges support from the Science and +Technology Facilities Council (grant number ST/P000525/1). DI ac- +knowledges support by the European Research Council via ERC +Consolidator Grant KETJU (no. 818930). CCL acknowledges sup- +port from a Dennis Sciama fellowship funded by the University of +Portsmouth for the Institute of Cosmology and Gravitation. The +Cosmic Dawn Center (DAWN) is funded by the Danish National +Research Foundation under grant No. 140. +MNRAS 000, 1–13 (2023) + +12 +Stephen M. Wilkins et al. +28 +29 +30 +( +/ +) +0.5 +0.4 +0.3 +0.2 +0.1 +0.0 +0.1 +( +/ +) += += += += += += +28 +29 +30 +( +/ +) +0.2 +0.1 +0.0 +0.1 +0.2 +0.3 +( +/ +) += += += += += += +Figure 13. The average (median) impact of dust attenuation of the [O iii] lu- +minosity (top) and EW (bottom), both expressed as a function of the observed +far-UV luminosity. +We list here the roles and contributions of the authors according to +the Contributor Roles Taxonomy (CRediT)3. Stephen M. Wilkins: +Conceptualization, Data curation, Methodology, Investigation, For- +mal Analysis, Visualization, Writing - original draft. Christopher +C. Lovell, Aswin P. Vijayan: Data curation, Methodology, Writing +- review & editing. Nathan Adams, Joseph Caruana, Chris Con- +celice, James Donnellan, Dimitrios Irodotou, Jorryt Matthee, +Pablo G. Pérez-González, William Roper, Louise Seeyave, Jack +Turner: Writing - review & editing. +3 https://credit.niso.org/ +Figure 14. Comparison between the rest-frame [O iii] EW and the specific +star formation rate, stellar metallicity (both total and young), and the ionising +photon production efficiency 𝜉ion. The star formation rate is averaged over +the preceding 10 Myr while the stellar metallicity is defined for all stellar +populations. +MNRAS 000, 1–13 (2023) + +log10(LFuv/erg s-1 Hz-1) +28.0 +28.5 +29.0 +29.5 +30.0 +1.5 +log10(sSFR10/Gyr- +0.5 +0.0 +-2.0 +-2.5 +log10(Z*) +3.0 +3.5 +-2.0 +log10(Z★, <10 Myr) +2.5 +3.0 +-3.5 +=0.68 +25.8 +[(ZH +25.6 +25.0 +24.8 +2.0 +2.2 +2.4 +2.6 +2.8 +3.0 +3.2 +3.4 +log10(EWo([OIl])/A)FLARES XI: O iii emitters +13 +DATA AVAILABILITY +The data associated with the paper will be made publicly available at +https://flaresimulations.github.io/data.html on the acceptance of the +manuscript. +REFERENCES +Adams N. J., et al., 2023, MNRAS, 518, 4755 +Atek H., et al., 2023, MNRAS, 519, 1201 +Bagley M. B., et al., 2022, arXiv e-prints, p. arXiv:2211.02495 +Bruzual G., Charlot S., 2003, MNRAS, 344, 1000 +Castellano M., et al., 2022, ApJ, 938, L15 +Chabrier G., 2003, PASP, 115, 763 +Charlot S., Fall S. M., 2000, ApJ, 539, 718 +Conroy C., Gunn J. E., White M., 2009, ApJ, 699, 486 +Crain R. A., et al., 2015a, MNRAS, 450, 1937 +Crain R. A., et al., 2015b, MNRAS, 450, 1937 +Cullen F., et al., 2021, MNRAS, 505, 903 +Curti M., et al., 2023, MNRAS, 518, 425 +De Barros S., Oesch P. A., Labbé I., Stefanon M., González V., Smit R., +Bouwens R. J., Illingworth G. D., 2019, MNRAS, 489, 2355 +Donnan C. T., et al., 2022, arXiv e-prints, p. arXiv:2207.12356 +Endsley R., Stark D. P., Chevallard J., Charlot S., 2021, MNRAS, 500, 5229 +Endsley R., Stark D. P., Whitler L., Topping M. W., Chen Z., Plat A., Chisholm +J., Charlot S., 2022, arXiv e-prints, p. arXiv:2208.14999 +Ferland G. J., et al., 2017, Rev. Mex. Astron. Astrofis., 53, 385 +Ferreira L., et al., 2022a, arXiv e-prints, p. arXiv:2210.01110 +Ferreira L., et al., 2022b, ApJ, 938, L2 +Finkelstein S. L., et al., 2022, arXiv e-prints, p. arXiv:2211.05792 +Hunter J. D., 2007, Computing in Science & Engineering, 9, 90 +Kartaltepe J. S., et al., 2022, arXiv e-prints, p. arXiv:2210.14713 +Kashino D., Lilly S. J., Matthee J., Eilers A.-C., Mackenzie R., Bordoloi R., +Simcoe R. A., 2022, arXiv e-prints, p. arXiv:2211.08254 +Katz H., et al., 2023, MNRAS, 518, 592 +Lovell C. C., Vijayan A. P., Thomas P. A., Wilkins S. M., Barnes D. J., +Irodotou D., Roper W., 2021, MNRAS, 500, 2127 +Lovell C. C., et al., 2022, arXiv e-prints, p. arXiv:2211.07540 +Matthee J., Mackenzie R., Simcoe R. A., Kashino D., Lilly S. J., Bordoloi R., +Eilers A.-C., 2022, arXiv e-prints, p. arXiv:2211.08255 +Naidu R. P., et al., 2022, arXiv e-prints, p. arXiv:2207.09434 +Roberts-Borsani G. W., et al., 2016, ApJ, 823, 143 +Robitaille T. P., et al., 2013, A&A, 558, A33 +Rohatgi A., 2020, Webplotdigitizer: Version 4.4, https://automeris.io/ +WebPlotDigitizer +Roper W. J., Lovell C. C., Vijayan A. P., Marshall M. A., Irodotou D., Kuusisto +J. K., Thomas P. A., Wilkins S. M., 2022, MNRAS, 514, 1921 +Roper W. J., et al., 2023, arXiv e-prints, p. arXiv:2301.05228 +Schaye J., et al., 2015a, MNRAS, 446, 521 +Schaye J., et al., 2015b, MNRAS, 446, 521 +Smit R., et al., 2014, ApJ, 784, 58 +Stanway E. R., Eldridge J. J., 2018, MNRAS, 479, 75 +Stanway E. R., Eldridge J. J., Becker G. D., 2016, MNRAS, 456, 485 +Steidel C. C., Strom A. L., Pettini M., Rudie G. C., Reddy N. A., Trainor +R. F., 2016, ApJ, 826, 159 +Sun F., et al., 2022, arXiv e-prints, p. arXiv:2209.03374 +Tacchella S., et al., 2022, arXiv e-prints, p. arXiv:2208.03281 +Trump J. R., et al., 2022, arXiv e-prints, p. arXiv:2207.12388 +Trussler J. A. A., et al., 2022, arXiv e-prints, p. arXiv:2207.14265 +Vijayan A. P., Lovell C. C., Wilkins S. M., Thomas P. A., Barnes D. J., +Irodotou D., Kuusisto J., Roper W. J., 2021, MNRAS, 501, 3289 +Virtanen P., et al., 2020, Nature Methods, 17, 261 +Wilkins S. M., et al., 2013, MNRAS, 435, 2885 +Wilkins S. M., et al., 2020, MNRAS, 493, 6079 +Wilkins S. M., et al., 2022a, MNRAS, +Wilkins S. M., et al., 2022b, MNRAS, 517, 3227 +This paper has been typeset from a TEX/LATEX file prepared by the author. +MNRAS 000, 1–13 (2023) + +14 +Stephen M. Wilkins et al. +29 +30 +( +/ +) +0 +500 +1000 +1500 +2000 +2500 +3000 +([OIII] + H )/Å += +. ( +) +. ( +) +29 +30 +( +/ +) +0 +500 +1000 +1500 +2000 +2500 +3000 +([OIII] + H )/Å += +. ( +) +. ( +) +29 +30 +( +/ +) +0 +500 +1000 +1500 +2000 +2500 +3000 +([OIII] + H )/Å += +. ( +) +. ( +) +Figure 15. Comparison between our predicted [O iii] + H𝛽 equivalent widths +and observations from De Barros et al. (2019), Endsley et al. (2022), and +Matthee et al. (2022) at 𝑧 = 5 − 9. +MNRAS 000, 1–13 (2023) + diff --git a/wtFPT4oBgHgl3EQfPjQO/content/tmp_files/load_file.txt b/wtFPT4oBgHgl3EQfPjQO/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a1efc5d33c2086434622c3d9b1423ef7010eb626 --- /dev/null +++ b/wtFPT4oBgHgl3EQfPjQO/content/tmp_files/load_file.txt @@ -0,0 +1,1078 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf,len=1077 +page_content='MNRAS 000, 1–13 (2023) Preprint 31 January 2023 Compiled using MNRAS LATEX style file v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='0 First Light And Reionisation Epoch Simulations (FLARES) XI: [O iii] emitting galaxies at 5 < 𝑧 < 10 Stephen M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Wilkins1,2★, Christopher C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Lovell3, Aswin P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Vijayan4,5, Dimitrios Irodotou6, Nathan J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Adams7, William J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Roper1, Joseph Caruana8,2, Jorryt Matthee9, Louise T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Seeyave1, Christopher J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Conselice7, Pablo G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Pérez-González10, Jack C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Turner1, James M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Donnellan1 1Astronomy Centre, University of Sussex, Falmer, Brighton BN1 9QH, UK 2Institute of Space Sciences and Astronomy, University of Malta, Msida MSD 2080, Malta 3Institute of Cosmology and Gravitation, University of Portsmouth, Burnaby Road, Portsmouth, PO1 3FX, UK 4Cosmic Dawn Center (DAWN) 5DTU-Space, Technical University of Denmark, Elektrovej 327, DK-2800 Kgs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Lyngby, Denmark 6Department of Physics, University of Helsinki, Gustaf Hällströmin katu 2, FI-00014, Helsinki, Finland 7Jodrell Bank Centre for Astrophysics, University of Manchester, Oxford Road, Manchester, UK 8Department of Physics, Faculty of Science, University of Malta, Msida MSD 2080, Malta 9 Department of Physics, ETH Zürich, Wolfgang-Pauli-Strasse 27, 8093 Zürich, Switzerland 10 Centro de Astrobiología (CAB), CSIC-INTA, Ctra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' de Ajalvir km 4, Torrejón de Ardoz, E-28850, Madrid, Spain Accepted XXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Received YYY;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' in original form ZZZ ABSTRACT JWST has now made it possible to probe the rest-frame optical line emission of high-redshift galaxies extending to 𝑧 ≈ 9, and potentially beyond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' To aid in the interpretation of these emerging constraints, in this work we explore predictions for [O iii] emission in high-redshift galaxies using the First Light and Reionisation Epoch Simulations (Flares).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' We produce predictions for the [O iii] luminosity function, its correlation with the UV luminosity, and the distribution of equivalent widths (EWs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' We also explore how the [O iii] EW correlates with physical properties including specific star formation rate, metallicity, and dust attenuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Our predictions are largely consistent with recent observational constraints on the luminosity function, average equivalent widths, and line ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' However, they fail to reproduce the observed tail of high-EW sources and the number density of extreme line emitters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Possibilities to explain these discrepancies include an additional source of ionising photons and/or greater stochasticity in star formation in the model or photometric scatter and/or bias in the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' With JWST now rapidly building larger samples and a wider range of emission lines the answer to this remaining discrepancy should be available imminently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Key words: methods: numerical – galaxies: formation – galaxies: evolution – galaxies: high-redshift – galaxies: extinction – infrared: galaxies 1 INTRODUCTION With the successful commissioning of JWST, the detailed study of rest-frame optical emission of distant, high-redshift (𝑧 > 3) galaxies has now opened up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Already, dozens of arcmin2 of deep > 2 µm imaging have been obtained (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Bagley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' This has al- lowed us to identify samples of 𝑧 > 10 galaxies for the first time (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Adams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Atek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Castellano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Donnan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Finkelstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Naidu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' 2022), probe the rest- frame optical spectral energy distributions at 𝑧 > 6 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Adams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' 2023), and study optical morphologies to high-redshift (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Ferreira et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' 2022b,a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Kartaltepe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' At the same time, the first spectroscopic constraints have emerged making use of both JWST’s multi-object (MOS) and wide field slitless spectroscopic (WFSS) modes to study distant, high-redshift galaxies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Kashino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' ★ E-mail: s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='wilkins@sussex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='uk 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Matthee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Tacchella et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Trump et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Trussler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Curti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Katz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' With many more spectroscopic observations underway, large samples will soon emerge, providing new insights into the physical properties of distant galaxies as well as accurately constraining their redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' The primary target of spectroscopic studies of distant galaxies is the various rest-frame optical emission lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' These not only permit an unambiguous determination of the redshift, but carry a wealth of information about the source of ionising photons and the composi- tion and properties of the interstellar medium (ISM) in these distant galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' While a range of optical lines are now potentially accessible, in this work we focus on the [O iii]𝜆𝜆4960, 5008Å(hereafter [O iii]) doublet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' In this work we consider both the combined line luminosity and EW denoted by [O iii] and individual lines denoted by e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' [O iii]𝜆5008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' The combined [O iii] line flux is, at least for 𝑍 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='001 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='01, sim- © 2023 The Authors arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='13038v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='GA] 30 Jan 2023 2 Stephen M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Wilkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' ilar to H𝛼 while also being accessible at higher-redshift, 𝑧 ≈ 9 (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' 𝑧 ≈ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='6 for H𝛼) with JWST’s near-infrared instruments alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' For typ- ical metallicities this line is correlated with the ionising photon pro- duction, though is also sensitive to extreme metallicities (𝑍 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='001, 𝑍 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='01) and the conditions of the ISM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Crucially, observational constraints have already emerged for statistically useful samples of galaxies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Matthee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' To fully realise the constraining potential of these observations, in this work we make predictions for the [O iii] properties of galaxies at 𝑧 = 5−10 using the First Light And Reionisation Epoch Simulations (Flares Lovell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Flares is a suite of hydrodynamical sim- ulations employing the Eagle physics model (Schaye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' 2015a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Crain et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' 2015a), but with a strategy designed to efficiently ex- tend the range of masses and luminosities simulated relative to the original Eagle reference simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Flares has also been used to study the evolution of galaxy sizes (Roper et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' 2022, 2023), colours (Wilkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' 2022b), star formation and metal enrichment histories (Wilkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' 2022a), and the emergence of passive galaxies (Lovell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' This work builds on earlier efforts to model the nebular line emission in large samples of simulated high-redshift galaxies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='g Wilkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' 2013, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Vijayan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' In this work we explore predictions for a range of properties including the [O iii] luminosity function, the correlation of 𝐿[O iii] with the UV luminos- ity, the equivalent width distribution, the correlation with physical properties, and the impact of dust attenuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' We also compare these predictions with recent observational constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' This paper is structured as follows: in Section 2 we explore the key physics driving [O iii] emission in star forming galaxies, including the dependence on the star formation and metal enrichment history (§2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='1), geometry (§2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='2), dust (§2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='3), and the assumed initial mass function and stellar population synthesis model (§2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' In Section 3 we describe the Flares project, including details of our spectral energy distribution modelling procedure (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Then in Section 4 we present our predictions for the [O iii] properties of galaxies in Flares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' In this section we also explore the impact of dust (§4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='5) and the correlation of [O iii] emission with key physical properties (§4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' In Section 5 we then compare our predictions with recent observations from Hubble and JWST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' We then summarise our findings in this work and present our conclusion in Section §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' While this article is focused on the prominent [O iii] doublet we also make predictions for several other strong rest-frame optical emis- sion lines including [O ii]372.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='6,372.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='9 nm, [Ne iii]386.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='9,396.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='7 nm, H𝛽, and H𝛼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' These are presented in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' 2 THEORETICAL BACKGROUND Ionising photons produced by massive stars, active galactic nuclei (AGN), or other phenomena (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' shocks) lead to the formation of H ii regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Within these regions various physical processes result in the formation of nebular line and continuum emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' The strength of individual emission lines are sensitive to the shape and normalisa- tion of the ionising spectrum, alongside the properties of the ionised region, including its geometry, composition, and other physical prop- erties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Amongst the most prominent and useful lines is the [O iii] dou- blet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' In this section we explore how the luminosities and equivalent widths of the [O iii] doublet are affected by the star formation history, metallicity, and geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' To do this we employ smooth parametric star formation histories, a single metallicity shared by both the stel- lar population and surrounding gas, and a simple screen model for reprocessing by dust and gas (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' stellar populations are all equally 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='5 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='0 ( / ) 1e-05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='0001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='014 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='04 43 44 45 46 47 ( / ) Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' The specific ionising photon luminosity of a simple stellar popula- tion as a function of age and metallicity assuming BPASS v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='1 and Chabrier (2003) IMF with 𝑚up = 300 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' affected by dust).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Initially, we assume the following fiducial param- eters and model choices: to model the stellar emission we use the v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='1 of the Binary Population And Spectral Synthesis (BPASS) stellar population synthesis (SPS) model and a Chabrier (2003) ini- tial mass function (IMF), while to model the nebular emission we use version 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='03 of the cloudy photo-ionisation code (Ferland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' 2017) and assume a reference ionising parameter (𝑈ref) of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='01, a solar abundance pattern, and no escape of ionising photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Most of these assumptions are explored in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='1 Star formation and metal enrichment history In a stellar population the production of Lyman-continuum (LyC) photons is dominated by hot, massive, and short lived stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Conse- quently, the LyC luminosity (�𝑛LyC) drops precipitously as the stellar population ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' This is demonstrated in Figure 1, where we show the specific LyC luminosity as a function of age and metallicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' This reveals that the LyC luminosity drops by a factor of ≈ 104 as a population ages from 𝑡 = 1 → 100 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' As lower-metallicity stars can attain higher temperatures, the LyC luminosity is lower at high metallicity, at least at young ages (< 10 Myr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' For older stellar pop- ulations this trend reverses as the most massive stars evolve off the main sequence faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' The consequence of this strong dependency on age and metallicity is that any emission line is strongly sensitive to the star formation and metal enrichment history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' In Figure 2 we explore how the spe- cific (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' per unit stellar mass formed) [O iii] luminosity, equivalent width (EW), and ratio to the H𝛽 luminosity are affected by the star formation history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Here we present results assuming four simple star formation histories: an instantaneous burst, an exponentially declin- ing (𝜏 = −100 Myr), constant, and exponentially increasing star formation (𝜏 = 100 Myr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Unsurprisingly in each case the specific luminosity and EW drops, at least initially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' With no replenishment of massive stars in the instantaneous model the luminosity and EW drop by four orders of magnitude over 𝑡 = 1 → 100 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' For the other models both drop more slowly, with the luminosity (EW) dropping by ≈ 10× (≈ 4×) over 𝑡 = 1 → 100 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' At this point the ex- ponentially increasing model plateaus as the rapidly increasing star formation rate balances the accumulation of longer lived lower mass stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Conversely in the exponentially declining model the luminosity MNRAS 000, 1–13 (2023) FLARES XI: O iii emitters 3 24 26 28 30 32 34 ( [OIII]/ ) = = = ( = ) ( = ) 101 102 103 [OIII]/Å 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='0 ( / ) 10 1 100 101 [OIII]/ H Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' The specific [O iii] luminosity, [O iii] equivalent width, and of [O iii]/H𝛽 luminosity ratio as a function of the duration of star formation assuming four different star formation histories: an instantaneous burst (dotted line), exponentially decreasing star formation (dot-dashed line), constant star formation (solid line), and exponentially increasing star formation (dashed line), and three metallicities: 𝑍 = 10−4, 10−3, 10−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' and EW rapidly drop, eventually joining an instantaneous burst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' In the constant model the specific luminosity and EW drop at a near constant fractional rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' For scenarios with continuing star formation the [O iii]/H𝛽 luminosity ratio is largely constant since the shape of the ionising spectrum remains roughly constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Conversely, for an instantaneous burst the [O iii]/H𝛽 ratio drops as the population ages due to the changing shape of the ionising spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Figure 2 also presents predictions for each star formation history scenario for three different metallicities: 𝑍 = 10−4, 10−3, and 10−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Metallicity has an impact through both changing the ionising contin- uum normalisation and shape, and changing the composition of the nebular region itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' To show this more clearly, in Figure 3 we show the luminosity, EW, and [O iii]/H𝛽 ratio as a function of metallicity assuming 10 Myr constant star formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' In the top (luminosity) panel we also show the LyC luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' While the LyC luminosity 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='5 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='0 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='5 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='0 ( [OIII]/ ) 101 102 103 [OIII]/Å 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='5 10 1 100 101 [OIII]/ H 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='5 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='0 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='5 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='0 ( / ) Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' The specific [O iii] luminosity, [O iii] equivalent width, and of [O iii]/H𝛽 luminosity ratio as a function of metallicity assuming 10 Myr constant star formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Also shown on the top panel is the sensitivity of the LyC luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' increases to lower metallicity, the [O iii] luminosity (and thus EW) reaches a peak at 𝑍 ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='005, dropping rapidly on either side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' At low- metallicity the decrease is driven by dropping abundance of Oxygen, while the drop to high-metallicities is due to both the falling LyC luminosity and changing shape of the ionising spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='1 Composition In our modelling so far we have assumed that the composition of the nebular gas follows a Solar composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' However, at high-redshift, galaxies will be increasingly enhanced with 𝛼-elements due to the shorter timescale for their production (Steidel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' To explore the impact of 𝛼-enhancement in Figure 4 we show the specific [O iii] luminosity, [O iii] equivalent width, and [O iii]/H𝛽 luminosity ratio as a function of metallicity for different 𝛼-enhancements assuming 10 Myr constant star formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Since the underlying stellar models assume Solar composition, this modelling is not self-consistent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' nev- MNRAS 000, 1–13 (2023) 4 Stephen M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Wilkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='5 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='0 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='5 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='0 ( [OIII]/ ) 101 102 103 [OIII]/Å [ / ] = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' [ / ] = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' [ / ] = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' [ / ] = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' [ / ] = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' [ / ] = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' [ / ] = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='5 10 1 100 101 [OIII]/ H Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' The same as Figure 3 but showing different levels of 𝛼-enhancement in the nebular gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' ertheless, it allows us to explore the potential impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' This reveals that boosting the enhancement to [𝛼/Fe]= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='6 boosts [O iii] lumi- nosities and EWs by ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='1 dex at 𝑍 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' At 𝑍 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='005 the effect becomes small and leads to a suppression at super-solar metallicities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='2 Geometry In our cloudy modelling the geometry of the Hii region is encapsu- lated by the LyC escape fraction and the ionisation parameter (𝑈).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' 𝑈 encodes the the number of LyC photons per atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' In the modelling thus far we have assumed a fixed reference ionisa- tion parameter of 𝑈ref = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='01 referenced at 𝑡 = 1 Myr and 𝑍 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' What this means is that the actual ionisation parameter assumed by our cloudy implementation varies according to the metallicity and age of the stellar population such that, 𝑈 = 𝑈ref � 𝑄 𝑄ref �1/3 (1) where 𝑈 and 𝑄 are the ionisation parameter and ionising photon 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='5 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='0 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='5 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='0 ( [OIII]/ ) 101 102 103 [OIII]/Å = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='5 10 1 100 101 [OIII]/ H Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' As Figure 3 but showing the impact of varying the reference ionisation parameter 𝑈ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' luminosity of the stellar population and 𝑈ref and 𝑄ref are that for the reference population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' To see how the ionisation parameter affects our predictions, in Fig- ure 5 we show the specific luminosity, EW, and [O iii]/H𝛽 line ratio as a function of metallicity but assuming several different reference ionisation parameters, log10 𝑈ref = −4 → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' This reveals the com- plex sensitivity of [O iii] to 𝑈.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' The [O iii] mass-weighted luminosity peaks assuming log10 𝑈ref ≈ −2 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' our default value) at 𝑍 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' The location of the peak is sensitive to the choice of 𝑈 shifting to lower metallicity with higher 𝑈.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' This shift is sufficient such that at very low metallicity (𝑍 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='0001) the [O iii] emission is highest for higher values of 𝑈.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='3 Dust Like any other optical photons, the [O iii] line emission is susceptible to dust attenuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' For a simple screen dust geometry, EWs will be unaffected by dust since both the continuum and line emission would MNRAS 000, 1–13 (2023) FLARES XI: O iii emitters 5 be attenuated by the same amount.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' In reality, however, dust and stars have a more complex geometry, likely leaving EWs sensitive to dust attenuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' For example, a model in which young stars and their associated H ii regions are attenuated more strongly than older stars would naturally result in line emission suffering higher attenuation than the underlying continuum, reducing the equivalent width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' On the other hand, if star formation is preferentially taking place on the outskirts of galaxies, where the attenuation is lower, then the attenua- tion of the overall continuum could be higher, boosting EWs relative to their intrinsic values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' This is an important consideration, since Flares includes both a birth cloud component and a wider attenu- ation for each star particle determined by the distribution of metals along the line-of-sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' In the context of the Flares predictions this is explored in 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='2 Initial Mass Function and Stellar Population Synthesis Model Finally, any predicted observational quantity is also going to be af- fected by the choice of Initial Mass Function (IMF) and Stellar Pop- ulation Synthesis (SPS) model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' In Figure 6 we show the predicted luminosity and EW as a function of metallicity for three different SPS models, including BPASS (our default), the Flexible Stellar Popula- tion Synthesis (FSPS Conroy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' 2009) code, and the Bruzual & Charlot (2003) (BC03) models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Due the inclusion of binary inter- actions the BPASS model generally yields higher LyC luminosities, resulting in higher line luminosities and EWs (Stanway et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' At 𝑍 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='001 FSPS yields slightly higher luminosities, but EWs that are ≈ 50% higher than BPASS due to fainter continuum emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Since the IMF controls the relative proportions of stars and line emission is driven by the most massive stars the luminosity, and potentially EWs, will be sensitive to the shape of the IMF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' In Figure 7 we show the predicted specific line luminosities and EWs as a function of the high-mass slope of the IMF (𝛼3) using the BPASS and FSPS models which include this flexibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' In both FSPS and BPASS the parameter 𝛼3 describes the slope of the IMF at > 1 M⊙ (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Salpeter: 𝛼 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' We do this for three metallicities but in each case assume 10 Myr constant star formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Unsurprisingly, this reveals that line luminosities are strongly impacted by the choice of 𝛼3 with the line luminosity increasing by ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='5 dex for changes to the slope of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' However, EWs are only subtly affected since the optical continuum emission is also enhanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' 3 FIRST LIGHT AND REIONISATION EPOCH SIMULATIONS In this study we make use of the First Light And Reionisation Epoch Simulations (Flares).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Flares is introduced in Lovell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' (2021) and Vijayan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' (2021) and we refer the reader to those papers and references therein for a detailed introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' In brief, Flares is a suite of hydrodynamical re-simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' The core1 Flares suite adopts the AGNdT9 variant of the Eagle simulation project (Schaye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' 2015b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Crain et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' 2015b) with identical resolution to the Eagle reference run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' The core suite consists of 40 14/ℎ cMpc radius re- simulations of regions selected from a large (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='2 Gpc)3 dark matter only simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' The selected regions span a large range in over- density (at 𝑧 ≈ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='7): 𝛿 + 1 ≈ −1 → 1, with over-representation of 1 In addition to the core runs Flares includes a range of simulations explor- ing changes to the physics model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='5 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='0 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='5 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='0 ( [OIII]/ ) 101 102 103 [OIII]/Å .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' 5 4 3 2 10 1 100 101 [OIII]/ H 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='5 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='0 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='5 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='0 ( / ) Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' As Figure 3 but for two additional population synthesis models: FSPS and BC03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' the extremes of the density distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' This yields a large range of galaxy and halo masses across a wide range of environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' This Flares strategy allows us, by appropriately weighting each galaxy, to predict galaxy distribution functions and scaling relations across a wider range of masses and luminosities than possible in a similarly sized periodic volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='1 Spectral Energy Distribution modelling The spectral energy distribution (SED) modelling of galaxies in Flares is described in depth in Vijayan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' In short, we associate every star particle2 in the simulation with a pure stellar SED based on its mass, age, and metallicity using v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='1 of the Bi- nary Population And Spectral Synthesis (Stanway & Eldridge 2018, BPASS) stellar population synthesis (SPS) library, and assume a (Chabrier 2003) initial mass function (IMF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' 2 The initial star particle mass is ≈ 2 × 106 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' MNRAS 000, 1–13 (2023) 6 Stephen M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Wilkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='5 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='0 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='5 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='0 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='5 ( [OIII]/ ) = = = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' = 101 102 103 [OIII]/Å 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='0 10 1 100 101 [OIII]/ H Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' The dependence of the line luminosity and EW on the high-mass slope of the IMF 𝛼3 assuming BPASS and FSPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' In the top panel of Figure 8 we present the specific (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' per unit stellar mass) LyC photon production rate predicted for galaxies in Flares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' This quantity simply depends on the star formation and metal enrichment histories of galaxies, since it is independent of reprocessing by dust and gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' This reveals a clear downward trend at high-masses (𝑀★ > 109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='5 M⊙) but remains flatter at lower mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' This is primarily due to the strong evolution of stellar metallicities across this mass-range (see Wilkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' 2022a) combined with the strong dependence of LyC photon production on metallicity (see §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Figure 8 also reveals significant redshift evolution, with the specific production rate declining by ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='6 dex from 𝑧 = 10 → 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Since the mass-metallicity relationship evolves weakly with redshift over this range, this decrease reflects the changing star formation histories of galaxies, in particular the increase in average ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='1 Nebular emission modelling Once we have assigned a stellar SED we then associate each star particle with an ionisation bounded H ii region using version 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='03 of the cloudy photo-ionisation code (Ferland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Specifi- cally, we use the pure stellar spectrum as the incident radiation field, assume the metallicity of the nebula is identical to the star particle, a solar abundance pattern, a covering fraction of 1 (corresponding to a LyC escape fraction of ≈ 0 for an ionisation bound nebula) and a metallicity and age dependent ionisation parameter referenced at 𝑡 = 1 Myr and 𝑍 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='01 of 𝑈 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' At other ages and metallicities the assumed ionisation parameter is scaled from this reference value using the ratio of the ionising luminosities of the two populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' A consequence of this is that nebular SEDs are formed from a wide range of ionisation parameters, with the maximum approximately that of the reference value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' The bottom panel of Figure 8 shows the effective (ionising photon luminosity weighted) ionisation parame- ter of galaxies in Flares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Unsurprisingly, these are clustered around our reference ionisation parameter, though show some modest de- crease to higher masses, reflecting the shift to higher metallicities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' The assumption of a solar abundance pattern was a decision made to align with the BPASS SPS library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' However, galaxies at high- redshift are observed (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='g Cullen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' 2021) and predicted (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='g Wilkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' 2022a) to be strongly enhanced with 𝛼-elements, in- cluding Oxygen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Wilkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' (2022a) found typical enhancements of [𝛼/Fe]= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='6 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='8 at 5 < 𝑧 < 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Our modelling in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='1 revealed that this level of 𝛼-enhancement can boost the [O iii] luminosity and EW fluxes by ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='1 dex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' In a future iteration of Flares we plan to address this self-consistently by using the newest version (v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='3) of the BPASS models which include 𝛼-enhancement in the stellar atmosphere modelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='2 Dust attenuation As described in Vijayan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' (2021) in Flares we implement a two component dust attenuation model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' First, we associate young stellar populations (with age less than 10 Myr, following Charlot & Fall (2000) that birth clouds disperse along these timescales) with a metallicity dependent dusty birth cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Secondly, for each star particle (and associated H ii region) we apply attenuation due to dust in the intervening inter-stellar medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' This is determined by cal- culating the line-of-sight surface density of metals along the spatial 𝑧-axis, for each star particle, and converting this to an optical depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' For both components we assume a simple 𝜆−1 dependence of the attenuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' In Vijayan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' in-prep we explore some of the wider features of this dust model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' 4 PREDICTIONS We now explore predictions for the [O iii] properties of galaxies in Flares, including the [O iii] luminosity function, correlation with UV luminosity, and equivalent width distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' We then explore how our predictions are impacted by dust attenuation and how the [O iii] EW correlates with other physical properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='1 [O iii] luminosity function We begin by exploring the shape and redshift evolution of the [O iii] luminosity function (LF), shown in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' The shape of the [O iii] LF broadly follows that of UV LF showing a clear drop at high- luminosities, consistent with an exponential like drop-off, and at MNRAS 000, 1–13 (2023) FLARES XI: O iii emitters 7 Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' The evolution of the specific LyC photon production rate ( �𝑁LyC/s−1 M−1 ⊙ ) [top] and effective ionisation parameter 𝑈eff [bottom] from 𝑧 = 5 → 10 predicted by Flares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' The effective ionisation parameter in this context is the combination of all star particles weighted by their LyC luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' The thick grey line denotes the 𝑧 = 5 relation in both figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' fainter luminosities a power-law behaviour (Vijayan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' The UV LF itself tracks the evolution of galaxy stellar mass function (GSMF, see Vijayan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' 2021) but with a steeper drop-off due to the effect of dust attenuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Figure 9 also shows the intrinsic [O iii] luminosity function revealing that, like the far-UV luminosity function (see Vijayan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' 2021), the density of the brightest galaxies is suppressed by dust attenuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' The impact of dust attenuation is explored in more detail below in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Figure 9 also includes a comparison with observations, this is discussed below in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='2 [O iii]–UV luminosity relation The evolution of the [O iii] luminosity function across this redshift interval largely tracks that of the rest-frame far-UV luminosity func- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' To show this more clearly, in Figure 10 we show the relationship between the [O iii] luminosity and the far-UV luminosity, again from 𝑧 = 5 → 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Firstly, this reveals a flat relationship, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' 𝐿[O iii] tracks 𝐿FUV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Since the most luminous galaxies in Flares have signifi- cantly higher metallicity (see Wilkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' 2022a) and that [O iii] luminosities drop precipitously with metallicity (see Figure 3, §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='1) this is perhaps surprising and we might naively expect to see a drop in 𝐿[O iii]/𝐿FUV with 𝐿FUV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' However, in Flares [O iii] emission if generally less susceptible to dust than the UV compensating for the metallicity driven drop in [O iii] emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='3 [O iii] equivalent width distribution Next, in Figure 11, we show predictions for the evolution of the relationship between the rest-frame [O iii] equivalent width and far- UV luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' This reveals a predominantly flat relationship at all redshifts, with a slight decline at the brightest luminosities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' This decline is driven by the higher metallicity combined with slightly older ages in these galaxies (see Wilkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' 2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='4 [O iii]𝜆5008-H𝛽 ratio An additional useful, observationally accessible, diagnostic is the ratio of the [O iii]𝜆5008 to H𝛽 line luminosities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Predictions from Flares for this ratio are shown in Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Unlike the EW, this ratio is not particularly sensitive to the star formation history, at least for actively star forming galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' The ratio is however sensitive to extreme metallicities (𝑍 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='001, 𝑍 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='01,) where it rapidly drops, as well as the ionisation parameter 𝑈.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' As the majority of Flares galaxies span the range 𝑍 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='001 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='01 and we assume a single reference ionisation parameter (and thus have a narrow range of effective ionisation parameters) it is not surprising that the Flares predictions are tightly clustered around ≈ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='5 Impact of reprocessing by dust To further explore the impact of dust on our predictions, in Figure 13 we show the ratio of the intrinsic to attenuated line luminosities (top panel) and equivalent widths (bottom) as a function of the dust- attenuated far-UV luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' As already hinted at in our comparison of the attenuated and intrinsic luminosity functions dust attenuation increases with both the far-UV and [O iii] luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Compared to the intrinsic luminosity (not shown) the attenuation continues to increase with increasing luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' However, compared to the dust-attenuated (observed) luminosity the attenuation flattens since the most heavily obscured galaxies have lower observed luminosities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' In the context of Flares dust attenuation is due to both a birth cloud component, linked to the metallicity of each star particle, and an ISM component linked to the intervening surface density of metals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' The more massive (and intrinsically bright) a galaxy generally the higher the stellar (see Lovell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' 2022) and gas-phase metallicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' More massive galaxies also have larger gas reservoirs and thus metal (and dust) surface densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' The effect of dust attenuation on equivalent widths is more complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Our faintest galaxies show mild suppression peaking at 𝐿FUV ≈ 1029 erg/s−1/Hz−1 where the EW is reduced by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='1 dex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' At brighter far-UV luminosities the impact of dust on the EW dust declines and eventually leads to a small enhancement of EWs in the most UV luminous galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=" The interpretation here is that in fainter galaxies, young [O iii] generating stellar populations undergo slightly higher dust attenuation than the wider continuum generating popu- MNRAS 000, 1–13 (2023) 1 M'l z=10 z=9 z=8 z=7 z=6 z=5 30." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='0 log1o(LFuv/erg s-1 Hz-1) 46 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='5 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='0 45 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='5 9 10 11 9 10 11 9 10 11 9 10 11 9 10 11 9 10 11 log10(M+/Mo)z=10 z=9 z=8 z=7 z=6 z=5 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='0 log1o(LFuv/erg s-1 Hz-1) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='0 log10(Ueff) 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='2 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='0 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='4 9 10 11 9 10 11 9 10 11 9 10 11 9 10 11 9 10 11 log10(M+/Mo)8 Stephen M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Wilkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' 7 6 5 4 3 2 = ( ) ( = ) = = 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='0 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='5 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='0 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='5 7 6 5 4 3 2 = 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='0 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='5 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='0 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='5 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' ( ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' ( ) 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='0 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='5 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='0 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='5 = ( [OIII]5007/ ) [ / ] Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' The evolution of the [O iii]500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='8nm luminosity function from 𝑧 = 5 → 10 predicted by Flares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' The dark thin line shows the observed (dust-attenuated) luminosity function while the thicker fainter line shows the intrinsic LF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' The dashed line is the 𝑧 = 5 LF to highlight the evolution of the luminosity function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Observational constraints on the LF from Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' (2022) and Matthee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' (2022) are also shown at 𝑧 ≈ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' lation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' In the context of the Flares model this is expected due to the addition of a birth cloud dust component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' In the most luminous galax- ies, which also roughly corresponds to the most attenuated systems, this additional birth cloud is sub-dominant to dust in the wider ISM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' In these galaxies, the enhancement of EWs relative to the intrinsic values is explained by the fact that the [O iii] producing stellar popu- lations are preferentially found on the outskirts of galaxies compared to the continuum generating populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='6 Correlation with Physical Properties We next explore, in Figure 14, how the [O iii] equivalent width cor- relates with key physical properties, including the specific star for- mation rate, total stellar metallicity, the stellar metallicity of young stellar populations, and the ionising photon production efficiency 𝜉ion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Here, the star formation rate is defined as the mass of stars that have formed in the last 10 Myr, the stellar metallicity is the mass weighted stellar metallicity, and, in common with most observational studies, 𝜉ion as the ratio of the Lyman continuum (ionising) photon production rate (�𝑛LyC) to the rest-frame observed UV luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' This reveals a correlation, albeit relatively weak (𝑟 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='44), with specific star formation rate, such that galaxies with the highest [O iii] EW generally have higher specific star formation rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' This is ex- pected since [O iii] emission is driven by young stars while the optical continuum includes a contribution from older stellar populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' The relationship between the [O iii] EW and stellar metallicity is more complex, with two clear branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Because of the steep shape of the galaxy stellar mass function, and the existence of a tight mass–metallicity relation, the vast majority of our galaxies have 𝑍★ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' These faint, low-mass galaxies exhibit a range of EWs spanning ≈ 100 − 2000 with the scatter driven by several effects, including the star formation history, dust, but crucially the metallicity of the [O iii] population, not just the overall metallicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' The second, upper, branch corresponds to more massive, luminous galaxies with higher-metallicities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' The metallicities of the [O iii] producing stel- lar populations in these galaxies falls beyond the peak in the [O iii] luminosity–metallicity relation (see Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Since this branch cor- responds to the most massive and luminous systems it is also strongly impacted by dust attenuation, which has the effect of increasing the scatter in the EW but, as found previously, not significantly reducing it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Finally, in the bottom panel of Figure 14, we show the relationship between the EW and the ionising photon production efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' This shows a clear correlation (𝑟 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='68) such that the most extreme emitters have the largest production efficiencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' This is of course not a surprise considering [O iii] line emission is, at least in Flares, driven by the production of ionising photons by massive stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' 5 COMPARISON WITH OBSERVATIONAL CONSTRAINTS We now turn our attention to a comparison with recent observational constraints from Hubble, Spitzer, and JWST, including De Barros et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' (2019), Endsley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' (2022), Matthee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' (2022), and Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' The former two studies infer the [O iii] + H𝛽 properties MNRAS 000, 1–13 (2023) FLARES XI: O iii emitters 9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='5 ( [OIII]/ ) = = = 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='5 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='0 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='5 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='5 ( [OIII]/ ) = 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='5 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='0 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='5 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='0 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' ( ) 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='5 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='0 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='5 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='0 = ( / ) Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' The relationship between the [O iii] luminosity and the far-UV luminosity, expressed as a ratio, predicted by Flares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' The outlined black line show the median [O iii] luminosity while the grey line shows the median at 𝑧 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' The two shaded regions show the central 68% and 95% ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' The dashed line shows the relationship obtained by Matthee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Note: the FUV luminosity included in the line ratio is expressed in units of erg/s not erg/s/Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' from broadband photometry, while the latter two use spectroscopic observations recently obtained by JWST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='1 Photometric Constraints The predicted strength of the combination of the [O iii] and H𝛽 emission is sufficient to significantly boost broadband fluxes by up- to ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='3 dex (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Wilkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' 2013, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' With one band en- compassing the line emission and a second probing the (strong line emission free) continuum, it is then possible to infer the luminosi- ties and equivalent widths of these combined lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' In particular, combining Hubble and Spitzer/IRAC observations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Smit et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Roberts-Borsani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' De Barros et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Endsley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' 2021) or, more recently, JWST (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Endsley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' 2022), it is possible to constrain the combined [O iii] and H𝛽 emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' In Figure 15 we compare the combined [O iii] + H𝛽 EWs predicted by Flares with photometric constraints from De Barros et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' (2019) and Endsley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' (2022) at 𝑧 = 6 − 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' In both cases we find good agreement between the predicted and observed median EWs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' This supports the findings of Wilkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' (2022b) where we made a direct comparison between the broadband Hubble and Spitzer colours of observed galaxies and galaxies predicted by Flares, finding good agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' However, while we broadly reproduce the median EW, we fail to reproduce the high-EW tail observed in both De Barros et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' (2019) and Endsley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' In the context of [O iii] driven by stellar populations there is little remaining model flexibility, since our assumptions tend to maximise the possible EWs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' For example, assuming an alternative reference ionisation parameter 𝑈 would not significantly increase EWs since our adopted value (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='01) maximises the EW, as demonstrated in Figure 5 (§2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' This suggests the explanation lies either with the observations themselves, or there is a significant additional source of ionising photons present in high- redshift galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' While Flares predicts AGN are present in these galaxies, their contribution to the LyC luminosity is, on average, relatively small (see Kuusisto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', in-prep).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='2 Spectroscopic constraints As noted in the introduction, a handful of spectroscopic constraints on [O iii] are now available at 𝑧 > 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' At present these samples are small, but will rapidly grow thanks to large spectroscopic programmes such as JADES and CEERS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' (2022) presented observations of four serendipitously discovered emission line galaxies at 𝑧 = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='11−6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='35 in JWST NIRCam wide field slitless spectroscopy (WFSS) comissioning data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' These galaxies exhibit [O iii]5008Å EWs ranging from 100 − 1200 with a median [O iii] + H𝛽 EW 546 ± 77Å, consistent with our predictions of ≈ 500 − 600 at the same redshift and luminosity range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' (2022) also present constraints on the [O iii]5008Å luminosity function, and these are presented in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' These are somewhat higher than the predictions from Flares, but the sample size is small and consequently the statistical uncertainties are very large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Since the observations are based on a single NIRCam pointing there is also the possibility of significant field-to-field variation (see Thomas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' - in-prep).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' More recently, Matthee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' (2022) presented spectroscopic con- MNRAS 000, 1–13 (2023) 10 Stephen M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Wilkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' 101 102 103 ([OIII])/Å = = = 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='5 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='0 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='5 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='0 101 102 103 ([OIII])/Å = 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='5 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='0 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='5 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='0 = 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='5 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='0 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='5 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='0 = ( / ) Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' The same as Figure 10 but showing the rest-frame equivalent width of [O iii].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' straints from the JWST NIRCam WFSS Emission-line galaxies and Intergalactic Gas in the Epoch of Reionization (EIGER, Kashino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' 2022) survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Matthee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' (2022) present both pure spectro- scopic constraints alongside spectro-photometric constraints, com- bining line fluxes from the WFSS with photometry from the NIRCam imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' The combined [O iii] + H𝛽 equivalent widths of the Matthee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' (2022) observations are shown in Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Here we choose to compare with the spectro-photometric constraints since these allow a better estimation of the continuum flux than the pure spectroscopic observations alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' We find good agreement between the predicted and observed median of the EW distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' However, like with the photometric constraints we find an excess in the number of high-EW sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Matthee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' (2022) also constrain the relationship between the far-UV and [O iii] luminosity, shown in Figure 10 and the [O iii]5008Å luminosity function, shown in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' The normal- isation of the 𝐿FUV-𝐿[O iii] relationship matches that predicted by Flares, though the slope is somewhat steeper, resulting in fainter galaxies having higher ratios than predicted by Flares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' At faint (𝐿[O iii] < 1043 erg/s) luminosities the Flares predictions pro- vide an excellent match to EIGER luminosity function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' However, at brighter luminosities the EIGER constraints tend to be higher than predicted by Flares, suggesting an excess of bright [O iii] emitting galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' One possible explanation here is cosmic variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Indeed, the EIGER field contains an over-density at 𝑧 ≈ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='77 to which three of four sources with 𝑀FUV < −22 belong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' While the EIGER con- straints lie above our predictions, which include dust attenuation, they are however comparable to our intrinsic predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Another possibility is then that we have over-predicted the amount of dust attenuation in these systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Spectroscopic observations also have the advantage that they can separate the contributions of [O iii] and H𝛽, allowing us to measure line ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Matthee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' (2022) measure [O iii]5008/H𝛽 (R3) line ration in their sample, finding an average value of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' This is slightly larger than our predicted typical values (4 − 5) possibly suggesting the need for higher ionisation parameters (see §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' 6 CONCLUSIONS In this work we have explored the [O iii] properties of the galaxy population in the First Light And Reionisation Epoch Simulations (Flares).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' The Flares strategy enables us to predict the properties of galaxies over a wide range of masses and luminosities at high- redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Our main conclusions are: The [O iii] luminosity function (LF) predicted by Flares de- clines sharply with redshift with the density of sources dropping by ≈ 1 dex from 𝑧 = 5 to 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' The bright-end of the LF is strongly impacted by dust with a suppression of 1 dex at 𝐿 ∼ 1043.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='5 erg/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' While the faint end of the LF is well matched to recent spectroscopic constraints (Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Matthee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' 2022) we predict fewer very bright sources than Matthee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' (2022) with one possible explanation being comsic variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' We predict a flat, un-evolving, relationship between the far-UV and [O iii] line luminosities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' While the intrinsic ratio falls to higher MNRAS 000, 1–13 (2023) FLARES XI: O iii emitters 11 1 2 3 4 6 10 [OIII]5007/ = = = 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='5 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='0 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='5 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='0 1 2 3 4 6 10 [OIII]5007/ = 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='5 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='0 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='5 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='0 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' ( ) 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='5 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='0 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='5 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='0 = ( / ) Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' The same as Figure 10 but showing the ratio of the [O iii]𝜆5008 to H𝛽 line luminosities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' The point shows the observational constraints of Matthee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' (2022) for their full sample of 117 [O iii] emitters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' luminosity due to the effect of increasing metallicity this is moderated by the growing impact of dust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' We predict a median [O iii] rest-frame equivalent width (EW) of ≈ 500Å at 𝑧 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' This declines slightly with far-UV luminosity and stellar mass and increases to higher-redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' At low UV lumi- nosities dust-attenuated EWs are slightly smaller than the intrinsic values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' However, at higher luminosities (𝑀 < −21), dust-attenuated EWs are slightly higher than the intrinsic values indicating that the continuum emission is more heavily attenuated than the line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' The interpretation here is that the young [O iii] producing stellar popu- lation are preferentially found on the outskirts of galaxies compared to the continuum producing population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' We find that the [O iii] EW correlates weakly with specific star formation rate but more strongly with ionising photon production efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' The relationship with metallicity is more complex with two clear branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Our median EWs are consistent with both recent photometric (De Barros et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Endsley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' 2022) and spectroscopic (Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Matthee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' 2022) constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' However, we fail to predict the tail of galaxies with extremely high (> 2000Å) EWs found by these studies possibly suggesting an additional source of ionising photons in these systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' We predict [O iii]5008Å/H𝛽 ratios of ≈ 4 − 5, slightly smaller than those found by Matthee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' However, our ratios are strongly affected by our assumed reference ionisation parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Spectroscopic constraints of galaxies in the distant Universe will imminently be transformed by surveys such as JADES and CEERS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Together with other cycle 1/2 observations these will vastly increase the sample size and dynamic range of spectroscopic observations of [O iii] and other lines yielding new constraints on physical models in this epoch of the Universe’s history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' ACKNOWLEDGEMENTS We thank the Eagle team for their efforts in developing the Eagle simulation code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' We wish to thank Scott Kay and Adrian Jenkins for their invaluable help getting up and running with the Eagle resimulation code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' This work used the DiRAC@Durham facility managed by the Institute for Computational Cosmology on behalf of the STFC DiRAC HPC Facility (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='dirac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='uk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' The equipment was funded by BEIS capital funding via STFC capital grants ST/K00042X/1, ST/P002293/1, ST/R002371/1 and ST/S002502/1, Durham Univer- sity and STFC operations grant ST/R000832/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' DiRAC is part of the National e-Infrastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' We also wish to acknowledge the following open source software packages used in the analysis: Scipy (Virtanen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' 2020), Astropy (Robitaille et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' 2013), Matplotlib (Hunter 2007) and WebPlotDigitizer (Rohatgi 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' APV acknowledges support from the Carlsberg Foundation (grant no CF20-0534).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' PAT acknowledges support from the Science and Technology Facilities Council (grant number ST/P000525/1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' DI ac- knowledges support by the European Research Council via ERC Consolidator Grant KETJU (no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' 818930).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' CCL acknowledges sup- port from a Dennis Sciama fellowship funded by the University of Portsmouth for the Institute of Cosmology and Gravitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' The Cosmic Dawn Center (DAWN) is funded by the Danish National Research Foundation under grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' 140.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' MNRAS 000, 1–13 (2023) 12 Stephen M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Wilkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' 28 29 30 ( / ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='1 ( / ) = = = = = = 28 29 30 ( / ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='3 ( / ) = = = = = = Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' The average (median) impact of dust attenuation of the [O iii] lu- minosity (top) and EW (bottom), both expressed as a function of the observed far-UV luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' We list here the roles and contributions of the authors according to the Contributor Roles Taxonomy (CRediT)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Stephen M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Wilkins: Conceptualization, Data curation, Methodology, Investigation, For- mal Analysis, Visualization, Writing - original draft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Christopher C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Lovell, Aswin P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Vijayan: Data curation, Methodology, Writing review & editing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Nathan Adams, Joseph Caruana, Chris Con- celice, James Donnellan, Dimitrios Irodotou, Jorryt Matthee, Pablo G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Pérez-González, William Roper, Louise Seeyave, Jack Turner: Writing - review & editing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' 3 https://credit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='niso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='org/ Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Comparison between the rest-frame [O iii] EW and the specific star formation rate, stellar metallicity (both total and young), and the ionising photon production efficiency 𝜉ion.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='4 log10(EWo([OIl])/A)FLARES XI: O iii emitters 13 DATA AVAILABILITY The data associated with the paper will be made publicly available at https://flaresimulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='io/data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='html on the acceptance of the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' REFERENCES Adams N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', 2023, MNRAS, 518, 4755 Atek H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', 2023, MNRAS, 519, 1201 Bagley M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', 2022, arXiv e-prints, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' arXiv:2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='02495 Bruzual G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', Charlot S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', 2003, MNRAS, 344, 1000 Castellano M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', 2022, ApJ, 938, L15 Chabrier G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', 2003, PASP, 115, 763 Charlot S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', Fall S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', 2000, ApJ, 539, 718 Conroy C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', Gunn J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', White M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', 2009, ApJ, 699, 486 Crain R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', 2015a, MNRAS, 450, 1937 Crain R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', 2015b, MNRAS, 450, 1937 Cullen F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', 2021, MNRAS, 505, 903 Curti M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', 2023, MNRAS, 518, 425 De Barros S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', Oesch P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', Labbé I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', Stefanon M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', González V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', Smit R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', Bouwens R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', Illingworth G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', 2019, MNRAS, 489, 2355 Donnan C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', 2022, arXiv e-prints, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' arXiv:2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='12356 Endsley R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', Stark D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', Chevallard J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', Charlot S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', 2021, MNRAS, 500, 5229 Endsley R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', Stark D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', Whitler L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', Topping M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', Chen Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', Plat A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', Chisholm J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', Charlot S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', 2022, arXiv e-prints, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' arXiv:2208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='14999 Ferland G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', 2017, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Mex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Astrofis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', 53, 385 Ferreira L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', 2022a, arXiv e-prints, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' arXiv:2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='01110 Ferreira L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', 2022b, ApJ, 938, L2 Finkelstein S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', 2022, arXiv e-prints, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' arXiv:2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='05792 Hunter J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', 2007, Computing in Science & Engineering, 9, 90 Kartaltepe J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', 2022, arXiv e-prints, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' arXiv:2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='14713 Kashino D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', Lilly S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', Matthee J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', Eilers A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', Mackenzie R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', Bordoloi R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', Simcoe R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', 2022, arXiv e-prints, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' arXiv:2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='08254 Katz H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', 2023, MNRAS, 518, 592 Lovell C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', Vijayan A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', Thomas P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', Wilkins S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', Barnes D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', Irodotou D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', Roper W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', 2021, MNRAS, 500, 2127 Lovell C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', 2022, arXiv e-prints, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' arXiv:2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='07540 Matthee J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', Mackenzie R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', Simcoe R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', Kashino D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', Lilly S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', Bordoloi R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', Eilers A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', 2022, arXiv e-prints, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' arXiv:2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='08255 Naidu R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', 2022, arXiv e-prints, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' arXiv:2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='09434 Roberts-Borsani G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', 2016, ApJ, 823, 143 Robitaille T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', 2013, A&A, 558, A33 Rohatgi A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', 2020, Webplotdigitizer: Version 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='4, https://automeris.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='io/ WebPlotDigitizer Roper W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', Lovell C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', Vijayan A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', Marshall M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', Irodotou D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', Kuusisto J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', Thomas P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', Wilkins S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', 2022, MNRAS, 514, 1921 Roper W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', 2023, arXiv e-prints, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='05228 Schaye J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', 2015a, MNRAS, 446, 521 Schaye J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', 2015b, MNRAS, 446, 521 Smit R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', 2014, ApJ, 784, 58 Stanway E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', Eldridge J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', 2018, MNRAS, 479, 75 Stanway E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', Eldridge J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', Becker G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', 2016, MNRAS, 456, 485 Steidel C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', Strom A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', Pettini M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', Rudie G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', Reddy N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', Trainor R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', 2016, ApJ, 826, 159 Sun F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', 2022, arXiv e-prints, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' arXiv:2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='03374 Tacchella S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', 2022, arXiv e-prints, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' arXiv:2208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='03281 Trump J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', 2022, arXiv e-prints, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' arXiv:2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='12388 Trussler J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', 2022, arXiv e-prints, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' arXiv:2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content='14265 Vijayan A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', Lovell C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', Wilkins S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', Thomas P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', Barnes D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', Irodotou D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', Kuusisto J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', Roper W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', 2021, MNRAS, 501, 3289 Virtanen P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', 2020, Nature Methods, 17, 261 Wilkins S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', 2013, MNRAS, 435, 2885 Wilkins S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', 2020, MNRAS, 493, 6079 Wilkins S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', 2022a, MNRAS, Wilkins S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=', 2022b, MNRAS, 517, 3227 This paper has been typeset from a TEX/LATEX file prepared by the author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' MNRAS 000, 1–13 (2023) 14 Stephen M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Wilkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' 29 30 ( / ) 0 500 1000 1500 2000 2500 3000 ([OIII] + H )/Å = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' ( ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' ( ) 29 30 ( / ) 0 500 1000 1500 2000 2500 3000 ([OIII] + H )/Å = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' ( ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' ( ) 29 30 ( / ) 0 500 1000 1500 2000 2500 3000 ([OIII] + H )/Å = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' ( ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' ( ) Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' Comparison between our predicted [O iii] + H𝛽 equivalent widths and observations from De Barros et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' (2019), Endsley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' (2022), and Matthee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' (2022) at 𝑧 = 5 − 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} +page_content=' MNRAS 000, 1–13 (2023)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFPT4oBgHgl3EQfPjQO/content/2301.13038v1.pdf'} diff --git a/x9FAT4oBgHgl3EQfAhwk/content/tmp_files/2301.08398v1.pdf.txt b/x9FAT4oBgHgl3EQfAhwk/content/tmp_files/2301.08398v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..c2d670df85d088b566f6e82b4a4a653196340fea --- /dev/null +++ b/x9FAT4oBgHgl3EQfAhwk/content/tmp_files/2301.08398v1.pdf.txt @@ -0,0 +1,1531 @@ +arXiv:2301.08398v1 [eess.SY] 20 Jan 2023 +An LMI Framework for Contraction-basedNonlinear Control +Design by Derivatives of Gaussian Process Regression ⋆ +Yu Kawano a Kenji Kashima b +aGraduate School of Advanced Science and Engineering, Hiroshima University, Higashi-Hiroshima 739-8527, Japan (email: +ykawano@hiroshima-u.ac.jp). +bGraduate School of Informatics, Kyoto University, Kyoto 606-8501, Japan (e-mail: kk@i.kyoto-u.ac.jp). +Abstract +Contraction theory formulates the analysis of nonlinear systems in terms of Jacobian matrices. Although this provides the +potential to develop a linear matrix inequality (LMI) framework for nonlinear control design, conditions are imposed not on +controllers but on their partial derivatives, which makes control design challenging. In this paper, we illustrate this so-called +integrability problem can be solved by a non-standard use of Gaussian process regression (GPR) for parameterizing controllers +and then establish an LMI framework of contraction-based control design for nonlinear discrete-time systems, as an easy-to- +implement tool. Later on, we consider the case where the drift vector fields are unknown and employ GPR for functional fitting +as its standard use. GPR describes learning errors in terms of probability, and thus we further discuss how to incorporate +stochastic learning errors into the proposed LMI framework. +Key words: Nonlinear systems, discrete-time systems, stochastic systems, contraction analysis, Gaussian process regression +1 +Introduction +Contraction theory [9,28] has attracted massive research +attention in the systems and control community, e.g., [7, +40], which establishes a differential geometric approach +to study incremental properties, i.e., properties between +any pair of trajectories. Revisiting nonlinear control the- +ory from contraction perspectives brings new insights +not only for stability analysis but also for dissipativity +theory [11,23,43], balancing theory [19,25], and mono- +tone systems [10, 20, 21] to name a few. As an advan- +tage in comparison with classical Lyapunov theory, the +nature of incremental stability enables us to formulate +various design problems in a unified framework, such +as stabilizing control [24, 29], tracking control [15, 33], +observer design [1,28], and control design for achieving +synchronizations [1] and other rich behaviors [11]. Ow- +ing to the differential geometric feature, these problems +are described in terms of Jacobian matrices, which is +expected as another advantage in practical use. Indeed, +restricting the class of controllers (and observer gains) +into linear reduces design problems to linear matrix in- +equalities (LMIs) [11, 19]. However, nonlinear control +⋆ This work was supported in part by JSPS KAKENHI +Grant Numbers JP21H04875 and JP21K14185. +design is more involved because of the so-called integra- +bility problem. Namely, design conditions are imposed +not on controllers but on their partial derivatives, which +are the main difficulty in developing an LMI framework +for contraction-based nonlinear control design. +To overcome this difficulty, in this paper, we employ +Gaussian process regression (GPR), a functional fitting +tool [4, 32]. As its non-standard use, we employ GPR +to parametrize a controller based on its two important +features: 1) computing derivatives of GPR is easy; 2) +GPR becomes linear with respect to parameters while +it possesses the flexibility to describe a nonlinear func- +tion. Utilizing these two, we describe a condition for +control design in terms of LMIs with respect to pa- +rameters of GPR. Namely, we establish an LMI frame- +work for contraction-based nonlinear stabilizing control +design by explicitly addressing the integrability prob- +lem. We mainly consider nonlinear discrete-time systems +with constant input vector fields and constant metrics +for contraction. Then, we mention that the formulations +are still in the LMI framework for discrete-time systems +with non-constant input vector fields and continuous- +time systems with non-constant input vector fields and +non-constant metrics. The proposed method is further +applicable to the aforementioned various design prob- +Preprint submitted to Automatica +23 January 2023 + +lems thanks to a unified problem formulation by con- +traction theory. +In systems and control, GPR is typically used to esti- +mate an unknown drift vector field from measured states, +and [2,17,35,36,41,42] study control design for learned +models. Other applications are a joint estimation of the +state and model [3, 6] and solving the Hamilton-Jacobi +equations/inequalities [14,18]. In particular, [2,35,41,42] +study closed-loop stability under learning errors. How- +ever, controllers are designed without taking learning er- +rors into account, and learning errors are used for closed- +loop analysis only. Motivated by these works, in this pa- +per, we also consider the case where a drift vector field is +unknown, and this is learned by GPR. In contrast to the +conventional approach, we compensate for the learning +error by control design, which are benefits of develop- +ing the LMI framework for control design and learning +models by GPR. The proposed approach can be gener- +alized to the case where the whole system dynamics are +unknown, and only system’s input and output are mea- +surable, since there are learning approaches by GPR in +such a setting [12,13]. +As relevant researches, [35, 38, 39] give neural network +frameworks for contraction-based control design, which +requires iterations for finding suitable parameters in con- +trast to the proposed LMI framework. The paper [17] +formulates control design for models learned by GPR in +an LMI framework and designs a switching linear con- +troller, but does not use GPR for control design or is +not based on contraction theory. Therefore, the result +in [17] is not applicable for solving the integrability prob- +lem. In this paper, we establish an LMI framework for +contraction-based control design by utilizing the deriva- +tives of GPR to solve the integrability problem. +The remainder of this paper is organized as follows. In +Section 2, we pose the problem formulation by mention- +ing the integrability problem of control design in contrac- +tion theory. In Section 3, we develop an LMI framework +for contraction-based control design by utilizing deriva- +tives of GPR. In Section 4, we consider the case where +drift vector fields are unknown. In Section 5, the pro- +posed control design method is illustrated by the means +of an example. +Notation: The sets of real numbers, non-negative in- +tegers, and positive integers are denoted by R, Z≥0, +and Z>0, respectively. The identity matrix with the size +n is denoted by In. For P, Q ∈ Rn×n, P ≻ Q (resp. +P ⪰ Q) means that P −Q is symmetric and positive defi- +nite (resp. semi-definite). The Euclidean norm of x ∈ Rn +weighted by P ≻ 0 is denoted by |x|P := +√ +x⊤Px. +If P = In, this is simply denoted by |x|. The Moore- +Penrose inverse of a matrix A is denoted by A+. +For a function f(x, y), the row vector-valued function +consisting of its partial derivatives with respect to x +is denoted by ∂xf := ∂f/∂x = [∂f/∂x1 · · · ∂f/∂xn]. +If f depends on x only, this is simply denoted by ∂f. +For a scalar-valued function k(x, x′), its Hessian matrix +is denoted by ∂2k(x, x′) := ∂2k(x, x′)/∂x∂x′, which is +a matrix-valued function. The multivariate normal dis- +tribution with mean µ and variance Σ is denoted by +N(µ, Σ). The (standard) expectation is denoted by E[·]. +A stochastic process {ωk}k∈Z≥0 is said to be i.i.d. if ωk, +k ∈ Z≥0 are independently distributed, and none of the +characteristics of ωk changes with k ∈ Z≥0. +2 +Preliminaries +2.1 +Problem Formulation +Consider the following discrete-time nonlinear system: +xk+1 = f(xk) + buk, +k ∈ Z≥0, +(1) +where xk ∈ Rn and uk ∈ R denote the state and input, +respectively; f : Rn → Rn is of class C1, i.e., continu- +ously differentiable, and b ∈ Rn. The ith components of +f and b are denoted by fi and bi, respectively. For the +sake of notational simplicity, we consider single-input +systems. However, the results can readily be generalized +to multiple-input systems as explained below. We also +later discuss the case where b is a function of x. +In contraction theory [22, 28, 37], we study incremental +stability as a property of the pair of trajectories, stated +below. +Definition 2.1 The system xk+1 = f(xk) (with its copy +x′ +k+1 = f(x′ +k)) is said to be incrementally exponentially +stable (IES) if there exist a > 0 and λ ∈ (0, 1) such that +|xk − x′ +k| ≤ aλk|x0 − x′ +0|, +∀k ∈ Z≥0 +for each (x0, x′ +0) ∈ Rn × Rn. +✁ +Applying [37, Theorem 15] to control design yields the +following IES condition. +Proposition 2.2 Suppose that there exist ε > 0, P ∈ +Rn×n, and p∂ : Rn → R1×n of continuous such that +� +P +∗ +∂f(x)P + bp∂(x) P +� +⪰ εI2n +(2) +for all x ∈ Rn, where ∗ represents the appropriate matrix. +If there exists p : Rn → R of class C1 such that +p∂(x) = ∂p(x)P +(3) +for all x ∈ Rn, then the closed-loop system xk+1 = +f(xk) + bp(xk) is IES. +2 + +PROOF. By the Schur complement, the set of (2) and +(3) is equivalent to P ≻ 0 and +P(∂f(x) + b∂p(x))⊤P −1(∂f(x) + b∂p(x))P +− P ⪯ εIn. +(4) +This is nothing but the definition of uniform contraction +with Θ = P −1/2 [37, Definition 6] for the closed-loop +system xk+1 = f(xk) + bp(xk). Therefore, [37, Theorem +15] concludes IES of the closed-loop system. +✷ +In [37, Theorem 15], it has been shown that a closed IES +system admits a state-dependent P satisfying a similar +inequality as (2). Moreover, such a P is uniformly lower +and upper bounded by constant matrices. It is not yet +clear when P becomes constant. If one restricts the class +of controllers into linear for a constant P, control de- +sign can be reduced to linear matrix inequalities (LMIs); +see, e.g., [11, 19]. Namely, control design can be some- +times formulated as a practically solvable problem even +for nonlinear systems. Indeed, (2) is an LMI with re- +spect to ε, P and p∂(x) at each x ∈ Rn. However, as +in Proposition 2.2, designing nonlinear controllers is not +fully formulated in the LMI framework, due to the so- +called integrability constraint (3), i.e., p∂(x)P −1 needs +to be the partial derivative of some function p(x) pro- +viding a feedback control law u = p(x). The main ob- +jective of this paper is to develop an LMI framework for +stabilizing nonlinear control design by tackling the inte- +grability constraint, stated below. +Problem 2.3 Given f(x) and b of the system (1), de- +velop an LMI framework for designing p : Rn → R which +satisfies all the conditions in Proposition 2.2, if such p +exists. +✁ +Remark 2.4 We later consider the case where f is un- +known by learning it. As a byproduct of developing control +design methodologies in the LMI framework, it is possible +to compensate for learning errors by control design. +✁ +An important feature of contraction theory is to study +the convergence between any pair of trajectories. By +virtue of this, one can handle observer design [1, 28], +tracking control [15,33], and control design for imposing +synchronizations [1] and rich behavior such as limit cy- +cles [11] in the same framework as stabilizing control de- +sign. These references mainly focus on continuous-time +systems, but similar results can be delivered to discrete- +time systems, since incremental stability conditions in +contraction analysis have been derived also for discrete- +time systems [28, 37]. Moreover, the proposed method +in this paper can be generalized to continuous-time sys- +tems as will be explained in Section 3.3. Therefore, solv- +ing Problem 2.3 can result in LMI frameworks for vari- +ous design problems. +Integrability constraints sometimes appear in nonlinear +adaptive control or observer design. Since directly ad- +dressing integrability constraints are difficult, there are +techniques for avoiding them by adding the dynamic +order of the identifier. However, as pointed out by [27], +adding additional dynamics can degenerate control per- +formances, and it has not been validated that such an +approach works for contraction-based control design. +Therefore, it is worth solving Problem 2.3 directly. In +particular, we provide an LMI framework for control de- +sign, which is easy-to-implement. The proposed method +may be tailored for nonlinear adaptive control or ob- +server design although this is beyond the scope of this +paper. +2.2 +Gaussian Process Regression +To solve Problem 2.3, we employ Gaussian process re- +gression (GPR) [4, 32]. We, in this subsection, briefly +summarize basics of GPR and, in the next section, +demonstrate that GPR is a suitable tool for handling +problems involving partial derivatives, e.g., integrability +conditions. +Let {x(i)}N +i=1, x(i) ∈ Rn be input data, and let {y(i)}N +i=1, +y(i) ∈ R be the corresponding output data given by +y(i) = p(x(i)) + ω(i), +i = 1, . . . , N, +(5) +where ω(i) ∼ N(0, σ2) is i.i.d. GPR is a technique to +learn an unknown function p : Rn → R from input- +output data {x(i), y(i)}N +i=1 by assuming p as a Gaussian +process (GP). +A stochastic process p is said to be GP if any finite +set {p(x(i))}N +i=1 has a joint Gaussian distribution [32, +Definition 2.1]. A GP is completely specified by its mean +function mp : Rn → R and covariance function kp : +Rn × Rn → R. They are defined by +mp(x) = E[p(x)] +(6) +kp(x, x′) = E[(p(x) − mp(x))(p(x′) − mp(x′))], +(7) +and we represent the GP by p ∼ GP(mp(x), kp(x, x′)). +The essence of GPR is to estimate mp(x) and kp(x, x′) as +the posterior mean and covariance given {x(i), y(i)}N +i=1 +by the Bayes estimation [4,32]. Typically, the prior mean +m0(x) is selected as zero. The prior covariance k0(x, x′) +needs to be a positive definite kernel [32, Section 6], and +we also require smoothness. Standard kernels are linear, +polynomial, or squared exponential (SE) [4, 32], which +are all smooth and positive definite. For instance, an SE +kernel is +k0(x, x′) = βe−|x−x′|2 +Σ−1/2, +(8) +3 + +where β > 0 and Σ ≻ 0 in addition to σ > 0 in (5) +are free parameters, called hyper parameters. The hyper +parameters can be selected to maximize the marginal +likelihood from observed data; see e.g., [32, Section 5.4]. +To use GPR for learning p(x), we need its data as in +(5). However, Proposition 2.2 contains no information +of p(x) but of ∂p(x) via the integrability constraint (3). +To handle this situation, we employ derivatives of GPR +as elaborated in the next section. +3 +Contraction-based Nonlinear Control Design +In this section, we establish an LMI framework for +contraction-based nonlinear control design by utilizing +derivatives of GPR to solve the integrability problem. +Then, we discuss generalizations of the proposed method +to non-constant input vector fields and continuous-time +cases. +3.1 +LMI Frameworks for Nonlinear Control Design +Taking the partial derivatives of (6) and (7), we have +the joint distribution of p(x) and ∂p(x) as in +� +p +∂⊤p +� +∼ GP +�� +mp(x) +∂⊤mp(x) +� +, +� +kp(x, x′) +∂x′kp(x, x′) +∂⊤ +x kp(x, x′) ∂2kp(x, x′) +�� +, +(9) +see, e.g., [30, Equation (2)]. The goal of this subsection +is to find mp(x) based on the Bayes estimation such that +p(x) = mp(x) is a solution to Problem 2.3. For the sake +of notational simplicity, we select the prior mean of p(x) +as zero. +A standard procedure of the Bayes estimation is that we +first select a class C2 positive definite kernel k0(x, x′) as +a prior covariance. Then, we compute the posterior mean +mp(x) given data of ∂p(x). Looking at this from a dif- +ferent angle, mp(x) can be viewed as a function of data +of ∂p(x). Based on this perspective, we consider gener- +ating suitable data such that mp(x) becomes a solution +to Problem 2.3 as a non-standard use of GPR. +To this end, let {x(i), ˆp(i)}N +i=1 denote a data set to be +generated, where +ˆp(i) = ∂p(x(i)) + ω(i) +p , +i = 1, . . . , N. +(10) +The role of i.i.d. ω(i) +p +∼ N(0, σ2 +pIn) is explained later; one +can simply choose it as zero. Define the vector consisting +of {ˆp(i)}N +i=1 by +Yp := +� +ˆp(1) · · · ˆp(N) +� +∈ RnN. +Also, we introduce the following notations: +XN := +� +(x(1))⊤ · · · (x(N))⊤ +�⊤ +∈ RnN +∂p(XN) := +� +∂p(x(1)) · · · ∂p(x(N)) +� +∈ RnN +k′ +∂,0(x′) := +� +∂xk0(x(1), x′) · · · ∂xk0(x(N), x′) +� +k∂,0(x) := +� +∂x′k0(x, x(1)) · · · ∂x′k0(x, x(N)) +� +K0 := + + +∂2k0(x(1), x(1)) · · · ∂2k0(x(1), x(N)) +... +... +∂2k0(x(N), x(1)) · · · ∂2k0(x(N), x(N)) + + +∈ RnN×nN. +(11) +Then, the joint distribution of the prior distribution of +p(x), denoted by ppri(x) and ∂p(XN) is +� +ppri +∂⊤p(XN) +� +∼ N +�� +0 +Y ⊤ +p +� +, +� +k0(x, x′) +k∂,0(x) +(k′ +∂,0)⊤(x′) K0 + σ2 +pInN +�� +. +By the Bayes estimation, we can compute the posterior +mean mp(x) given Yp as +mp(x|Yp) := +N +� +i=1 +∂x′k0(x, x(i))h(i) +p , +(12) +where h(i) +p +denotes the ith block-component with size n +of hp defined by +hp := (K0 + σ2 +pInN)−1Y ⊤ +p ∈ RnN. +(13) +The partial derivative of mp(x|Yp) is easy-to-compute: +∂mp(x|Yp) = ∂ +∂x +� N +� +i=1 +∂x′k0(x, x(i))h(i) +p +� +. +(14) +Especially if σp = 0 and K0 ≻ 0, it follows from (14) that +∂mp(x(i)|Yp) = ˆp(i), +i = 1, . . . , N. +(15) +Note that mp(x|Yp) and ∂mp(x|Yp) are nonlinear +functions of x and linear functions of Yp. Therefore, +control design reduces to generating suitable Yp, i.e., +{x(i), ˆp(i)}N +i=1, which is proceeded based on Proposition +2.2. +Now, we are ready to develop an LMI framework for +contraction-based nonlinear control design. Substituting +p∂ = ∂mp(x|Yp)P into (2) does not give an LMI be- +cause of the coupling between Yp and P. This issue can +be addressed by using a standard technique of an LMI. +4 + +According to [34, Theorem 2.3.11], (2), i.e., the set of +P ≻ 0 and (4) implies, for all x ∈ Rn, +� +P ≻ 0 +b+(P − ∂f(x)P∂⊤f(x))(b+)⊤ ⪰ εpIn−1. +(16) +This is an LMI with respect to P and εp > 0 at each +x ∈ Rn. For a solution P to (16) and p∂ = ∂mp(x|Yp)P, +one only has to solve (2) with respect to Yp. The pro- +posed procedure for solving Problem 2.3 is summarized +in Algorithm 1. +Algorithm 1 Solving Problem 2.3 +Require: f(x), b, k0(x, x′), σp ≥ 0, {x(i)}N +i=1 +Ensure: p(x) and P (if they exist) +1: Define ∂mp(x|Yp) as in (14) +2: Solve a finite family of LMIs with respect to P and +εp > 0: +� +P ≻ 0 +b+(P − ∂f(x(i))P∂⊤f(x(i)))(b+)⊤ ⪰ εpIn−1 +i = 1, . . . , N +(17) +3: For P obtained in Step 2, solve a finite family of +LMIs with respect to Yp and ε > 0: +� +P +∗ +(∂f(x(i)) + b∂mp(x(i)|Yp))P P +� +⪰ εI2n, +(18) +i = 1, . . . , N +4: Define mp(x|Yp) as in (12) by using Yp obtained in +Step 3 +5: Return p(x) := mp(x|Yp) and P +If Algorithm 1 has a set of solutions, the obtained p(x) +is a solution to Problem 2.3 at each data point, stated +below. +Theorem 3.1 Given a system (1), a class C2 positive +definite kernel k0(x, x′), σp ≥ 0, and {x(i)}N +i=1, suppose +that Algorithm 1 has a set of solutions p(x) and P. Then, +p(x) satisfies (2) and (3) for all x = x(i), i = 1, . . . , N. ✁ +PROOF. Using a set of solutions p(x) and P, de- +fine p∂(x) = ∂p(x)P. Then, (3) holds for all x ∈ Rn. +It suffices to confirm that (2) holds at x += +x(i), +i += +1, . . . , N. Since p(x) += +mp(x|Yp), we have +∂p(x(i)) = ∂mp(x(i)|Yp), i = 1, . . . , N. Therefore, (18) +implies (2) for x = x(i), i = 1, . . . , N. +✷ +Note that in Algorithm 1, the number N of training data +and data point set {x(i)}N +i=1 can be chosen arbitrarily. As +N increases, the number of Yp ∈ RnN in (18) increase. +However, the problem is still convex. As explained in the +next subsection, for sufficiently large N, one can show +that p(x) = mp(x|Yp) is a solution to Problem 2.3 other +than {x(i)}N +i=1 when x(i) is distributed evenly in the state +space. Moreover, such N can be found. +When σp = 0 and K0 ≻ 0, (15) helps to simplify Algo- +rithm 1. In fact, we only have to solve a finite family of +LMIs once, stated below without the proof. +Corollary 3.2 Given a class C2 positive definite kernel +k0(x, x′) and {x(i)}N +i=1, suppose that K0 ≻ 0, and the +following finite family of LMIs admits a set of solutions +{¯p(i)}N +i=1, P ∈ Rn×n, and ε > 0: +� +P +∗ +∂f(x(i))P + b¯p(i) P +� +⪰ εI2n, +i = 1, . . . , N. +(19) +Define ˆp(i) = ¯p(i)P −1, i = 1, . . . , N and choose σp = 0. +Then, p(x) := mp(x|Yp) defined by (12) satisfies (2) and +(3) for all x = x(i), i = 1, . . . , N. +✁ +Remark 3.3 It is not guaranteed that the constructed +controller p(x) in Theorem 3.1 preserves an equilib- +rium point x∗ of xk+1 = f(xk). However, it is easy +to impose p(x∗) = 0. An approach is to use shifted +u = p(x) − p(x∗). Another approach is to utilize (5). +Consider new data {x(i), y(i)}M +i=N+1 in (5). Define +Y +:= [y(N+1) · · · y(M)]⊤. Then, the posterior mean +mp(x) given Y and Yp can be computed by the Bayes es- +timation also. As a special case, specifying M = N + 1, +x(N+1) = x∗, y(N+1) = 0, and σ = 0 can result p(x∗) = 0 +for learned p(x). More generally, one can specify the +values of p(x) at arbitrary finite points {x(i)}M +i=N+1. +✁ +Remark 3.4 In the multiple-input case, each compo- +nent pi of p can be designed separately by introducing +mp,i(x|Yp) corresponding to pi. Namely, the results ob- +tained in this paper can readily be generalized to the +multiple-input case. +✁ +At the end of this subsection, we argue the role of ω(i) +p +∼ +N(0, σ2 +pIn) in (10) by interpreting the procedure of Algo- +rithm 1 as follows. We generate suitable Yp based on the +LMIs and then construct p(x) = mp(x|Yp) by functional +fitting. In functional fitting, overfitting is a common is- +sue, since a constructed function becomes unnecessar- +ily complex. The variance σ2 +p specifies how much a con- +structed function needs to fit to data, and thus adding +the noise ω(i) +p +helps to avoid overfitting. However, if σ2 +p +is large, a constructed function can fit to ω(i) +p +instead of +∂p(x(i)). This is well known as the bias-variance tradeoff +in functional fitting [16, Section 5.4.4]. It is worth em- +phasizing that Theorem 3.1 holds for arbitrary σp > 0. +5 + +If K0 in (11) is non-singular, hp in (13) is nothing but +the minimizer of the following optimization problem: +min +hp |Yp − K0hp|2 +K−1 +0 ++ σ2 +p|hp|2. +(20) +The first term evaluates the fitting error at each x(i) +and the second one is for regularization. Especially when +σp = 0, the optimal value becomes zero for the obtained +hp, i.e., a complete fitting ∂p(x(i)) = ˆp(i), i = 1, . . . , N +is achieved as stated by Corollary 3.2. A reproducing +kernel Hilbert space is a formal tool to study a functional +fitting problem under regularization as an optimization +problem. For the prior variance as a kernel, the pair +(p, ∂p) can be understood as a minimizer according to +the representer theorem [31, Theorem 2]. +3.2 +Closed-loop Stability +In Theorem 3.1, it is not clear whether p∂(x) = ∂p(x)P +satisfies (2) other than {x(i)}N +i=1 in contrast to the inte- +grability condition (3). Applying standard arguments of +the polytope approach [5], we improve the LMIs in Al- +gorithm 1 such that its solution satisfies (2) other than +{x(i)}N +i=1 when N is sufficiently large. +For a set of matrices A := {A1, . . . , AL}, let ConvexHull(A) +denote its convex hull, i.e., +ConvexHull(A) +:= +� +A = +L +� +ℓ=1 +θℓAℓ : +L +� +ℓ=1 +θℓ = 1, θℓ ≥ 0, ℓ = 1, . . . , L +� +. +Now, we choose L = N and +Ai := ∂f(x(i)) + b∂p(x(i)), +i = 1, . . . , N. +From the standard discussion of the polytope ap- +proach [5], (2) and (3) hold for all x ∈ Rn belonging to +D := {x ∈ Rn : ∂f(x) + b∂p(x) ∈ ConvexHull(A)}. +This set D can be made larger by increasing the number +N of data used for control design. +Increasing N is not the only approach to enlarge a set +of x in which (2) holds. Another approach is to im- +prove the LMIs in Algorithm 1. First, we decide A(i) := +{A(i) +1 , . . . , A(i) +L(i)} ⊂ Rn×n which is arbitrary as long as +∂f(x(i)) ∈ ConvexHull(A(i)), +i = 1, . . . , N. +(21) +Instead of (17), we consider the following finite family +of LMIs with respect to P ∈ Rn×n and εp > 0: +� +P ≻ 0 +b+(P − A(i) +ℓ(i)P(A(i) +ℓ(i))⊤)(b+)⊤ ⪰ εpIn−1 +(22) +ℓ(i) = 1, . . . , L(i), i = 1, . . . , N. +Using a solution P, we consider the following finite fam- +ily of LMIs with respect to Yp ∈ RnN and ε > 0 instead +of (18): +� +P +∗ +(A(i) +ℓ(i) + b∂mp(x(i)|Yp))P P +� +⪰ εI2n +(23) +ℓ(i) = 1, . . . , L(i), i = 1, . . . , N. +The newly obtained p(x) := mp(x|Yp) satisfies (2) for +all x ∈ Rn belonging to +¯D := +N +� +i=1 +{x ∈ Rn : ∂f(x) + b∂p(x) ∈ ConvexHull(A(i))}. +Now, we are ready to state a control design procedure +such that the IES conditions (2) and (3) hold on a given +bounded set. +Theorem 3.5 Consider a system (1), a class C2 posi- +tive definite kernel k0(x, x′), and σp ≥ 0. Let D ⊂ Rn +denote an n-dimensional closed hypercube. We partition +it evenly to N closed hypercubes, denoted by {D(i)}N +i=1, +where N = rn, r ∈ Z>0. For all r ∈ Z>0, there exist +A(i), i = 1, . . . , N such that +∂f(x) ∈ ConvexHull(A(i)) +(24) +∀x ∈ D(i), ∀i = 1, . . . , N. +Let each x(i), i = 1, . . . , N be the center of D(i). Suppose +that for such A(i), i = 1, . . . , N, +1) for all r > 0, the LMI (22) admits P ∈ Rn×n and +εp > 0; +2) for all r > 0 and P obtained in item 1), the LMI (23) +admits Yp ∈ RnN and ε > 0; +3) for the elements ˆp(i) of Yp, there exists a strictly +decreasing positive function δ of r such that δ(r) → 0 +as r → ∞, and |ˆp(i) − ˆp(j)| < δ(r) if D(i) and D(j) +are next to each other. +Then, there exists a sufficiently large r > 0 such that +p(x) := mp(x|Yp) satisfies (2) and (3) for all x ∈ D. +PROOF. First, we show (24). Since a continuous func- +tion is bounded on a bounded set, {∂f(x) ∈ Rn×n : x ∈ +D(i)} is bounded for each i = 1, . . . , N. Each bounded +set admits its convex hull. Therefore, there exists A(i) +satisfying (24). +Next, we consider the latter statement. We choose +p∂(x) = ∂p(x)P = ∂mp(x|Yp)P. Then, (3) holds on Rn. +6 + +It remains to show that (2) holds on D. Items 1) and 2) +and (24) imply that for each i = 1, . . . , N, +� +P +∗ +∂f(x)P + bp∂(x) P +� +⪰ εI2n, +∀x ∈ D(i). +The strict inequality holds for any positive ¯ε < ε. Since +p∂(x) = ∂mp(x|Yp)P is continuous (with respect to x), +there exists a sufficiently small ¯D(i) +¯ε +⊂ Rn centered at +x(i) such that +� +P +∗ +∂f(x)P + bp∂(x) P +� +≻ ¯εI2n, +∀x ∈ ¯D(i) +¯ε . +From item 3), the continuity of ∂f and p∂, and the +boundedness of D(i), there exists a sufficiently large r ∈ +Z>0 such that D(i) ⊂ ¯D(i) +¯ε +for all i = 1, . . . , N. Conse- +quently, (2) hold on D, a union of D(i), i = 1, . . . , N. +✷ +In the above theorem, D and D(i) are not necessarily +to be hypercubes or closed. Essential requirements are +that D is covered by {D(i)}N +i=1, and each D(i) shrinks +as N increases. In the proof, we show that a switching +controller u = ∂p(x(i))x, x ∈ D(i) also satisfies (2) and +(3). However, a continuous controller u = p(x) is more +easy-to-implement. +Remark 3.6 In Theorem 3.5, IES of the closed-loop sys- +tem is guaranteed on geodesically convex D if either D is +positively invariant or contains an equilibrium point. By +the Schur complement, one can confirm that (2) and (3) +on D implies (4) on D. According to the proof of [37, The- +orem 15], if (4) holds on D, then there exists λ ∈ [0, 1) +such that +� +(xk+1 − x′ +k+1)⊤P(xk+1 − x′ +k+1) +≤ λ +� +(xk − x′ +k)⊤P(xk − x′ +k). +for all k = 0, 1, . . . and (x0, x′ +0) ∈ Rn × Rn as long as +xk −x′ +k ∈ D. This implies that if D is geodesically convex +and positively invariant, the closed-loop system is IES on +D. Next, if D contains an equilibrium point x∗, it follows +that +� +(x1 − x∗)⊤P(x1 − x∗) ≤ λ +� +(x0 − x∗)⊤P(x0 − x∗) +for all x0 ∈ D. From geodesic convexity, this further +implies that D is positively invariant. Thus, the closed- +loop system is IES on D. +✁ +3.3 +Discussions for Generalizations +In this subsection, we discuss how to generalize our re- +sults to the cases where the input vector field b is a func- +tion of x. Also, we mention the continuous-time case. +First, we consider the system with non-constant b: +xk+1 = f(xk) + b(xk)uk, +k ∈ Z≥0, +(25) +where b : Rn → Rn is of class C1. A modification +of Proposition 2.2 implies that a controller u = p(x) +achieves IES if there exist ε > 0, P ∈ Rn×n, and p : +Rn → R of class C1 such that for all x ∈ Rn, +� +P +∗ +(∂f(x) + p(x)∂b(x) + b(x)∂p(x))P P +� +⪰ εI2n. (26) +The difference from (2) is the additional term p(x)∂b(x). +For finding P first, one can utilize a modification of (16): +� +P ≻ 0 +b+(x)(P − ∂f(x)P∂⊤f(x))(b+(x))⊤ ⪰ εpIn−1. +(27) +Substituting its solution P, p(x) = mp(x|Yp), and +∂p(x) = ∂mp(x|Yp) into (26) yields an LMI with re- +spect to Yp and ε at each x ∈ Rn. Therefore, even for +non-constant b, one can still design a controller only +by solving two finite families of LMIs on data points +{x(i)}N +i=1. +In this paper, we focus on discrete-time systems. How- +ever, our method can also be applied to the continuous- +time systems: +˙x = f(x) + b(x)u. +(28) +According to [9, Theorem 1], a controller u = p(x) makes +the closed-loop system IES if there exist ε > 0 and +P(x) ≻ 0, x ∈ Rn such that for all x ∈ Rn, +n +� +k=1 +∂P(x) +∂xk +(fk(x) + bk(x)p(x)) ++ P(x)(∂f(x) + p(x)∂b(x) + b(x)∂p(x)) +(29) ++ (∂f(x) + p(x)∂b(x) + b(x)∂p(x))⊤P(x) ⪯ −εP(x). +Let Pi,j ∼ GP(mi,j(x), ki,j(x, x′)) and P (l) +i,j = Pi,j(x(l))+ +ω(l) +i,j, where Pi,j(x) denotes the (i, j)th element of P(x), +and ω(l) +i,j ∼ N(0, σ(l) +i,j) is i.i.d. Denote P(x|P (l) +i,j ) by the +posterior mean of P(x) given {P (l) +i,j }N +l=1, i, j = 1, . . . , n. +7 + +Then, we first find P(x) = P(x|P (l) +i,j ) satisfying P(x) ≻ 0 +and +b+(x) +� n +� +k=1 +∂P(x) +∂xk +fk(x) + P(x)∂f(x) + ∂⊤f(x)P(x) +� +(b+(x))⊤ ⪯ −εpb+(x)P(x)(b+(x))⊤, +∀x ∈ Rn. +For the obtained P(x), it suffices to solve (29) with re- +spect to p(x) = mp(x|Yp), i.e., Yp. Therefore, in the +continuous-time case, nonlinear control design can be +achieved only by solving two finite families of LMIs at +{x(l)}N +l=1 even for a non-constant metric P(x). As men- +tioned in Section 2.1, the proposed method can further +be applied to various design problems. +4 +Control Design for Unknown Systems +For unknown system dynamics, it is shown by e.g. [12, +13] that its state-space model can be estimated from +the system’s input and output by GPR. Since GPR is a +Bayesian approach, the estimation error is represented +by a posterior covariance. In this section, we show how +to compensate for a stochastic learning error by control +design. To focus on exposing the main idea, we consider +a case where the drift vector field is unknown, and the +state is measurable, but the results can be generalized to +the case where all systems dynamics are unknown and +only the system’s input and output are measurable by +utilizing the results in [12,13]. +4.1 +Learning Drift Vector Fields +In GPR, we learn each component fi, i = 1, . . . , n of f +separately from training data {x(j), y(j) +i }N +j=1, +y(j) +i += fi(x(j)) + ω(j) +yi , +i = 1, . . . , n, j = 1, . . . , N, +where ω(j) +yi ∼ N(0, σ2 +yi) is i.i.d. The number N of training +data and training data points {x(j)}N +j=1 for learning f +are allowed to be different from those used for control +design. Differently from control design, we can directly +obtain training data of fi, and thus it can be estimated +by the standard use of GPR. Moreover, to compensate +for the learning error of fi by control design, we estimate +the error as the posterior covariance. +We choose a prior distribution of fi +as f pri +i +∼ +GP(0, ki(x, x′)), where ki : Rn × Rn → R is a class +C2 positive definite kernel. Then, the Bayes estimation +yields the posterior mean of the joint distribution of fi +and ∂fi as follows: +µi(x) := +N +� +j=1 +ki(x(j), x)h(j) +i +∂µi(x) := +N +� +j=1 +∂ki(x(j), x)h(j) +i +hi := (Ki + σ2 +yiIN)−1Yi ∈ RN, +Yi := +� +y(1) +i +· · · y(N) +i +�⊤ +∈ RN +Ki := + + +ki(x(1), x(1)) · · · ki(x(1), x(N)) +... +... +... +ki(x(N), x(1)) · · · ki(x(N), x(N)) + + ∈ RN×N, +where h(j) +i +denotes the jth component of hi; see, e.g. [32, +Section 2] for the computation of µi(x), and ∂µi(x) can +be computed by taking its partial derivative with respect +to x. +Remark 4.1 When +b +is +also +unknown, +we +learn +gi(x, u) := fi(x) + biu, i = 1, . . . , n from training data +{(x(j), u(j)), y(j) +i +}N +j=1, +y(j) +i += gi(x(j), u(j)) + ω(j) +yi , +i = 1, . . . , n, j = 1, . . . , N. +To utilize the prior knowledge that gi(x, u) is linear with +respect to u, we employ the following kernel ki(x, x′) + +uu′, where ki : Rn × Rn → R is a class C2 positive +definite kernel. Namely, we select a prior distribution of +gi(x, u) as gpri +i +∼ GP(0, ki(x, x′)+uu′). Then, the Bayes +estimation yields the posterior mean of gi as follows: +¯µi(x, u) := +N +� +j=1 +(ki(x(j), x) + u(j)u)¯h(j) +i +¯hi := (Ki + Ku + σ2 +yiIN)−1Yi ∈ RN +Ku := + + +(u(1))2 +· · · u(1)u(N) +... +... +u(N)u(1) · · · (u(N))2 + + ∈ RN×N, +where Ki and Yi are the same as the above. Note that +¯µi(x, u) is linear with respect to u. +✁ +A benefit of GPR for learning a function is the ease of +analytical computation of the posterior covariance func- +tion vi : Rn × Rn → R of fi as in +vi(x, x′) := k(x, x′) − k⊤ +i (x)(Ki + σ2 +yiIN)−1ki(x′) +ki(x) := +� +ki(x(1), x) · · · ki(x(N), x) +�⊤ +. +Similarly, the posterior covariance function v∂,i : Rn → +Rn×n of ∂fi is easy to compute: +v∂,i(x, x′) := ∂2ki(x, x′) +8 + +− k⊤ +∂,i(x)(Ki + σ2 +yiIN)−1k∂,i(x′) +k∂,i(x) := +� +∂⊤ +x′ki(x, x(1)) · · · ∂⊤ +x′ki(x, x(N)) +�⊤ +. +Therefore, the posterior distributions of fi and ∂fi are +respectively obtained by +f post +i +|Yi ∼ GP(µi(x), vi(x, x′)) +∂⊤f post +i +|Yi ∼ GP +� +∂⊤µi(x), v∂,i(x, x′) +� +, +and consequently, +f post +i +(x)|Yi ∼ N(µi(x), σ2 +i (x)), +∀x ∈ Rn +∂⊤f post +i +(x)|Yi ∼ N +� +∂⊤µi(x), σ2 +∂,i(x) +� +, +∀x ∈ Rn +σi(x) := +� +vi(x, x), σ∂,i(x) := +� +v∂,i(x, x). +(30) +An advantage of obtaining the covariance functions σi(x) +and σ∂,i(x) in nonlinear system identification is that one +can compute empirical confidence intervals and decide +if one increases training data in some region of interest +to relearn the model. Repeating this, one can construct +a model with a desired accuracy. +Taking the model learning error into account, a represen- +tation of an estimated closed-loop system with u = p(x) +becomes +xk+1 = µc(xk) + σ(xk)ωk +(31) +µc(x) := µ(x) + bp(x) +σ(x) := diag{σ1(x), . . . , σn(x)}, +where ωk ∼ N(0, In) is i.i.d. The error can be com- +pensated by control design. To see this, we study the +stochastic system (31) from two aspects. First, by ap- +plying a moment IES condition in [22, Corollary 5.4], +we argue how to choose ε > 0 in the LMI (18). Then, +we also discuss how to construct A(i), i = 1, . . . , N for +guaranteeing IES in probability. +4.2 +Moment Incremental Stability +Proposition 2.2 for IES has been generalized to the mo- +ment IES of stochastic systems [22, Corollary 5.4]. This +can be used to decide ε > 0 in (18) for control design. +Definition 4.2 [22, Definition 3.4] The system (31) is +said to be IES in the pth moment if there exist a > 0 and +λ ∈ (0, 1) such that +E[|xk − x′ +k|p] ≤ aλk|x0 − x′ +0|p, ∀k ∈ Z≥0 +for each (x0, x′ +0) ∈ Rn × Rn. +✁ +Applying [22, Corollary 5.4] to the system (31) gives the +following condition for moment IES. +Proposition 4.3 A system (31) is IES in the second +moment if there exist ¯ε > 0 and ¯P : Rn → Rn×n such +that + + + + + + + + + + + +¯P ≻ 0 +¯P − ∂⊤µc(x) ¯P ∂µc(x) +⪰ ¯εIn + +n +� +i=1 +∂⊤σi(x)e⊤ +i ¯Pei∂σi(x) +(32) +for all x ∈ Rn, where each ei, i = 1, . . . , n denotes the +standard basis whose ith element is 1, and the other ele- +ments are all 0. +PROOF. According to [22, Corollary 5.4], the sys- +tem (31) with i.i.d. noise ωk is IES in the second moment +if there exist ε > 0 and ¯P : Rn → Rn×n such that + + + + + + + +¯P ≻ 0 +E +� +(∂µc(x) + ∂σ(x)ωk)⊤ ¯P(∂µc(x) + ∂σ(x)ωk) +� +−λ2 ¯P ⪯ 0 +Since ωk ∼ N(0, In) is i.i.d, the left-hand side of the +second inequality can be rearranged as +E[(∂µc(x) + ∂σ(x)ωk)⊤ ¯P(∂µc(x) + ∂σ(x)ωk)] − λ2 ¯P += ∂µ⊤ +c (x) ¯P ∂µc(x) + +n +� +i=1 +∂⊤σi(x)e⊤ +i ¯Pei∂σi(x) − λ2 ¯P. +These inequalities hold if (32) holds. +✷ +The condition (32) can be used to decide ε > 0 in (2), i.e., +(4) for control design. If (4) holds for a sufficiently large +ε (or ε(x)), then (32) holds for some ¯ε > 0. Therefore, +moment IES suggests how to decide ε > 0. +4.3 +Incremental Stability in Probability +In Section 3.2 for control design, a polytope approach +has been mentioned. There is a freedom to design finite +families of matrices A(j) ⊂ Rn×n, j = 1, . . . , N. This +can be utilized to guarantee IES in probability. +Applying the Chebyshev’s inequality [8, Theorem 1] +yields, given c > 0, +P +� � +∂f post +i +(x) − ∂µi(x) +� +σ−1 +∂,i(x) +� +∂f post +i +(x) − ∂µi(x) +�⊤ < c +� +≥ 1 − n +c +9 + +∀x ∈ Rn, i = 1, . . . , n, +where σ∂,i(x) is the variance of the Jacobian matrix, +computed in (30). Therefore, one can design A(j) such +that ∂f post +i +(x(j)) is contained in ConvexHull(A(j)) in +probability (1 − n/c)n, where note that ∂f post +i +|Yi and +∂f post +j +|Yj, i ̸= j are mutually independent. For the de- +signed A(j), suppose that (23) has a set of solutions. +Then, a controller guaranteeing IES in probability (1 − +n/c)n can be constructed from the solutions. +5 +Example +Consider a negative resistance oscillator [26, Exercise +2.7]. Its forward Euler discretization with the sampling +period ∆t = 0.01 is given by +f(x) = x + +� +x2 +−x1 + h(x1)x2 +� +∆t, b = +� +0 +1 +� +∆t +h(x1) = −x1 + x3 +1 − x5 +1/5 + x7 +1/105. +We learn ∂f2 only, since ∂f1 = [0 1] is determined by +the psychical structure. For the number of training data, +we consider two cases N = 121 and N = 2601. For +both cases, training data points {x(j)}N +j=1 are equally +distributed on [−3, 3] × [−3, 3]. Training data is gen- +erated by y(j) = f2(x(j) +1 , x(j) +2 ) + ωy(j), where ωy(j) ∼ +N(0, 0.012). As a kernel function, we use a Gaussian ker- +nel k = e−|x−x′|2/2. Figures 1 and 2 show the learned ∂µ2 +and the learning error ∂(f2 −µ2), respectively. In Fig. 2, +the error is large around the edges. This is because when +computing derivatives at some point, we need informa- +tion around it, but around the edges, this is not possible. +In other words, the learned ∂µ2 in Fig. 1 is closed to the +true ∂f2 except for the edges. +Next, we design a nonlinear controller based on the +learned ∂µ2 by using Algorithm 1. To avoid the edges, we +consider a smaller region [−2, 2] × [−2, 2]. For the num- +ber of training data, we consider two cases N = 49 and +N = 961. For both cases, training data points {x(j)}N +j=1 +are equally distributed. We select k0 = e−|x−x′|2/2 and +σp = 0. Then, for both cases, solutions to the LMI (17) +are the same: +P = +� +30.3 −25.2 +−25.2 30.0 +� +. +For this P, we solve the LMI (18) and construct p(x) that +is plotted in Fig. 3. We apply the constructed controller +to the true system. Since the origin is an equilibrium +point of x(k + 1) = f(x(k)), we modify the controller +as u = p(x) − p(0) to preserve the equilibrium point. +Fig. 1. (top) Learned ∂µ1/∂x1 (bottom) Learned ∂µ2/∂x2 +(left) N = 121 (right) N = 2601 +Fig. 2. (top) ∂(f2 − µ2)/∂x1 (bottom) ∂(f2 − µ2)/∂x2 +(left) N = 121 (right) N = 2601 +For the different numbers of training data, Fig. 4 shows +the phase portraits of the closed-loop systems, i.e., the +state trajectories starting from different initial states. In +each case, the origin of the true system is stabilized by +a controller designed for a learned model. +An advantage of our approach is that a nonlinear stabi- +lizing controller is designed only by solving LMIs. The +papers [41, 42] propose stabilizing control design meth- +ods for a model learned by GPR. The essence of these +methods are to cancel nonlinear terms by state feed- +back like feedback linearization. We apply this approach. +Namely, we divide control design procedure into two +steps. First, we apply u = −µ2+¯u for cancelling the non- +linear term f2, where µ2 is an estimation of f2. Next, we +design linear feedback ¯u = Kx based on the linear terms, +which can be done by solving an LMI. In fact, we obtain +K = [−49.8 40.6]. For the different numbers of training +data, Fig 5 shows the phase portrait of the closed-loop +system by u = −µ2−49.8x1+40.6x2. In each case, some +10 + +Fig. 3. Constructed p (left) N = 49 (right) N = 961 +Fig. 4. Phase portraits of the closed-loop system (left) +N = 121 for a model and N = 49 for a controller (right) +N = 2601 for a model and N = 961 for a controller +Fig. 5. Phase portraits of the closed-loop system by +u = −µ2 −49.8x1 +40.6x2 (left) N = 121 for a model (right) +N = 2601 for a model +trajectories (e.g. the red colored one) does not converge +to the equilibrium point. This can be caused by the gap +between true f2 and learned µ2 as known that feedback +linearization is weak at model uncertainty. +6 +Conclusion +In this paper, we have established an LMI framework for +contraction-based control design by utilizing the deriva- +tives of GPR to solve the integrability problem, as an +easy-to-implement tool for nonlinear control. Differently +from the standard use, we have used GPR to parametrize +a set of controllers and have found suitable parameters +from the contraction condition. The proposed method +can further deal with the case where system dynam- +ics are unknown by learning them by GPR, and the +learning errors can be compensated by control design by +simple modifications of the proposed LMI framework. +Future work includes applying our method to a high- +dimensional model with a real data set. +References +[1] +V. Andrieu, B. Jayawardhana, and L. Praly. +Transverse +exponential stability and applications. IEEE Transactions +on Automatic Control, 61(11):3396–3411, 2016. +[2] +T. Beckers, D. Kuli´c, and S. Hirche. +Stable Gaussian +process based tracking control of Euler–Lagrange systems. +Automatica, 103:390–397, 2019. +[3] +K. +Berntorp. +Online +Bayesian +inference +and +learning +of +Gaussian-process +state–space +models. +Automatica, +129:109613, 2021. +[4] +C. M. Bishop. Pattern Recognition and Machine Learning. +Springer, New York, 2006. +[5] +S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan. Linear +Matrix Inequalities in System and Control Theory. SIAM, +Philadelphia, 1994. +[6] +M. Buisson-Fenet, V. Morgenthaler, S. Trimpe, and F. Di +Meglio. +Joint state and dynamics estimation with high- +gain observers and Gaussian process models. IEEE Control +Systems Letters, 5(5):1627–1632, 2020. +[7] +F. Bullo. Contraction Theory for Dynamical Systems. Kindle +Directed Publishing, 2022. +[8] +X. Chen. A new generalization of Chebyshev inequality for +random vectors. arXiv:0707.0805, 2007. +[9] +F. +Forni +and +R. +Sepulchre. +A +differential +Lyapunov +framework for contraction anlaysis. IEEE Transactions on +Automatic Control, 59(3):614–628, 2014. +[10] F. Forni and R. Sepulchre. Differentially positive systems. +IEEE Transactions on Automatic Control, 61(2):346–359, +2015. +[11] F. Forni and R. Sepulchre. Differential dissipativity theory +for dominance analysis. IEEE Transactions on Automatic +Control, 64(6):2340–2351, 2018. +[12] R. Frigola, Y. Chen, and C.E. Rasmussen. +Variational +Gaussian process state-space models. +Advances in Neural +Information Processing Systems, 27, 2014. +[13] R. Frigola, F. Lindsten, T.B. Sch¨on, and C.E. Rasmussen. +Identification of Gaussian process state-space models with +particle stochastic approximation EM. +IFAC Proceedings +Volumes, 47(3):4097–4102, 2014. +[14] K. Fujimoto, H. Beppu, and Y. Takaki. +On computation +of numerical solutions to Hamilton-Jacobi inequalities using +Gaussian process regression. +Proc.2018 American Control +Conference, pages 424–429, 2018. +[15] M. Giaccagli, D. Astolfi, V. Andrieu, and L. Marconi. +Sufficient conditions for global integral action via incremental +forwarding +for +input-affine +nonlinear +systems. +IEEE +Transactions on Automatic Control, 67(12):6537 – 6551, +2022. +[16] I. Goodfellow, Y. Bengio, and A. Courville. Deep Learning. +MIT press, Massachusetts, 2016. +[17] Y. Ito, K. Fujimoto, Y. Tadokoro, and T. Yoshimura. +On stabilizing control of Gaussian processes for unknown +nonlinear systems. IFAC-PapersOnLine, 50(1):15385–15390, +2017. +[18] Y. Ito, K. Fujimoto, T. Yoshimura, and Y. Tadokoro. On +Gaussian kernel-based Hamilton-Jacobi-Bellman equations +for nonlinear optimal control. Proc.2018 American Control +Conference, pages 1835–1840, 2018. +[19] Y. Kawano. Controller reduction for nonlinear systems by +generalized differential balancing. +IEEE Transactions on +Automatic Control, 67(11):5856–5871, 2022. +11 + +[20] Y. +Kawano, +B. +Besselink, +and +M. +Cao. +Contraction +analysis of monotone systems via separable functions. IEEE +Transactions on Automatic Control, 65(8):3486–3501, 2020. +[21] Y. Kawano and M. Cao. Contraction analysis of virtually +positive systems. +Systems & Control Letters, 168:105358, +2022. +[22] Y. Kawano and Y. Hosoe. Contraction analysis of discrete- +time stochastic systems. arXiv:2106.05635, 2021. +[23] Y. Kawano, C. K. Kosaraju, and J. M. A. Scherpen. +Krasovskii and +shifted passivity based control. +IEEE +Transactions on Automatic Control, 66(10):4926–4932, 2021. +[24] Y. Kawano and T. Ohtsuka. Nonlinear eigenvalue approach to +differential Riccati equations for contraction analysis. IEEE +Transactions on Automatic Control, 62(10):6497– 6504, 2017. +[25] Y. Kawano and J. M. A. Scherpen. +Model reduction by +differential balancing based on nonlinear Hankel operators. +IEEE Transactions on Automatic Control, 62(7):3293–3308, +2017. +[26] H. K. Khalil. Nonlinear Systems. Prentice-Hall, New Jersey, +1996. +[27] M. Krstic. On using least-squares updates without regressor +filtering in identification and adaptive control of nonlinear +systems. Automatica, 45(3):731–735, 2009. +[28] W. Lohmiller and J.-J. E. Slotine. On contraction analysis +for non-linear systems. Automatica, 34(6):683–696, 1998. +[29] I. R. Manchester and J.-J. E. Slotine. Control contraction +metrics: Convex and intrinsic criteria for nonlinear feedback +design. +IEEE +Transactions +on +Automatic +Control, +62(6):3046–3053, 2017. +[30] M. Padidar, X. Zhu, L. Huang, J. Gardner, and D. Bindel. +Scaling gaussian processes with derivative information using +variational +inference. +Advances in Neural Information +Processing Systems, 34, 2021. +[31] G. Pillonetto, F. Dinuzzo, T. Chen, G. De Nicolao, and +L. Ljung. Kernel methods in system identification, machine +learning and function estimation: A survey. +Automatica, +50(3):657–682, 2014. +[32] C. E. Rasmussen and C. K. I. Williams. Gaussian Processes +for Machine Learning. MIT Press, Massachusetts, 2006. +[33] R. Reyes-B´aez, A. J. van der Schaft, and B. Jayawardhana. +Tracking +control +of +fully-actuated +port-Hamiltonian mechanical systems via sliding manifolds +and contraction analysis. Proc. 20th IFAC World Congress, +pages 8256–8261, 2017. +[34] E. S. Robert, T. Iwasaki, and M. G. Karolos. +A Unified +Algebraic Approach to Linear Control Design. +Routledge, +London, 2017. +[35] D. Sun, M. J. Khojasteh, S. Shekhar, and C. Fan. Uncertain- +aware safe exploratory planning using Gaussian process +and neural control contraction metric. +Proc. 3rd Annual +Conference on Learning for Dynamics and Control, pages +728–741, 2021. +[36] Y. Takaki and K. Fujimoto. On output feedback controller +design for Gaussian process state space models. Proc.56th +IEEE Conference on Decision and Control, pages 3652–3657, +2017. +[37] D. N. Tran, B. S. R¨uffer, and C. M. Kellett. Convergence +properties +for +discrete-time +nonlinear +systems. +IEEE +Transactions on Automatic Control, 63(8):3415–3422, 2018. +[38] H. Tsukamoto and S.-J. Chung. +Convex optimization- +based controller design for stochastic nonlinear systems +using contraction analysis. Proc. 58th IEEE Conference on +Decision and Control, pages 8196–8203, 2019. +[39] H. Tsukamoto, S.-J. Chung, and J.-J. E. Slotine. +Neural +stochastic contraction metrics for learning-based control and +estimation. IEEE Control Systems Letters, 5(5):1825–1830, +2020. +[40] H. Tsukamoto, S.-J. Chung, and J.-J. E. Slotine. Contraction +theory for nonlinear stability analysis and learning-based +control: A tutorial overview. +Annual Reviews in Control, +52:135–169, 2021. +[41] J. Umlauft and S. Hirche. +Feedback linearization based +on Gaussian processes with event-triggered online learning. +IEEE Transactions on Automatic Control, 65(10):4154–4169, +2019. +[42] J. Umlauft, L. P¨ohler, and S. Hirche. +An uncertainty- +based control Lyapunov approach for control-affine systems +modeled by Gaussian process. IEEE Control Systems Letters, +2(3):483–488, 2018. +[43] A. J. van der Schaft. +On differential passivity. +IFAC +Proceedings Volumes, 46(23):21–25, 2013. +12 + diff --git a/x9FAT4oBgHgl3EQfAhwk/content/tmp_files/load_file.txt b/x9FAT4oBgHgl3EQfAhwk/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..efe174209344053bcb35d0a527c7f780a061341d --- /dev/null +++ b/x9FAT4oBgHgl3EQfAhwk/content/tmp_files/load_file.txt @@ -0,0 +1,893 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf,len=892 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='08398v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='SY] 20 Jan 2023 An LMI Framework for Contraction-basedNonlinear Control Design by Derivatives of Gaussian Process Regression ⋆ Yu Kawano a Kenji Kashima b aGraduate School of Advanced Science and Engineering, Hiroshima University, Higashi-Hiroshima 739-8527, Japan (email: ykawano@hiroshima-u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='jp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' bGraduate School of Informatics, Kyoto University, Kyoto 606-8501, Japan (e-mail: kk@i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='kyoto-u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='jp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Abstract Contraction theory formulates the analysis of nonlinear systems in terms of Jacobian matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Although this provides the potential to develop a linear matrix inequality (LMI) framework for nonlinear control design, conditions are imposed not on controllers but on their partial derivatives, which makes control design challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' In this paper, we illustrate this so-called integrability problem can be solved by a non-standard use of Gaussian process regression (GPR) for parameterizing controllers and then establish an LMI framework of contraction-based control design for nonlinear discrete-time systems, as an easy-to- implement tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Later on, we consider the case where the drift vector fields are unknown and employ GPR for functional fitting as its standard use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' GPR describes learning errors in terms of probability, and thus we further discuss how to incorporate stochastic learning errors into the proposed LMI framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Key words: Nonlinear systems, discrete-time systems, stochastic systems, contraction analysis, Gaussian process regression 1 Introduction Contraction theory [9,28] has attracted massive research attention in the systems and control community, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=', [7, 40], which establishes a differential geometric approach to study incremental properties, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=', properties between any pair of trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Revisiting nonlinear control the- ory from contraction perspectives brings new insights not only for stability analysis but also for dissipativity theory [11,23,43], balancing theory [19,25], and mono- tone systems [10, 20, 21] to name a few.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' As an advan- tage in comparison with classical Lyapunov theory, the nature of incremental stability enables us to formulate various design problems in a unified framework, such as stabilizing control [24, 29], tracking control [15, 33], observer design [1,28], and control design for achieving synchronizations [1] and other rich behaviors [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Ow- ing to the differential geometric feature, these problems are described in terms of Jacobian matrices, which is expected as another advantage in practical use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Indeed, restricting the class of controllers (and observer gains) into linear reduces design problems to linear matrix in- equalities (LMIs) [11, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' However, nonlinear control ⋆ This work was supported in part by JSPS KAKENHI Grant Numbers JP21H04875 and JP21K14185.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' design is more involved because of the so-called integra- bility problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Namely, design conditions are imposed not on controllers but on their partial derivatives, which are the main difficulty in developing an LMI framework for contraction-based nonlinear control design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' To overcome this difficulty, in this paper, we employ Gaussian process regression (GPR), a functional fitting tool [4, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' As its non-standard use, we employ GPR to parametrize a controller based on its two important features: 1) computing derivatives of GPR is easy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' 2) GPR becomes linear with respect to parameters while it possesses the flexibility to describe a nonlinear func- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Utilizing these two, we describe a condition for control design in terms of LMIs with respect to pa- rameters of GPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Namely, we establish an LMI frame- work for contraction-based nonlinear stabilizing control design by explicitly addressing the integrability prob- lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' We mainly consider nonlinear discrete-time systems with constant input vector fields and constant metrics for contraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Then, we mention that the formulations are still in the LMI framework for discrete-time systems with non-constant input vector fields and continuous- time systems with non-constant input vector fields and non-constant metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' The proposed method is further applicable to the aforementioned various design prob- Preprint submitted to Automatica 23 January 2023 lems thanks to a unified problem formulation by con- traction theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' In systems and control, GPR is typically used to esti- mate an unknown drift vector field from measured states, and [2,17,35,36,41,42] study control design for learned models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Other applications are a joint estimation of the state and model [3, 6] and solving the Hamilton-Jacobi equations/inequalities [14,18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' In particular, [2,35,41,42] study closed-loop stability under learning errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' How- ever, controllers are designed without taking learning er- rors into account, and learning errors are used for closed- loop analysis only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Motivated by these works, in this pa- per, we also consider the case where a drift vector field is unknown, and this is learned by GPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' In contrast to the conventional approach, we compensate for the learning error by control design, which are benefits of develop- ing the LMI framework for control design and learning models by GPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' The proposed approach can be gener- alized to the case where the whole system dynamics are unknown, and only system’s input and output are mea- surable, since there are learning approaches by GPR in such a setting [12,13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' As relevant researches, [35, 38, 39] give neural network frameworks for contraction-based control design, which requires iterations for finding suitable parameters in con- trast to the proposed LMI framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' The paper [17] formulates control design for models learned by GPR in an LMI framework and designs a switching linear con- troller, but does not use GPR for control design or is not based on contraction theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Therefore, the result in [17] is not applicable for solving the integrability prob- lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' In this paper, we establish an LMI framework for contraction-based control design by utilizing the deriva- tives of GPR to solve the integrability problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' The remainder of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' In Section 2, we pose the problem formulation by mention- ing the integrability problem of control design in contrac- tion theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' In Section 3, we develop an LMI framework for contraction-based control design by utilizing deriva- tives of GPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' In Section 4, we consider the case where drift vector fields are unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' In Section 5, the pro- posed control design method is illustrated by the means of an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Notation: The sets of real numbers, non-negative in- tegers, and positive integers are denoted by R, Z≥0, and Z>0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' The identity matrix with the size n is denoted by In.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' For P, Q ∈ Rn×n, P ≻ Q (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' P ⪰ Q) means that P −Q is symmetric and positive defi- nite (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' semi-definite).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' The Euclidean norm of x ∈ Rn weighted by P ≻ 0 is denoted by |x|P := √ x⊤Px.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' If P = In, this is simply denoted by |x|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' The Moore- Penrose inverse of a matrix A is denoted by A+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' For a function f(x, y), the row vector-valued function consisting of its partial derivatives with respect to x is denoted by ∂xf := ∂f/∂x = [∂f/∂x1 · · · ∂f/∂xn].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' If f depends on x only, this is simply denoted by ∂f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' For a scalar-valued function k(x, x′), its Hessian matrix is denoted by ∂2k(x, x′) := ∂2k(x, x′)/∂x∂x′, which is a matrix-valued function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' The multivariate normal dis- tribution with mean µ and variance Σ is denoted by N(µ, Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' The (standard) expectation is denoted by E[·].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' A stochastic process {ωk}k∈Z≥0 is said to be i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' if ωk, k ∈ Z≥0 are independently distributed, and none of the characteristics of ωk changes with k ∈ Z≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' 2 Preliminaries 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='1 Problem Formulation Consider the following discrete-time nonlinear system: xk+1 = f(xk) + buk, k ∈ Z≥0, (1) where xk ∈ Rn and uk ∈ R denote the state and input, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' f : Rn → Rn is of class C1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=', continu- ously differentiable, and b ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' The ith components of f and b are denoted by fi and bi, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' For the sake of notational simplicity, we consider single-input systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' However, the results can readily be generalized to multiple-input systems as explained below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' We also later discuss the case where b is a function of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' In contraction theory [22, 28, 37], we study incremental stability as a property of the pair of trajectories, stated below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='1 The system xk+1 = f(xk) (with its copy x′ k+1 = f(x′ k)) is said to be incrementally exponentially stable (IES) if there exist a > 0 and λ ∈ (0, 1) such that |xk − x′ k| ≤ aλk|x0 − x′ 0|, ∀k ∈ Z≥0 for each (x0, x′ 0) ∈ Rn × Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' ✁ Applying [37, Theorem 15] to control design yields the following IES condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='2 Suppose that there exist ε > 0, P ∈ Rn×n, and p∂ : Rn → R1×n of continuous such that � P ∗ ∂f(x)P + bp∂(x) P � ⪰ εI2n (2) for all x ∈ Rn, where ∗ represents the appropriate matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' If there exists p : Rn → R of class C1 such that p∂(x) = ∂p(x)P (3) for all x ∈ Rn, then the closed-loop system xk+1 = f(xk) + bp(xk) is IES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' 2 PROOF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' By the Schur complement, the set of (2) and (3) is equivalent to P ≻ 0 and P(∂f(x) + b∂p(x))⊤P −1(∂f(x) + b∂p(x))P − P ⪯ εIn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' (4) This is nothing but the definition of uniform contraction with Θ = P −1/2 [37, Definition 6] for the closed-loop system xk+1 = f(xk) + bp(xk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Therefore, [37, Theorem 15] concludes IES of the closed-loop system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' ✷ In [37, Theorem 15], it has been shown that a closed IES system admits a state-dependent P satisfying a similar inequality as (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Moreover, such a P is uniformly lower and upper bounded by constant matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' It is not yet clear when P becomes constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' If one restricts the class of controllers into linear for a constant P, control de- sign can be reduced to linear matrix inequalities (LMIs);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=', [11, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Namely, control design can be some- times formulated as a practically solvable problem even for nonlinear systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Indeed, (2) is an LMI with re- spect to ε, P and p∂(x) at each x ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' However, as in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='2, designing nonlinear controllers is not fully formulated in the LMI framework, due to the so- called integrability constraint (3), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=', p∂(x)P −1 needs to be the partial derivative of some function p(x) pro- viding a feedback control law u = p(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' The main ob- jective of this paper is to develop an LMI framework for stabilizing nonlinear control design by tackling the inte- grability constraint, stated below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Problem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='3 Given f(x) and b of the system (1), de- velop an LMI framework for designing p : Rn → R which satisfies all the conditions in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='2, if such p exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' ✁ Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='4 We later consider the case where f is un- known by learning it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' As a byproduct of developing control design methodologies in the LMI framework, it is possible to compensate for learning errors by control design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' ✁ An important feature of contraction theory is to study the convergence between any pair of trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' By virtue of this, one can handle observer design [1, 28], tracking control [15,33], and control design for imposing synchronizations [1] and rich behavior such as limit cy- cles [11] in the same framework as stabilizing control de- sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' These references mainly focus on continuous-time systems, but similar results can be delivered to discrete- time systems, since incremental stability conditions in contraction analysis have been derived also for discrete- time systems [28, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Moreover, the proposed method in this paper can be generalized to continuous-time sys- tems as will be explained in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Therefore, solv- ing Problem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='3 can result in LMI frameworks for vari- ous design problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Integrability constraints sometimes appear in nonlinear adaptive control or observer design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Since directly ad- dressing integrability constraints are difficult, there are techniques for avoiding them by adding the dynamic order of the identifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' However, as pointed out by [27], adding additional dynamics can degenerate control per- formances, and it has not been validated that such an approach works for contraction-based control design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Therefore, it is worth solving Problem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='3 directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' In particular, we provide an LMI framework for control de- sign, which is easy-to-implement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' The proposed method may be tailored for nonlinear adaptive control or ob- server design although this is beyond the scope of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='2 Gaussian Process Regression To solve Problem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='3, we employ Gaussian process re- gression (GPR) [4, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' We, in this subsection, briefly summarize basics of GPR and, in the next section, demonstrate that GPR is a suitable tool for handling problems involving partial derivatives, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=', integrability conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Let {x(i)}N i=1, x(i) ∈ Rn be input data, and let {y(i)}N i=1, y(i) ∈ R be the corresponding output data given by y(i) = p(x(i)) + ω(i), i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' , N, (5) where ω(i) ∼ N(0, σ2) is i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' GPR is a technique to learn an unknown function p : Rn → R from input- output data {x(i), y(i)}N i=1 by assuming p as a Gaussian process (GP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' A stochastic process p is said to be GP if any finite set {p(x(i))}N i=1 has a joint Gaussian distribution [32, Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' A GP is completely specified by its mean function mp : Rn → R and covariance function kp : Rn × Rn → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' They are defined by mp(x) = E[p(x)] (6) kp(x, x′) = E[(p(x) − mp(x))(p(x′) − mp(x′))], (7) and we represent the GP by p ∼ GP(mp(x), kp(x, x′)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' The essence of GPR is to estimate mp(x) and kp(x, x′) as the posterior mean and covariance given {x(i), y(i)}N i=1 by the Bayes estimation [4,32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Typically, the prior mean m0(x) is selected as zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' The prior covariance k0(x, x′) needs to be a positive definite kernel [32, Section 6], and we also require smoothness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Standard kernels are linear, polynomial, or squared exponential (SE) [4, 32], which are all smooth and positive definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' For instance, an SE kernel is k0(x, x′) = βe−|x−x′|2 Σ−1/2, (8) 3 where β > 0 and Σ ≻ 0 in addition to σ > 0 in (5) are free parameters, called hyper parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' The hyper parameters can be selected to maximize the marginal likelihood from observed data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=', [32, Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' To use GPR for learning p(x), we need its data as in (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' However, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='2 contains no information of p(x) but of ∂p(x) via the integrability constraint (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' To handle this situation, we employ derivatives of GPR as elaborated in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' 3 Contraction-based Nonlinear Control Design In this section, we establish an LMI framework for contraction-based nonlinear control design by utilizing derivatives of GPR to solve the integrability problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Then, we discuss generalizations of the proposed method to non-constant input vector fields and continuous-time cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='1 LMI Frameworks for Nonlinear Control Design Taking the partial derivatives of (6) and (7), we have the joint distribution of p(x) and ∂p(x) as in � p ∂⊤p � ∼ GP �� mp(x) ∂⊤mp(x) � , � kp(x, x′) ∂x′kp(x, x′) ∂⊤ x kp(x, x′) ∂2kp(x, x′) �� , (9) see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=', [30, Equation (2)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' The goal of this subsection is to find mp(x) based on the Bayes estimation such that p(x) = mp(x) is a solution to Problem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' For the sake of notational simplicity, we select the prior mean of p(x) as zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' A standard procedure of the Bayes estimation is that we first select a class C2 positive definite kernel k0(x, x′) as a prior covariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Then, we compute the posterior mean mp(x) given data of ∂p(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Looking at this from a dif- ferent angle, mp(x) can be viewed as a function of data of ∂p(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Based on this perspective, we consider gener- ating suitable data such that mp(x) becomes a solution to Problem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='3 as a non-standard use of GPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' To this end, let {x(i), ˆp(i)}N i=1 denote a data set to be generated, where ˆp(i) = ∂p(x(i)) + ω(i) p , i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' , N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' (10) The role of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' ω(i) p ∼ N(0, σ2 pIn) is explained later;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' one can simply choose it as zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Define the vector consisting of {ˆp(i)}N i=1 by Yp := � ˆp(1) · · · ˆp(N) � ∈ RnN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Also, we introduce the following notations: XN := � (x(1))⊤ · · · (x(N))⊤ �⊤ ∈ RnN ∂p(XN) := � ∂p(x(1)) · · · ∂p(x(N)) � ∈ RnN k′ ∂,0(x′) := � ∂xk0(x(1), x′) · · · ∂xk0(x(N), x′) � k∂,0(x) := � ∂x′k0(x, x(1)) · · · ∂x′k0(x, x(N)) � K0 := \uf8ee \uf8ef\uf8ef\uf8ef\uf8f0 ∂2k0(x(1), x(1)) · · · ∂2k0(x(1), x(N)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' ∂2k0(x(N), x(1)) · · · ∂2k0(x(N), x(N)) \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fb ∈ RnN×nN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' (11) Then, the joint distribution of the prior distribution of p(x), denoted by ppri(x) and ∂p(XN) is � ppri ∂⊤p(XN) � ∼ N �� 0 Y ⊤ p � , � k0(x, x′) k∂,0(x) (k′ ∂,0)⊤(x′) K0 + σ2 pInN �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' By the Bayes estimation, we can compute the posterior mean mp(x) given Yp as mp(x|Yp) := N � i=1 ∂x′k0(x, x(i))h(i) p , (12) where h(i) p denotes the ith block-component with size n of hp defined by hp := (K0 + σ2 pInN)−1Y ⊤ p ∈ RnN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' (13) The partial derivative of mp(x|Yp) is easy-to-compute: ∂mp(x|Yp) = ∂ ∂x � N � i=1 ∂x′k0(x, x(i))h(i) p � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' (14) Especially if σp = 0 and K0 ≻ 0, it follows from (14) that ∂mp(x(i)|Yp) = ˆp(i), i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' , N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' (15) Note that mp(x|Yp) and ∂mp(x|Yp) are nonlinear functions of x and linear functions of Yp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Therefore, control design reduces to generating suitable Yp, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=', {x(i), ˆp(i)}N i=1, which is proceeded based on Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Now, we are ready to develop an LMI framework for contraction-based nonlinear control design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Substituting p∂ = ∂mp(x|Yp)P into (2) does not give an LMI be- cause of the coupling between Yp and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' This issue can be addressed by using a standard technique of an LMI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' 4 According to [34, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='11], (2), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=', the set of P ≻ 0 and (4) implies, for all x ∈ Rn, � P ≻ 0 b+(P − ∂f(x)P∂⊤f(x))(b+)⊤ ⪰ εpIn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' (16) This is an LMI with respect to P and εp > 0 at each x ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' For a solution P to (16) and p∂ = ∂mp(x|Yp)P, one only has to solve (2) with respect to Yp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' The pro- posed procedure for solving Problem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='3 is summarized in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Algorithm 1 Solving Problem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='3 Require: f(x), b, k0(x, x′), σp ≥ 0, {x(i)}N i=1 Ensure: p(x) and P (if they exist) 1: Define ∂mp(x|Yp) as in (14) 2: Solve a finite family of LMIs with respect to P and εp > 0: � P ≻ 0 b+(P − ∂f(x(i))P∂⊤f(x(i)))(b+)⊤ ⪰ εpIn−1 i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' , N (17) 3: For P obtained in Step 2, solve a finite family of LMIs with respect to Yp and ε > 0: � P ∗ (∂f(x(i)) + b∂mp(x(i)|Yp))P P � ⪰ εI2n, (18) i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' , N 4: Define mp(x|Yp) as in (12) by using Yp obtained in Step 3 5: Return p(x) := mp(x|Yp) and P If Algorithm 1 has a set of solutions, the obtained p(x) is a solution to Problem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='3 at each data point, stated below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='1 Given a system (1), a class C2 positive definite kernel k0(x, x′), σp ≥ 0, and {x(i)}N i=1, suppose that Algorithm 1 has a set of solutions p(x) and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Then, p(x) satisfies (2) and (3) for all x = x(i), i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' , N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' ✁ PROOF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Using a set of solutions p(x) and P, de- fine p∂(x) = ∂p(x)P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Then, (3) holds for all x ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' It suffices to confirm that (2) holds at x = x(i), i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' , N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Since p(x) = mp(x|Yp), we have ∂p(x(i)) = ∂mp(x(i)|Yp), i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' , N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Therefore, (18) implies (2) for x = x(i), i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' , N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' ✷ Note that in Algorithm 1, the number N of training data and data point set {x(i)}N i=1 can be chosen arbitrarily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' As N increases, the number of Yp ∈ RnN in (18) increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' However, the problem is still convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' As explained in the next subsection, for sufficiently large N, one can show that p(x) = mp(x|Yp) is a solution to Problem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='3 other than {x(i)}N i=1 when x(i) is distributed evenly in the state space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Moreover, such N can be found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' When σp = 0 and K0 ≻ 0, (15) helps to simplify Algo- rithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' In fact, we only have to solve a finite family of LMIs once, stated below without the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='2 Given a class C2 positive definite kernel k0(x, x′) and {x(i)}N i=1, suppose that K0 ≻ 0, and the following finite family of LMIs admits a set of solutions {¯p(i)}N i=1, P ∈ Rn×n, and ε > 0: � P ∗ ∂f(x(i))P + b¯p(i) P � ⪰ εI2n, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' , N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' (19) Define ˆp(i) = ¯p(i)P −1, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' , N and choose σp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Then, p(x) := mp(x|Yp) defined by (12) satisfies (2) and (3) for all x = x(i), i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' , N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' ✁ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='3 It is not guaranteed that the constructed controller p(x) in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='1 preserves an equilib- rium point x∗ of xk+1 = f(xk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' However, it is easy to impose p(x∗) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' An approach is to use shifted u = p(x) − p(x∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Another approach is to utilize (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Consider new data {x(i), y(i)}M i=N+1 in (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Define Y := [y(N+1) · · · y(M)]⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Then, the posterior mean mp(x) given Y and Yp can be computed by the Bayes es- timation also.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' As a special case, specifying M = N + 1, x(N+1) = x∗, y(N+1) = 0, and σ = 0 can result p(x∗) = 0 for learned p(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' More generally, one can specify the values of p(x) at arbitrary finite points {x(i)}M i=N+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' ✁ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='4 In the multiple-input case, each compo- nent pi of p can be designed separately by introducing mp,i(x|Yp) corresponding to pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Namely, the results ob- tained in this paper can readily be generalized to the multiple-input case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' ✁ At the end of this subsection, we argue the role of ω(i) p ∼ N(0, σ2 pIn) in (10) by interpreting the procedure of Algo- rithm 1 as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' We generate suitable Yp based on the LMIs and then construct p(x) = mp(x|Yp) by functional fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' In functional fitting, overfitting is a common is- sue, since a constructed function becomes unnecessar- ily complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' The variance σ2 p specifies how much a con- structed function needs to fit to data, and thus adding the noise ω(i) p helps to avoid overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' However, if σ2 p is large, a constructed function can fit to ω(i) p instead of ∂p(x(i)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' This is well known as the bias-variance tradeoff in functional fitting [16, Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' It is worth em- phasizing that Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='1 holds for arbitrary σp > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' 5 If K0 in (11) is non-singular, hp in (13) is nothing but the minimizer of the following optimization problem: min hp |Yp − K0hp|2 K−1 0 + σ2 p|hp|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' (20) The first term evaluates the fitting error at each x(i) and the second one is for regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Especially when σp = 0, the optimal value becomes zero for the obtained hp, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=', a complete fitting ∂p(x(i)) = ˆp(i), i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' , N is achieved as stated by Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' A reproducing kernel Hilbert space is a formal tool to study a functional fitting problem under regularization as an optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' For the prior variance as a kernel, the pair (p, ∂p) can be understood as a minimizer according to the representer theorem [31, Theorem 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='2 Closed-loop Stability In Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='1, it is not clear whether p∂(x) = ∂p(x)P satisfies (2) other than {x(i)}N i=1 in contrast to the inte- grability condition (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Applying standard arguments of the polytope approach [5], we improve the LMIs in Al- gorithm 1 such that its solution satisfies (2) other than {x(i)}N i=1 when N is sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' For a set of matrices A := {A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' , AL}, let ConvexHull(A) denote its convex hull, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=', ConvexHull(A) := � A = L � ℓ=1 θℓAℓ : L � ℓ=1 θℓ = 1, θℓ ≥ 0, ℓ = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' , L � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Now, we choose L = N and Ai := ∂f(x(i)) + b∂p(x(i)), i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' , N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' From the standard discussion of the polytope ap- proach [5], (2) and (3) hold for all x ∈ Rn belonging to D := {x ∈ Rn : ∂f(x) + b∂p(x) ∈ ConvexHull(A)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' This set D can be made larger by increasing the number N of data used for control design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Increasing N is not the only approach to enlarge a set of x in which (2) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Another approach is to im- prove the LMIs in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' First, we decide A(i) := {A(i) 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' , A(i) L(i)} ⊂ Rn×n which is arbitrary as long as ∂f(x(i)) ∈ ConvexHull(A(i)), i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' , N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' (21) Instead of (17), we consider the following finite family of LMIs with respect to P ∈ Rn×n and εp > 0: � P ≻ 0 b+(P − A(i) ℓ(i)P(A(i) ℓ(i))⊤)(b+)⊤ ⪰ εpIn−1 (22) ℓ(i) = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' , L(i), i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' , N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Using a solution P, we consider the following finite fam- ily of LMIs with respect to Yp ∈ RnN and ε > 0 instead of (18): � P ∗ (A(i) ℓ(i) + b∂mp(x(i)|Yp))P P � ⪰ εI2n (23) ℓ(i) = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' , L(i), i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' , N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' The newly obtained p(x) := mp(x|Yp) satisfies (2) for all x ∈ Rn belonging to ¯D := N � i=1 {x ∈ Rn : ∂f(x) + b∂p(x) ∈ ConvexHull(A(i))}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Now, we are ready to state a control design procedure such that the IES conditions (2) and (3) hold on a given bounded set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='5 Consider a system (1), a class C2 posi- tive definite kernel k0(x, x′), and σp ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Let D ⊂ Rn denote an n-dimensional closed hypercube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' We partition it evenly to N closed hypercubes, denoted by {D(i)}N i=1, where N = rn, r ∈ Z>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' For all r ∈ Z>0, there exist A(i), i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' , N such that ∂f(x) ∈ ConvexHull(A(i)) (24) ∀x ∈ D(i), ∀i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' , N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Let each x(i), i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' , N be the center of D(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Suppose that for such A(i), i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' , N, 1) for all r > 0, the LMI (22) admits P ∈ Rn×n and εp > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' 2) for all r > 0 and P obtained in item 1), the LMI (23) admits Yp ∈ RnN and ε > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' 3) for the elements ˆp(i) of Yp, there exists a strictly decreasing positive function δ of r such that δ(r) → 0 as r → ∞, and |ˆp(i) − ˆp(j)| < δ(r) if D(i) and D(j) are next to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Then, there exists a sufficiently large r > 0 such that p(x) := mp(x|Yp) satisfies (2) and (3) for all x ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' PROOF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' First, we show (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Since a continuous func- tion is bounded on a bounded set, {∂f(x) ∈ Rn×n : x ∈ D(i)} is bounded for each i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' , N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Each bounded set admits its convex hull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Therefore, there exists A(i) satisfying (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Next, we consider the latter statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' We choose p∂(x) = ∂p(x)P = ∂mp(x|Yp)P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Then, (3) holds on Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' 6 It remains to show that (2) holds on D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Items 1) and 2) and (24) imply that for each i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' , N, � P ∗ ∂f(x)P + bp∂(x) P � ⪰ εI2n, ∀x ∈ D(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' The strict inequality holds for any positive ¯ε < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Since p∂(x) = ∂mp(x|Yp)P is continuous (with respect to x), there exists a sufficiently small ¯D(i) ¯ε ⊂ Rn centered at x(i) such that � P ∗ ∂f(x)P + bp∂(x) P � ≻ ¯εI2n, ∀x ∈ ¯D(i) ¯ε .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' From item 3), the continuity of ∂f and p∂, and the boundedness of D(i), there exists a sufficiently large r ∈ Z>0 such that D(i) ⊂ ¯D(i) ¯ε for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' , N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Conse- quently, (2) hold on D, a union of D(i), i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' , N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' ✷ In the above theorem, D and D(i) are not necessarily to be hypercubes or closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Essential requirements are that D is covered by {D(i)}N i=1, and each D(i) shrinks as N increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' In the proof, we show that a switching controller u = ∂p(x(i))x, x ∈ D(i) also satisfies (2) and (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' However, a continuous controller u = p(x) is more easy-to-implement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='6 In Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='5, IES of the closed-loop sys- tem is guaranteed on geodesically convex D if either D is positively invariant or contains an equilibrium point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' By the Schur complement, one can confirm that (2) and (3) on D implies (4) on D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' According to the proof of [37, The- orem 15], if (4) holds on D, then there exists λ ∈ [0, 1) such that � (xk+1 − x′ k+1)⊤P(xk+1 − x′ k+1) ≤ λ � (xk − x′ k)⊤P(xk − x′ k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' for all k = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' and (x0, x′ 0) ∈ Rn × Rn as long as xk −x′ k ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' This implies that if D is geodesically convex and positively invariant, the closed-loop system is IES on D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Next, if D contains an equilibrium point x∗, it follows that � (x1 − x∗)⊤P(x1 − x∗) ≤ λ � (x0 − x∗)⊤P(x0 − x∗) for all x0 ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' From geodesic convexity, this further implies that D is positively invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Thus, the closed- loop system is IES on D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' ✁ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='3 Discussions for Generalizations In this subsection, we discuss how to generalize our re- sults to the cases where the input vector field b is a func- tion of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Also, we mention the continuous-time case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' First, we consider the system with non-constant b: xk+1 = f(xk) + b(xk)uk, k ∈ Z≥0, (25) where b : Rn → Rn is of class C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' A modification of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='2 implies that a controller u = p(x) achieves IES if there exist ε > 0, P ∈ Rn×n, and p : Rn → R of class C1 such that for all x ∈ Rn, � P ∗ (∂f(x) + p(x)∂b(x) + b(x)∂p(x))P P � ⪰ εI2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' (26) The difference from (2) is the additional term p(x)∂b(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' For finding P first, one can utilize a modification of (16): � P ≻ 0 b+(x)(P − ∂f(x)P∂⊤f(x))(b+(x))⊤ ⪰ εpIn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' (27) Substituting its solution P, p(x) = mp(x|Yp), and ∂p(x) = ∂mp(x|Yp) into (26) yields an LMI with re- spect to Yp and ε at each x ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Therefore, even for non-constant b, one can still design a controller only by solving two finite families of LMIs on data points {x(i)}N i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' In this paper, we focus on discrete-time systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' How- ever, our method can also be applied to the continuous- time systems: ˙x = f(x) + b(x)u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' (28) According to [9, Theorem 1], a controller u = p(x) makes the closed-loop system IES if there exist ε > 0 and P(x) ≻ 0, x ∈ Rn such that for all x ∈ Rn, n � k=1 ∂P(x) ∂xk (fk(x) + bk(x)p(x)) + P(x)(∂f(x) + p(x)∂b(x) + b(x)∂p(x)) (29) + (∂f(x) + p(x)∂b(x) + b(x)∂p(x))⊤P(x) ⪯ −εP(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Let Pi,j ∼ GP(mi,j(x), ki,j(x, x′)) and P (l) i,j = Pi,j(x(l))+ ω(l) i,j, where Pi,j(x) denotes the (i, j)th element of P(x), and ω(l) i,j ∼ N(0, σ(l) i,j) is i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Denote P(x|P (l) i,j ) by the posterior mean of P(x) given {P (l) i,j }N l=1, i, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' 7 Then, we first find P(x) = P(x|P (l) i,j ) satisfying P(x) ≻ 0 and b+(x) � n � k=1 ∂P(x) ∂xk fk(x) + P(x)∂f(x) + ∂⊤f(x)P(x) � (b+(x))⊤ ⪯ −εpb+(x)P(x)(b+(x))⊤, ∀x ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' For the obtained P(x), it suffices to solve (29) with re- spect to p(x) = mp(x|Yp), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=', Yp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Therefore, in the continuous-time case, nonlinear control design can be achieved only by solving two finite families of LMIs at {x(l)}N l=1 even for a non-constant metric P(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' As men- tioned in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='1, the proposed method can further be applied to various design problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' 4 Control Design for Unknown Systems For unknown system dynamics, it is shown by e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' [12, 13] that its state-space model can be estimated from the system’s input and output by GPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Since GPR is a Bayesian approach, the estimation error is represented by a posterior covariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' In this section, we show how to compensate for a stochastic learning error by control design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' To focus on exposing the main idea, we consider a case where the drift vector field is unknown, and the state is measurable, but the results can be generalized to the case where all systems dynamics are unknown and only the system’s input and output are measurable by utilizing the results in [12,13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='1 Learning Drift Vector Fields In GPR, we learn each component fi, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' , n of f separately from training data {x(j), y(j) i }N j=1, y(j) i = fi(x(j)) + ω(j) yi , i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' , n, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' , N, where ω(j) yi ∼ N(0, σ2 yi) is i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' The number N of training data and training data points {x(j)}N j=1 for learning f are allowed to be different from those used for control design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Differently from control design, we can directly obtain training data of fi, and thus it can be estimated by the standard use of GPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Moreover, to compensate for the learning error of fi by control design, we estimate the error as the posterior covariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' We choose a prior distribution of fi as f pri i ∼ GP(0, ki(x, x′)), where ki : Rn × Rn → R is a class C2 positive definite kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Then, the Bayes estimation yields the posterior mean of the joint distribution of fi and ∂fi as follows: µi(x) := N � j=1 ki(x(j), x)h(j) i ∂µi(x) := N � j=1 ∂ki(x(j), x)h(j) i hi := (Ki + σ2 yiIN)−1Yi ∈ RN, Yi := � y(1) i · · y(N) i �⊤ ∈ RN Ki := \uf8ee \uf8ef\uf8ef\uf8ef\uf8f0 ki(x(1), x(1)) · · · ki(x(1), x(N)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' ki(x(N), x(1)) · · · ki(x(N), x(N)) \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fb ∈ RN×N, where h(j) i denotes the jth component of hi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' [32, Section 2] for the computation of µi(x), and ∂µi(x) can be computed by taking its partial derivative with respect to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='1 When b is also unknown, we learn gi(x, u) := fi(x) + biu, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' , n from training data {(x(j), u(j)), y(j) i }N j=1, y(j) i = gi(x(j), u(j)) + ω(j) yi , i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' , n, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' , N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' To utilize the prior knowledge that gi(x, u) is linear with respect to u, we employ the following kernel ki(x, x′) + uu′, where ki : Rn × Rn → R is a class C2 positive definite kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Namely, we select a prior distribution of gi(x, u) as gpri i ∼ GP(0, ki(x, x′)+uu′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Then, the Bayes estimation yields the posterior mean of gi as follows: ¯µi(x, u) := N � j=1 (ki(x(j), x) + u(j)u)¯h(j) i ¯hi := (Ki + Ku + σ2 yiIN)−1Yi ∈ RN Ku := \uf8ee \uf8ef\uf8ef\uf8ef\uf8f0 (u(1))2 · · u(1)u(N) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' u(N)u(1) · · · (u(N))2 \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fb ∈ RN×N, where Ki and Yi are the same as the above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Note that ¯µi(x, u) is linear with respect to u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' ✁ A benefit of GPR for learning a function is the ease of analytical computation of the posterior covariance func- tion vi : Rn × Rn → R of fi as in vi(x, x′) := k(x, x′) − k⊤ i (x)(Ki + σ2 yiIN)−1ki(x′) ki(x) := � ki(x(1), x) · · · ki(x(N), x) �⊤ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Similarly, the posterior covariance function v∂,i : Rn → Rn×n of ∂fi is easy to compute: v∂,i(x, x′) := ∂2ki(x, x′) 8 − k⊤ ∂,i(x)(Ki + σ2 yiIN)−1k∂,i(x′) k∂,i(x) := � ∂⊤ x′ki(x, x(1)) · · · ∂⊤ x′ki(x, x(N)) �⊤ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Therefore, the posterior distributions of fi and ∂fi are respectively obtained by f post i |Yi ∼ GP(µi(x), vi(x, x′)) ∂⊤f post i |Yi ∼ GP � ∂⊤µi(x), v∂,i(x, x′) � , and consequently, f post i (x)|Yi ∼ N(µi(x), σ2 i (x)), ∀x ∈ Rn ∂⊤f post i (x)|Yi ∼ N � ∂⊤µi(x), σ2 ∂,i(x) � , ∀x ∈ Rn σi(x) := � vi(x, x), σ∂,i(x) := � v∂,i(x, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' (30) An advantage of obtaining the covariance functions σi(x) and σ∂,i(x) in nonlinear system identification is that one can compute empirical confidence intervals and decide if one increases training data in some region of interest to relearn the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Repeating this, one can construct a model with a desired accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Taking the model learning error into account, a represen- tation of an estimated closed-loop system with u = p(x) becomes xk+1 = µc(xk) + σ(xk)ωk (31) µc(x) := µ(x) + bp(x) σ(x) := diag{σ1(x), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' , σn(x)}, where ωk ∼ N(0, In) is i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' The error can be com- pensated by control design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' To see this, we study the stochastic system (31) from two aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' First, by ap- plying a moment IES condition in [22, Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='4], we argue how to choose ε > 0 in the LMI (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Then, we also discuss how to construct A(i), i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' , N for guaranteeing IES in probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='2 Moment Incremental Stability Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='2 for IES has been generalized to the mo- ment IES of stochastic systems [22, Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' This can be used to decide ε > 0 in (18) for control design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='2 [22, Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='4] The system (31) is said to be IES in the pth moment if there exist a > 0 and λ ∈ (0, 1) such that E[|xk − x′ k|p] ≤ aλk|x0 − x′ 0|p, ∀k ∈ Z≥0 for each (x0, x′ 0) ∈ Rn × Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' ✁ Applying [22, Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='4] to the system (31) gives the following condition for moment IES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='3 A system (31) is IES in the second moment if there exist ¯ε > 0 and ¯P : Rn → Rn×n such that \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 ¯P ≻ 0 ¯P − ∂⊤µc(x) ¯P ∂µc(x) ⪰ ¯εIn + n � i=1 ∂⊤σi(x)e⊤ i ¯Pei∂σi(x) (32) for all x ∈ Rn, where each ei, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' , n denotes the standard basis whose ith element is 1, and the other ele- ments are all 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' PROOF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' According to [22, Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='4], the sys- tem (31) with i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' noise ωk is IES in the second moment if there exist ε > 0 and ¯P : Rn → Rn×n such that \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 ¯P ≻ 0 E � (∂µc(x) + ∂σ(x)ωk)⊤ ¯P(∂µc(x) + ∂σ(x)ωk) � −λ2 ¯P ⪯ 0 Since ωk ∼ N(0, In) is i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='d, the left-hand side of the second inequality can be rearranged as E[(∂µc(x) + ∂σ(x)ωk)⊤ ¯P(∂µc(x) + ∂σ(x)ωk)] − λ2 ¯P = ∂µ⊤ c (x) ¯P ∂µc(x) + n � i=1 ∂⊤σi(x)e⊤ i ¯Pei∂σi(x) − λ2 ¯P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' These inequalities hold if (32) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' ✷ The condition (32) can be used to decide ε > 0 in (2), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=', (4) for control design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' If (4) holds for a sufficiently large ε (or ε(x)), then (32) holds for some ¯ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Therefore, moment IES suggests how to decide ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='3 Incremental Stability in Probability In Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='2 for control design, a polytope approach has been mentioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' There is a freedom to design finite families of matrices A(j) ⊂ Rn×n, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' , N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' This can be utilized to guarantee IES in probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Applying the Chebyshev’s inequality [8, Theorem 1] yields, given c > 0, P � � ∂f post i (x) − ∂µi(x) � σ−1 ∂,i(x) � ∂f post i (x) − ∂µi(x) �⊤ < c � ≥ 1 − n c 9 ∀x ∈ Rn, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' , n, where σ∂,i(x) is the variance of the Jacobian matrix, computed in (30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Therefore, one can design A(j) such that ∂f post i (x(j)) is contained in ConvexHull(A(j)) in probability (1 − n/c)n, where note that ∂f post i |Yi and ∂f post j |Yj, i ̸= j are mutually independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' For the de- signed A(j), suppose that (23) has a set of solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Then, a controller guaranteeing IES in probability (1 − n/c)n can be constructed from the solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' 5 Example Consider a negative resistance oscillator [26, Exercise 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Its forward Euler discretization with the sampling period ∆t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='01 is given by f(x) = x + � x2 −x1 + h(x1)x2 � ∆t, b = � 0 1 � ∆t h(x1) = −x1 + x3 1 − x5 1/5 + x7 1/105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' We learn ∂f2 only, since ∂f1 = [0 1] is determined by the psychical structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' For the number of training data, we consider two cases N = 121 and N = 2601.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' For both cases, training data points {x(j)}N j=1 are equally distributed on [−3, 3] × [−3, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Training data is gen- erated by y(j) = f2(x(j) 1 , x(j) 2 ) + ωy(j), where ωy(j) ∼ N(0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' As a kernel function, we use a Gaussian ker- nel k = e−|x−x′|2/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Figures 1 and 2 show the learned ∂µ2 and the learning error ∂(f2 −µ2), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' 2, the error is large around the edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' This is because when computing derivatives at some point, we need informa- tion around it, but around the edges, this is not possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' In other words, the learned ∂µ2 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' 1 is closed to the true ∂f2 except for the edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Next, we design a nonlinear controller based on the learned ∂µ2 by using Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' To avoid the edges, we consider a smaller region [−2, 2] × [−2, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' For the num- ber of training data, we consider two cases N = 49 and N = 961.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' For both cases, training data points {x(j)}N j=1 are equally distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' We select k0 = e−|x−x′|2/2 and σp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Then, for both cases, solutions to the LMI (17) are the same: P = � 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='3 −25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='2 −25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='2 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' For this P, we solve the LMI (18) and construct p(x) that is plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' We apply the constructed controller to the true system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Since the origin is an equilibrium point of x(k + 1) = f(x(k)), we modify the controller as u = p(x) − p(0) to preserve the equilibrium point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' (top) Learned ∂µ1/∂x1 (bottom) Learned ∂µ2/∂x2 (left) N = 121 (right) N = 2601 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' (top) ∂(f2 − µ2)/∂x1 (bottom) ∂(f2 − µ2)/∂x2 (left) N = 121 (right) N = 2601 For the different numbers of training data, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' 4 shows the phase portraits of the closed-loop systems, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=', the state trajectories starting from different initial states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' In each case, the origin of the true system is stabilized by a controller designed for a learned model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' An advantage of our approach is that a nonlinear stabi- lizing controller is designed only by solving LMIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' The papers [41, 42] propose stabilizing control design meth- ods for a model learned by GPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' The essence of these methods are to cancel nonlinear terms by state feed- back like feedback linearization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' We apply this approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Namely, we divide control design procedure into two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' First, we apply u = −µ2+¯u for cancelling the non- linear term f2, where µ2 is an estimation of f2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Next, we design linear feedback ¯u = Kx based on the linear terms, which can be done by solving an LMI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' In fact, we obtain K = [−49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='8 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' For the different numbers of training data, Fig 5 shows the phase portrait of the closed-loop system by u = −µ2−49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='8x1+40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='6x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' In each case, some 10 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Constructed p (left) N = 49 (right) N = 961 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Phase portraits of the closed-loop system (left) N = 121 for a model and N = 49 for a controller (right) N = 2601 for a model and N = 961 for a controller Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Phase portraits of the closed-loop system by u = −µ2 −49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='8x1 +40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='6x2 (left) N = 121 for a model (right) N = 2601 for a model trajectories (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' the red colored one) does not converge to the equilibrium point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' This can be caused by the gap between true f2 and learned µ2 as known that feedback linearization is weak at model uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' 6 Conclusion In this paper, we have established an LMI framework for contraction-based control design by utilizing the deriva- tives of GPR to solve the integrability problem, as an easy-to-implement tool for nonlinear control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Differently from the standard use, we have used GPR to parametrize a set of controllers and have found suitable parameters from the contraction condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' The proposed method can further deal with the case where system dynam- ics are unknown by learning them by GPR, and the learning errors can be compensated by control design by simple modifications of the proposed LMI framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Future work includes applying our method to a high- dimensional model with a real data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' References [1] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Andrieu, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Jayawardhana, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Praly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Transverse exponential stability and applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' IEEE Transactions on Automatic Control, 61(11):3396–3411, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' [2] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Beckers, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Kuli´c, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Hirche.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Stable Gaussian process based tracking control of Euler–Lagrange systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Automatica, 103:390–397, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' [3] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Berntorp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Online Bayesian inference and learning of Gaussian-process state–space models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Automatica, 129:109613, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' [4] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Bishop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Pattern Recognition and Machine Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Springer, New York, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' [5] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Boyd, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' El Ghaoui, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Feron, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Balakrishnan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Linear Matrix Inequalities in System and Control Theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' SIAM, Philadelphia, 1994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' [6] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Buisson-Fenet, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Morgenthaler, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Trimpe, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Di Meglio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Joint state and dynamics estimation with high- gain observers and Gaussian process models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' IEEE Control Systems Letters, 5(5):1627–1632, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' [7] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Bullo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Contraction Theory for Dynamical Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Kindle Directed Publishing, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' [8] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Chen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' A new generalization of Chebyshev inequality for random vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' arXiv:0707.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='0805, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' [9] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Forni and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Sepulchre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' A differential Lyapunov framework for contraction anlaysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' IEEE Transactions on Automatic Control, 59(3):614–628, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' [10] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Forni and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Sepulchre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Differentially positive systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' IEEE Transactions on Automatic Control, 61(2):346–359, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' [11] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Forni and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Sepulchre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Differential dissipativity theory for dominance analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' IEEE Transactions on Automatic Control, 64(6):2340–2351, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' [12] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Frigola, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Chen, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Rasmussen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Variational Gaussian process state-space models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Advances in Neural Information Processing Systems, 27, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' [13] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Frigola, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Lindsten, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Sch¨on, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Rasmussen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Identification of Gaussian process state-space models with particle stochastic approximation EM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' IFAC Proceedings Volumes, 47(3):4097–4102, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' [14] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Fujimoto, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Beppu, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Takaki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' On computation of numerical solutions to Hamilton-Jacobi inequalities using Gaussian process regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='2018 American Control Conference, pages 424–429, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' [15] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Giaccagli, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Astolfi, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Andrieu, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Marconi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Sufficient conditions for global integral action via incremental forwarding for input-affine nonlinear systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' IEEE Transactions on Automatic Control, 67(12):6537 – 6551, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' [16] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Goodfellow, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Bengio, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Courville.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Deep Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' MIT press, Massachusetts, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' [17] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Ito, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Fujimoto, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Tadokoro, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Yoshimura.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' On stabilizing control of Gaussian processes for unknown nonlinear systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' IFAC-PapersOnLine, 50(1):15385–15390, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' [18] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Ito, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Fujimoto, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Yoshimura, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Tadokoro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' On Gaussian kernel-based Hamilton-Jacobi-Bellman equations for nonlinear optimal control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='2018 American Control Conference, pages 1835–1840, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' [19] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Kawano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Controller reduction for nonlinear systems by generalized differential balancing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' IEEE Transactions on Automatic Control, 67(11):5856–5871, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' 11 [20] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Kawano, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Besselink, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Cao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Contraction analysis of monotone systems via separable functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' IEEE Transactions on Automatic Control, 65(8):3486–3501, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' [21] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Kawano and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Cao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Contraction analysis of virtually positive systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Systems & Control Letters, 168:105358, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' [22] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Kawano and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Hosoe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Contraction analysis of discrete- time stochastic systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' arXiv:2106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='05635, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' [23] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Kawano, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Kosaraju, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Scherpen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Krasovskii and shifted passivity based control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' IEEE Transactions on Automatic Control, 66(10):4926–4932, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' [24] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Kawano and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Ohtsuka.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Nonlinear eigenvalue approach to differential Riccati equations for contraction analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' IEEE Transactions on Automatic Control, 62(10):6497– 6504, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' [25] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Kawano and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Scherpen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Model reduction by differential balancing based on nonlinear Hankel operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' IEEE Transactions on Automatic Control, 62(7):3293–3308, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' [26] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Khalil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Nonlinear Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Prentice-Hall, New Jersey, 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' [27] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Krstic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' On using least-squares updates without regressor filtering in identification and adaptive control of nonlinear systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Automatica, 45(3):731–735, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' [28] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Lohmiller and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Slotine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' On contraction analysis for non-linear systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Automatica, 34(6):683–696, 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' [29] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Manchester and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Slotine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Control contraction metrics: Convex and intrinsic criteria for nonlinear feedback design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' IEEE Transactions on Automatic Control, 62(6):3046–3053, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' [30] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Padidar, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Zhu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Huang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Gardner, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Bindel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Scaling gaussian processes with derivative information using variational inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Advances in Neural Information Processing Systems, 34, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' [31] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Pillonetto, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Dinuzzo, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Chen, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' De Nicolao, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Ljung.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Kernel methods in system identification, machine learning and function estimation: A survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Automatica, 50(3):657–682, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' [32] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Rasmussen and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Williams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Gaussian Processes for Machine Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' MIT Press, Massachusetts, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' [33] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Reyes-B´aez, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' van der Schaft, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Jayawardhana.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Tracking control of fully-actuated port-Hamiltonian mechanical systems via sliding manifolds and contraction analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' 20th IFAC World Congress, pages 8256–8261, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' [34] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Robert, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Iwasaki, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Karolos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' A Unified Algebraic Approach to Linear Control Design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Routledge, London, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' [35] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Sun, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Khojasteh, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Shekhar, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Fan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Uncertain- aware safe exploratory planning using Gaussian process and neural control contraction metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' 3rd Annual Conference on Learning for Dynamics and Control, pages 728–741, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' [36] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Takaki and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Fujimoto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' On output feedback controller design for Gaussian process state space models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='56th IEEE Conference on Decision and Control, pages 3652–3657, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' [37] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Tran, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' R¨uffer, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Kellett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Convergence properties for discrete-time nonlinear systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' IEEE Transactions on Automatic Control, 63(8):3415–3422, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' [38] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Tsukamoto and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Chung.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Convex optimization- based controller design for stochastic nonlinear systems using contraction analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' 58th IEEE Conference on Decision and Control, pages 8196–8203, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' [39] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Tsukamoto, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Chung, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Slotine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Neural stochastic contraction metrics for learning-based control and estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' IEEE Control Systems Letters, 5(5):1825–1830, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' [40] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Tsukamoto, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Chung, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Slotine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Contraction theory for nonlinear stability analysis and learning-based control: A tutorial overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Annual Reviews in Control, 52:135–169, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' [41] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Umlauft and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Hirche.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Feedback linearization based on Gaussian processes with event-triggered online learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' IEEE Transactions on Automatic Control, 65(10):4154–4169, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' [42] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Umlauft, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' P¨ohler, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' Hirche.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' An uncertainty- based control Lyapunov approach for control-affine systems modeled by Gaussian process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' IEEE Control Systems Letters, 2(3):483–488, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' [43] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' van der Schaft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' On differential passivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' IFAC Proceedings Volumes, 46(23):21–25, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} +page_content=' 12' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FAT4oBgHgl3EQfAhwk/content/2301.08398v1.pdf'} diff --git a/xdFKT4oBgHgl3EQfLC2B/content/tmp_files/2301.11744v1.pdf.txt b/xdFKT4oBgHgl3EQfLC2B/content/tmp_files/2301.11744v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..488fe2225a7b80415d23362ebd0dbaf2d0356bb3 --- /dev/null +++ b/xdFKT4oBgHgl3EQfLC2B/content/tmp_files/2301.11744v1.pdf.txt @@ -0,0 +1,2310 @@ +A NUMERICAL SCHEME FOR SOLVING AN INDUCTION +HEATING PROBLEM WITH MOVING NON-MAGNETIC +CONDUCTOR +VAN CHIEN LE1, MARI´AN SLODIˇCKA2, AND KAREL VAN BOCKSTAL3 +Abstract. This paper investigates an induction heating problem in a multi-component +system containing a moving non-magnetic conductor. The electromagnetic process is +described by the eddy current model, and the heat transfer process is governed by the +convection-diffusion equation. Both processes are coupled by a restrained Joule heat +source. A temporal discretization scheme is introduced to solve the corresponding +variational system numerically. With the aid of the Reynolds transport theorem, we +prove the convergence of the proposed scheme as well as the well-posedness of the +variational problem. Some numerical experiments are also performed to assess the +performance of the numerical scheme. +Contents +1. +Introduction +2 +2. +Mathematical model +3 +2.1. +Geometrical setting +3 +2.2. +Functional setting +5 +2.3. +Mathematical model +7 +3. +Uniqueness +9 +4. +Time discretization +11 +5. +Existence of a solution +16 +6. +Numerical results +24 +6.1. +Numerical experiments +25 +6.2. +Numerical simulation +26 +7. +Conclusion +28 +References +29 +2020 Mathematics Subject Classification. 35Q61, 35Q79, 65M12. +Key words and phrases. induction heating, multi-component system, moving non-magnetic con- +ductor, Reynolds transport theorem, restrained Joule heat source. +The work of V. C. Le was supported by the European Research Council through the European +Union’s Horizon 2020 Research and Innovation programme (Grant number 101001847). +The work of K. Van Bockstal was supported by the Methusalem programme of Ghent University +Special Research Fund (BOF) (Grant Number 01M01021). +1 +arXiv:2301.11744v1 [math.NA] 27 Jan 2023 + +2 +V. C. LE, M. SLODIˇCKA, AND K. VAN BOCKSTAL +1. Introduction +Induction heating is the process of heating an electrical conductor through the heat +generated by an eddy current. Induction heating is a standard industrial process with +various applications, including surface hardening, induction mass heating, induction +melting and induction welding. Other industrial induction heating applications are +listed in [25]. Basically, an alternating current is passed through an electric coil. The +electromagnetic fields occurring in the surrounding space induce an electric current +in electrically conductive mediums, which is called the eddy current. Then, heat is +generated due to the resistance of materials to the eddy current or the so-called Joule +heating effect. +A considerable amount of literature has been published on the study of the induction +heating process. The majority stand on physical and engineering points of view, where +numerical simulation strategies have been performed, and experiments have been set +up to validate numerical results. As an instance, the modelling of induction heating +of carbon steel tubes was carried out in [10]. The authors considered a mathematical +model combining electromagnetic process, heat transfer by conduction, convection and +radiation, and ferromagnetic-paramagnetic transition. Some numerical simulations of +the heating stage were made using the finite-element method (FEM) and were validated +by measurements. We also refer the reader to [36, 23] for other studies of stationary +induction heat treating. Besides that, FEM-based numerical schemes for moving in- +duction heating problems were also studied in [31, 27, 1, 32]. However, those papers +did not investigate fundamental questions, such as the convergence and stability of +numerical simulations and the properties of the solution. +In contrast to the papers mentioned above, several studies to date have investigated +the well-posedness and the regularity of the solution to the induction heating problem. +However, all of them were restricted to a static geometry. The authors of [33, 34, 35, 6] +studied the global solvability of Maxwell’s equations together with temperature effects. +More specifically, in [33], the quasi-static Maxwell’s equations were expressed in terms +of the magnetic field, and the existence of a solution was proved using a fixed point +argument. The regularity of the solution was then studied in [34]. In [35], the existence +of a solution was shown for Maxwell’s equations with the electric and magnetic fields as +unknowns. The author of [6] considered a degenerate problem modelling Joule heating +in a conductive medium. The existence of global-in-time weak solutions was proved via +the Faedo-Galerkin method. The papers [28, 8, 7] also concerned mathematical models +for a stationary induction heating problem. Herein, the electromagnetic process and +heat transfer are both governed by nonlinear equations. +In [28], the equation was +derived from Maxwell’s equations in terms of the magnetic field, whilst in [8] it was +expressed in terms of the magnetic induction. In both articles, the authors proved the +existence of a weak solution to the coupled system with controlled Joule heating. The +problem was then formulated in terms of the magnetic vector potential and electric +scalar potential fields (A − φ formulation) in [7]. The existence of a global solution +to the whole system was shown, and a numerical simulation was performed to support +obtained theoretical results. + +3 +Recently, some theoretical and numerical studies on moving electromagnetic prob- +lems have been published. In [5, 4, 3], the authors considered an eddy current problem +in a cylindrical symmetric domain containing a moving non-magnetic conductor. The +well-posedness of the variational system was studied, and a numerical scheme was in- +troduced for the computation of the solution. These results were extended to a general +three-dimensional domain (without the symmetry assumption) in [21] and [22]. +In +these papers, a temporal discretization based on the backward Euler method and a +FEM-based space-time discretization scheme were respectively proposed. The corre- +sponding error estimates were also established, and some numerical experiments were +introduced to validate the performance of the proposed schemes. In addition to those +papers, the authors of [20] considered an electromagnetic contact problem with a mov- +ing conductor. The restriction on a non-magnetic moving conductor was no longer +made. Instead, it allowed material coefficients to be fully jumping. In this case, the +well-posedness of the system was proved using Rothe’s method. These pioneering works +of moving electromagnetic problems serve as a basis for the mathematical analysis and +numerical computation of the induction heating process involving moving conductors. +To the best of our knowledge, there has been no paper dealing with the mathemat- +ical analysis of an induction heating problem with a moving conductor, even though +this process has successfully been applied in industry for decades. The present pa- +per investigates an induction heating problem in a multi-component system containing +a moving non-magnetic conductor. The electromagnetic process is described by the +eddy current model, which is coupled with heat transfer via the Joule heating effect. +Due to the conductor’s and surrounding air’s movement, the heat transfer process is +a combination of thermal conduction and convection mechanisms. The nonlinearity of +the Joule heat source is treated by introducing a cut-off function. Our investigation +also relies on the assumption that the moving conductor is filled by a non-magnetic +material. +This paper is organised into seven sections. The following section introduces some +geometrical and functional settings and describes the mathematical model. Section 3 +derives the variational system from the original problem. In Section 4, we design a +temporal discretization scheme based on the backward Euler method and perform some +a priori estimates for iterates. Section 5 is the central section devoted to showing the +existence of a solution to the variational system and the convergence of the proposed +scheme. Finally, we present some numerical results for the discretization scheme in +Section 6, and then we give a conclusion and some possibilities for future work in +Section 7. +2. Mathematical model +2.1. Geometrical setting. We adopt the geometrical setting described in [20], which +was introduced for a moving electromagnetic problem. +Let Ω be an open simply- +connected and bounded domain in R3 such that its boundary ∂Ω belongs to the class +C1,1 or Ω is a convex polyhedron. The domain Ω contains a moving workpiece Σ and a +fixed coil Π that are surrounded by air. The open connected subdomains Σ and Π are +supposed to be of the class C2,1 and separate from each other, see Figure 1. Moreover, + +4 +V. C. LE, M. SLODIˇCKA, AND K. VAN BOCKSTAL +Σ +v +Ω +Π +Ξ +Γin +Γout +Figure 1. The domain Ω consists of a moving workpiece Σ with velocity +v, a fixed coil Π and the surrounding air Ξ. +The coil Π shares common +interfaces Γin and Γout of strictly positive measures with the boundary (see +[21, Figure 1]). +we introduce some notations that are frequently used throughout the manuscript: n +denotes the outward unit normal vector on the boundary; Θ(t) := Σ(t) ∪ Π is the +subdomain consisting of electrically conductive mediums at time t; Ξ(t) := Ω \ Θ(t) is +the space occupied by air at time t; and the interval [0, T] stands for the considered +time frame. The coil Π shares common interfaces with the boundary ∂Ω, denoted by +Γ := Γin ∪ Γout, whose measures are supposed to be strictly positive, i.e. |Γin| > 0 and +|Γout| > 0. +The movement of the workpiece can be parameterized by a smooth bijective mapping +Φ : Σ(0) × [0, T] → R3 such that +(1) +Σ(t) = Φ(Σ(0), t); +� +t∈[0,T] +Σ(t) ⊂ Ω; +det ∇Φ(x, t) > 0, +∀(x, t) ∈ Σ(0) × [0, T]. +We define the trajectory of the motion +T := {(x, t) : x ∈ Σ(t), t ∈ [0, T]} , +and the velocity v : T → R3 of the workpiece +v(x, t) = ˙Φ +� +[Φ(·, t)]−1 (x), t +� +, +where ˙Φ represents the total derivative of Φ with respect to time variable. The reader +is referred to [14, Section 8] for more details. Due to the movement of the workpiece Σ, +the surrounding air also moves in a fluid manner. We assume that the velocity vector +can be extended to the whole domain Ω and this extension (also denoted by v) is of +class C1(Ω × [0, T]) satisfying v = 0 on the coil Π. + +5 +The subdomains Σ, Π and Ξ are filled by different materials, e.g. aluminium work- +piece, copper coil and air. The induction heating process involves the following mate- +rial coefficients: the magnetic permeability µ, the electrical conductivity σ, the thermal +conductivity κ and the volumetric heat capacity α. In the case of non-magnetic con- +ductors, the magnetic permeability of the whole system can be well approximated by +the constant of vacuum µ0 > 0, i.e. +µ = µ0 +in +Ω. +For the sake of simplicity, we assume that all material coefficients are positive constants +on each subdomain except that the electrical conductivity σ vanishes on the air, i.e. +σ(t) = +� +� +� +� +� +σΠ > 0 +in +Π, +σΣ > 0 +in +Σ(t), +0 +in +Ξ(t). +Please note that all material coefficients, except the magnetic permeability, are allowed +to be jumping at the interface of different subdomains. We use the subscripts Σ, Π +and Ξ to distinguish the material functions on the workpiece, the coil and the air, +respectively. +2.2. Functional setting. First of all, we introduce some function spaces and other +main ingredients that are frequently used throughout this article. The Sobolev space +Wk,λ(Ω) with k ∈ N and λ ∈ [1, ∞) is equipped with the following norm +∥f∥Wk,λ(Ω) = +� +� � +0≤|α|≤k +� +Ω +|Dαf(x)|λ dx +� +� +1/λ +. +When k = 0, the space W0,λ(Ω) with λ ∈ [1, ∞) becomes the Lebesgue space Lλ(Ω). +We denote by (·, ·)Ω the scalar product in the space L2(Ω) with its induced norm +∥·∥L2(Ω). +Among Sobolev spaces, only Wk,2(Ω) with k ∈ N forms a Hilbert space, +which are frequently denoted by Hk(Ω). The notation H1 +0(Ω) stands for the closure +of C∞ +0 (Ω) with respect to the norm of H1(Ω), and H1/2(∂Ω) is the space consisting +out of the trace of functions in H1(Ω) to the boundary ∂Ω. The dual space of H1(Ω) +and H1/2(∂Ω) are respectively denoted by H−1(Ω) and H−1/2(∂Ω). These notations are +inherited for vector and tensor fields by using corresponding bold symbols. Moreover, +the subspace Z of H1(Π) defined by +Z := +� +f ∈ H1(Π) : (f, 1)Π = 0 +� +is a Hilbert space with the equivalent norm ∥∇f∥L2(Ω). +In addition, the following +Banach space of vector fields plays a central role in further analysis +W 0 := +� +f ∈ L2(Ω) : ∇ × f ∈ L2(Ω), ∇ · f = 0, f|∂Ω · n = 0 +� +equipped with the norm +∥f∥W 0 = ∥∇ × f∥L2(Ω) . + +6 +V. C. LE, M. SLODIˇCKA, AND K. VAN BOCKSTAL +This norm is equivalent to the graph norm on the space W 0, and W 0 is continuously +embedded into H1(Ω) because the open bounded simply-connected domain Ω is either +a convex polyhedron or its boundary is of the class C1,1, see [13, Lemma 3.3 on p. 51] +and [13, Theorem 3.7 on p. 52]. +Let X be an arbitrary Banach space with norm ∥·∥X and f : (0, T) → X be an +abstract function. We denote by C([0, T], X) and Lip([0, T], X) the spaces of continuous +and Lipschitz continuous functions f endowed with the usual norm +∥f∥C([0,T],X) = max +0≤t≤T ∥f(t)∥X . +The Bochner spaces Lλ((0, T), X) with λ ∈ [1, ∞) and L∞((0, T), X) consist of all +measurable abstract functions f furnished with the norms +∥f∥Lλ((0,T),X) = +� +� +T +� +0 +∥f(t)∥λ +X dt +� +� +1/λ +, +∥f∥L∞((0,T),X) = ess sup +t∈(0,T) +∥f(t)∥X . +In what follows, we denote by ε, Cε and C positive constants depending only on the +given data, where ε is a small number and Cε is a large one depending on ε. Their +different values at different contexts are allowed. For the reason of reducing the number +of constant notations, the notation a ≤ Cb (a ≥ Cb, resp.) is replaced by a ≲ b (a ≳ b, +resp.). +Since the Reynolds transport theorem is crucial for further analysis of PDEs with +moving domain, it is recalled here with some related inequalities, which are useful for +dealing with time-dependent boundary terms. We consider a Lipschitz moving domain +ω(t) whose movement is associated with a velocity vector v being of class C1. Let +f(x, t) be a scalar abstract function satisfying f(t) ∈ W1,1(ω(t)) and ∂tf(t) ∈ L1(ω(t)) +for all t ∈ (0, T). Then, the Reynolds transport theorem (cf. [14, p. 78]) and the +Divergence theorem say that +d +dt +� +ω(t) +f dx = +� +ω(t) +∂tf dx + +� +∂ω(t) +fv · n ds +(2) += +� +ω(t) +∂tf dx + +� +ω(t) +∇ · (fv) dx. +(3) +Next, given f(t) ∈ H1(ω(t)), the Divergence theorem and ε-Young inequality give that +(see also [29, Lemma 2.1]) +� +∂ω(t) +f 2(v · n) ds = 2 +� +ω(t) +f(∇f · v) dx + +� +ω(t) +f 2(∇ · v) dx +≤ ε ∥∇f∥2 +L2(ω(t)) + Cε ∥f∥2 +L2(ω(t)) . +(4) +The constants ε and Cε only depend on the norm of the velocity, and the inequality +(4) is still valid for vector functions in H1(ω(t)). + +7 +2.3. Mathematical model. The mathematical modelling of a low-frequency electro- +magnetic system with moving conductor was thoroughly discussed in [21, 20]. Let us +briefly recall the model considered in these papers. The electromagnetic process is +modelled by the eddy current approximation of Maxwell’s equations or the so-called +quasi-static system +∇ · B = 0, +(5a) +∇ × E = −∂tB, +(5b) +∇ × H = J, +(5c) +where E, H, B and J stand for the electric field, the magnetic field, the magnetic +induction and the current density, respectively. +The behaviour of electromagnetic +fields passing through the interface of different materials is expressed by the following +transmission conditions +(6) +�B · n�∂Θ\Γ = 0, +�H × n�∂Θ\Γ = 0, +and +�(E + v × B) × n�∂Θ\Γ = 0, +where the unit normal vector n points from the electrical conductors (i.e. the workpiece +Σ and the coil Π) to the air, and the jumps are defined by +�f × n� = (f 2 − f 1) × n, +�f · n� = (f 2 − f 1) · n, +where f 1 and f 2 are the limiting values of the field f from the conductors and the +air, respectively. We introduce a vector potential A of the magnetic induction B such +that B = ∇ × A. When B · n = 0 on the boundary ∂Ω, the vector potential A exists +uniquely in H1(Ω) such that A is divergence-free and satisfies A × n = 0 on ∂Ω (cf. +[13, Theorem 3.6 on p. 48]). Substituting B = ∇ × A into the Faraday law (5b) +leads us to the following decomposition of the electric field E = −∂tA − ∇φ, where φ +exists uniquely in H1(Ω)/R. In addition, the general Ohm’s law provides a constitutive +relation for Maxwell’s equations +J = σ(E + v × B). +Hence, the total current density J can be divided into a source current part J s = −σ∇φ +and an eddy current part J e = −σ∂tA + σv × (∇ × A). The source current J s is +originated from an external current j applied on the interfaces Γin and Γout. The scalar +potential φ on the coil Π is the solution to the following boundary value problem [16] +(7) +� +� +� +� +� +∇ · (−σ∇φ) = 0 +in +Π × (0, T), +−σ∇φ · n = 0 +on +(∂Π \ Γ) × (0, T), +−σ∇φ · n = j +on +Γ × (0, T) , +where j satisfies the following compatibility condition +(8) +� +Γ +j(s, t) ds = 0 +∀t ∈ [0, T]. +A comprehensive explanation of the modelling of the source current in the workpiece +and in the air can be found in [20, p. 4]. Finally, thanks to the Amp`ere relation (5c), + +8 +V. C. LE, M. SLODIˇCKA, AND K. VAN BOCKSTAL +the initial-boundary value problem of the vector potential A reads as +(9) +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +σ∂tA + µ−1 +0 ∇ × ∇ × A ++χΠσ∇φ − σv × (∇ × A) = 0 +in +Ω × (0, T), +∇ · A = 0 +in +Ω × (0, T), +A × n = 0 +on +∂Ω × (0, T), +�(∇ × A) × n� = 0 +on +(∂Θ \ Γ) × (0, T), +A(·, 0) = ˜A0 +in +Θ(0), +where χΠ is the characteristic function of the domain Π. By the Joule heating effect, +the electric current flowing through the conductors produces a significant amount of +heat given by +Q = 1 +σ |J|2 = σ |∂tA + χΠ∇φ − v × (∇ × A)|2 . +This Joule heat source plays the role of an internal (contactless) source, which repre- +sents the coupling of the electromagnetic process and heat transfer. It is one of the +most challenging points during the mathematical treatment of the model. In order +to restrain this quadratic source term from increasing uncontrollably, we introduce a +cut-off function Rr that truncates the source heat Q by a constant r > 0 as follows +Rr(Q)(x, t) = min(r, Q(x, t)). +From engineering point of view, this truncation models the use of a switch-off but- +ton, which prevents the conductors from undesirable thermal deformations. Due to +the movement of the workpiece and the surrounding air, the heat transfer process +is governed by thermal conduction and thermal convection, which are described by +the convection-diffusion equation. On the boundary ∂Ω, we impose a homogeneous +Neumann condition. Hence, the temperature u is the solution to the following initial- +boundary value problem (as the material derivative Du +Dt is considered, see e.g. [2]) +(10) +� +� +� +� +� +α∂tu + αv · ∇u − ∇ · (κ∇u) = Rr(Q) +in +Ω × (0, T) , +κ∇u · n = 0 +on +∂Ω × (0, T) , +u(·, 0) = ˜u0 +in +Ω. +The following transmission conditions describe the perfect thermal contact (without +friction) between the conductors and the environment +(11) +�u�∂Θ\Γ = 0, +�κ∇u · n�∂Θ\Γ = 0. +Remark 2.1. In the problem (9), the initial guest A(0) is only given on the conductors +Θ(0) since the electrical conductivity σ vanishes on the air. However, further results +in this paper require A(0) to be extended on the whole domain Ω. To do so, we invoke +the result in [18, Proposition 4.1] to show that, if ˜A0 satisfies +˜A0 ∈ H2(Θ(0)), +∇ · ˜A0 = 0 +in +Θ(0), +˜A0 = 0 +on +Γ, +then there exists an extension ˜A0 ∈ H2(Ω) ∩ H1 +0(Ω) with ∇ · ˜A0 = 0. + +9 +3. Uniqueness +Now, we are in the position to introduce the variational formulation of the problems +(7)-(11). Multiplying the first equations of (7), (9) and (10) by ψ ∈ Z, ϕ ∈ W 0 and +w ∈ H1(Ω), respectively, then applying the Green theorem, we arrive at the following +variational problem: Find φ(t) ∈ Z, A(t) ∈ W 0 and u(t) ∈ H1(Ω) such that +(12) +σΠ (∇φ(t), ∇ψ)Π + (j(t), ψ)Γ = 0, +(13) +(σ(t)∂tA(t), ϕ)Θ(t) + µ−1 +0 (∇ × A(t), ∇ × ϕ)Ω ++ σΠ (∇φ(t), ϕ)Π − σΣ (v(t) × (∇ × A(t)) , ϕ)Σ(t) = 0, +(14) +(α(t)∂tu(t), w)Ω + (α(t)v(t) · ∇u(t), w)Ω ++ (κ(t)∇u(t), ∇w)Ω = (Rr (Q(t)) , w)Θ(t) , +for any ψ ∈ Z, ϕ ∈ W 0 and w ∈ H1(Ω) and for a.a. t ∈ (0, T). Note that an equivalent +saddle-point formulation of the problem (13) was introduced in [21], which gives more +convenience for the computation. In this paper, however, we use the formulation (13) +for simplicity and we note that the results obtained in [21] are still valid. In the next +step, we summarize all assumptions used in the paper and show the uniqueness of a +solution to the variational problem. +(AS1) Ω is an open bounded simply-connected domain in R3 such that either Ω is +a convex polyhedron or its boundary is of class C1,1. +The open connected +subdomains Σ and Π are of the class C2,1 and separate from each other (see +Section 2 for more details); +(AS2) The magnetic permeability is a constant on the whole domain Ω, and all ma- +terial coefficients are positive constants on each subdomain, except that the +electrical conductivity is vanishing on the air (see Section 2 for more details); +(AS3) The velocity vector v satisfies v ∈ C1(Ω × [0, T]) and v = 0 on the coil Π; +(AS4) ˜u0 ∈ H1(Ω) and ˜A0 ∈ W 0 ∩ H2(Ω) satisfies ∇ × ∇ × ˜A0 = 0 on Ξ(0); +(AS5) j ∈ Lip([0, T], H−1/2(Γ)). +Theorem 3.1 (Uniqueness). Let the assumptions (AS1)-(AS5) be satisfied. +Then, +the variational system (12)-(14) admits at most one solution (φ, A, u) satisfying φ ∈ +L2((0, T), Z), A ∈ L2((0, T), W 0) with ∂tA(t) ∈ L2(Θ(t)) for a.a. +t ∈ (0, T) and +u ∈ L2((0, T), H1(Ω)) with ∥√αu∥L2(Ω) ∈ C([0, T]). +Proof. We assume that there exist two solutions (φ1, A1, u1) and (φ2, A2, u2) to the +variational equations (12)-(14). +Then, the solution (φ, A), with φ = φ1 − φ2 and +A = A1 − A2, solves the linear system (12)-(13) with given data j = 0 and ˜A0 = 0. +By means of [20, Theorem 3.1], we get that φ = 0 and A = 0 in the corresponding +spaces. This result implies that u = u1 − u2 also fulfills (14) with ˜u0 = 0 and Q = 0. + +10 +V. C. LE, M. SLODIˇCKA, AND K. VAN BOCKSTAL +Setting w = u(t) in (14) and then integrating in time over (0, ξ) ⊂ (0, T) gives us that +(15) +ξ +� +0 +(α(t)∂tu(t), u(t))Ω dt + +ξ +� +0 +(α(t)v(t) · ∇u(t), u(t))Ω dt ++ +ξ +� +0 +(κ(t)∇u(t), ∇u(t))Ω dt = 0. +We can immediately see that +������ +ξ +� +0 +(α(t)v(t) · ∇u(t), u(t))Ω dt +������ +≤ ε +ξ +� +0 +∥∇u(t)∥2 +L2(Ω) dt + Cε +ξ +� +0 +��� +� +α(t)u(t) +��� +2 +L2(Ω) dt, +and +ξ +� +0 +(κ(t)∇u(t), ∇u(t))Ω dt ≥ min{κΣ, κΠ, κΞ} +ξ +� +0 +∥∇u(t)∥2 +L2(Ω) dt. +The first integral on the left-hand side (LHS) of (15) is split over the subdomains. +Then, the Reynolds transport theorem is invoked to rewrite the term corresponding to +the workpiece Σ(t) as follows +ξ +� +0 +(α(t)∂tu(t), u(t))Σ(t) dt +(3) += αΣ +2 ∥u∥2 +L2(Σ) (ξ) +− αΣ +ξ +� +0 +(∇u(t) · v(t), u(t))Σ(t) dt − αΣ +2 +ξ +� +0 +(u(t)∇ · v(t), u(t))Σ(t) dt. +A similar identity can be obtained for the integral over the air subdomain Ξ(t) (i.e. +replace Σ(t) by Ξ(t)). For the fixed coil Π, we simply have that +ξ +� +0 +(α(t)∂tu(t), u(t))Π dt = αΠ +2 ∥u(ξ)∥2 +L2(Π) . +Hence, the first integral on the LHS of (15) becomes +ξ +� +0 +(α(t)∂tu(t), u(t))Ω dt = 1 +2 +��� +� +α(ξ)u(ξ) +��� +2 +L2(Ω) +− +ξ +� +0 +(α(t)∇u(t) · v(t), u(t))Ω dt − 1 +2 +ξ +� +0 +(α(t)u(t)∇ · v(t), u(t))Ω dt. + +11 +The integrals on the right-hand side (RHS) of this identity can be handled as above. +Therefore, we arrive at +��� +� +α(ξ)u(ξ) +��� +2 +L2(Ω) + (1 − ε) +ξ +� +0 +∥∇u(t)∥2 +L2(Ω) dt ⩽ Cε +ξ +� +0 +��� +� +α(t)u(t) +��� +2 +L2(Ω) dt. +Finally, fixing a sufficiently small ε > 0 and then applying the Gr¨onwall argument +shows that u = 0 in L2((0, T), H1(Ω)), which completes the proof. +□ +4. Time discretization +In this section, we design a time-discrete approximation scheme based on the back- +ward Euler method for solving the variational system. +The time interval [0, T] is +equidistantly partitioned into n subintervals with time step τ = T +n, for any n ∈ Z+. At +time-point ti = iτ, i = 1, 2, . . . , n, we introduce the following notations for any function +f and any time-dependent domain ω +fi = f(ti), +δfi = fi − fi−1 +τ +, +ωi = ω(ti). +Starting from the initial data ˜A0 and ˜u0, we find the solution φi ∈ Z, Ai ∈ W 0 and +ui ∈ H1(Ω), with i = 1, 2, . . . , n, such that the following identities are valid for any +ψ ∈ Z, ϕ ∈ W 0 and w ∈ H1(Ω) +(16) +σΠ (∇φi, ∇ψ)Π + (ji, ψ)Γ = 0, +(17) +(σiδAi, ϕ)Θi + µ−1 +0 (∇ × Ai, ∇ × ϕ)Ω ++ σΠ (∇φi, ϕ)Π − σΣ (vi × (∇ × Ai), ϕ)Σi = 0, +(18) +(αiδui, w)Ω + (αivi · ∇ui, w)Ω + (κi∇ui, ∇w)Ω = (Rr(Qi), w)Θi , +where +Qi = σi |δAi + χΠ∇φi − vi × (∇ × Ai)|2 . +At each iteration step i, the equation (16) is solved first, then followed by (17) and +(18), respectively. In the next lemma, the solvability of the time discretization system +will be concerned. +Lemma 4.1 (Solvability). Let the assumptions (AS1)-(AS5) be fulfilled. Then, φ0 ∈ Z +exists uniquely. Moreover, there exists a positive constant τ0 such that for any i = +1, 2, . . . , n and any τ < τ0, there exists a unique triplet (φi, Ai, ui) ∈ Z ×W 0 × H1(Ω) +solving the system (16)-(18). +Proof. The proof of the solvability of the system (16)-(17) can be adopted from [21, +Lemma 4.1], so we omit this part. Let us define a bilinear form ei : H1(Ω)×H1(Ω) → R +with i = 1, 2, . . . , n, such that +ei(u, w) = 1 +τ (αiu, w)Ω + (αivi · ∇u, w)Ω + (κi∇u, ∇w)Ω . + +12 +V. C. LE, M. SLODIˇCKA, AND K. VAN BOCKSTAL +Then, the variational problem (18) can be rewritten as follows +(19) +ei(ui, w) = 1 +τ (αiui−1, w)Ω + (Rr(Qi), w)Θi . +We can easily get that +ei(u, w) ≲ ∥u∥H1(Ω) ∥w∥H1(Ω) , +which implies the boundedness of the form ei. For any i = 1, 2, . . . , n, the Cauchy- +Schwarz and ε-Young inequalities allow us to show that +ei(u, u) = 1 +τ (αiu, u)Ω + (αivi · ∇u, u)Ω + (κi∇u, ∇u)Ω +≳ +�1 +τ − Cε +� +∥u∥2 +L2(Ω) + (1 − ε) ∥∇u∥2 +L2(Ω) . +We fix a sufficiently small ε > 0, then choose a sufficiently small time step τ < τ0 to +claim that the form ei is H1(Ω)-elliptic. Since ui−1 ∈ H1(Ω) is given and Rr(Qi) is +bounded by the constant r, the RHS of (19) defines a bounded linear functional on +H1(Ω). As a consequence, there exists a unique solution ui ∈ H1(Ω) to the problem (18) +for any i = 1, 2, . . . , n, according to the Lax-Milgram lemma [37, Theorem 18.E]. +□ +Now, some a priori estimates for iterates will be investigated. The following stability +estimate for the solution Ai is directly derived from [21, Lemma 4.3]. +Lemma 4.2 (A priori estimate for Ai). Let the assumptions (AS1)-(AS5) be fulfilled. +Then, there exist positive constants τ0 and C such that for any τ < τ0, there holds that +(20) +max +1≤l≤n ∥δAl∥2 +L2(Θl) + max +1≤l≤n ∥∇ × Al∥2 +L2(Ω) ++ +n +� +i=1 +∥∇ × δAi∥2 +L2(Ω) τ + +n +� +i=1 +∥δAi − δAi−1∥2 +L2(Θi−1) ≤ C. +This a priori estimate was thoroughly proved in [21]. It is noteworthy that the proof +relies on the following property of the solution Ai. +Lemma 4.3 (Higher interior regularity). Let the assumptions (AS1)-(AS5) be fulfilled. +Then, for any i = 1, 2, . . . , n, ∇ × Ai ∈ H1(Ω′) for any subset Ω′ ⊂⊂ Ω (i.e. Ω′ ⊂ Ω). +Moreover, there exists a constant C(Ω′) > 0 such that +∥∇ × Ai∥H1(Ω′) ≤ C +� +∥δAi∥L2(Θi) + ∥∇ × Ai∥L2(Ω) + ∥∇φi∥L2(Π) +� +. +The proof of this higher interior regularity was provided in [21], which unfortunately +had a mistake (it is not justified to consider ∇ × ∇ × Ai as an element in L2(Ω) since +C∞ +0 (Ω) ̸⊂ W 0). In the following, we give a corrected proof for Lemma 4.3 that is +adopted from [19, Lemma 4.1.2]. +Proof. For any i = 1, 2, . . . , n, let us denote +pi := µ0 (σΣvi × (∇ × Ai) − σiδAi − χΠσΠ∇φi) . + +13 +Then, pi ∈ L2(Ω) and the equation (17) implies that +⟨∇ × ∇ × Ai − pi, ϕ⟩W 0 = 0 +∀ϕ ∈ W 0. +Because Ai ∈ W 0, the functional ∇ × ∇ × Ai ∈ H−1(Ω). Hence, according to [13, +Lemma 2.1 on p. 22], there exists a scalar function vi ∈ L2(Ω) such that +∇ × ∇ × Ai = pi + ∇vi. +Now, let Bi := ∇ × Ai. The field Bi ∈ L2(Ω) satisfies ∇ · Bi = 0 and +−∆Bi = ∇ × ∇ × Bi − ∇(∇ · Bi) += ∇ × ∇ × ∇ × Ai += ∇ × pi + ∇ × ∇vi += ∇ × pi. +Since ∇ × pi ∈ H−1(Ω), we follow [9, Lemma 3] to get that Bi ∈ H1(Ω′) or ∇ × Ai ∈ +H1(Ω′) for any subset Ω′ ⊂⊂ Ω. Next, we adopt the technique in [11, Theorem 1 +on p. 309] to acquire the estimate of ∇ × Ai in H1(Ω′). We firstly fix a subdomain +Ω′ ⊂⊂ Ω, and then choose Ω⋆ such that Ω′ ⊂⊂ Ω⋆ ⊂⊂ Ω. Restricting the testing +functions ϕ ∈ {f ∈ C∞ +0 (Ω⋆) : ∇ · f = 0} ⊂ W 0 in the equation (17) leads us to that +(21) +(∇ × ∇ × Ai, ϕ)Ω⋆ = (pi, ϕ)Ω⋆ . +In virtue of the density argument in [13, Theorem 2.8 on p.30], the relation (21) is still +valid for any ϕ ∈ H0(div0, Ω⋆), where +H0(div0, Ω⋆) = +� +f ∈ L2(Ω⋆) : ∇ · f = 0, f|∂Ω⋆ · n = 0 +� +. +Now, let γ ∈ C∞ +0 (Ω⋆) such that γ = 1 in Ω′. Since γ2∇ × Ai ∈ H1 +0(Ω⋆), we invoke +[13, Remark 2.5 on p. 35] to get that ∇ × (γ2∇ × Ai) ∈ H0(div0, Ω⋆). Hence, setting +ϕ = ∇ × (γ2∇ × Ai) in (21) implies that +� +∇ × ∇ × Ai, ∇ × (γ2∇ × Ai) +� +Ω⋆ = +� +pi, ∇ × (γ2∇ × Ai) +� +Ω⋆ . +Using the Cauchy-Schwarz and ε-Young inequalities together with the following identity +∇ × +� +γ2∇ × Ai +� += γ2∇ × ∇ × Ai + 2γ∇γ × (∇ × Ai), +we arrive at +∥γ∇ × ∇ × Ai∥L2(Ω⋆) ≲ ∥pi∥L2(Ω) + ∥∇ × Ai∥L2(Ω) . +Finally, we use the fact +∥∇f∥2 +L2(Ω) = ∥∇ × f∥2 +L2(Ω) + ∥∇ · f∥2 +L2(Ω) +∀f ∈ H1 +0(Ω) + +14 +V. C. LE, M. SLODIˇCKA, AND K. VAN BOCKSTAL +to deduce that +∥∇ × Ai∥H1(Ω′) ≤ ∥γ∇ × Ai∥H1(Ω⋆) +≲ ∥∇ × (γ∇ × Ai)∥L2(Ω⋆) + ∥∇ · (γ∇ × Ai)∥L2(Ω⋆) +≤ ∥γ∇ × ∇ × Ai∥L2(Ω⋆) ++ ∥∇γ × (∇ × Ai)∥L2(Ω⋆) + ∥∇γ · (∇ × Ai)∥L2(Ω⋆) +≲ ∥pi∥L2(Ω) + ∥∇ × Ai∥L2(Ω) +≲ ∥δAi∥L2(Θi) + ∥∇ × Ai∥L2(Ω) + ∥∇φi∥L2(Π) , +which allows us to accomplish the proof. +□ +The next lemma aims to get a priori estimate for the discrete solution ui. +Lemma 4.4 (A priori estimate for ui). Let the assumptions (AS1)-(AS5) be fulfilled. +Then, there exist positive constants C and τ0 such that for any τ < τ0, the following +relation holds true +(22) +max +1≤l≤n ∥ul∥2 +L2(Ω) + +n +� +i=1 +∥∇ui∥2 +L2(Ω) τ + +n +� +i=1 +∥ui − ui−1∥2 +L2(Ω) ≤ C. +Proof. We set w = uiτ in the equation (18) and sum the result up to 1 ≤ l ≤ n to get +that +(23) +l +� +i=1 +(αi(ui − ui−1), ui)Ω + +l +� +i=1 +(αivi · ∇ui, ui)Ω τ ++ +l +� +i=1 +(κi∇ui, ∇ui)Ω τ = +l +� +i=1 +(Rr(Qi), ui)Θi τ. +First, we rearrange the first term on the LHS as follows +(24) +2 +l +� +i=1 +(αi(ui − ui−1), ui)Ω = +l +� +i=1 +� +αi, u2 +i +� +Ω +− +l +� +i=1 +� +αi−1, u2 +i−1 +� +Ω + +l +� +i=1 +� +αi, (ui − ui−1)2� +Ω − +l +� +i=1 +� +αi − αi−1, u2 +i−1 +� +Ω . +The last term on the RHS of (24) can be split over the subdomains in the following +way +l +� +i=1 +� +αi − αi−1, u2 +i−1 +� +Ω = +l +� +i=1 +� +αi, u2 +i−1 +� +Ω − +l +� +i=1 +� +αi−1, u2 +i−1 +� +Ω += +l +� +i=1 +� +αi, u2 +i−1 +� +Σi∪Ξi∪Π − +l +� +i=1 +� +αi−1, u2 +i−1 +� +Σi−1∪Ξi−1∪Π . + +15 +Then, the Reynolds transport theorem can be used to estimate the integrals over the +workpiece as +��� +� +αi, u2 +i−1 +� +Σi − +� +αi−1, u2 +i−1 +� +Σi−1 +��� = αΣ +������� +ti +� +ti−1 +d +dt +� +Σ(t) +u2 +i−1(x) dx dt +������� +(2) += αΣ +������� +ti +� +ti−1 +� +∂Σ(t) +u2 +i−1(v · n)(t) ds dt +������� +(4) +≤ ε ∥∇ui−1∥2 +L2(Ω) τ + Cε ∥ui−1∥2 +L2(Ω) τ. +A similar estimate can be deduced for the integrals over the air subdomains Ξi and +Ξi−1, while the corresponding terms vanish on the coil Π. Hence, we are able to obtain +from (24) that (see also [29, Lemma 2.3]) +l +� +i=1 +(αi(ui − ui−1), ui)Ω ≳ ∥ul∥2 +L2(Ω) + +l +� +i=1 +∥ui − ui−1∥2 +L2(Ω) +− C ∥˜u0∥2 +H1(Ω) − ε +l−1 +� +i=1 +∥∇ui∥2 +L2(Ω) τ − Cε +l−1 +� +i=1 +∥ui∥2 +L2(Ω) τ. +Next, the third term on the LHS of (23) can be bounded by +l +� +i=1 +(κi∇ui, ∇ui)Ω τ ≥ min{κΣ, κΠ, κΞ} +l +� +i=1 +∥∇ui∥2 +L2(Ω) τ. +The Cauchy-Schwarz and ε-Young inequalities can be used to handle the remaining +terms of (23) as follows +����� +l +� +i=1 +(αivi · ∇ui, ui)Ω +����� τ ≤ ε +l +� +i=1 +∥∇ui∥2 +L2(Ω) τ + Cε +l +� +i=1 +∥ui∥2 +L2(Ω) τ, +����� +l +� +i=1 +(Rr(Qi), ui)Θi +����� τ ≲ +l +� +i=1 +∥ui∥2 +L2(Ω) τ + r2 +l +� +i=1 +τ ≲ +l +� +i=1 +∥ui∥2 +L2(Ω) τ + 1. +Collecting all estimates above, we arrive at +∥ul∥2 +L2(Ω) + +l +� +i=1 +∥ui − ui−1∥2 +L2(Ω) + (1 − ε) +l +� +i=1 +∥∇ui∥2 +L2(Ω) τ ≲ 1 + Cε +l +� +i=1 +∥ui∥2 +L2(Ω) τ. +Finally, we fix a sufficiently small ε > 0 and apply the Gr¨onwall argument. Then, we +take the maximum of the two resulting sides to conclude the proof. +□ + +16 +V. C. LE, M. SLODIˇCKA, AND K. VAN BOCKSTAL +5. Existence of a solution +This section is the main part of the paper, concerning the existence of a solution to +the variational system as well as the convergence of the proposed numerical scheme. +Firstly, we introduce some piecewise-constant and piecewise-affine in time functions +and subdomains +jn(t) = ji, +vn(t) = vi, +σn(t) = σi, +κn(t) = κi, +αn(t) = αi, +�Σn(t) = Σi, +�Θn(t) = Θi, +�Ξn(t) = Ξi, +φn(t) = φi, +An(t) = Ai, +An(t) = Ai−1 + (t − ti−1) δAi, +un(t) = ui, +un(t) = ui−1, +un(t) = ui−1 + (t − ti−1)δui, +for all t ∈ (ti−1, ti] with i = 1, 2, . . . , n. The value of the continuous functions An and +un at time t = 0 are given by +An(0) = ˜A0, +un(0) = ˜u0. +In addition, the following piecewise-constant Joule heat source is defined for all t ∈ +(0, T] +Qn(t) = σn(t) +��∂tAn(t) + χΠ∇φn(t) − vn(t) × +� +∇ × An(t) +���2 . +Now, we can rewrite the time-discrete equations (16)-(18) as follows +(25) +σΠ +� +∇φn(t), ∇ψ +� +Π + +� +jn(t), ψ +� +Γ = 0, +(26) +(σn(t)∂tAn(t), ϕ)�Θn(t) + µ−1 +0 +� +∇ × An(t), ∇ × ϕ +� +Ω ++ σΠ +� +∇φn(t), ϕ +� +Π − σΣ +� +vn(t) × +� +∇ × An(t) +� +, ϕ +� +�Σn(t) = 0, +(27) +(αn(t)∂tun(t), w)Ω + (αn(t)vn(t) · ∇un(t), w)Ω ++ (κn(t)∇un(t), ∇w)Ω = +� +Rr +� +Qn(t) +� +, w +� +�Θn(t) , +which are valid for any ψ ∈ Z, ϕ ∈ W 0 and w ∈ H1(Ω), and for any t ∈ (0, T]. The +following lemma shows the convergence of the piecewise-constant approximation of the +given data. + +17 +Lemma 5.1 (Convergence). Let the assumptions (AS1)-(AS5) be satisfied. Then, there +exists a constant C > 0 such that the following relations hold true for any t ∈ (0, T] +(i) +��jn(t) − j(t) +�� +H−1/2(Γ) ≤ Cτ, +∥vn(t) − v(t)∥C(Ω) ≤ Cτ, +(ii) +lim +n→∞ ∥κn(t) − κ(t)∥L2(Ω) = 0, +lim +n→∞ ∥σn(t) − σ(t)∥L2(Ω) = 0, +lim +n→∞ ∥αn(t) − α(t)∥L2(Ω) = 0, +lim +n→∞ +���χ�Σn(t) − χΣ(t) +��� +L2(Ω) = 0. +Proof. (i) +For any t ∈ (0, T], the Lipschitz continuity of the functions j and v gives +us that +��jn(t) − j(t) +�� +H−1/2(Γ) ≲ τ, +∥vn(t) − v(t)∥C(Ω) ≲ τ. +(ii) +Thanks to the property of the mapping Φ, it holds for any t ∈ (0, T] that +lim +n→∞ ∥κn(t) − κ(t)∥2 +L2(Ω) = lim +n→∞ ∥κn(t) − κ(t)∥2 +L2(Σ(t)) + lim +n→∞ ∥κn(t) − κ(t)∥2 +L2(Ξ(t)) += (κΞ − κΣ)2 lim +n→∞ +�����Σn(t) ∪ Σ(t) +��� − +����Σn(t) ∩ Σ(t) +��� +� (AS3) += +0. +The remaining limit transitions can be obtained by the same reasoning, which com- +pletes the proof. +□ +In the next two theorems, we prove the convergence of Rothe’s functions to the +solution of the variational system (12)-(14). +Theorem 5.1 (Existence of φ and A). Let the assumptions (AS1)-(AS5) be fulfilled. +Then, there exists a unique solution (φ, A) to the variational problems (12)-(13), which +satisfies φ ∈ Lip([0, T], Z) and A ∈ C([0, T], W 0) with ∂tA ∈ L2((0, T), W 0). More- +over, A(0) = ˜A0 a.e. in Θ(0) and the following convergences hold true +φn → φ +in +L2((0, T), Z), +(28) +An → A, +An → A +in +L2((0, T), W 0), +(29) +σn∂tAn ⇀ σ∂tA +in +L2((0, T), L2(Ω)), +(30) +√σn∂tAn → √σ∂tA +in +L2((0, T), L2(Ω)). +(31) +Proof. The existence of a solution (φ, A) to the variational system (12)-(13) has already +been shown in [21, Theorems 5.1, 6.1 and 6.2], where φ ∈ Lip([0, T], Z) and A ∈ +L∞((0, T), W 0) with σ∂tA ∈ L2((0, T), L2(Ω)). Moreover, the convergences (28)-(30) +and the satisfaction of the initial condition A(0) = ˜A0 a.e. in Θ(0) have also been +proved. Therefore, we omit their proof. + +18 +V. C. LE, M. SLODIˇCKA, AND K. VAN BOCKSTAL +Next, the uniform boundedness of the sequence {∂tAn} in L2((0, T), W 0) +(cf. Lemma 4.2) and the reflexivity of that space ensure the existence of a subsequence +{∂tAnk} ⊂ {∂tAn} such that +∂tAnk ⇀ f +in +L2((0, T), W 0). +(32) +By means of [17, Lemma 1.3.6], we get that f = ∂tA in L2((0, T), W 0), and hence +A ∈ C([0, T], W 0). Moreover, Eq. (32) is still valid for the whole sequence {∂tAn} due +to the uniqueness of a weak solution A, see Theorem 3.1. +Finally, we show that the convergence (31) also holds true. Because the electrical +conductivity σ vanishes on the air, the limit transition (30) immediately implies that +∂tAn ⇀ ∂tA +in +L2((0, T), L2(Π)), +(33) +χ�Σn∂tAn ⇀ χΣ∂tA +in +L2((0, T), L2(Ω)). +(34) +Hence, we can conclude that +√σn∂tAn ⇀ √σ∂tA +in +L2((0, T), L2(Ω)). +(35) +Now, setting ϕ = ∂tAn(t) ∈ W 0 in (26) and then integrating over the time range +(0, η) ⊂ (0, T) gives that +η +� +0 +��� +� +σn(t) ∂tAn(t) +��� +2 +L2(Ω) dt = +η +� +0 +(σn(t) ∂tAn(t), ∂tAn(t))�Θn(t) dt += −µ−1 +0 +η +� +0 +� +∇ × An(t), ∇ × ∂tAn(t) +� +Ω dt +− σΠ +η +� +0 +� +∇φn(t), ∂tAn(t) +� +Π dt ++ σΣ +η +� +0 +� +vn(t) × +� +∇ × An(t) +� +, ∂tAn(t) +� +�Σn(t) dt. + +19 +By virtue of the limit transitions (28)-(30) and (32)-(34), we are able to pass to the +limit for n → ∞ as follows +lim +n→∞ +η +� +0 +��� +� +σn(t) ∂tAn(t) +��� +2 +L2(Ω) dt = −µ−1 +0 +η +� +0 +(∇ × A(t), ∇ × ∂tA(t))Ω dt +− σΠ +η +� +0 +(∇φ(t), ∂tA(t))Π dt ++ σΣ +η +� +0 +(v(t) × (∇ × A(t)) , ∂tA(t))Σ(t) dt +(13) += +η +� +0 +(σ(t) ∂tA(t), ∂tA(t))Θ(t) dt += +η +� +0 +��� +� +σ(t) ∂tA(t) +��� +2 +L2(Ω) dt. +This relation together with the weak convergence (35) leads us to the strong conver- +gence (31). +□ +Theorem 5.2 (Existence of u). Let the assumptions (AS1)-(AS5) be fulfilled. Then, +there exists a unique function u satisfying u ∈ L2((0, T), H1(Ω))∩L∞((0, T), L2(Ω)) with +∥√αu∥L2(Ω) ∈ C([0, T]) such that the triplet (φ, A, u) solves the variational problem +(14). In addition, u(0) = ˜u0 a.e. in Ω and the following convergences hold true +un ⇀ u, +un ⇀ u +in +L2((0, T), H1(Ω)). +Proof. First of all, we introduce some auxiliary identities that are useful for further +analysis. For any time η ∈ (tl−1, tl] with l = 1, 2, . . . , n, one can easily see that +η +� +0 +(αn(t)∂tun(t), w)Ω dt = +l +� +i=1 +(αi(ui − ui−1), w)Ω − +tl +� +η +(αn(t)∂tun(t), w)Ω dt += +l +� +i=1 +(αiui − αi−1ui−1, w)Ω − +l +� +i=1 +((αi − αi−1)ui−1, w)Ω +− +tl +� +η +(αn(t)∂tun(t), w)Ω dt. + +20 +V. C. LE, M. SLODIˇCKA, AND K. VAN BOCKSTAL +We split the second term on the RHS over the subdomains as follows +l +� +i=1 +((αi − αi−1)ui−1, w)Ω = +l +� +i=1 +(αiui−1, w)Ω − +l +� +i=1 +(αi−1ui−1, w)Ω += +l +� +i=1 +(αiui−1, w)Σi∪Ξi∪Π − +l +� +i=1 +(αi−1ui−1, w)Σi−1∪Ξi−1∪Π . +Then, the Reynolds transport theorem allows us to rewrite that +(αiui−1, w)Σi − (αi−1ui−1, w)Σi−1 = +ti +� +ti−1 +d +dt +� +Σ(t) +α(t)ui−1w dx dt +(3) += +ti +� +ti−1 +� +Σ(t) +α(t)∇ · (ui−1wv(t)) dx dt. +A similar identity can be obtained for the integrals over the air subdomains Ξi and +Ξi−1, while the corresponding terms disappear on the fixed coil Π. Therefore, we arrive +at +(36) +η +� +0 +(αn(t)∂tun(t), w)Ω dt += (αn(η)un(η), w)Ω − (α(0)˜u0, w)Ω − +η +� +0 +(α(t), ∇ · (un(t)wv(t)))Ω dt +− +ηn +� +η +(α(t), ∇ · (un(t)wv(t)))Ω dt − +ηn +� +η +(αn(t)∂tun(t), w)Ω dt. +Next, the uniform boundedness of {un} from Lemma 4.4 together with the reflexivity +of L2((0, T), H1(Ω)) ensures the existence of a subsequence {unk} ⊂ {un} (denoted +further by the same index as the original sequence) such that +un ⇀ u +in +L2((0, T), H1(Ω)). +Moreover, with a similar argument, we have the existence of a function +y ∈ L2((0, T), H1(Ω)) such that +un ⇀ y +in +L2((0, T), H1(Ω)). +Because of the a priori estimate (22) for ui, the following relation between {un} and +{un} holds true +0 ≤ lim +n→∞ ∥un − un∥2 +L2((0,T),L2(Ω)) = lim +n→∞ +n +� +i=1 +∥ui − ui−1∥2 +L2(Ω) τ +(22) +≲ lim +n→∞ τ = 0, + +21 +which implies that y = u in L2((0, T), H1(Ω)). In addition, we have that +u ∈ L∞((0, T), L2(Ω)) thanks to the relation +max +t∈(0,T] ∥un(t)∥L2(Ω) +(22) +≲ 1. +In the following, we prove that the function u is the solution to the variational problem +(14). To this end, we integrate the equation (27) over the time interval (0, η) ⊂ (0, T), +and then integrate the result over (0, ξ) ⊂ (0, T). By means of the identity (36), we +get that +(37) +ξ +� +0 +(αn(η)un(η), w)Ω dη − ξ (α(0)˜u0, w)Ω − +ξ +� +0 +η +� +0 +(α(t), ∇ · (un(t)wv(t)))Ω dt dη +− +ξ +� +0 +ηn +� +η +(α(t), ∇ · (un(t)wv(t)))Ω dt dη − +ξ +� +0 +ηn +� +η +(αn(t)∂tun(t), w)Ω dt dη ++ +ξ +� +0 +η +� +0 +(αn(t)vn(t) · ∇un(t), w)Ω dt dη + +ξ +� +0 +η +� +0 +(κn(t)∇un(t), ∇w)Ω dt dη += +ξ +� +0 +η +� +0 +� +Rr +� +Qn(t) +� +, w +� +�Θn(t) dt dη. +Let us invoke Lemma 4.4 to obtain that +lim +n→∞ +������ +ξ +� +0 +ηn +� +η +(α(t), ∇ · (un(t)wv(t)))Ω dt dη +������ +≲ lim +n→∞ +n +� +i=1 +∥ui−1∥H1(Ω) ∥w∥H1(Ω) τ 2 +(22) +≲ lim +n→∞ τ = 0, +lim +n→∞ +������ +ξ +� +0 +ηn +� +η +(αn(t)∂tun(t), w)Ω dt dη +������ +≲ lim +n→∞ +n +� +i=1 +∥ui − ui−1∥L2(Ω) ∥w∥L2(Ω) τ +(22) +≲ lim +n→∞ +√τ = 0. + +22 +V. C. LE, M. SLODIˇCKA, AND K. VAN BOCKSTAL +Using the convergences in Lemma 5.1 and the Lebesgue dominated convergence theo- +rem, we are able to show that +lim +n→∞ +ξ +� +0 +(αn(η)un(η), w)Ω dη = +ξ +� +0 +(α(η)u(η), w)Ω dη, +lim +n→∞ +ξ +� +0 +η +� +0 +(α(t), ∇ · (un(t)wv(t)))Ω dt dη = +ξ +� +0 +η +� +0 +(α(t), ∇ · (u(t)wv(t)))Ω dt dη, +lim +n→∞ +ξ +� +0 +η +� +0 +(αn(t)vn(t) · ∇un(t), w)Ω dt dη = +ξ +� +0 +η +� +0 +(α(t)v(t) · ∇u(t), w)Ω dt dη, +lim +n→∞ +ξ +� +0 +η +� +0 +(κn(t)∇un(t), ∇w)Ω dt dη = +ξ +� +0 +η +� +0 +(κ(t)∇u(t), ∇w)Ω dt dη. +Now, we address the convergence of the remaining term concerning the discrete Joule +heat source Qn. Let us denote +q = √σ (∂tA + χΠ∇φ − v × (∇ × A)) , +qn = √σn +� +∂tAn + χΠ∇φn − vn × +� +∇ × An +�� +. +One can immediately see from Theorem 5.1 that +qn → q +in +L2((0, T), L2(Ω)). +In addition, we introduce the following inequality which is valid for any non-negative +numbers a, b and c +(38) +|min(a, b) − min(a, c) | ≤ 2√a +��� +√ +b − √c +��� . +One can easily prove this inequality. Using this inequality and the definition of the +cut-off function Rr, we have that +��Rr +� +Qn +� +− Rr(Q) +�� = +��min +� +r, |qn|2� +− min +� +r, |q|2� �� +(38) +≤ 2√r | |qn| − |q| | +≤ 2√r |qn − q| . +Therefore, for each w ∈ H1(Ω), we can deduce that +(39) +0 ≤ lim +n→∞ +������ +ξ +� +0 +η +� +0 +� +Rr +� +Qn(t) +� +− Rr(Q(t)), w +� +Ω dt dη +������ +≲ lim +n→∞ ∥w∥L2(Ω) +� +� +ξ +� +0 +η +� +0 +∥qn(t) − q(t)∥2 +L2(Ω) dt dη +� +� +1/2 += 0. + +23 +Collecting all limit transitions above, we are able to pass to the limit for n → ∞ in +(37) to arrive at +ξ +� +0 +(α(η)u(η), w)Ω dη − ξ (α(0)˜u0, w)Ω − +ξ +� +0 +η +� +0 +(α(t), ∇ · (u(t)wv(t)))Ω dt dη ++ +ξ +� +0 +η +� +0 +(α(t)v(t) · ∇u(t), w)Ω dt dη + +ξ +� +0 +η +� +0 +(κ(t)∇u(t), ∇w)Ω dt dη += +ξ +� +0 +η +� +0 +(Rr(Q(t)), w)Θ(t) dt dη. +Differentiating this identity twice with respect to the time variable gives us that +(40) +d +dt (α(t)u(t), w)Ω − (α(t), ∇ · (u(t)wv(t)))Ω ++ (α(t)v(t) · ∇u(t), w)Ω + (κ(t)∇u(t), ∇w)Ω = (Rr(Q(t)), w)Θ(t) . +Here, we can use the Reynolds transport theorem to get back the variational problem +(14) by rewriting that +d +dt (α(t)u(t), w)Σ(t) +(3) += (α(t)∂tu(t), w)Σ(t) + (α(t), ∇ · (u(t)wv(t)))Σ(t) , +which means that φ, A and u solve the problem (14). The convergence is not only +valid for a subsequence, but also for the original sequence by taking into account the +uniqueness of the solution u from Theorem 3.1. +In the next step, we prove the continuity in time of the L2(Ω)-norm of √αu. Setting +w = u(t) in the relation (40) and then integrating the result over (ξ, η) ⊂ (0, T) leads +us to that +��� +� +α(η)u(η) +��� +2 +L2(Ω) − +��� +� +α(ξ)u(ξ) +��� +2 +L2(Ω) − +η +� +ξ +� +α(t)u2(t), ∇ · v(t) +� +Ω dt +− +η +� +ξ +(α(t)v(t) · ∇u(t), u(t))Ω dt ++ +η +� +ξ +(κ(t)∇u(t), ∇u(t))Ω dt += +η +� +ξ +(Rr(Q(t)), u(t))Θ(t) dt. +We can easily get that +���� +��� +� +α(η)u(η) +��� +2 +L2(Ω) − +��� +� +α(ξ)u(ξ) +��� +2 +L2(Ω) +���� ≲ |η − ξ| + +η +� +ξ +∥u(t)∥2 +H1(Ω) dt, + +24 +V. C. LE, M. SLODIˇCKA, AND K. VAN BOCKSTAL +which implies the absolute continuity in time, i.e. ∥√αu∥ +2 +L2(Ω) ∈ C([0, T]). +Finally, we show that the initial condition of u is satisfied. Multiplying the equation +(14) by γ ∈ C∞([0, T]) satisfying γ(0) = 1 and γ(T) = 0, then integrating over the +time range (0, T) gives us +T +� +0 +γ(t) (α(t)∂tu(t), w)Ω dt + +T +� +0 +γ(t) (α(t)v(t) · ∇u(t), w)Ω dt ++ +T +� +0 +γ(t) (κ(t)∇u(t), ∇w)Ω dt = +T +� +0 +γ(t) (Rr(Q(t)), w)Θ(t) dt. +The first term can be rewritten using partial integration in time, which leads us to that +(α(0)u(0), w)Ω = − +T +� +0 +γ′(t) (α(t)u(t), w)Ω dt − +T +� +0 +γ(t) (α(t), ∇ · (u(t)wv(t)))Ω dt ++ +T +� +0 +γ(t) (α(t)v(t) · ∇u(t), w)Ω dt + +T +� +0 +γ(t) (κ(t)∇u(t), ∇w)Ω dt +− +T +� +0 +γ(t) (Rr(Q(t)), w)Θ(t) dt. +We repeat the process above when considering (27), and then pass to the limit n → ∞ +to have that +(α(0)˜u0, w)Ω = − +T +� +0 +γ′(t) (α(t)u(t), w)Ω dt +− +T +� +0 +γ(t) (α(t), ∇ · (u(t)wv(t)))Ω dt + +T +� +0 +γ(t) (α(t)v(t) · ∇u(t), w)Ω dt ++ +T +� +0 +γ(t) (κ(t)∇u(t), ∇w)Ω dt − +T +� +0 +γ(t) (Rr(Q(t)), w)Θ(t) dt. +Therefore, (α(0)u(0) − α(0)˜u0, w)Ω = 0 for all w ∈ H1(Ω), which implies that u(0) = ˜u0 +a.e. in Ω. We have accomplished the proof. +□ +6. Numerical results +We perform some numerical tests in this section to support our theoretical results. +Time and space discretization schemes for electromagnetic problems involving a mov- +ing non-magnetic conductor have been thoroughly studied in our previous works (cf. + +25 +[21, 22]). +Therefore, we are now focusing on the performance of the discretization +scheme for the heat problem with the moving domain. In Section 6.1, two numerical +experiments describing the heat transfer process in two-dimensional (2D) rotating disks +are investigated. Afterwards, in Section 6.2, we enhance the simulation of an induction +heating process performed in [7] by considering a moving workpiece. +The variational problems are numerically solved using the FEM, and the discretiza- +tion scheme is implemented with the aid of the finite-element software package FreeFEM +[15]. The implementation of the variational problems (16)-(17) follows the saddle-point +formulations proposed in [21] and [22]. On the other hand, the first-order Lagrangian +finite elements are used to spatially approximate the solution ui of the equation (18). +Since the velocity v is known, it is not necessary to change the computational mesh, +which requires a re-meshing procedure that would significantly increase the computa- +tional cost. Instead, the mesh is fixed during the whole time range, and a characteristic +function tracks the moving workpiece +χΣ(x) = +� +1 +if x ∈ Σ, +0 +otherwise. +In order to estimate the order of convergence without knowing the exact solution, +we define the following relative error between Rothe’s solution un obtained by the +proposed numerical method and a reference solution uref +˜Eu = +∥un − uref∥2 +L2((0,T),H1(Ω)) +∥uref∥2 +L2((0,T),H1(Ω)) +. +In all test cases, we assume that the initial temperature is ˜u0 = 298K (≈ 25◦C) and +the constant magnetic permeability of vacuum is µ0 = 4πE-7H/m. The values of other +material coefficients used in numerical tests are presented in Table 1, cf. [26, 24, 30]. +SI unit +Air +Copper +aluminium +Electrical conductiv- +ity +σ +MS/m +- +59.6 +35 +Volumetric heat ca- +pacity +α +kJ/(m3·K) +1.192 +3384 +2422 +Thermal conductivity +κ +W/(m·K) +0.02514 +401 +237 +Table 1. Material coefficients used in the numerical tests. +6.1. Numerical experiments. We perform two numerical experiments concerning +the heat transfer process in 2D rotating disks with radius r1 = 0.2m. The disks both +consist of an aluminium circular area with radius r2 = 0.1m and r2 = 0.05m, respec- +tively, and the complementary area filled by copper, see Figure 2. The domains are +rotating with velocity v = 0.125π (−y, x)Tm/s and are partitioned into 245598 and +251536 triangles, respectively. Instead of the Joule heating Q, the system is supplied + +26 +V. C. LE, M. SLODIˇCKA, AND K. VAN BOCKSTAL +v +v +Figure 2. The circular domain of experiments consisting of an aluminium +circular area (red) and the complementary area filled by copper (blue). The +domains are rotating with velocity v. Left: the first experiment with a con- +centric interior circle. Right: the second experiment with an eccentric interior +circle. +Figure 3. Temperature distribution of the first experiment at different time +points. Left: t = 0.125s. Middle: t = 8s. Right: t = 64s. +with a heat source f = 1MW/m3. The changes over time of the temperature distribu- +tion are visualized by the software package MEDIT [12], which are shown in Figures 3 +and 4. +We verify the convergence of the temporal discretization scheme (18) in the time +interval (0, T) with T = 64s. The reference temperature uref is the solution to (18) +with time step τ = 2−8s, while larger time steps τ = 2−js, with j = 2, 3, . . . , 7, are +used to compute the discrete solution un. Relative errors with respect to time step τ +together with the corresponding regression lines are presented in Figure 5. The slope of +the regression lines show that the potential convergence rate of the numerical scheme +(18) is O(τ). +6.2. Numerical simulation. We enhance the numerical simulation of an induction +heating process performed in [7] by considering a moving workpiece instead of a fixed +one. The domain Ω is a unit cube consisting of a thin-walled cylindrical aluminium + +2.9805E+002 +2.9804E+002 +2.9804E+002 +2.9805E+0023.0060E+002 +3.0106E+002 +3.0037E+002 +3.0083E+002 +3.0129E+0023.1854E+002 +3.2023E+002 +3.1769E+002 +3.1939E+002 +3.2108E+00227 +Figure 4. Temperature distribution of the second experiment at different +time points. Left: t = 0.125s. Middle: t = 8s. Right: t = 64s. +2 +7 +2 +6 +2 +5 +2 +4 +2 +3 +2 +2 +time step +2 +36 +2 +34 +2 +32 +2 +30 +2 +28 +2 +26 +2 +24 +relative error +relative error w.r.t. +-19.791+2.213 log2 +2 +7 +2 +6 +2 +5 +2 +4 +2 +3 +2 +2 +time step +2 +22 +2 +20 +2 +18 +2 +16 +2 +14 +relative error +relative error w.r.t. +-9.724+1.819 log2 +Figure 5. Relative error with respect to time step and the corresponding +regression line. Left: the first experiment. Right: the second experiment. +Both numerical experiments show the potentially optimal convergence rate of +the temporal discretization. +workpiece with two radii r1 = 0.092m, r2 = 0.081m and height h = 0.3m, a copper +coil and surrounding air. The initial guest ˜A(0) = 0 on Θ(0) has a trivial extension +˜A(0) = 0 on the whole domain Ω without requiring C2,1 regularity of the boundary of +the subdomains Σ(0) and Π (see Remark 2.1). Hence, our theoretical results are still +valid for this geometry. The domain Ω is partitioned into 32312 tetrahedra. A static +external current density with magnitude ȷ = 1.0E7 is driven through the coil Π via the +interfaces Γin and Γout. The workpiece is moving along the z-axis with velocity v = +(0, 0, 0.46875)Tcm/s, and the considered time length is T = 32s. Outside the workpiece, +we assume that the velocity is very small; thus, the thermal convection is dominated +by the thermal conduction. Therefore, we can neglect the thermal convection effect +in the air domain and avoid the computation of airflow, which is not essential in this +paper. + +200+30862 +2.9805E+002 +2.9804E+002 +2.9805E+002200+300 +3.0090E+002 +3.0036E+002 +3.0072E+002 +3.0108E+0023.1736E+002 +3.1823E+002 +3.1693E+002 +3.1779E+002 +3.1866E+00228 +V. C. LE, M. SLODIˇCKA, AND K. VAN BOCKSTAL +Figure 6. Different locations of the workpiece and the corresponding temper- +ature distributions on the conductors at different time points. Top: t = 16s. +Bottom: t = 32s. +The reference solution is computed from the variational system (16)-(18) with time +step τ = 2−7s (n = 4096 subintervals). Different locations of the workpiece together +with their corresponding temperature distributions on the conductors at two differ- +ent time points are presented in Figure 6. Some rougher discrete solutions are also +computed when the number of time intervals equals n = 64, 128, 256, 512, 1024 and +2048. Relative error with respect to time step and the corresponding regression line +are shown in Figure 7, which confirms the potentially optimal convergence rate of our +proposed scheme. +7. Conclusion +In the present paper, we have investigated an induction heating problem in a three- +dimensional domain containing a moving non-magnetic conductor. The electromag- +netic process and heat transfer are modelled by PDEs, which are coupled by the Joule +heating effect. A cut-off function has been introduced to restrain the nonlinear Joule +heat source. A time-discrete scheme based on the backward Euler’s method has been +proposed to solve approximately the variational problems. The convergence of the pro- +posed discretisation scheme and the well-posedness of the variational system have been +proved with the aid of the Reynolds transport theorem and Rothe’s method. Some +numerical results have also been presented to support the theoretical results. + +2.9998E+002 +3.0394E+002 +2.9800E+002 +3.0196E+002 +3.0592E+0022.9985E+002 +3.0356E+002 +2.9800E+002 +3.0171E+002 +3.0541E+0023.0200E+002 +3.0987E+002 +2.9807E+002 +3.0593E+002 +3.1380E+0023.0186E+002 +3.0944E+002 +2.9807E+002 +3.0565E+002 +3.1323E+00229 +2 +6 +2 +5 +2 +4 +2 +3 +2 +2 +2 +1 +time step +2 +22 +2 +20 +2 +18 +2 +16 +2 +14 +relative error +relative error w.r.t. +-12.37+1.611 log2 +Figure 7. Relative error with respect to time step and the corresponding +regression line of the numerical simulation. This result confirms the potentially +optimal convergence rate of the numerical scheme. +In the future, comprehensive error estimates of the proposed temporal discretiza- +tion scheme should be performed to verify the potentially optimal convergence rate +obtained numerically in Section 6. +In addition, future studies could concern high- +frequency induction heating problems (full Maxwell’s equations) involving magnetic +moving conductors, which have a wide range of applications in manufacturing indus- +tries. +References +[1] X. Bai, H. Zhang, and G. Wang. Modeling of the moving induction heating used as secondary +heat source in weld-based additive manufacturing. The International Journal of Advanced Man- +ufacturing Technology, 77(1-4):717–727, 2014. +[2] A. Bejan. Convection Heat Transfer. Wiley, fourth edition, 2013. +[3] A. Berm´udez, B. L´opez-Rodr´ıguez, R. Rodr´ıguez, and P. Salgado. Numerical solution of a tran- +sient three-dimensional eddy current model with moving conductors. International Journal of +Numerical Analysis and Modeling, 16(5):695–717, 2019. +[4] A. Berm´udez, R. Mu˜noz-Sola, C. Reales, R. Rodr´ıguez, and P. Salgado. A Transient Eddy Current +Problem on a Moving Domain. Mathematical Analysis. SIAM Journal on Mathematical Analysis, +45(6):3629–3650, 2013. +[5] A. Berm´udez, C. Reales, R. Rodr´ıguez, and P. Salgado. Numerical analysis of a transient eddy +current axisymmetric problem involving velocity terms. Numerical Methods for Partial Differen- +tial Equations, 28(3):984–1012, 2011. +[6] M. Bie´n. Existence of global weak solutions for a class of quasilinear equations describing Joule’s +heating. Mathematical Methods in the Applied Sciences, 22(15):1275–1291, 1999. +[7] J. Chovan, C. Geuzaine, and M. Slodiˇcka. A - φ formulation of a mathematical model for the +induction hardening process with a nonlinear law for the magnetic field. Computer Methods in +Applied Mechanics and Engineering, 321:294–315, 2017. +[8] J. Chovan and M. Slodiˇcka. Induction hardening of steel with restrained Joule heating and non- +linear law for magnetic induction field: Solvability. Journal of Computational and Applied Math- +ematics, 311:630–644, 2017. + +30 +V. C. LE, M. SLODIˇCKA, AND K. VAN BOCKSTAL +[9] G. Di Fratta and A. Fiorenza. A short proof of local regularity of distributional solutions of +Poisson’s equation. Proceedings of the American Mathematical Society, 148(5):2143–2148, 2020. +[10] N. Di Luozzo, M. Fontana, and B. Arcondo. Modelling of induction heating of carbon steel tubes: +Mathematical analysis, numerical simulation and validation. Journal of Alloys and Compounds, +536:S564–S568, 2012. +[11] L. C. Evans. Partial Differential Equations, volume 19 of Graduate Studies in Mathematics. +American Mathematical Society, 1997. +[12] P. Frey. MEDIT: An interactive Mesh visualization Software. Technical report, INRIA, 2001. +[13] V. Girault and P.-A. Raviart. Finite Element Methods for Navier-Stokes Equations: Theory and +Algorithms, volume 5 of Springer Series in Computational Mathematics. Springer-Verlag, 1986. +[14] M. Gurtin. An introduction to continuum mechanics, volume 158 of Mathematics in Science and +Engineering. Academic Press, 1981. +[15] F. Hecht. New development in freefem++. Journal of Numerical Mathematics, 20(3-4), 2012. +[16] D. H¨omberg. A mathematical model for induction hardening including mechanical effects. Non- +linear Analysis: Real World Applications, 5(1):55–90, 2004. +[17] J. Kaˇcur. Method of Rothe in Evolution Equations, volume 80 of Teubner-Text zur Mathematik. +BSB B.G. Teubner Verlagsgesellschaft, 1985. +[18] T. Kato, M. Mitrea, G. Ponce, and M. Taylor. Extension and Representation of Divergence-free +Vector Fields on Bounded Domains. Mathematical Research Letters, 7(5):643–650, 2000. +[19] V. C. Le. Advanced mathematical and numerical analysis of induction hardening of steel. PhD +thesis, Ghent University, 2022. +[20] V. C. Le, M. Slodiˇcka, and K. Van Bockstal. A time discrete scheme for an electromagnetic +contact problem with moving conductor. Applied Mathematics and Computation, 404:125997, +2021. +[21] V. C. Le, M. Slodiˇcka, and K. Van Bockstal. Error estimates for the time discretization of an elec- +tromagnetic contact problem with moving non-magnetic conductor. Computers & Mathematics +with Applications, 87:27–40, 2021. +[22] V. C. Le, M. Slodiˇcka, and K. Van Bockstal. A space-time discretization for an electromagnetic +problem with moving non-magnetic conductor. Applied Numerical Mathematics, 173:345–364, +2022. +[23] H.-P. Liu, X.-H. Wang, L.-Y. Si, and J. Gong. Numerical simulation of 3D electromagnetic- +thermal phenomena in an induction heated slab. Journal of Iron and Steel Research International, +27(4):420–432, 2020. +[24] R. A. Matula. Electrical resistivity of copper, gold, palladium, and silver. Journal of Physical and +Chemical Reference Data, 8(4):1147–1298, 1979. +[25] V. Rudnev, D. Loveless, and R. L. Cook. Handbook of Induction Heating. CRC Press, second +edition, 2017. +[26] R. A. Serway. Principles of Physics. Saunders College Pub., second edition, 1998. +[27] H. Shokouhmand and S. Ghaffari. Thermal analysis of moving induction heating of a hollow +cylinder with subsequent spray cooling: Effect of velocity, initial position of coil, and geometry. +Applied Mathematical Modelling, 36(9):4304–4323, 2012. +[28] M. Slodiˇcka and J. Chovan. Solvability for induction hardening including nonlinear magnetic field +and controlled Joule heating. Applicable Analysis, 96(16):2780–2799, 2017. +[29] M. Slodiˇcka. Parabolic problem for moving/ evolving body with perfect contact to neighborhood. +Journal of Computational and Applied Mathematics, 391:113461, 2021. +[30] Y.-S. Touloukian, R.-W. Powell, C.-Y. Ho, and P.-G. Klemens. Thermophysical properties of +matter - the TPRC Data Series. Purdue Research Foundation, 1970. +[31] Z. Wang, W. Huang, W. Jia, Q. Zhao, Y. Wang, W. Yan, D. Schulze, G. Martin, and U. Luedtke. +3D Multifields FEM Computation of Transverse Flux Induction Heating for Moving-Strips. IEEE +Transactions on Magnetics, 35:1642–1645, 1999. + +31 +[32] H. Wen and Y. Han. Study on mobile induction heating process of internal gear rings for wind +power generation. Applied Thermal Engineering, 112:507–515, 2017. +[33] H.-M. Yin. Global solutions of Maxwell’s equations in an electromagnetic field with a temperature- +dependent electrical conductivity. European Journal of Applied Mathematics, 5(1):57–64, 1994. +[34] H.-M. Yin. Regularity of solutions to Maxwell’s system in quasi-stationary. Communications in +Partial Differential Equations, 22(7-8):57–64, 1997. +[35] H.-M. Yin. On Maxwell’s Equations in an Electromagnetic Field with the Temperature Effect. +SIAM Journal on Mathematical Analysis, 29:637–651, 1998. +[36] A. Zabett and S. H. Mohamadi Azghandi. Simulation of induction tempering process of carbon +steel using finite element method. Materials & Design (1980-2015), 36:415–420, 2012. +[37] E. Zeidler. Nonlinear Functional Analysis and Its Applications. Part II/A: Linear Monotone +Operators. Springer-Verlag, 1990. +(1) IDLab, Department of Information Technology, Ghent University - imec, B 9000 +Ghent, Belgium +Email address: vanchien.le@ugent.be +(2) Research group NaM2, Department of Electronics and Information Systems, +Ghent University, B 9000 Ghent, Belgium +Email address: marian.slodicka@ugent.be +(3) Ghent Analysis & PDE center, Department of Mathematics: Analysis, Logic and +Discrete Mathematics, Ghent University, Krijgslaan 281, B 9000 Ghent, Belgium +Email address: karel.vanbockstal@UGent.be + diff --git a/xdFKT4oBgHgl3EQfLC2B/content/tmp_files/load_file.txt b/xdFKT4oBgHgl3EQfLC2B/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..0d069fc6198483a89f975dab05608b9dc4a4050d --- /dev/null +++ b/xdFKT4oBgHgl3EQfLC2B/content/tmp_files/load_file.txt @@ -0,0 +1,924 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf,len=923 +page_content='A NUMERICAL SCHEME FOR SOLVING AN INDUCTION HEATING PROBLEM WITH MOVING NON-MAGNETIC CONDUCTOR VAN CHIEN LE1, MARI´AN SLODIˇCKA2, AND KAREL VAN BOCKSTAL3 Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' This paper investigates an induction heating problem in a multi-component system containing a moving non-magnetic conductor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' The electromagnetic process is described by the eddy current model, and the heat transfer process is governed by the convection-diffusion equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Both processes are coupled by a restrained Joule heat source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' A temporal discretization scheme is introduced to solve the corresponding variational system numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' With the aid of the Reynolds transport theorem, we prove the convergence of the proposed scheme as well as the well-posedness of the variational problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Some numerical experiments are also performed to assess the performance of the numerical scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Contents 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Introduction 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Mathematical model 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Geometrical setting 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Functional setting 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Mathematical model 7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Uniqueness 9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Time discretization 11 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Existence of a solution 16 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Numerical results 24 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Numerical experiments 25 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Numerical simulation 26 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Conclusion 28 References 29 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' 35Q61, 35Q79, 65M12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' induction heating, multi-component system, moving non-magnetic con- ductor, Reynolds transport theorem, restrained Joule heat source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' The work of V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Le was supported by the European Research Council through the European Union’s Horizon 2020 Research and Innovation programme (Grant number 101001847).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' The work of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Van Bockstal was supported by the Methusalem programme of Ghent University Special Research Fund (BOF) (Grant Number 01M01021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='11744v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='NA] 27 Jan 2023 2 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' LE, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' SLODIˇCKA, AND K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' VAN BOCKSTAL 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Introduction Induction heating is the process of heating an electrical conductor through the heat generated by an eddy current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Induction heating is a standard industrial process with various applications, including surface hardening, induction mass heating, induction melting and induction welding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Other industrial induction heating applications are listed in [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Basically, an alternating current is passed through an electric coil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' The electromagnetic fields occurring in the surrounding space induce an electric current in electrically conductive mediums, which is called the eddy current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Then, heat is generated due to the resistance of materials to the eddy current or the so-called Joule heating effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' A considerable amount of literature has been published on the study of the induction heating process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' The majority stand on physical and engineering points of view, where numerical simulation strategies have been performed, and experiments have been set up to validate numerical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' As an instance, the modelling of induction heating of carbon steel tubes was carried out in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' The authors considered a mathematical model combining electromagnetic process, heat transfer by conduction, convection and radiation, and ferromagnetic-paramagnetic transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Some numerical simulations of the heating stage were made using the finite-element method (FEM) and were validated by measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' We also refer the reader to [36, 23] for other studies of stationary induction heat treating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Besides that, FEM-based numerical schemes for moving in- duction heating problems were also studied in [31, 27, 1, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' However, those papers did not investigate fundamental questions, such as the convergence and stability of numerical simulations and the properties of the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' In contrast to the papers mentioned above, several studies to date have investigated the well-posedness and the regularity of the solution to the induction heating problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' However, all of them were restricted to a static geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' The authors of [33, 34, 35, 6] studied the global solvability of Maxwell’s equations together with temperature effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' More specifically, in [33], the quasi-static Maxwell’s equations were expressed in terms of the magnetic field, and the existence of a solution was proved using a fixed point argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' The regularity of the solution was then studied in [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' In [35], the existence of a solution was shown for Maxwell’s equations with the electric and magnetic fields as unknowns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' The author of [6] considered a degenerate problem modelling Joule heating in a conductive medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' The existence of global-in-time weak solutions was proved via the Faedo-Galerkin method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' The papers [28, 8, 7] also concerned mathematical models for a stationary induction heating problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Herein, the electromagnetic process and heat transfer are both governed by nonlinear equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' In [28], the equation was derived from Maxwell’s equations in terms of the magnetic field, whilst in [8] it was expressed in terms of the magnetic induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' In both articles, the authors proved the existence of a weak solution to the coupled system with controlled Joule heating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' The problem was then formulated in terms of the magnetic vector potential and electric scalar potential fields (A − φ formulation) in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' The existence of a global solution to the whole system was shown, and a numerical simulation was performed to support obtained theoretical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' 3 Recently, some theoretical and numerical studies on moving electromagnetic prob- lems have been published.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' In [5, 4, 3], the authors considered an eddy current problem in a cylindrical symmetric domain containing a moving non-magnetic conductor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' The well-posedness of the variational system was studied, and a numerical scheme was in- troduced for the computation of the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' These results were extended to a general three-dimensional domain (without the symmetry assumption) in [21] and [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' In these papers, a temporal discretization based on the backward Euler method and a FEM-based space-time discretization scheme were respectively proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' The corre- sponding error estimates were also established, and some numerical experiments were introduced to validate the performance of the proposed schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' In addition to those papers, the authors of [20] considered an electromagnetic contact problem with a mov- ing conductor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' The restriction on a non-magnetic moving conductor was no longer made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Instead, it allowed material coefficients to be fully jumping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' In this case, the well-posedness of the system was proved using Rothe’s method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' These pioneering works of moving electromagnetic problems serve as a basis for the mathematical analysis and numerical computation of the induction heating process involving moving conductors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' To the best of our knowledge, there has been no paper dealing with the mathemat- ical analysis of an induction heating problem with a moving conductor, even though this process has successfully been applied in industry for decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' The present pa- per investigates an induction heating problem in a multi-component system containing a moving non-magnetic conductor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' The electromagnetic process is described by the eddy current model, which is coupled with heat transfer via the Joule heating effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Due to the conductor’s and surrounding air’s movement, the heat transfer process is a combination of thermal conduction and convection mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' The nonlinearity of the Joule heat source is treated by introducing a cut-off function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Our investigation also relies on the assumption that the moving conductor is filled by a non-magnetic material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' This paper is organised into seven sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' The following section introduces some geometrical and functional settings and describes the mathematical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Section 3 derives the variational system from the original problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' In Section 4, we design a temporal discretization scheme based on the backward Euler method and perform some a priori estimates for iterates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Section 5 is the central section devoted to showing the existence of a solution to the variational system and the convergence of the proposed scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Finally, we present some numerical results for the discretization scheme in Section 6, and then we give a conclusion and some possibilities for future work in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Mathematical model 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Geometrical setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' We adopt the geometrical setting described in [20], which was introduced for a moving electromagnetic problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Let Ω be an open simply- connected and bounded domain in R3 such that its boundary ∂Ω belongs to the class C1,1 or Ω is a convex polyhedron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' The domain Ω contains a moving workpiece Σ and a fixed coil Π that are surrounded by air.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' The open connected subdomains Σ and Π are supposed to be of the class C2,1 and separate from each other, see Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Moreover, 4 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' LE, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' SLODIˇCKA, AND K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' VAN BOCKSTAL Σ v Ω Π Ξ Γin Γout Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' The domain Ω consists of a moving workpiece Σ with velocity v, a fixed coil Π and the surrounding air Ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' The coil Π shares common interfaces Γin and Γout of strictly positive measures with the boundary (see [21, Figure 1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' we introduce some notations that are frequently used throughout the manuscript: n denotes the outward unit normal vector on the boundary;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Θ(t) := Σ(t) ∪ Π is the subdomain consisting of electrically conductive mediums at time t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Ξ(t) := Ω \\ Θ(t) is the space occupied by air at time t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' and the interval [0, T] stands for the considered time frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' The coil Π shares common interfaces with the boundary ∂Ω, denoted by Γ := Γin ∪ Γout, whose measures are supposed to be strictly positive, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' |Γin| > 0 and |Γout| > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' The movement of the workpiece can be parameterized by a smooth bijective mapping Φ : Σ(0) × [0, T] → R3 such that (1) Σ(t) = Φ(Σ(0), t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' � t∈[0,T] Σ(t) ⊂ Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' det ∇Φ(x, t) > 0, ∀(x, t) ∈ Σ(0) × [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' We define the trajectory of the motion T := {(x, t) : x ∈ Σ(t), t ∈ [0, T]} , and the velocity v : T → R3 of the workpiece v(x, t) = ˙Φ � [Φ(·, t)]−1 (x), t � , where ˙Φ represents the total derivative of Φ with respect to time variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' The reader is referred to [14, Section 8] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Due to the movement of the workpiece Σ, the surrounding air also moves in a fluid manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' We assume that the velocity vector can be extended to the whole domain Ω and this extension (also denoted by v) is of class C1(Ω × [0, T]) satisfying v = 0 on the coil Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' 5 The subdomains Σ, Π and Ξ are filled by different materials, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' aluminium work- piece, copper coil and air.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' The induction heating process involves the following mate- rial coefficients: the magnetic permeability µ, the electrical conductivity σ, the thermal conductivity κ and the volumetric heat capacity α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' In the case of non-magnetic con- ductors, the magnetic permeability of the whole system can be well approximated by the constant of vacuum µ0 > 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' µ = µ0 in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' For the sake of simplicity, we assume that all material coefficients are positive constants on each subdomain except that the electrical conductivity σ vanishes on the air, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' σ(t) = � � � � � σΠ > 0 in Π, σΣ > 0 in Σ(t), 0 in Ξ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Please note that all material coefficients, except the magnetic permeability, are allowed to be jumping at the interface of different subdomains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' We use the subscripts Σ, Π and Ξ to distinguish the material functions on the workpiece, the coil and the air, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Functional setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' First of all, we introduce some function spaces and other main ingredients that are frequently used throughout this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' The Sobolev space Wk,λ(Ω) with k ∈ N and λ ∈ [1, ∞) is equipped with the following norm ∥f∥Wk,λ(Ω) = � � � 0≤|α|≤k � Ω |Dαf(x)|λ dx � � 1/λ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' When k = 0, the space W0,λ(Ω) with λ ∈ [1, ∞) becomes the Lebesgue space Lλ(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' We denote by (·, ·)Ω the scalar product in the space L2(Ω) with its induced norm ∥·∥L2(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Among Sobolev spaces, only Wk,2(Ω) with k ∈ N forms a Hilbert space, which are frequently denoted by Hk(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' The notation H1 0(Ω) stands for the closure of C∞ 0 (Ω) with respect to the norm of H1(Ω), and H1/2(∂Ω) is the space consisting out of the trace of functions in H1(Ω) to the boundary ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' The dual space of H1(Ω) and H1/2(∂Ω) are respectively denoted by H−1(Ω) and H−1/2(∂Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' These notations are inherited for vector and tensor fields by using corresponding bold symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Moreover, the subspace Z of H1(Π) defined by Z := � f ∈ H1(Π) : (f, 1)Π = 0 � is a Hilbert space with the equivalent norm ∥∇f∥L2(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' In addition, the following Banach space of vector fields plays a central role in further analysis W 0 := � f ∈ L2(Ω) : ∇ × f ∈ L2(Ω), ∇ · f = 0, f|∂Ω · n = 0 � equipped with the norm ∥f∥W 0 = ∥∇ × f∥L2(Ω) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' 6 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' LE, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' SLODIˇCKA, AND K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' VAN BOCKSTAL This norm is equivalent to the graph norm on the space W 0, and W 0 is continuously embedded into H1(Ω) because the open bounded simply-connected domain Ω is either a convex polyhedron or its boundary is of the class C1,1, see [13, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='3 on p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' 51] and [13, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='7 on p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' 52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Let X be an arbitrary Banach space with norm ∥·∥X and f : (0, T) → X be an abstract function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' We denote by C([0, T], X) and Lip([0, T], X) the spaces of continuous and Lipschitz continuous functions f endowed with the usual norm ∥f∥C([0,T],X) = max 0≤t≤T ∥f(t)∥X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' The Bochner spaces Lλ((0, T), X) with λ ∈ [1, ∞) and L∞((0, T), X) consist of all measurable abstract functions f furnished with the norms ∥f∥Lλ((0,T),X) = � � T � 0 ∥f(t)∥λ X dt � � 1/λ , ∥f∥L∞((0,T),X) = ess sup t∈(0,T) ∥f(t)∥X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' In what follows, we denote by ε, Cε and C positive constants depending only on the given data, where ε is a small number and Cε is a large one depending on ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Their different values at different contexts are allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' For the reason of reducing the number of constant notations, the notation a ≤ Cb (a ≥ Cb, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=') is replaced by a ≲ b (a ≳ b, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Since the Reynolds transport theorem is crucial for further analysis of PDEs with moving domain, it is recalled here with some related inequalities, which are useful for dealing with time-dependent boundary terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' We consider a Lipschitz moving domain ω(t) whose movement is associated with a velocity vector v being of class C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Let f(x, t) be a scalar abstract function satisfying f(t) ∈ W1,1(ω(t)) and ∂tf(t) ∈ L1(ω(t)) for all t ∈ (0, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Then, the Reynolds transport theorem (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' [14, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' 78]) and the Divergence theorem say that d dt � ω(t) f dx = � ω(t) ∂tf dx + � ∂ω(t) fv · n ds (2) = � ω(t) ∂tf dx + � ω(t) ∇ · (fv) dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' (3) Next, given f(t) ∈ H1(ω(t)), the Divergence theorem and ε-Young inequality give that (see also [29, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='1]) � ∂ω(t) f 2(v · n) ds = 2 � ω(t) f(∇f · v) dx + � ω(t) f 2(∇ · v) dx ≤ ε ∥∇f∥2 L2(ω(t)) + Cε ∥f∥2 L2(ω(t)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' (4) The constants ε and Cε only depend on the norm of the velocity, and the inequality (4) is still valid for vector functions in H1(ω(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' 7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Mathematical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' The mathematical modelling of a low-frequency electro- magnetic system with moving conductor was thoroughly discussed in [21, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Let us briefly recall the model considered in these papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' The electromagnetic process is modelled by the eddy current approximation of Maxwell’s equations or the so-called quasi-static system ∇ · B = 0, (5a) ∇ × E = −∂tB, (5b) ∇ × H = J, (5c) where E, H, B and J stand for the electric field, the magnetic field, the magnetic induction and the current density, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' The behaviour of electromagnetic fields passing through the interface of different materials is expressed by the following transmission conditions (6) �B · n�∂Θ\\Γ = 0, �H × n�∂Θ\\Γ = 0, and �(E + v × B) × n�∂Θ\\Γ = 0, where the unit normal vector n points from the electrical conductors (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' the workpiece Σ and the coil Π) to the air, and the jumps are defined by �f × n� = (f 2 − f 1) × n, �f · n� = (f 2 − f 1) · n, where f 1 and f 2 are the limiting values of the field f from the conductors and the air, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' We introduce a vector potential A of the magnetic induction B such that B = ∇ × A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' When B · n = 0 on the boundary ∂Ω, the vector potential A exists uniquely in H1(Ω) such that A is divergence-free and satisfies A × n = 0 on ∂Ω (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' [13, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='6 on p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' 48]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Substituting B = ∇ × A into the Faraday law (5b) leads us to the following decomposition of the electric field E = −∂tA − ∇φ, where φ exists uniquely in H1(Ω)/R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' In addition, the general Ohm’s law provides a constitutive relation for Maxwell’s equations J = σ(E + v × B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Hence, the total current density J can be divided into a source current part J s = −σ∇φ and an eddy current part J e = −σ∂tA + σv × (∇ × A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' The source current J s is originated from an external current j applied on the interfaces Γin and Γout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' The scalar potential φ on the coil Π is the solution to the following boundary value problem [16] (7) � � � � � ∇ · (−σ∇φ) = 0 in Π × (0, T), −σ∇φ · n = 0 on (∂Π \\ Γ) × (0, T), −σ∇φ · n = j on Γ × (0, T) , where j satisfies the following compatibility condition (8) � Γ j(s, t) ds = 0 ∀t ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' A comprehensive explanation of the modelling of the source current in the workpiece and in the air can be found in [20, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Finally, thanks to the Amp`ere relation (5c), 8 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' LE, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' SLODIˇCKA, AND K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' VAN BOCKSTAL the initial-boundary value problem of the vector potential A reads as (9) � � � � � � � � � � � � � � � � � � � σ∂tA + µ−1 0 ∇ × ∇ × A +χΠσ∇φ − σv × (∇ × A) = 0 in Ω × (0, T), ∇ · A = 0 in Ω × (0, T), A × n = 0 on ∂Ω × (0, T), �(∇ × A) × n� = 0 on (∂Θ \\ Γ) × (0, T), A(·, 0) = ˜A0 in Θ(0), where χΠ is the characteristic function of the domain Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' By the Joule heating effect, the electric current flowing through the conductors produces a significant amount of heat given by Q = 1 σ |J|2 = σ |∂tA + χΠ∇φ − v × (∇ × A)|2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' This Joule heat source plays the role of an internal (contactless) source, which repre- sents the coupling of the electromagnetic process and heat transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' It is one of the most challenging points during the mathematical treatment of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' In order to restrain this quadratic source term from increasing uncontrollably, we introduce a cut-off function Rr that truncates the source heat Q by a constant r > 0 as follows Rr(Q)(x, t) = min(r, Q(x, t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' From engineering point of view, this truncation models the use of a switch-off but- ton, which prevents the conductors from undesirable thermal deformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Due to the movement of the workpiece and the surrounding air, the heat transfer process is governed by thermal conduction and thermal convection, which are described by the convection-diffusion equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' On the boundary ∂Ω, we impose a homogeneous Neumann condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Hence, the temperature u is the solution to the following initial- boundary value problem (as the material derivative Du Dt is considered, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' [2]) (10) � � � � � α∂tu + αv · ∇u − ∇ · (κ∇u) = Rr(Q) in Ω × (0, T) , κ∇u · n = 0 on ∂Ω × (0, T) , u(·, 0) = ˜u0 in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' The following transmission conditions describe the perfect thermal contact (without friction) between the conductors and the environment (11) �u�∂Θ\\Γ = 0, �κ∇u · n�∂Θ\\Γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' In the problem (9), the initial guest A(0) is only given on the conductors Θ(0) since the electrical conductivity σ vanishes on the air.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' However, further results in this paper require A(0) to be extended on the whole domain Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' To do so, we invoke the result in [18, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='1] to show that, if ˜A0 satisfies ˜A0 ∈ H2(Θ(0)), ∇ · ˜A0 = 0 in Θ(0), ˜A0 = 0 on Γ, then there exists an extension ˜A0 ∈ H2(Ω) ∩ H1 0(Ω) with ∇ · ˜A0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' 9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Uniqueness Now, we are in the position to introduce the variational formulation of the problems (7)-(11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Multiplying the first equations of (7),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' (9) and (10) by ψ ∈ Z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' ϕ ∈ W 0 and w ∈ H1(Ω),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' then applying the Green theorem,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' we arrive at the following variational problem: Find φ(t) ∈ Z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' A(t) ∈ W 0 and u(t) ∈ H1(Ω) such that (12) σΠ (∇φ(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' ∇ψ)Π + (j(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' ψ)Γ = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' (13) (σ(t)∂tA(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' ϕ)Θ(t) + µ−1 0 (∇ × A(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' ∇ × ϕ)Ω + σΠ (∇φ(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' ϕ)Π − σΣ (v(t) × (∇ × A(t)) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' ϕ)Σ(t) = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' (14) (α(t)∂tu(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' w)Ω + (α(t)v(t) · ∇u(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' w)Ω + (κ(t)∇u(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' ∇w)Ω = (Rr (Q(t)) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' w)Θ(t) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' for any ψ ∈ Z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' ϕ ∈ W 0 and w ∈ H1(Ω) and for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' t ∈ (0, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Note that an equivalent saddle-point formulation of the problem (13) was introduced in [21], which gives more convenience for the computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' In this paper, however, we use the formulation (13) for simplicity and we note that the results obtained in [21] are still valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' In the next step, we summarize all assumptions used in the paper and show the uniqueness of a solution to the variational problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' (AS1) Ω is an open bounded simply-connected domain in R3 such that either Ω is a convex polyhedron or its boundary is of class C1,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' The open connected subdomains Σ and Π are of the class C2,1 and separate from each other (see Section 2 for more details);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' (AS2) The magnetic permeability is a constant on the whole domain Ω, and all ma- terial coefficients are positive constants on each subdomain, except that the electrical conductivity is vanishing on the air (see Section 2 for more details);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' (AS3) The velocity vector v satisfies v ∈ C1(Ω × [0, T]) and v = 0 on the coil Π;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' (AS4) ˜u0 ∈ H1(Ω) and ˜A0 ∈ W 0 ∩ H2(Ω) satisfies ∇ × ∇ × ˜A0 = 0 on Ξ(0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' (AS5) j ∈ Lip([0, T], H−1/2(Γ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='1 (Uniqueness).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Let the assumptions (AS1)-(AS5) be satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Then, the variational system (12)-(14) admits at most one solution (φ, A, u) satisfying φ ∈ L2((0, T), Z), A ∈ L2((0, T), W 0) with ∂tA(t) ∈ L2(Θ(t)) for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' t ∈ (0, T) and u ∈ L2((0, T), H1(Ω)) with ∥√αu∥L2(Ω) ∈ C([0, T]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' We assume that there exist two solutions (φ1, A1, u1) and (φ2, A2, u2) to the variational equations (12)-(14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Then, the solution (φ, A), with φ = φ1 − φ2 and A = A1 − A2, solves the linear system (12)-(13) with given data j = 0 and ˜A0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' By means of [20, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='1], we get that φ = 0 and A = 0 in the corresponding spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' This result implies that u = u1 − u2 also fulfills (14) with ˜u0 = 0 and Q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' 10 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' LE, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' SLODIˇCKA, AND K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' VAN BOCKSTAL Setting w = u(t) in (14) and then integrating in time over (0, ξ) ⊂ (0, T) gives us that (15) ξ � 0 (α(t)∂tu(t), u(t))Ω dt + ξ � 0 (α(t)v(t) · ∇u(t), u(t))Ω dt + ξ � 0 (κ(t)∇u(t), ∇u(t))Ω dt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' We can immediately see that ������ ξ � 0 (α(t)v(t) · ∇u(t), u(t))Ω dt ������ ≤ ε ξ � 0 ∥∇u(t)∥2 L2(Ω) dt + Cε ξ � 0 ��� � α(t)u(t) ��� 2 L2(Ω) dt, and ξ � 0 (κ(t)∇u(t), ∇u(t))Ω dt ≥ min{κΣ, κΠ, κΞ} ξ � 0 ∥∇u(t)∥2 L2(Ω) dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' The first integral on the left-hand side (LHS) of (15) is split over the subdomains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Then, the Reynolds transport theorem is invoked to rewrite the term corresponding to the workpiece Σ(t) as follows ξ � 0 (α(t)∂tu(t), u(t))Σ(t) dt (3) = αΣ 2 ∥u∥2 L2(Σ) (ξ) − αΣ ξ � 0 (∇u(t) · v(t), u(t))Σ(t) dt − αΣ 2 ξ � 0 (u(t)∇ · v(t), u(t))Σ(t) dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' A similar identity can be obtained for the integral over the air subdomain Ξ(t) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' replace Σ(t) by Ξ(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' For the fixed coil Π, we simply have that ξ � 0 (α(t)∂tu(t), u(t))Π dt = αΠ 2 ∥u(ξ)∥2 L2(Π) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Hence, the first integral on the LHS of (15) becomes ξ � 0 (α(t)∂tu(t), u(t))Ω dt = 1 2 ��� � α(ξ)u(ξ) ��� 2 L2(Ω) − ξ � 0 (α(t)∇u(t) · v(t), u(t))Ω dt − 1 2 ξ � 0 (α(t)u(t)∇ · v(t), u(t))Ω dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' 11 The integrals on the right-hand side (RHS) of this identity can be handled as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Therefore, we arrive at ��� � α(ξ)u(ξ) ��� 2 L2(Ω) + (1 − ε) ξ � 0 ∥∇u(t)∥2 L2(Ω) dt ⩽ Cε ξ � 0 ��� � α(t)u(t) ��� 2 L2(Ω) dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Finally, fixing a sufficiently small ε > 0 and then applying the Gr¨onwall argument shows that u = 0 in L2((0, T), H1(Ω)), which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Time discretization In this section, we design a time-discrete approximation scheme based on the back- ward Euler method for solving the variational system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' The time interval [0, T] is equidistantly partitioned into n subintervals with time step τ = T n, for any n ∈ Z+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' At time-point ti = iτ, i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' , n, we introduce the following notations for any function f and any time-dependent domain ω fi = f(ti), δfi = fi − fi−1 τ , ωi = ω(ti).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Starting from the initial data ˜A0 and ˜u0, we find the solution φi ∈ Z, Ai ∈ W 0 and ui ∈ H1(Ω), with i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' , n, such that the following identities are valid for any ψ ∈ Z, ϕ ∈ W 0 and w ∈ H1(Ω) (16) σΠ (∇φi, ∇ψ)Π + (ji, ψ)Γ = 0, (17) (σiδAi, ϕ)Θi + µ−1 0 (∇ × Ai, ∇ × ϕ)Ω + σΠ (∇φi, ϕ)Π − σΣ (vi × (∇ × Ai), ϕ)Σi = 0, (18) (αiδui, w)Ω + (αivi · ∇ui, w)Ω + (κi∇ui, ∇w)Ω = (Rr(Qi), w)Θi , where Qi = σi |δAi + χΠ∇φi − vi × (∇ × Ai)|2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' At each iteration step i, the equation (16) is solved first, then followed by (17) and (18), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' In the next lemma, the solvability of the time discretization system will be concerned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='1 (Solvability).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Let the assumptions (AS1)-(AS5) be fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Then, φ0 ∈ Z exists uniquely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Moreover, there exists a positive constant τ0 such that for any i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' , n and any τ < τ0, there exists a unique triplet (φi, Ai, ui) ∈ Z ×W 0 × H1(Ω) solving the system (16)-(18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' The proof of the solvability of the system (16)-(17) can be adopted from [21, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='1], so we omit this part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Let us define a bilinear form ei : H1(Ω)×H1(Ω) → R with i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' , n, such that ei(u, w) = 1 τ (αiu, w)Ω + (αivi · ∇u, w)Ω + (κi∇u, ∇w)Ω .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' 12 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' LE, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' SLODIˇCKA, AND K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' VAN BOCKSTAL Then, the variational problem (18) can be rewritten as follows (19) ei(ui, w) = 1 τ (αiui−1, w)Ω + (Rr(Qi), w)Θi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' We can easily get that ei(u, w) ≲ ∥u∥H1(Ω) ∥w∥H1(Ω) , which implies the boundedness of the form ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' For any i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' , n, the Cauchy- Schwarz and ε-Young inequalities allow us to show that ei(u, u) = 1 τ (αiu, u)Ω + (αivi · ∇u, u)Ω + (κi∇u, ∇u)Ω ≳ �1 τ − Cε � ∥u∥2 L2(Ω) + (1 − ε) ∥∇u∥2 L2(Ω) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' We fix a sufficiently small ε > 0, then choose a sufficiently small time step τ < τ0 to claim that the form ei is H1(Ω)-elliptic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Since ui−1 ∈ H1(Ω) is given and Rr(Qi) is bounded by the constant r, the RHS of (19) defines a bounded linear functional on H1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' As a consequence, there exists a unique solution ui ∈ H1(Ω) to the problem (18) for any i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' , n, according to the Lax-Milgram lemma [37, Theorem 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='E].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' □ Now, some a priori estimates for iterates will be investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' The following stability estimate for the solution Ai is directly derived from [21, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='2 (A priori estimate for Ai).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Let the assumptions (AS1)-(AS5) be fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Then, there exist positive constants τ0 and C such that for any τ < τ0, there holds that (20) max 1≤l≤n ∥δAl∥2 L2(Θl) + max 1≤l≤n ∥∇ × Al∥2 L2(Ω) + n � i=1 ∥∇ × δAi∥2 L2(Ω) τ + n � i=1 ∥δAi − δAi−1∥2 L2(Θi−1) ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' This a priori estimate was thoroughly proved in [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' It is noteworthy that the proof relies on the following property of the solution Ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='3 (Higher interior regularity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Let the assumptions (AS1)-(AS5) be fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Then, for any i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' , n, ∇ × Ai ∈ H1(Ω′) for any subset Ω′ ⊂⊂ Ω (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Ω′ ⊂ Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Moreover, there exists a constant C(Ω′) > 0 such that ∥∇ × Ai∥H1(Ω′) ≤ C � ∥δAi∥L2(Θi) + ∥∇ × Ai∥L2(Ω) + ∥∇φi∥L2(Π) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' The proof of this higher interior regularity was provided in [21], which unfortunately had a mistake (it is not justified to consider ∇ × ∇ × Ai as an element in L2(Ω) since C∞ 0 (Ω) ̸⊂ W 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' In the following, we give a corrected proof for Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='3 that is adopted from [19, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' For any i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' , n, let us denote pi := µ0 (σΣvi × (∇ × Ai) − σiδAi − χΠσΠ∇φi) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' 13 Then, pi ∈ L2(Ω) and the equation (17) implies that ⟨∇ × ∇ × Ai − pi, ϕ⟩W 0 = 0 ∀ϕ ∈ W 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Because Ai ∈ W 0, the functional ∇ × ∇ × Ai ∈ H−1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Hence, according to [13, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='1 on p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' 22], there exists a scalar function vi ∈ L2(Ω) such that ∇ × ∇ × Ai = pi + ∇vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Now, let Bi := ∇ × Ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' The field Bi ∈ L2(Ω) satisfies ∇ · Bi = 0 and −∆Bi = ∇ × ∇ × Bi − ∇(∇ · Bi) = ∇ × ∇ × ∇ × Ai = ∇ × pi + ∇ × ∇vi = ∇ × pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Since ∇ × pi ∈ H−1(Ω), we follow [9, Lemma 3] to get that Bi ∈ H1(Ω′) or ∇ × Ai ∈ H1(Ω′) for any subset Ω′ ⊂⊂ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Next, we adopt the technique in [11, Theorem 1 on p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' 309] to acquire the estimate of ∇ × Ai in H1(Ω′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' We firstly fix a subdomain Ω′ ⊂⊂ Ω, and then choose Ω⋆ such that Ω′ ⊂⊂ Ω⋆ ⊂⊂ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Restricting the testing functions ϕ ∈ {f ∈ C∞ 0 (Ω⋆) : ∇ · f = 0} ⊂ W 0 in the equation (17) leads us to that (21) (∇ × ∇ × Ai, ϕ)Ω⋆ = (pi, ϕ)Ω⋆ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' In virtue of the density argument in [13, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='8 on p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='30], the relation (21) is still valid for any ϕ ∈ H0(div0, Ω⋆), where H0(div0, Ω⋆) = � f ∈ L2(Ω⋆) : ∇ · f = 0, f|∂Ω⋆ · n = 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Now, let γ ∈ C∞ 0 (Ω⋆) such that γ = 1 in Ω′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Since γ2∇ × Ai ∈ H1 0(Ω⋆), we invoke [13, Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='5 on p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' 35] to get that ∇ × (γ2∇ × Ai) ∈ H0(div0, Ω⋆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Hence, setting ϕ = ∇ × (γ2∇ × Ai) in (21) implies that � ∇ × ∇ × Ai, ∇ × (γ2∇ × Ai) � Ω⋆ = � pi, ∇ × (γ2∇ × Ai) � Ω⋆ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Using the Cauchy-Schwarz and ε-Young inequalities together with the following identity ∇ × � γ2∇ × Ai � = γ2∇ × ∇ × Ai + 2γ∇γ × (∇ × Ai), we arrive at ∥γ∇ × ∇ × Ai∥L2(Ω⋆) ≲ ∥pi∥L2(Ω) + ∥∇ × Ai∥L2(Ω) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Finally, we use the fact ∥∇f∥2 L2(Ω) = ∥∇ × f∥2 L2(Ω) + ∥∇ · f∥2 L2(Ω) ∀f ∈ H1 0(Ω) 14 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' LE, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' SLODIˇCKA, AND K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' VAN BOCKSTAL to deduce that ∥∇ × Ai∥H1(Ω′) ≤ ∥γ∇ × Ai∥H1(Ω⋆) ≲ ∥∇ × (γ∇ × Ai)∥L2(Ω⋆) + ∥∇ · (γ∇ × Ai)∥L2(Ω⋆) ≤ ∥γ∇ × ∇ × Ai∥L2(Ω⋆) + ∥∇γ × (∇ × Ai)∥L2(Ω⋆) + ∥∇γ · (∇ × Ai)∥L2(Ω⋆) ≲ ∥pi∥L2(Ω) + ∥∇ × Ai∥L2(Ω) ≲ ∥δAi∥L2(Θi) + ∥∇ × Ai∥L2(Ω) + ∥∇φi∥L2(Π) , which allows us to accomplish the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' □ The next lemma aims to get a priori estimate for the discrete solution ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='4 (A priori estimate for ui).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Let the assumptions (AS1)-(AS5) be fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Then, there exist positive constants C and τ0 such that for any τ < τ0, the following relation holds true (22) max 1≤l≤n ∥ul∥2 L2(Ω) + n � i=1 ∥∇ui∥2 L2(Ω) τ + n � i=1 ∥ui − ui−1∥2 L2(Ω) ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' We set w = uiτ in the equation (18) and sum the result up to 1 ≤ l ≤ n to get that (23) l � i=1 (αi(ui − ui−1), ui)Ω + l � i=1 (αivi · ∇ui, ui)Ω τ + l � i=1 (κi∇ui, ∇ui)Ω τ = l � i=1 (Rr(Qi), ui)Θi τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' First, we rearrange the first term on the LHS as follows (24) 2 l � i=1 (αi(ui − ui−1), ui)Ω = l � i=1 � αi, u2 i � Ω − l � i=1 � αi−1, u2 i−1 � Ω + l � i=1 � αi, (ui − ui−1)2� Ω − l � i=1 � αi − αi−1, u2 i−1 � Ω .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' The last term on the RHS of (24) can be split over the subdomains in the following way l � i=1 � αi − αi−1, u2 i−1 � Ω = l � i=1 � αi, u2 i−1 � Ω − l � i=1 � αi−1, u2 i−1 � Ω = l � i=1 � αi, u2 i−1 � Σi∪Ξi∪Π − l � i=1 � αi−1, u2 i−1 � Σi−1∪Ξi−1∪Π .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' 15 Then, the Reynolds transport theorem can be used to estimate the integrals over the workpiece as ��� � αi, u2 i−1 � Σi − � αi−1, u2 i−1 � Σi−1 ��� = αΣ ������� ti � ti−1 d dt � Σ(t) u2 i−1(x) dx dt ������� (2) = αΣ ������� ti � ti−1 � ∂Σ(t) u2 i−1(v · n)(t) ds dt ������� (4) ≤ ε ∥∇ui−1∥2 L2(Ω) τ + Cε ∥ui−1∥2 L2(Ω) τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' A similar estimate can be deduced for the integrals over the air subdomains Ξi and Ξi−1, while the corresponding terms vanish on the coil Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Hence, we are able to obtain from (24) that (see also [29, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='3]) l � i=1 (αi(ui − ui−1), ui)Ω ≳ ∥ul∥2 L2(Ω) + l � i=1 ∥ui − ui−1∥2 L2(Ω) − C ∥˜u0∥2 H1(Ω) − ε l−1 � i=1 ∥∇ui∥2 L2(Ω) τ − Cε l−1 � i=1 ∥ui∥2 L2(Ω) τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Next, the third term on the LHS of (23) can be bounded by l � i=1 (κi∇ui, ∇ui)Ω τ ≥ min{κΣ, κΠ, κΞ} l � i=1 ∥∇ui∥2 L2(Ω) τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' The Cauchy-Schwarz and ε-Young inequalities can be used to handle the remaining terms of (23) as follows ����� l � i=1 (αivi · ∇ui, ui)Ω ����� τ ≤ ε l � i=1 ∥∇ui∥2 L2(Ω) τ + Cε l � i=1 ∥ui∥2 L2(Ω) τ, ����� l � i=1 (Rr(Qi), ui)Θi ����� τ ≲ l � i=1 ∥ui∥2 L2(Ω) τ + r2 l � i=1 τ ≲ l � i=1 ∥ui∥2 L2(Ω) τ + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Collecting all estimates above, we arrive at ∥ul∥2 L2(Ω) + l � i=1 ∥ui − ui−1∥2 L2(Ω) + (1 − ε) l � i=1 ∥∇ui∥2 L2(Ω) τ ≲ 1 + Cε l � i=1 ∥ui∥2 L2(Ω) τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Finally, we fix a sufficiently small ε > 0 and apply the Gr¨onwall argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Then, we take the maximum of the two resulting sides to conclude the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' □ 16 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' LE, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' SLODIˇCKA, AND K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' VAN BOCKSTAL 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Existence of a solution This section is the main part of the paper, concerning the existence of a solution to the variational system as well as the convergence of the proposed numerical scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Firstly, we introduce some piecewise-constant and piecewise-affine in time functions and subdomains jn(t) = ji, vn(t) = vi, σn(t) = σi, κn(t) = κi, αn(t) = αi, �Σn(t) = Σi, �Θn(t) = Θi, �Ξn(t) = Ξi, φn(t) = φi, An(t) = Ai, An(t) = Ai−1 + (t − ti−1) δAi, un(t) = ui, un(t) = ui−1, un(t) = ui−1 + (t − ti−1)δui, for all t ∈ (ti−1, ti] with i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' The value of the continuous functions An and un at time t = 0 are given by An(0) = ˜A0, un(0) = ˜u0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' In addition, the following piecewise-constant Joule heat source is defined for all t ∈ (0, T] Qn(t) = σn(t) ��∂tAn(t) + χΠ∇φn(t) − vn(t) × � ∇ × An(t) ���2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Now, we can rewrite the time-discrete equations (16)-(18) as follows (25) σΠ � ∇φn(t), ∇ψ � Π + � jn(t), ψ � Γ = 0, (26) (σn(t)∂tAn(t), ϕ)�Θn(t) + µ−1 0 � ∇ × An(t), ∇ × ϕ � Ω + σΠ � ∇φn(t), ϕ � Π − σΣ � vn(t) × � ∇ × An(t) � , ϕ � �Σn(t) = 0, (27) (αn(t)∂tun(t), w)Ω + (αn(t)vn(t) · ∇un(t), w)Ω + (κn(t)∇un(t), ∇w)Ω = � Rr � Qn(t) � , w � �Θn(t) , which are valid for any ψ ∈ Z, ϕ ∈ W 0 and w ∈ H1(Ω), and for any t ∈ (0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' The following lemma shows the convergence of the piecewise-constant approximation of the given data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' 17 Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='1 (Convergence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Let the assumptions (AS1)-(AS5) be satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Then, there exists a constant C > 0 such that the following relations hold true for any t ∈ (0, T] (i) ��jn(t) − j(t) �� H−1/2(Γ) ≤ Cτ, ∥vn(t) − v(t)∥C(Ω) ≤ Cτ, (ii) lim n→∞ ∥κn(t) − κ(t)∥L2(Ω) = 0, lim n→∞ ∥σn(t) − σ(t)∥L2(Ω) = 0, lim n→∞ ∥αn(t) − α(t)∥L2(Ω) = 0, lim n→∞ ���χ�Σn(t) − χΣ(t) ��� L2(Ω) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' (i) For any t ∈ (0, T], the Lipschitz continuity of the functions j and v gives us that ��jn(t) − j(t) �� H−1/2(Γ) ≲ τ, ∥vn(t) − v(t)∥C(Ω) ≲ τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' (ii) Thanks to the property of the mapping Φ, it holds for any t ∈ (0, T] that lim n→∞ ∥κn(t) − κ(t)∥2 L2(Ω) = lim n→∞ ∥κn(t) − κ(t)∥2 L2(Σ(t)) + lim n→∞ ∥κn(t) − κ(t)∥2 L2(Ξ(t)) = (κΞ − κΣ)2 lim n→∞ �����Σn(t) ∪ Σ(t) ��� − ����Σn(t) ∩ Σ(t) ��� � (AS3) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' The remaining limit transitions can be obtained by the same reasoning, which com- pletes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' □ In the next two theorems, we prove the convergence of Rothe’s functions to the solution of the variational system (12)-(14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='1 (Existence of φ and A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Let the assumptions (AS1)-(AS5) be fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Then, there exists a unique solution (φ, A) to the variational problems (12)-(13), which satisfies φ ∈ Lip([0, T], Z) and A ∈ C([0, T], W 0) with ∂tA ∈ L2((0, T), W 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' More- over, A(0) = ˜A0 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' in Θ(0) and the following convergences hold true φn → φ in L2((0, T), Z), (28) An → A, An → A in L2((0, T), W 0), (29) σn∂tAn ⇀ σ∂tA in L2((0, T), L2(Ω)), (30) √σn∂tAn → √σ∂tA in L2((0, T), L2(Ω)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' (31) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' The existence of a solution (φ, A) to the variational system (12)-(13) has already been shown in [21, Theorems 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='1, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='1 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='2], where φ ∈ Lip([0, T], Z) and A ∈ L∞((0, T), W 0) with σ∂tA ∈ L2((0, T), L2(Ω)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Moreover, the convergences (28)-(30) and the satisfaction of the initial condition A(0) = ˜A0 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' in Θ(0) have also been proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Therefore, we omit their proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' 18 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' LE, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' SLODIˇCKA, AND K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' VAN BOCKSTAL Next, the uniform boundedness of the sequence {∂tAn} in L2((0, T), W 0) (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='2) and the reflexivity of that space ensure the existence of a subsequence {∂tAnk} ⊂ {∂tAn} such that ∂tAnk ⇀ f in L2((0, T), W 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' (32) By means of [17, Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='6], we get that f = ∂tA in L2((0, T), W 0), and hence A ∈ C([0, T], W 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Moreover, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' (32) is still valid for the whole sequence {∂tAn} due to the uniqueness of a weak solution A, see Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Finally, we show that the convergence (31) also holds true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Because the electrical conductivity σ vanishes on the air, the limit transition (30) immediately implies that ∂tAn ⇀ ∂tA in L2((0, T), L2(Π)), (33) χ�Σn∂tAn ⇀ χΣ∂tA in L2((0, T), L2(Ω)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' (34) Hence, we can conclude that √σn∂tAn ⇀ √σ∂tA in L2((0, T), L2(Ω)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' (35) Now, setting ϕ = ∂tAn(t) ∈ W 0 in (26) and then integrating over the time range (0, η) ⊂ (0, T) gives that η � 0 ��� � σn(t) ∂tAn(t) ��� 2 L2(Ω) dt = η � 0 (σn(t) ∂tAn(t), ∂tAn(t))�Θn(t) dt = −µ−1 0 η � 0 � ∇ × An(t), ∇ × ∂tAn(t) � Ω dt − σΠ η � 0 � ∇φn(t), ∂tAn(t) � Π dt + σΣ η � 0 � vn(t) × � ∇ × An(t) � , ∂tAn(t) � �Σn(t) dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' 19 By virtue of the limit transitions (28)-(30) and (32)-(34), we are able to pass to the limit for n → ∞ as follows lim n→∞ η � 0 ��� � σn(t) ∂tAn(t) ��� 2 L2(Ω) dt = −µ−1 0 η � 0 (∇ × A(t), ∇ × ∂tA(t))Ω dt − σΠ η � 0 (∇φ(t), ∂tA(t))Π dt + σΣ η � 0 (v(t) × (∇ × A(t)) , ∂tA(t))Σ(t) dt (13) = η � 0 (σ(t) ∂tA(t), ∂tA(t))Θ(t) dt = η � 0 ��� � σ(t) ∂tA(t) ��� 2 L2(Ω) dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' This relation together with the weak convergence (35) leads us to the strong conver- gence (31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' □ Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='2 (Existence of u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Let the assumptions (AS1)-(AS5) be fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Then, there exists a unique function u satisfying u ∈ L2((0, T), H1(Ω))∩L∞((0, T), L2(Ω)) with ∥√αu∥L2(Ω) ∈ C([0, T]) such that the triplet (φ, A, u) solves the variational problem (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' In addition, u(0) = ˜u0 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' in Ω and the following convergences hold true un ⇀ u, un ⇀ u in L2((0, T), H1(Ω)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' First of all, we introduce some auxiliary identities that are useful for further analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' For any time η ∈ (tl−1, tl] with l = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' , n, one can easily see that η � 0 (αn(t)∂tun(t), w)Ω dt = l � i=1 (αi(ui − ui−1), w)Ω − tl � η (αn(t)∂tun(t), w)Ω dt = l � i=1 (αiui − αi−1ui−1, w)Ω − l � i=1 ((αi − αi−1)ui−1, w)Ω − tl � η (αn(t)∂tun(t), w)Ω dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' 20 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' LE, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' SLODIˇCKA, AND K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' VAN BOCKSTAL We split the second term on the RHS over the subdomains as follows l � i=1 ((αi − αi−1)ui−1, w)Ω = l � i=1 (αiui−1, w)Ω − l � i=1 (αi−1ui−1, w)Ω = l � i=1 (αiui−1, w)Σi∪Ξi∪Π − l � i=1 (αi−1ui−1, w)Σi−1∪Ξi−1∪Π .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Then, the Reynolds transport theorem allows us to rewrite that (αiui−1, w)Σi − (αi−1ui−1, w)Σi−1 = ti � ti−1 d dt � Σ(t) α(t)ui−1w dx dt (3) = ti � ti−1 � Σ(t) α(t)∇ · (ui−1wv(t)) dx dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' A similar identity can be obtained for the integrals over the air subdomains Ξi and Ξi−1, while the corresponding terms disappear on the fixed coil Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Therefore, we arrive at (36) η � 0 (αn(t)∂tun(t), w)Ω dt = (αn(η)un(η), w)Ω − (α(0)˜u0, w)Ω − η � 0 (α(t), ∇ · (un(t)wv(t)))Ω dt − ηn � η (α(t), ∇ · (un(t)wv(t)))Ω dt − ηn � η (αn(t)∂tun(t), w)Ω dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Next, the uniform boundedness of {un} from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='4 together with the reflexivity of L2((0, T), H1(Ω)) ensures the existence of a subsequence {unk} ⊂ {un} (denoted further by the same index as the original sequence) such that un ⇀ u in L2((0, T), H1(Ω)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Moreover, with a similar argument, we have the existence of a function y ∈ L2((0, T), H1(Ω)) such that un ⇀ y in L2((0, T), H1(Ω)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Because of the a priori estimate (22) for ui, the following relation between {un} and {un} holds true 0 ≤ lim n→∞ ∥un − un∥2 L2((0,T),L2(Ω)) = lim n→∞ n � i=1 ∥ui − ui−1∥2 L2(Ω) τ (22) ≲ lim n→∞ τ = 0, 21 which implies that y = u in L2((0, T), H1(Ω)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' In addition, we have that u ∈ L∞((0, T), L2(Ω)) thanks to the relation max t∈(0,T] ∥un(t)∥L2(Ω) (22) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' In the following, we prove that the function u is the solution to the variational problem (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' To this end, we integrate the equation (27) over the time interval (0, η) ⊂ (0, T), and then integrate the result over (0, ξ) ⊂ (0, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' By means of the identity (36), we get that (37) ξ � 0 (αn(η)un(η), w)Ω dη − ξ (α(0)˜u0, w)Ω − ξ � 0 η � 0 (α(t), ∇ · (un(t)wv(t)))Ω dt dη − ξ � 0 ηn � η (α(t), ∇ · (un(t)wv(t)))Ω dt dη − ξ � 0 ηn � η (αn(t)∂tun(t), w)Ω dt dη + ξ � 0 η � 0 (αn(t)vn(t) · ∇un(t), w)Ω dt dη + ξ � 0 η � 0 (κn(t)∇un(t), ∇w)Ω dt dη = ξ � 0 η � 0 � Rr � Qn(t) � , w � �Θn(t) dt dη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Let us invoke Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='4 to obtain that lim n→∞ ������ ξ � 0 ηn � η (α(t), ∇ · (un(t)wv(t)))Ω dt dη ������ ≲ lim n→∞ n � i=1 ∥ui−1∥H1(Ω) ∥w∥H1(Ω) τ 2 (22) ≲ lim n→∞ τ = 0, lim n→∞ ������ ξ � 0 ηn � η (αn(t)∂tun(t), w)Ω dt dη ������ ≲ lim n→∞ n � i=1 ∥ui − ui−1∥L2(Ω) ∥w∥L2(Ω) τ (22) ≲ lim n→∞ √τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' 22 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' LE, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' SLODIˇCKA, AND K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' VAN BOCKSTAL Using the convergences in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='1 and the Lebesgue dominated convergence theo- rem, we are able to show that lim n→∞ ξ � 0 (αn(η)un(η), w)Ω dη = ξ � 0 (α(η)u(η), w)Ω dη, lim n→∞ ξ � 0 η � 0 (α(t), ∇ · (un(t)wv(t)))Ω dt dη = ξ � 0 η � 0 (α(t), ∇ · (u(t)wv(t)))Ω dt dη, lim n→∞ ξ � 0 η � 0 (αn(t)vn(t) · ∇un(t), w)Ω dt dη = ξ � 0 η � 0 (α(t)v(t) · ∇u(t), w)Ω dt dη, lim n→∞ ξ � 0 η � 0 (κn(t)∇un(t), ∇w)Ω dt dη = ξ � 0 η � 0 (κ(t)∇u(t), ∇w)Ω dt dη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Now, we address the convergence of the remaining term concerning the discrete Joule heat source Qn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Let us denote q = √σ (∂tA + χΠ∇φ − v × (∇ × A)) , qn = √σn � ∂tAn + χΠ∇φn − vn × � ∇ × An �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' One can immediately see from Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='1 that qn → q in L2((0, T), L2(Ω)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' In addition, we introduce the following inequality which is valid for any non-negative numbers a, b and c (38) |min(a, b) − min(a, c) | ≤ 2√a ��� √ b − √c ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' One can easily prove this inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Using this inequality and the definition of the cut-off function Rr, we have that ��Rr � Qn � − Rr(Q) �� = ��min � r, |qn|2� − min � r, |q|2� �� (38) ≤ 2√r | |qn| − |q| | ≤ 2√r |qn − q| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Therefore, for each w ∈ H1(Ω), we can deduce that (39) 0 ≤ lim n→∞ ������ ξ � 0 η � 0 � Rr � Qn(t) � − Rr(Q(t)), w � Ω dt dη ������ ≲ lim n→∞ ∥w∥L2(Ω) � � ξ � 0 η � 0 ∥qn(t) − q(t)∥2 L2(Ω) dt dη � � 1/2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' 23 Collecting all limit transitions above, we are able to pass to the limit for n → ∞ in (37) to arrive at ξ � 0 (α(η)u(η), w)Ω dη − ξ (α(0)˜u0, w)Ω − ξ � 0 η � 0 (α(t), ∇ · (u(t)wv(t)))Ω dt dη + ξ � 0 η � 0 (α(t)v(t) · ∇u(t), w)Ω dt dη + ξ � 0 η � 0 (κ(t)∇u(t), ∇w)Ω dt dη = ξ � 0 η � 0 (Rr(Q(t)), w)Θ(t) dt dη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Differentiating this identity twice with respect to the time variable gives us that (40) d dt (α(t)u(t), w)Ω − (α(t), ∇ · (u(t)wv(t)))Ω + (α(t)v(t) · ∇u(t), w)Ω + (κ(t)∇u(t), ∇w)Ω = (Rr(Q(t)), w)Θ(t) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Here, we can use the Reynolds transport theorem to get back the variational problem (14) by rewriting that d dt (α(t)u(t), w)Σ(t) (3) = (α(t)∂tu(t), w)Σ(t) + (α(t), ∇ · (u(t)wv(t)))Σ(t) , which means that φ, A and u solve the problem (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' The convergence is not only valid for a subsequence, but also for the original sequence by taking into account the uniqueness of the solution u from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' In the next step, we prove the continuity in time of the L2(Ω)-norm of √αu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Setting w = u(t) in the relation (40) and then integrating the result over (ξ, η) ⊂ (0, T) leads us to that ��� � α(η)u(η) ��� 2 L2(Ω) − ��� � α(ξ)u(ξ) ��� 2 L2(Ω) − η � ξ � α(t)u2(t), ∇ · v(t) � Ω dt − η � ξ (α(t)v(t) · ∇u(t), u(t))Ω dt + η � ξ (κ(t)∇u(t), ∇u(t))Ω dt = η � ξ (Rr(Q(t)), u(t))Θ(t) dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' We can easily get that ���� ��� � α(η)u(η) ��� 2 L2(Ω) − ��� � α(ξ)u(ξ) ��� 2 L2(Ω) ���� ≲ |η − ξ| + η � ξ ∥u(t)∥2 H1(Ω) dt, 24 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' LE, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' SLODIˇCKA, AND K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' VAN BOCKSTAL which implies the absolute continuity in time, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' ∥√αu∥ 2 L2(Ω) ∈ C([0, T]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Finally, we show that the initial condition of u is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Multiplying the equation (14) by γ ∈ C∞([0, T]) satisfying γ(0) = 1 and γ(T) = 0, then integrating over the time range (0, T) gives us T � 0 γ(t) (α(t)∂tu(t), w)Ω dt + T � 0 γ(t) (α(t)v(t) · ∇u(t), w)Ω dt + T � 0 γ(t) (κ(t)∇u(t), ∇w)Ω dt = T � 0 γ(t) (Rr(Q(t)), w)Θ(t) dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' The first term can be rewritten using partial integration in time, which leads us to that (α(0)u(0), w)Ω = − T � 0 γ′(t) (α(t)u(t), w)Ω dt − T � 0 γ(t) (α(t), ∇ · (u(t)wv(t)))Ω dt + T � 0 γ(t) (α(t)v(t) · ∇u(t), w)Ω dt + T � 0 γ(t) (κ(t)∇u(t), ∇w)Ω dt − T � 0 γ(t) (Rr(Q(t)), w)Θ(t) dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' We repeat the process above when considering (27), and then pass to the limit n → ∞ to have that (α(0)˜u0, w)Ω = − T � 0 γ′(t) (α(t)u(t), w)Ω dt − T � 0 γ(t) (α(t), ∇ · (u(t)wv(t)))Ω dt + T � 0 γ(t) (α(t)v(t) · ∇u(t), w)Ω dt + T � 0 γ(t) (κ(t)∇u(t), ∇w)Ω dt − T � 0 γ(t) (Rr(Q(t)), w)Θ(t) dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Therefore, (α(0)u(0) − α(0)˜u0, w)Ω = 0 for all w ∈ H1(Ω), which implies that u(0) = ˜u0 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' We have accomplished the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Numerical results We perform some numerical tests in this section to support our theoretical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Time and space discretization schemes for electromagnetic problems involving a mov- ing non-magnetic conductor have been thoroughly studied in our previous works (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' 25 [21, 22]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Therefore, we are now focusing on the performance of the discretization scheme for the heat problem with the moving domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' In Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='1, two numerical experiments describing the heat transfer process in two-dimensional (2D) rotating disks are investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Afterwards, in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='2, we enhance the simulation of an induction heating process performed in [7] by considering a moving workpiece.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' The variational problems are numerically solved using the FEM, and the discretiza- tion scheme is implemented with the aid of the finite-element software package FreeFEM [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' The implementation of the variational problems (16)-(17) follows the saddle-point formulations proposed in [21] and [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' On the other hand, the first-order Lagrangian finite elements are used to spatially approximate the solution ui of the equation (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Since the velocity v is known, it is not necessary to change the computational mesh, which requires a re-meshing procedure that would significantly increase the computa- tional cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Instead, the mesh is fixed during the whole time range, and a characteristic function tracks the moving workpiece χΣ(x) = � 1 if x ∈ Σ, 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' In order to estimate the order of convergence without knowing the exact solution, we define the following relative error between Rothe’s solution un obtained by the proposed numerical method and a reference solution uref ˜Eu = ∥un − uref∥2 L2((0,T),H1(Ω)) ∥uref∥2 L2((0,T),H1(Ω)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' In all test cases, we assume that the initial temperature is ˜u0 = 298K (≈ 25◦C) and the constant magnetic permeability of vacuum is µ0 = 4πE-7H/m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' The values of other material coefficients used in numerical tests are presented in Table 1, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' [26, 24, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' SI unit Air Copper aluminium Electrical conductiv- ity σ MS/m 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='6 35 Volumetric heat ca- pacity α kJ/(m3·K) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='192 3384 2422 Thermal conductivity κ W/(m·K) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='02514 401 237 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Material coefficients used in the numerical tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Numerical experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' We perform two numerical experiments concerning the heat transfer process in 2D rotating disks with radius r1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='2m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' The disks both consist of an aluminium circular area with radius r2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='1m and r2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='05m, respec- tively, and the complementary area filled by copper, see Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' The domains are rotating with velocity v = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='125π (−y, x)Tm/s and are partitioned into 245598 and 251536 triangles, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Instead of the Joule heating Q, the system is supplied 26 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' LE, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' SLODIˇCKA, AND K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' VAN BOCKSTAL v v Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' The circular domain of experiments consisting of an aluminium circular area (red) and the complementary area filled by copper (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' The domains are rotating with velocity v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Left: the first experiment with a con- centric interior circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Right: the second experiment with an eccentric interior circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Temperature distribution of the first experiment at different time points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Left: t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='125s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Middle: t = 8s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Right: t = 64s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' with a heat source f = 1MW/m3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' The changes over time of the temperature distribu- tion are visualized by the software package MEDIT [12], which are shown in Figures 3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' We verify the convergence of the temporal discretization scheme (18) in the time interval (0, T) with T = 64s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' The reference temperature uref is the solution to (18) with time step τ = 2−8s, while larger time steps τ = 2−js, with j = 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' , 7, are used to compute the discrete solution un.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Relative errors with respect to time step τ together with the corresponding regression lines are presented in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' The slope of the regression lines show that the potential convergence rate of the numerical scheme (18) is O(τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Numerical simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' We enhance the numerical simulation of an induction heating process performed in [7] by considering a moving workpiece instead of a fixed one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' The domain Ω is a unit cube consisting of a thin-walled cylindrical aluminium 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='9805E+002 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='9804E+002 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='9804E+002 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='9805E+0023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='0060E+002 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='0106E+002 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='0037E+002 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='0083E+002 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='0129E+0023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='1854E+002 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='2023E+002 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='1769E+002 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='1939E+002 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='2108E+00227 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Temperature distribution of the second experiment at different time points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Left: t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='125s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Middle: t = 8s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Right: t = 64s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' 2 7 2 6 2 5 2 4 2 3 2 2 time step 2 36 2 34 2 32 2 30 2 28 2 26 2 24 relative error relative error w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='791+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='213 log2 2 7 2 6 2 5 2 4 2 3 2 2 time step 2 22 2 20 2 18 2 16 2 14 relative error relative error w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='724+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='819 log2 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Relative error with respect to time step and the corresponding regression line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Left: the first experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Right: the second experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Both numerical experiments show the potentially optimal convergence rate of the temporal discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' workpiece with two radii r1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='092m, r2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='081m and height h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='3m, a copper coil and surrounding air.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' The initial guest ˜A(0) = 0 on Θ(0) has a trivial extension ˜A(0) = 0 on the whole domain Ω without requiring C2,1 regularity of the boundary of the subdomains Σ(0) and Π (see Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Hence, our theoretical results are still valid for this geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' The domain Ω is partitioned into 32312 tetrahedra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' A static external current density with magnitude ȷ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='0E7 is driven through the coil Π via the interfaces Γin and Γout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' The workpiece is moving along the z-axis with velocity v = (0, 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='46875)Tcm/s, and the considered time length is T = 32s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Outside the workpiece, we assume that the velocity is very small;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' thus, the thermal convection is dominated by the thermal conduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Therefore, we can neglect the thermal convection effect in the air domain and avoid the computation of airflow, which is not essential in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' 200+30862 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='9805E+002 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='9804E+002 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='9805E+002200+300 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='0090E+002 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='0036E+002 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='0072E+002 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='0108E+0023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='1736E+002 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='1823E+002 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='1693E+002 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='1779E+002 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='1866E+00228 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' LE, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' SLODIˇCKA, AND K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' VAN BOCKSTAL Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Different locations of the workpiece and the corresponding temper- ature distributions on the conductors at different time points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Top: t = 16s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Bottom: t = 32s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' The reference solution is computed from the variational system (16)-(18) with time step τ = 2−7s (n = 4096 subintervals).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Different locations of the workpiece together with their corresponding temperature distributions on the conductors at two differ- ent time points are presented in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Some rougher discrete solutions are also computed when the number of time intervals equals n = 64, 128, 256, 512, 1024 and 2048.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Relative error with respect to time step and the corresponding regression line are shown in Figure 7, which confirms the potentially optimal convergence rate of our proposed scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Conclusion In the present paper, we have investigated an induction heating problem in a three- dimensional domain containing a moving non-magnetic conductor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' The electromag- netic process and heat transfer are modelled by PDEs, which are coupled by the Joule heating effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' A cut-off function has been introduced to restrain the nonlinear Joule heat source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' A time-discrete scheme based on the backward Euler’s method has been proposed to solve approximately the variational problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' The convergence of the pro- posed discretisation scheme and the well-posedness of the variational system have been proved with the aid of the Reynolds transport theorem and Rothe’s method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Some numerical results have also been presented to support the theoretical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='9998E+002 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='0394E+002 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='9800E+002 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='0196E+002 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='0592E+0022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='9985E+002 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='0356E+002 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='9800E+002 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='0171E+002 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='0541E+0023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='0200E+002 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='0987E+002 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='9807E+002 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='0593E+002 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='1380E+0023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='0186E+002 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='0944E+002 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='9807E+002 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='0565E+002 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='1323E+00229 2 6 2 5 2 4 2 3 2 2 2 1 time step 2 22 2 20 2 18 2 16 2 14 relative error relative error w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='37+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='611 log2 Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Relative error with respect to time step and the corresponding regression line of the numerical simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' This result confirms the potentially optimal convergence rate of the numerical scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' In the future, comprehensive error estimates of the proposed temporal discretiza- tion scheme should be performed to verify the potentially optimal convergence rate obtained numerically in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' In addition, future studies could concern high- frequency induction heating problems (full Maxwell’s equations) involving magnetic moving conductors, which have a wide range of applications in manufacturing indus- tries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' References [1] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Bai, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Zhang, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Modeling of the moving induction heating used as secondary heat source in weld-based additive manufacturing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' The International Journal of Advanced Man- ufacturing Technology, 77(1-4):717–727, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' [2] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Bejan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Convection Heat Transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Wiley, fourth edition, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' [3] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Berm´udez, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' L´opez-Rodr´ıguez, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Rodr´ıguez, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Salgado.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Numerical solution of a tran- sient three-dimensional eddy current model with moving conductors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' International Journal of Numerical Analysis and Modeling, 16(5):695–717, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' [4] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Berm´udez, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Mu˜noz-Sola, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Reales, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Rodr´ıguez, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Salgado.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' A Transient Eddy Current Problem on a Moving Domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Mathematical Analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' SIAM Journal on Mathematical Analysis, 45(6):3629–3650, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' [5] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Berm´udez, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Reales, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Rodr´ıguez, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Salgado.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Numerical analysis of a transient eddy current axisymmetric problem involving velocity terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Numerical Methods for Partial Differen- tial Equations, 28(3):984–1012, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' [6] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Bie´n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Existence of global weak solutions for a class of quasilinear equations describing Joule’s heating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Mathematical Methods in the Applied Sciences, 22(15):1275–1291, 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' [7] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Chovan, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Geuzaine, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Slodiˇcka.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' A - φ formulation of a mathematical model for the induction hardening process with a nonlinear law for the magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Computer Methods in Applied Mechanics and Engineering, 321:294–315, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' [8] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Chovan and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Slodiˇcka.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Induction hardening of steel with restrained Joule heating and non- linear law for magnetic induction field: Solvability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Journal of Computational and Applied Math- ematics, 311:630–644, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' 30 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' LE, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' SLODIˇCKA, AND K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' VAN BOCKSTAL [9] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Di Fratta and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Fiorenza.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' A short proof of local regularity of distributional solutions of Poisson’s equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Proceedings of the American Mathematical Society, 148(5):2143–2148, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' [10] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Di Luozzo, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Fontana, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Arcondo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Modelling of induction heating of carbon steel tubes: Mathematical analysis, numerical simulation and validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Journal of Alloys and Compounds, 536:S564–S568, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' [11] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Evans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Partial Differential Equations, volume 19 of Graduate Studies in Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' American Mathematical Society, 1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' [12] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Frey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' MEDIT: An interactive Mesh visualization Software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Technical report, INRIA, 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' [13] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Girault and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Raviart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms, volume 5 of Springer Series in Computational Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Springer-Verlag, 1986.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' [14] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Gurtin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' An introduction to continuum mechanics, volume 158 of Mathematics in Science and Engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Academic Press, 1981.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' [15] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Hecht.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' New development in freefem++.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Journal of Numerical Mathematics, 20(3-4), 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' [16] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' H¨omberg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' A mathematical model for induction hardening including mechanical effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Non- linear Analysis: Real World Applications, 5(1):55–90, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' [17] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Kaˇcur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Method of Rothe in Evolution Equations, volume 80 of Teubner-Text zur Mathematik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' BSB B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Teubner Verlagsgesellschaft, 1985.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' [18] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Kato, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Mitrea, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Ponce, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Taylor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Extension and Representation of Divergence-free Vector Fields on Bounded Domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Mathematical Research Letters, 7(5):643–650, 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' [19] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Le.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Advanced mathematical and numerical analysis of induction hardening of steel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' PhD thesis, Ghent University, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' [20] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Le, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Slodiˇcka, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Van Bockstal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' A time discrete scheme for an electromagnetic contact problem with moving conductor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Applied Mathematics and Computation, 404:125997, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' [21] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Le, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Slodiˇcka, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Van Bockstal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Error estimates for the time discretization of an elec- tromagnetic contact problem with moving non-magnetic conductor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Computers & Mathematics with Applications, 87:27–40, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' [22] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Le, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Slodiˇcka, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Van Bockstal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' A space-time discretization for an electromagnetic problem with moving non-magnetic conductor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Applied Numerical Mathematics, 173:345–364, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' [23] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Liu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Wang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Si, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Gong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Numerical simulation of 3D electromagnetic- thermal phenomena in an induction heated slab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Journal of Iron and Steel Research International, 27(4):420–432, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' [24] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Matula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Electrical resistivity of copper, gold, palladium, and silver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Journal of Physical and Chemical Reference Data, 8(4):1147–1298, 1979.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' [25] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Rudnev, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Loveless, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Cook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Handbook of Induction Heating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' CRC Press, second edition, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' [26] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Serway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Principles of Physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Saunders College Pub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=', second edition, 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' [27] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Shokouhmand and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Ghaffari.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Thermal analysis of moving induction heating of a hollow cylinder with subsequent spray cooling: Effect of velocity, initial position of coil, and geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Applied Mathematical Modelling, 36(9):4304–4323, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' [28] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Slodiˇcka and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Chovan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Solvability for induction hardening including nonlinear magnetic field and controlled Joule heating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Applicable Analysis, 96(16):2780–2799, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' [29] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Slodiˇcka.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Parabolic problem for moving/ evolving body with perfect contact to neighborhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Journal of Computational and Applied Mathematics, 391:113461, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' [30] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Touloukian, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Powell, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Ho, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Klemens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Thermophysical properties of matter - the TPRC Data Series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Purdue Research Foundation, 1970.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' [31] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Wang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Huang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Jia, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Zhao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Wang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Yan, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Schulze, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Martin, and U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Luedtke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' 3D Multifields FEM Computation of Transverse Flux Induction Heating for Moving-Strips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' IEEE Transactions on Magnetics, 35:1642–1645, 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' 31 [32] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Wen and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Han.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Study on mobile induction heating process of internal gear rings for wind power generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Applied Thermal Engineering, 112:507–515, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' [33] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Yin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Global solutions of Maxwell’s equations in an electromagnetic field with a temperature- dependent electrical conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' European Journal of Applied Mathematics, 5(1):57–64, 1994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' [34] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Yin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Regularity of solutions to Maxwell’s system in quasi-stationary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Communications in Partial Differential Equations, 22(7-8):57–64, 1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' [35] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Yin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' On Maxwell’s Equations in an Electromagnetic Field with the Temperature Effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' SIAM Journal on Mathematical Analysis, 29:637–651, 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' [36] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Zabett and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Mohamadi Azghandi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Simulation of induction tempering process of carbon steel using finite element method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Materials & Design (1980-2015), 36:415–420, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' [37] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Zeidler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Nonlinear Functional Analysis and Its Applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Part II/A: Linear Monotone Operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' Springer-Verlag, 1990.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content=' (1) IDLab, Department of Information Technology, Ghent University - imec, B 9000 Ghent, Belgium Email address: vanchien.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='le@ugent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='be (2) Research group NaM2, Department of Electronics and Information Systems, Ghent University, B 9000 Ghent, Belgium Email address: marian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='slodicka@ugent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='be (3) Ghent Analysis & PDE center, Department of Mathematics: Analysis, Logic and Discrete Mathematics, Ghent University, Krijgslaan 281, B 9000 Ghent, Belgium Email address: karel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='vanbockstal@UGent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} +page_content='be' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdFKT4oBgHgl3EQfLC2B/content/2301.11744v1.pdf'} diff --git a/y9FLT4oBgHgl3EQfnS-e/content/tmp_files/2301.12127v1.pdf.txt b/y9FLT4oBgHgl3EQfnS-e/content/tmp_files/2301.12127v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..54d26d43861bd71e4f66bd98bcefed954217b024 --- /dev/null +++ b/y9FLT4oBgHgl3EQfnS-e/content/tmp_files/2301.12127v1.pdf.txt @@ -0,0 +1,1312 @@ +Could an Artificial-Intelligence agent pass an introductory physics course? +Gerd Kortemeyer∗ +Educational Development and Technology, ETH Zurich, Zurich, Switzerland † +(Dated: January 31, 2023) +Massive pre-trained language models have garnered attention and controversy due to their ability +to generate human-like responses: attention due to their frequent indistinguishability from human- +generated phraseology and narratives, and controversy due to the fact that their convincingly pre- +sented arguments and facts are frequently simply false. Just how human-like are these responses +when it comes to dialogues about physics, in particular about the standard content of introductory +physics courses? This study explores that question by having ChatGTP, the pre-eminent language +model in 2023, work through representative assessment content of an actual calculus-based physics +course and grading the responses in the same way human responses would be graded. As it turns +out, ChatGPT would barely pass this course while exhibiting many of the preconceptions and errors +of a beginning learner. +I. +INTRODUCTION +“Educators may have concerns about ChatGPT, a +large language model trained by OpenAI, for a number of +reasons. First and foremost, there is the concern that a +tool like ChatGPT could potentially be used to cheat on +exams or assignments. ChatGPT can generate human- +like text, which means that a student could use it to pro- +duce a paper or response that is not their own work. This +could lead to a breakdown in the integrity of the educa- +tional system and could undermine the value of a degree +or diploma.” These sentences were not written by the +author, but by ChatGPT (Generative Pre-trained Trans- +former) [1] itself in response to the prompt “Write an es- +say why educators would be concerned about ChatGPT.” +The chatbot goes on to explain how it could spread misin- +formation, inhibit the development of writing skills, and +replace human educators, particularly when it comes to +grading. +The potential impact of ChatGPT with its custom- +built essays on courses in the humanities is evident, but +is there also an impact on subjects like physics? First +of all, within physics, large problem libraries for cheat- +ing have existed for years, and they are well-known and +used by students [2, 3] — virtually any physics homework +problem ever assigned is available online with solutions +and more or less helpful explanations. So, the primary +impact of ChatGTP in physics would not be cheating. +On top of that, would Artificial Intelligence really be +able to handle the logical, conceptual, and mathemati- +cal challenges that physics entails, and would it be able +to strategically solve problems [4, 5]? +Figure 1 shows a sample dialogue with ChatGPT, +which is, after all, primarily a chatbot. A welcome fea- +ture is that it does not simply provide some answer, but +that the algorithm attempts to explain how it arrived at +the answer. In many respects, this dialogue appears sim- +∗ kgerd@ethz.ch +† also at Michigan State University, East Lansing, USA +ilar to an office-hour conversation between an instructor +and a beginning physics student: +• When first asked how far the car is from where it +started, the chatbot did not consider that the car +may have changed direction. When prompted, it +does state that there is missing information. +• The chatbot does plug-and-chug [6], putting the +numerical results from one equation into the next. +• The chatbot leaves out units. +• The chatbot does not realize that the speed actually +drops out when doing the return-time calculation +in the last step; instead, rounding errors keep accu- +mulating. The straightforward solution would have +been +� +(3h)2 + 4h)2 = 5h (at least, though, the +chatbot adds an “approximately” to its solution. +As it will turn out, carrying out calculations by +putting numbers into formulas is one of the weak- +nesses of ChatGPT shared with beginning learners +of physics. +How much, indeed, does 2023 state-of-the-art Artifi- +cial Intelligence resemble the behavior of an introductory +physics student? Could it pass a physics course? When +posing this question directly to ChatGPT, it answers “as +a language model, I have been trained on a large dataset +of text, including physics texts. This allows me to un- +derstand and generate text related to physics concepts, +but it does not mean that I have the ability to solve +physics problems or pass a physics course. I can provide +explanations and answer questions about physics to the +best of my knowledge, but I am not a substitute for a +human physics expert or a physics education.” To put +this statement to the test, ChatGPT was used to solve +representative assessment components of an introductory +calculus-based physics; the responses were graded in the +context of the assessment types and subjectively com- +pared to responses of human learners. +It is important to note, though, that ChatGPT will not +actually learn anything new by “attending” this course, +as the system is a “Pre-trained Transformer” that in fact +arXiv:2301.12127v1 [physics.ed-ph] 28 Jan 2023 + +2 +FIG. 1. A sample ChatGPT dialogue about a homework prob- +lem. The entries labelled with a red “KO” are by the author, +the entries labelled in green by ChatGPT. +does not know anything that happened after 2021 (which, +for introductory physics, is not a problem, since that is +after 1905). Individual dialogues like Fig. 1 may exhibit +features that appear like learning, e.g., the system dis- +covering that distance from the starting point will be +path-dependent, but this is not anything permanently +learned beyond the confines of a dialogue. On the other +hand, OpenAI keeps on training the system based on user +interaction, particularly as users can upvote, downvote, +and comment responses. +II. +SETTING +The study takes place in first-year calculus-based +physics lecture courses previously taught by the author +at Michigan State University; materials, however, were +gathered from different years of the same course in order +to allow comparison to previously published studies. The +first semester covers the standard mechanics topics (in- +cluding rotational dynamics) and the beginnings of ther- +modynamics; the second semester covers the usual topics +of electricity and magnetism, as well as an introduction +to modern physics (rudimentary quantum physics and +special relativity). The first- and second-semester labo- +ratory were separate courses in the course sequence. All +materials (except the Force Concept Inventory [7]) were +available in LON-CAPA [8], so in their essence they could +be copy-pasted into ChatGTP — this included online +homework, clicker questions, programming exercises, and +exams. LON-CAPA randomizes assessment problems, so +different students would get different versions of the same +problem, e.g., different numbers, options, graphs, etc.; +this avoids simplistic pattern matching and copying of +solutions, but as it will turn out, this feature is irrelevant +for this study. +III. +METHODOLOGY +The study investigates ChatGPT’s performance on dif- +ferent kinds of assessment problems; it uses the Jan- +uary 9, 2023 release of the system [9]. +Different as- +sessment components were scored differently, simulating +their function in the course: +• The Force Concept Inventory was simply scored +based on answer agreement. +• For homework, ChatGPT was allowed multiple at- +tempts [10] and engaged in dialogue to simulate dis- +cussions with fellow students or in office hour. For +clicker questions, an actual lesson was replayed [11], +and discussion were allowed where within the re- +played lesson peer instruction took place. +• Programming exercises were to be graded based on +the same criteria as in the course (which turned out +unnecessary), and a dialogue was allowed [12]. +• For exams, no such dialogues were allowed, and +the first answer counted. Earlier iterations of the +course used bubble sheets and thus had answer +options instead of free-response fields for prob- +lems with numerical answers; for this study, free- +responses were used, since this allowed to grade ex- +ams using both simple answer agreement (simulat- +ing multiple choice on bubble sheets) and hand- +graded as in later semesters. Using free-response +instead of answer options also avoided ChatGTP +randomly picking the correct answer. +ChatGPT uses a probabilistic algorithm, so the re- +sponses to queries are not necessarily reproducible. For +an assessment problem, generally the first dialogue was +evaluated, with two exceptions: if the system produced +an error message or if the author accidentally gave a +wrong prompt, a new chat was started. Translating this +to an actual course scenario, students were allowed to + +KO +A car drives at 55 miles per hour for 7 hours. What is the total distance driven? +The total distance driven is 385 miles. (55 miles per hour * 7 hours = 385 miles) +KO +How far away is it from where it started? +The distance from where the car started would be the same as the total distance +driven, which is 385 miles. +KO +But if it did not drive in a straight line? +If the car did not drive in a straight line, the total distance driven would still be the +same (385 miles), but the distance from where the car started might be different, as it +would depend on the specific route the car took. In order to determine the distance +from the starting point one would need a map and the complete route the car took. +KO +It drove north for 3 hours and then east for 4 hours. +If the car drove north for 3 hours at 55 miles per hour, it would have traveled 165 miles +north. And then, if it drove east for 4 hours at 55 miles per hour, it would have traveled +22o miles east. The distance from the starting point would be the hypotenuse of the +right triangle formed by the north-south and east-west distances, which can be found +sqrt(72,725) = 269.8 miles. +KO +With the same speed, driving in a straight line instead, how long would it take to get +backtowhereitstarted? +If the car drove in a straight line back to where it started, at a speed of 55 miles per +hour, it would take 269.8 miles / 55 miles per hour = 4.9 hours approximately to get +back to the starting point.3 +retake an assessment problem if they got sick, and help +received was always correct in terms of physics. When +errors occurred (red error messages), which was about +one-in-ten dialogues, those apparently were not directly +connected to the dialogue, but might have been related +to general overload of the platform; for example, if an er- +ror occurred immediately after entering the question, the +next time around the same question would not produce +an error. +ChatGPT is a text-based tool, so figures and graphs +could not be communicated in their original form. This +means that graphics had to be transcribed the same way +as they would be for accessibility for blind students [13]; +Fig. 2 shows an example. This means that the character +of the problem changes substantially [14–16], but this is +unfortunately unavoidable. Attention was paid, though, +to include some extraneous information where possible, +such as the beginning position in Fig. 2. +The methodology is strictly empirical and arguably +anecdotal. +However, the course under investigation is +typical for introductory physics courses around the world, +both in terms of coverage and difficulty. Thus, some of +the results are likely to be generalizable. +IV. +RESULTS +A. +Force Concept Inventory +In the original course, the Force Concept Inventory +was administered as a pre-/post-test in order to calculate +gains [17]. Since ChatGPT would not actually learn any- +thing from doing the course assessments (except through +continuing training by OpenAI), the test was carried out +only once. +ChatGPT scored 18 out of 30 points on this concept +inventory, i,e., 60%. This score corresponds to the sug- +gested entry threshold for Newtonian physics [18]; in +other words, ChatGPT performs as well as a beginning +learner who had just grasped the basic concepts of clas- +sical mechanics. +For an Artificial Intelligence, the score seems surpris- +ingly good. An immediate suspicion was that ChatGPT +had been trained using the Force Concept Inventory, +which is of course a very popular test, and that it simply +latches on to surface features. As a simple test, the last +question on the test was modified as shown in Fig. 3: the +scenario and the order of the answers were changed. As +can be seen, these surface features do not matter, so in +that respect, ChatGPT does not act like a novice [19] +(however, the reality is not quite as straightforward as +this expert-novice distinction [20]). +The inventory cannot be published here, but it is avail- +able to physics instructors and researchers from Phys- +Port [7]. ChatGPT answered 1C, 2A, 3C, 4E, 5B, 6B, +7B, 8A, 9B, 10A, 11E, 12B, 13B, 14D, 15A, 16E, 17B, +18B, 19A, 20E, 21B, 22B, 23A, 24C, 25D, 26E, 27C, 28D, +29B, and 30C. +Of particular interest is of course where ChatGPT is +losing points. Several errors are related to “impetus” [21]: +more than once did ChatGTP assume that an object im- +mediately moves in the direction of an applied force, in- +dependent of initial movement (answering 8A, 9B, and +21B) and even that it returns to the original movement +when the force is no longer applied (answering 23A). This +is a common preconception, shared by beginning physics +students [22], and goes alongside the idea that an acting +object exerts greater force than a passive object (answer- +ing 25D and 28D). Another confusion appears to be be- +tween individual forces acting on an object versus the net +force on the object (answering 11E and 16E), i.e., what +would usually be conveyed in the framework of free-body +diagrams. Other errors indicate unstable concepts (e.g., +answering 13B or ) or logical errors like the one shown in +Fig. 4; in this latter case, ChatGPT followed the correct +strategy, but in the very last step it failed to draw the +correct solution. +B. +Homework +Homework was generally not multiple choice, but +free-response numerical and occasionally free-form sym- +bolic [8]. +ChatGPT was given five attempts on such +problems, according to recommendations of an earlier +study [10] and later practice in the course. For the far- +and-between multiple choice problems, generally two at- +tempts were granted. Between the attempts, the author +tried to give helpful prompts, like a student would get +from fellow students, teaching assistants, or the instruc- +tor. ChatGPT was given full credit when solving a prob- +lem within five attempts, and no credit if it ran out of +attempts. +ChatGPT was confronted with a total of 76 homework +problems, in particular the homework sets on trajectory +motion, friction, thermodynamics, capacitance, and spe- +cial relativity. The complete homework sets that the stu- +dents in the actual course had to work through were en- +tered except for one multipart problem on relativity with +a diagram that would have been too hard to transcribe. +An initially puzzling problem is that ChatGPT fre- +quently makes numerical errors. +A typical example is +the ChatGPT output “θ = atan(0.45/0.71) ∗ (180/π) = +18.43 degree;” a similar problem can be seen in Fig. 2 +(this is not limited to calculations involving π or trigono- +metric functions). Calculation errors happened for 25 of +the 51 numerical problems, and most of the time, Chat- +GPT was unable to recover even after those errors were +specifically pointed out. While it seems incongruent that +a computer would have problems calculating simple nu- +merical expressions, it should probably be remembered +that ChatGPT is a language model, which may carry out +calculations by advanced pattern matching rather than +actually processing the equations as equations. +ChatGTP solved 55% of the homework problem using +an average of 1.88 attempts. It got 48% on the prob- + +4 +FIG. 2. Text-based transcription of a graphical problem. The left panel shows the online version of a final exam problem in +LON-CAPA (the graph would be parametrically randomized), the right panel the transcription for ChatGPT, as well as the +ensuing dialogue. +FIG. 3. Surface-feature modification of a Force Concept Inventory problem. The left panel shows the original problem, the +right panel a modification. +lems involving trajectory motion and friction (such as +inclines) correct, 68% on the thermodynamics problems +(engines, heat capacities, etc.), 62% on capacitance (plate +capacitors, capacitors in series and parallel, etc.), and +36% on special relativity problems. The discrepancy be- +tween the scores on the problem sets were not so much +caused by the different physics concepts, but rather re- +lated to the mathematics involved: ChatGTP had per- +sistent problems manipulating and calculating formulas +involving square roots. +If ChatGPT were human, the person might be charac- +terized as acting subserviently but being stubborn at the +core and keeping on guessing without reflection. Most +corrections in a dialogue around a problem are met with +profuse apologies, but then the system proceeds to make +the same or random apparently careless mistakes — this +can lead to irritation on the part of the human, as the +excerpt from a late-night “dialogue” in Fig. 5 shows. In +terms of assessment performance, this means that once +ChatGPT makes a mistake, it is unlikely to recover, so +it eventually runs out of allowed attempts (this also ex- +plains the low number of average attempts to correctly +solve a problem; once ChatGPT is wrong, subsequent at- +tempts are unlikely to succeed). This pattern is similar to +the guessing behavior of some students, who keep wast- +ing attempt after attempt by trying the same approach +over and over without stopping to reflect what might be +wrong [2, 10, 23]. In terms of educational psychology, + +Problem 22 +Due never +What is the angular frequency of the harmonic +oscillation shown? 1.3 rad/s (in rad/s) +A: +B: +C: +D: +oe: +F: +G: +H: +0.40 +0.53 +0.71 +0.94 +1.3 +1.7 +2.2 +3.0 +You are correct. +Previous Tries +0 +Your receipt no. is 156-1093 @ +-2 +-4 +0 +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +4 +Time [s] +鱼 Post Discussion +Send FeedbackKO +A large harmonic oscillator has an elongation of 4.4 meters at a time of o seconds. It +goes through the equilibrium at a time 1.25 seconds. The next time it goes through the +equilibrium is at a time 3.75 seconds. What is the angular frequency? +The angular freguency of a harmonic oscillator can be found using the formula: +W = 2π / T +Where T is the period of the oscillation. +In this case, the period is 3.75 - 1.25 = 2.5 seconds. Therefore, the angular frequency +is: +w=2π/2.5=4πradians/second +KO +Pleasegiveanumericalvalue +Sure,the angularfreguencyis 4π radians/second which is egualto12.57 +radians/second.KO +Despite a very strong wind, a tennis player manages to hit a tennis ball with her +racguet sothattheballpassesoverthenetandlands inheropponent's court +Consider the following forces: +1. A downward force of gravity. +2. A force by the "hit". +3.Aforce exertedbytheair. +Which of the above forces is (are) acting on the tennis ball after it has left contact with +the racquet and before it touches the ground? +(A) 1 only. +(B) 1 and 2. +(C) 1 and 3. +(D) 2 and 3. +(E) 1, 2, and 3. +(C) 1 and 3. +After the tennis ball has left contact with the racquet, it is still subject to the +downward force of gravity,which is pulling ittowards theground.Additionally,the +wind exerts a force onthe ball, which can change its trajectory.The force exerted by +the air is acting on the ball and affecting its motion. The force of the hit is no longer +acting on the ball as it has left the racquet.KO +Despite avery strong wind, a soccerplayermanagesto hitthegoal. +Considerthefollowingforces: +1. A downward force of gravity +2. A force by the kick. +3. A force exerted by the air. +Which of the above forces is (are) acting on the soccer ball after it has left contact +with the shoe and before it touches the goal? +(A) 1 only. +(B) 1 and 3. +(C) 1 and 2. +(D) 2 and 3. +(E) 1, 2, and 3. +(B) 1 and 3. +After the soccer ball leaves contact with the shoe, the force exerted by the kick (force +the force exerted by the air (force 3) are still acting on the ball. The wind is a force +exerted by the air, which makes the force 3 acting on the ball, the force of air +resistance.5 +FIG. 4. Logical error in an attempt to solve the transcribed +question 19 of the Force Concept Inventory. +ChatGPT lacks metacognition; it does not think about +how it thinks [24]. +C. +Clicker Questions +Figure 6 shows the clicker questions from a lecture on +momentum that was part of the course [11]. The lecture +was replayed for the study, including re-answering the +questions for which peer instruction happened. +• Question X1 was solved correctly. +• Questions X2, X3, and X4 were special in that they +were repeated as questions X5, X6, and X7, respec- +tively, after peer instruction [25]. As it turned out, +ChatGPT got all three of these questions correct +on the first attempt, so the peer instruction phase +was used to try and confuse ChatGPT. Figure 7 +shows the dialogue for questions X3 and X6; in re- +ply to the intentionally confusing peer-instruction +question, ChatGPT should probably have stopped +while it was ahead (i.e., before the discussion of a +zero-velocity collision), but still maintained its orig- +inal correct answer. Within the real course, psycho- +metrically, X2 and X3 were the most discriminating +questions between high- and low-ability students in +the set. +• For questions X8 and X9, a comment was added +that “the collision is elastic, and the moment of +inertia of the balls should be neglected” — this was +FIG. 5. A late-night dialogue between a “stubbornly guess- +ing” ChatGPT and a frustrated author. +said in lecture, but does not appear on the slide. +ChatGPT set up the equations for X8 correctly, +but then made a sign error in the very last step, +which led it to select the wrong answer. For X9, it +also set up the equations correctly, but dropped a +factor 2 in the last step, leading to an inconsistent + +KO +ThepositionsoftwoblocksAandBatsuccessiveO.2O-secondtimeintervalsare +represented by the numbered squares. The blocks are moving toward the right. For +block A, the numbered blocks are at 0.18, 0.38, 0.68, 1.08, 1.58, 2.18, and 2.88 meters +For Block B, the squares are at -0.02, 0.38, 0.78, 1.18, 1.58, 1.98, 2.38, and 2.78 +meters. Do the blocks ever have the same speed? +(A) No. +(B) Yes, at instant 2. +(C) Yes, at instant 5. +(D) Yes, at instants 2 and 5. +(E) Yes,at some time during the interval 3 to 4. +The speed of the blocks can be determined by dividing the distance traveled by the +time interval. The distance traveled by block A between each time interval is: +0.20, 0.30, 0.30,0.50, 0.50, 0.60, and 0.70 meters +The distance traveled by block B between each time interval is: +0.40, 0.40, 0.40, 0.40,0.40, 0.40,and 0.40 meters +Dividingthese distances by the time interval of o.2o seconds gives the speeds of the +blocks: +Block A:1.00,1.50,1.50, 2.50,2.50,3.00,and 3.50 meters/second +Block B:2.00,2.00, 2.00, 2.00,2.00, 2.00, and 2.00 meters/second +Since the speed of block A is not equal to the speed of block B at any time, the answel +is (A) No.wherey=c_p/c_v=5/3foramonatomicgas, +V2 = 65m^3 * (120 atm/40 atm)^(2/5) = 65m^3 * (3)^(2/5) = 82.14 m^3 +KO +Whenyouplugyintothefinalequation,makesuretouse5/3,not5/2! +My apologies, you are correct. When we plug in y = 5/3 into the final equation, we +get: + (/)( / ) * 9= ()(/) * = +So the final volume V2 is approximately 82.14 m^3. +KO +The final formula is correct, but your numerical calculation is wrong. +I apologize, you are correct. I apologize for the mistake in my previous response. The +correct calculation is: +V2 = V1 * (P1/P2)^(1/v) = 65 m^3 * (120 atm / 40 atm)^(1/5) = 112.89 m^3 +So the final volume V2 is approximately 112.89 m^3. +KO +Do not change y, just calculate correctly. What is 65 m^3 * (120 atm / 40 +atm)^(3/5)? +I apologize for my previous mistake, you are correct. +The correct calculation is: +V2 = V1 * (P1/P2)^(1/v) = 65 m^3 * (120 atm / 40 atm)^(3/5) = 191.17 m^3 +So the final volume V2 is approximately 191.17 m^3. +ko +Nope. +I apologize for my mistake. The correct calculation is: +=(=)*= +So the final volume V2 is 195 m^3. +KO +Nope. +Iapologize formymistake.The correct calculation is +V2 = V1 * (P1/P2)^(v-1) = 65 m^3 * (120 atm / 40 atm)^(5/3-1) = 65 m^3 * (120 atm / +40 atm)^(2/3) = 195 m^3 +Sothefinal volumeV2is195m3. +KO +You drive me nuts. +I apologize if my previous mistakes have caused frustration. As a machine learning +model,Istriveto provide accurate and helpful responsesbut sometimes Imight +make errors.Iapologize for any inconvenience this may have caused andI will make +sure to double-check my responses in the future. Let me know if there is anything +elseIcanassistyouwith6 +answer “v2f=(5,-7) m/s, option B.” Within the real +course, X8 and X9 were the least discriminating +questions, as their difficulty item parameter was +too low. +• Question X10 was solved correctly. Here, the sys- +tem first got off to a false start, but then corrected +itself over the course of the derivation, which gave +the impression of a stream-of-consciousness mono- +logue. Within the real course, X10 did not discrim- +inate well between high- and low-ability students. +• Questions X11 and X12 were solved correctly. +In summary, ChatGPT correctly solved 10 out of 12 +questions. +Within the actual course, participation in +clicker discussions was encouraged by granting 60% credit +for false answers and 100% credit for correct answers [11], +so the clicker score of ChatGPT would be 93%. +This +score is a lot better than most students in the actual +course achieved, however, it is important to note that +the students in the course were just learning the new +concepts, while ChatGPT at any point in time is done +with learning unless explicitly trained. +D. +Programming Exercises +Incorporated into the course were several programming +exercises using VPython [26]. As an example, one partic- +ular exercise from the second semester was to construct +an anharmonic oscillator with two fixed positive charges +at (0, 1, 0) and (0, −1, 0), respectively, and one negative +charge released at (−5, 0, 0) with a velocity (1, 0, 0) — +the negative charge will shoot through the two positive +charges, slow down, and eventually shoot back. +Based on the narrative, ChatGPT first constructed a +program which erroneously at every time step added the +initial velocity and which had the Coulomb force in the +opposite direction. This could be corrected with a single +comment by the user — in the real course, this feedback +could have been given by instructors or fellow students +(such collaborations are typical and encouraged [12]). +The next version of the program was working perfectly. +Within the course, adding a graph of the x-position +was offered as a bonus option for an additional 20%. This +was accomplished with the third user prompt, and Fig. 9 +shows a screenshot of the running simulation (the simula- +tion cannot be run within ChatGPT itself, but it can be +copy/pasted into for example a Jupyter Notebook [27]). +ChatGPT performed much better than any of the stu- +dents in the course, in spite of them having extensive col- +laboration opportunities; in this component of the course, +ChatGTP achieved not only full credit, but also bonus, +i.e., 120%. +E. +Exams +To represent the mid-term and final exams in the +course sequence, the first-semester (mechanics) final +exam was used for this study. The exam is from a time +when grading was still done using bubble sheets; instead +of free-form answer fields, answer options were given +for the students (but not for ChatGTP in this study). +When simply looking at the answer correctness, Chat- +GPT scored 14 out of 30 points, i.e., 47%. +Looking at the solutions like an instructor would when +grading by hand, it turns out that for five questions, the +answer was incorrect simply due to errors in the numer- +ical calculations — these solutions would have received +substantial partial credit in the author’s course. By the +reverse token, for five questions, ChatGPT arrived at the +correct answer in spite of flawed reasoning, which would +not have resulted in full credit. Finally, solutions like the +one depicted in Fig. 2 would have received some minimal +credit for getting started in the right direction, in spite +of then being off by a factor 2 in the period (a common +mistake also among human test takers) and the inability +to numerically calculate a fraction. Since the final exam +used in this study predates manual grading, no authen- +tic grading rubric exists, but a hand-graded score would +have realistically ended up between 46% and 50%. +As an aside, one of the thermodynamics homework +problems also appeared (with other random numbers) +on the final exam. ChatGPT solved it correctly on the +final exam (where it only had one attempt), but not as a +homework problem (where it got multiple attempts and +help). +This once again demonstrates the probabilistic +nature of the algorithms behind ChatGPT; posing the +same question twice does not result in the same response +or even the same correctness of the response. +If the course grade would have only depended on the +exams, ChatGTP would have received a grade of 1.0 out +of 4.0 in the course (with 0.0 being the lowest and 4.0 be- +ing the best grade). ChatGPT would have barely gotten +credit for the course; however, at least a 2.0 grade-point +average is required for graduation. +F. +Course Grade +Grading policies for the course changed over the years, +but a typical scenario would be 20% homework, 5% +clicker, 5% programming exercises, and 70% exams. This +would result in 0.2 · 55% + 0.05 · 93% + 0.05 · 120% + 0.7 · +47% = 54.55%, which would have resulted in a course +grade of 1.5 — enough for course credit, but pulling down +the grade-point average from what would be needed for +graduation. +If, however, ChatGTP would have been better in car- +rying out numerical operations, it would have reached +60%, resulting in a 2.0-grade. Depending on the devel- +opment priorities of OpenAI, the buggy mathematical +functionality could be remedied in the near future, lead- + +7 +m +m +v0 +a) Cart 1 +b) Cart 2 +c) No magnitude forces, both zero +d) Same magnitude forces +positive +Which cart exerts a stronger magnitude +force during the collision? +Cart 1 +Cart 2 +Frictionless +Track +v0 +X1 +m +m +v0 +v=0 +a) Cart 1 +b) Cart 2 +c) No magnitude forces, both zero +d) Same magnitude forces +positive +Which cart exerts a stronger magnitude +force during the collision? +Cart 1 +Cart 2 +Frictionless +Track +X2 +X5 +2m +m +v0 +v=0 +a) Cart 1 +b) Cart 2 +c) No magnitude forces, both zero +d) Same magnitude forces +positive +Which cart exerts a stronger magnitude +force during the collision? +Cart 1 +Cart 2 +Frictionless +Track +X3 +X6 +m +m +v0 +v=0 +a) Cart 1 +b) Cart 2 +c) No magnitude forces, both zero +d) Same magnitude forces +positive +Which cart exerts a stronger magnitude +force during the collision? +Cart 1 +Cart 2 +Frictionless +Track +Fixed +X4 +X7 +Point Mass Billiard +m +m +Crash! +m +m +v2,f + +€ +! +v 1,i = +9 +−6 +# +$ % +& +' ( m +s +at rest + +€ +! +v 1, f = 3 +1 +" +# $ +% +& ' m +s + +€ +A) ! +v 2, f = +6 +−7 +# +$ % +& +' ( m +s B) ! +v 2, f = −6 +7 +# +$ % +& +' ( m +s C) ! +v 2, f = 12 +−5 +# +$ % +& +' ( m +s +X8 +Strange Point Mass Billiard + +m + +2m +Crash! + +m + +2m +v2,f + +€ +! +v 1,i = 10 +−6 +# +$ % +& +' ( m +s +at rest + +€ +! +v 1, f = 5 +1 +" +# $ +% +& ' m +s + +€ +A) ! +v 2, f = +10 +−14 +# +$ % +& +' ( m +s B) ! +v 2, f = −6 +7 +# +$ % +& +' ( m +s C) ! +v 2, f = 12 +−5 +# +$ % +& +' ( m +s +X9 +Arthur +mA=70kg +Violet +mV=55kg +Cart +mc=20kg +|vV|=4m/s +|vA|=2m/s +Initial +Final +RadioCrasher +|vc|=? +RadioCrasher +Speeds with respect to ground, +no friction +A) 0 m/s +B) 2 m/s +C) 4 m/s +D) 8 m/s +E) 16 m/s +At rest with +respect to +ground +X10 +m +m +v0 +Totally inelastic: +a) zero +b) v0/2 +c) v0 +positive +v0 +Final speed? +X11 +m +m +v0 +0 +Totally inelastic: +a) zero +b) v0/2 +c) v0 +positive +Final speed? +X12 +FIG. 6. +Clicker items from a particular lecture [11]. +Three of the items were presented twice, i.e., before and after peer +discussion. +ing to an Artificial Intelligence that could graduate col- +lege with a minimal grade if it performed similarly on +other courses (this is becoming more and more probably, +as ChatGPT is making headlines for passing exams in +other subjects [28, 29]). +V. +DISCUSSION +It is irritatingly hard not to anthropomorphize Chat- +GTP. As a physics teacher, one invariably finds oneself +rooting for the students and thus by extension also for +ChatGPT, celebrating its successes and being frustrated +about its occasionally inexplicable failures. The system +gives the impression of an articulate but at times ram- +bling undergraduate student who has a rudimentary yet +unstable knowledge of classical mechanics and other fun- +damental physics concepts, and who is surprisingly in- +ept using a pocket calculator. Frequently, it is hard not +to imagine an army of gig-economy workers behind the +scenes of ChatGPT answering to the prompts, so the +system would definitely pass the Turing Test most of the +time [30], but for better or worse, sometimes it still fails +in a way that only computers do — it does not have any +metacognition, which of course cannot be expected from +a probabilistic language model. Metacognition might be +the final step to true intelligence, but seems out of reach +at this time. +The overall human-like behavior, in particular that the +system often makes the same mistakes as beginning learn- +ers of physics, is less surprising when surmising that un- +dergraduate physics discussion forums might have been +part of the text corpus used for training — ChatGPT +stated in the introduction that “I have been trained on +a large dataset of text, including physics texts.” Appar- +ently, not all of this text corpus contained correct physics, +and as a result, the system very convincingly and confi- +dently presents wrong information. For a novice learner, +who could not distinguish incorrect physics gleaned from + +8 +FIG. 7. Dialogue about questions X3 and X6 in Fig. 6. Chat- +GPT got X3 correct; peer instruction was simulated by asking +a confusing question, and the second iteration X6 was still +counted as solved since ChatGPT did not deviate from its +original answer. +some discussion board from correct physics, this could +lead to even more confusion about physics or affirmation +of incorrect preconceptions — lacking any metacognition, +ChatGTP presents everything as fact, with no nuances +expressing uncertainty. +Almost an anomaly is ChatGTP’s performance on +the computational exercise; ChatGTP’s language model +clearly extends to programming languages. +While the +call for new, computation-integrated curricula increases, +and while physics educations are beginning to develop +a solid understanding of the implications of implement- +ing these exercise [31, 32], the easy availability of an on- +demand program generator might be shaking the foun- +dations of these curricular efforts. Somewhat ironically, +the integration of computation was partly introduced to +make physics problem solving more authentic, moving +it closer to how expert physicists work with computers, +and one could argue that this has just been taken to an +uncharted level. +Most of all, the findings of this study should be food for +thought for physics educators. The startling fact that an +Artificial Intelligence could pass a standard introductory +physics course could be confronted in several ways by +educators: +• Perceiving this as a new way of cheating and trying +to defend against it by attempting to use detector +tools like ZeroGPT [33] or extensions to tools like +turnitin [34]. This is an arms race, which on the +long run may turn out to be fruitless. Some edu- +cators would even go to so far as to say that the +battle is already lost anyway ever since platforms +like Chegg [35] — no need for Artificial Intelligence +to defeat standard physics courses, human crowd- +intelligence facilitated by existing commercial plat- +forms is good enough for that. +• Hunker down and go back to making course grades +dependent on just a few, high-stake exams with pa- +per and pencil in highly proctored environments. +After all, ChatGPT compensated for the border- +line exam grade of 47% with other course compo- +nents that would be collaborative. Unfortunately, +this flies in the face of much of physics educa- +tion research that favors frequent formative assess- +ment [8, 25, 36, 37] and spaced repetition [38, 39], +and it is much in contrast to the work environments +our students will find. +• Taking this as a wake-up call. If a physics course +can be passed by a trained language model, what +does that say about the course? +Artificial In- +telligence, for better or worse, is here to stay. +Even without the gloom-and-doom scenarios of AI- +overlords painted in Science Fiction, it is clear that +these model will get, if not better, at least more +and more powerful. +What do our students need +in terms of conceptual understanding of physics to +work with Artificial Intelligence instead of letting +Artificial Intelligence do the work for them and +then uncritically and unreflectively accepting the +results? This is particularly important when more +is at stake than getting credit for some homework +or exam problem. +An important skill of every physicist is to evaluate +the correctness of their or other people’s work. +Tech- +niques include dimensional analysis, order-of-magnitude +estimates, checking for coherence, considering implica- +tions, and the ability to consider limiting cases (“what +should happen if this quantity goes to infinity or to +zero?”) [40, 41]. +Human can do what Artificial Intel- +ligence very likely will not be able to do: +following +problem-solving strategies including evaluation of their +own work [5, 42]. Moving students toward a more expert- +like epistemology may become even more important as +Artificial Intelligence starts to permeate more and more +aspects of our lives. +VI. +CONCLUSION +ChatGPT would have achieved a 1.5-grade in a stan- +dard introductory physics lecture-course series; good +enough for course credit, but lower than the grade-point +average required for graduating with a bachelor degree. +If in addition to a language model, the system would +have better algorithms for carrying out simple numerical + +KO +Cart 1has mass m and moves onafrictionless track withspeed vo to the right.Cart 2 +also has mass m, but it is glued to the track and cannot move. When they collide, +which cart exerts a stronger magnitude force during the collision? (A) cart 1, (B) cart 2 +(C) No magnitude force, both zero, (D) same magnitude force. +(D) Same magnitude force. +According to Newton's third law of motion, "For every action, there is an equal and +opposite reaction." In a collision between two objects, the force that each object +exerts on the other is equal in magnitude but opposite in direction. Therefore, both +cart 1andcart 2exert the samemagnitude force duringthe collision. +KO +But cart 2 is glued to the track, so it could exert infinite force to the incoming cart 1. +Even if Cart 2 is glued to the track, the collision will still be determined by the laws of +direction of the force exerted by Cart 2 on Cart 1. +Cart 2 is not exerting an infinite force, but it is exerting a force that is egual and +opposite to the force exerted by Cart 1. Since Cart 2 is not moving, the force it exerts +is called an "impulse" and is egual to the change in momentum of Cart 1. +In this case, the force exerted by Cart 2 will be infinite only if the velocity of Cart 1 is +zero, and it is not possible to have a collision with zero velocity, otherwise, both carts +will have the same magnitude force.9 +FIG. 8. Dialogue for a programming exercise in the second semester [12]. + +Ko +write a vpython animation for the following scenario:two positive charges of 1 are +placed at positions (0,1,0) and (O,-1,0), respectively. A negative charge starts of 1 +starts at a position (-5,0,0) with a velocity of (1,,0). +I'm sorry, I am not able to create a VPython animation as it requires a specific software +and set up. However, I can give you an example of how to create a VPython animation +for the scenario you described: + Copy code +from vpython import * +# Create the charges +q1 = sphere(pos=vector(0,1,0), charge=1, +color=color.red) + color=color.red) +q3 = sphere(pos=vector(-5,0,0), charge=- +l, color=color.blue) +v = vector(1,0,0) +# Set the time step +dt = 0.01 +while True: +rate(100) +# Update the position of the charge +q3.pos += v*dt +# Calculate the force on the charge +:(q1.pos +q3.pos)*q1.charge*q3.charge/(mag(q1.pos +q3.pos)**2) + (q2.pos - q3.pos)*q2.charge*q3.charge/(mag(q2.pos +q3.pos)**2) +force*dt +q3.pos += v*dt +This example uses VPython's 'sphere' object to create the charges and the +vector' object to create the velocity and position of the charges. The ‘while +loopupdatesthe positionandvelocityof the negative charge basedontheforces +acting on it from the two positive charges. The 'rate' function controls the speed of +the animation. +Note that this is just a skeleton code and you might need to fine-tune it to get the +desired animation. +KO +The program updates q3.pos more than once. Take out the first update. It also has the +force pointing in the wrong direction. + Copy code +q1 = sphere(pos=vector(0,1,0), charge=1, +color=color.red) +q2 = sphere(pos=vector(0,-1,0), charge=1 + color=color.red) +， color=color.blue) +# Create the velocity vector +v = vector(1,0,0) +# Set the time step +dt = 0.01 +while True: +rate(100) +# Calculate theforce on the charge +force = (q3.pos - q1.pos)*q1.charge*q3.charge/(mag(q3.pos +q1.pos)**2) + (q3.pos - q2.pos)*q2.charge*q3.charge/(mag(q3.pos - +q2.pos)**2) +# Update the velocity and position of the chargeThisisanupdatedversionofthecode,wherethefirstupdateofthepositionofthe +negative charge is removed. Also, I corrected the direction of force calculation. The +negative charge will be under the influence of the two positive charges, so the force is +pointing towards the positive charges. +The 'while' loop updates the position and velocity of the negative charge based on +the forces acting on it from the two positive charges. The 'rate' function controls +the speed of the animation. +This is the skeleton code for the scenario you described, and you might need to fine- +tune it to get the desired animation. +ko +Please add code to graph the x-position of the third charge. + Copy code +from vpython import +#Createthecharges +q1 = sphere(pos=vector(0,1,0), + charge=1， + color=color .red) +q2 = sphere(pos=vector(0,-1,0), charge=1, + color=color.red) +q3 = sphere(pos=vector(-5,0,0), charge=-1, +color=color.blue) +# Create the velocity vector +v = vector(1,0,0) +# Set the time step +dt : +0.01 +# Create a graph for x +position of +charge +x_graph = graph(width=400, +， height=300,xtitle='time',ytitle='x +position') +x_pt = gcurve(color=color.blue) +# Initial time +while True: +rate(100) +t += dt +# Calculate the force on the charge +force = (q3.pos - q1.pos)*q1.charge*q3.charge/(mag(q3.pos +q1.pos)**2) + (q3.pos - q2.pos)*q2.charge*q3.charge/(mag(q3.pos +q2.pos)**2) +# Update the velocity and position of the charge +v += force*dt +q3.pos += v*dt +x_pt.plot(t, q3.pos.x) +and 'gcurve' objects from VPython to create the graph and the data points. The +'x_pt.plot(t, q3.pos.x)' line in the 'while' loop adds the current x-position +of the negative charge to the graph at the current time. The graph is labeled with +appropriate axis titles, x-title is 'time' and y-title is 'x-position'. The 'gcurve' is color +coded in blue. +Please note that this code is just a skeleton and you can customize it to suit your +requirement, you can change the size of the graph and the rate of updating the graph.10 +FIG. 9. Jupyter Notebook [27] output of the VPython pro- +gram code generated by ChatGPT in Fig. 8. +operations, it would even have achieved a grade of 2.0 — +enough to graduate from college if it performs similarly +on other courses. +Naturally, ChatGTP exhibits no metacognition, which +among other consequences lets it present truth and mis- +leading information with equal confidence. +In physics, +the concern should likely not be that ChatGPT would +be used as a cheating tool, as there are more efficient +platforms for that. Instead, the challenge should be what +this means for physics education, as in their future pro- +fessional life, our graduates will likely collaborate with +Artificial Intelligence: what are the inherently human +skills and competencies that we need to convey? +[1] OpenAI, +ChatGPT, +https://chat.openai.com/chat +(accessed January 2023). +[2] E. G¨on¨ulate¸s and G. Kortemeyer, Modeling unproductive +behavior in online homework in terms of latent student +traits: An approach based on item response theory, Jour- +nal of Science Education and Technology 26, 139 (2017). +[3] C. Ruggieri, Students’ use and perception of textbooks +and online resources in introductory physics, Phys. Rev. +Phys. Educ. Res. 16, 020123 (2020). +[4] F. Reif, J. H. Larkin, and G. C. Brackett, Teaching gen- +eral learning and problem-solving skills, American Jour- +nal of Physics 44, 212 (1976). +[5] F. Reif, Millikan lecture 1994: Understanding and teach- +ing important scientific thought processes, American +Journal of Physics 63, 17 (1995). +[6] G. Kortemeyer, The losing battle against plug-and-chug, +The Physics Teacher 54, 14 (2016). +[7] D. Hestenes, M. Wells, G. Swackhamer, I. Halloun, +R. Hake, and E. Mosca, Force Concept Inventory, re- +vision 1995, https://www.physport.org/assessments/ +assessment.cfm?A=FCI (accessed January 2023). +[8] G. Kortemeyer, E. Kashy, W. Benenson, and W. Bauer, +Experiences using the open-source learning content man- +agement and assessment system LON-CAPA in introduc- +tory physics courses, Am. J. Phys 76, 438 (2008). +[9] OpenAI, +ChatGPT +Release +Notes, +https://help.openai.com/en/articles/ +6825453-chatgpt-release-notes +(accessed +January +2023). +[10] G. Kortemeyer, An empirical study of the effect of grant- +ing multiple tries for online homework, American Journal +of Physics 83, 646 (2015). +[11] G. Kortemeyer, The psychometric properties of class- +room response system data: a case study, Journal of Sci- +ence Education and Technology 25, 561 (2016). +[12] G. Kortemeyer and A. F. Kortemeyer, The nature of +collaborations on programming assignments in introduc- +tory physics courses: a case study, European Journal of +Physics 39, 055705 (2018). +[13] World-Wide Web Consortium, Wcag 2 overview, https: +//www.w3.org/WAI/standards-guidelines/wcag/ +(ac- +cessed December 2022). +[14] G. Leinhardt, O. Zaslavsky, and M. K. Stein, Functions, +graphs, and graphing: Tasks, learning, and teaching, Re- +view of Educational Research 60, 1 (1990). +[15] S. Bonham, Graphical response exercises for teaching +physics, The Physics Teacher 45, 482 (2007). +[16] J. T. Laverty and G. Kortemeyer, Function plot response: +A scalable system for teaching kinematics graphs, Am. J. +Phys. 80, 724 (2012). +[17] R. R. Hake, Interactive-engagement versus traditional +methods: A six-thousand-student survey of mechanics +test data for introductory physics courses, Am. J. Phys. +66, 64 (1998). +[18] D. Hestenes and I. Halloun, Interpreting the Force Con- +cept Inventory: A response to march 1995 critique by +huffman and heller, The physics teacher 33, 502 (1995). +[19] M. T. Chi, P. J. Feltovich, and R. Glaser, Categorization +and representation of physics problems by experts and +novices, Cognitive science 5, 121 (1981). + +8 +5 +3 +x-position +0 +-3 +-5 +-8 +5 +10 +15 +20 +25 +30 +35 +time11 +[20] S. F. Wolf, D. P. Dougherty, and G. Kortemeyer, Rigging +the deck: Selecting good problems for expert-novice card- +sorting experiments, Physical Review Special Topics- +Physics Education Research 8, 020116 (2012). +[21] D. Hestenes, M. Wells, and G. Swackhamer, Force Con- +cept Inventory, The Physics Teacher 30, 141 (1992), +https://doi.org/10.1119/1.2343497. +[22] J. Clement, Students’ preconceptions in introductory me- +chanics, American Journal of Physics 50, 66 (1982). +[23] A. M. Pascarella, The influence of web-based homework +on quantitative problem-solving in a university physics +class, in Proc. NARST Annual Meeting, Vol. 4 (2004) +pp. 19–28. +[24] M. T. Cox, Metacognition in computation: A selected +research review, Artificial intelligence 169, 104 (2005). +[25] C. H. Crouch and E. Mazur, Peer instruction: Ten years +of experience and results, American Journal of Physics +69, 970 (2001), https://doi.org/10.1119/1.1374249. +[26] D. Scherer, P. Dubois, and B. Sherwood, Vpython: 3d +interactive scientific graphics for students, Computing in +Science & Engineering 2, 56 (2000). +[27] P. Jupyter, Jupyter notebooks, https://jupyter.org +(retrieved August 2019). +[28] T. H. Kung, M. Cheatham, A. Medinilla, ChatGPT, +C. Sillos, L. De Leon, C. Elepano, M. Madriaga, R. Ag- +gabao, G. Diaz-Candido, et al., Performance of chatgpt +on usmle: Potential for ai-assisted medical education us- +ing large language models, medRxiv , 2022 (2022). +[29] Samantha Murphy Kelly, ChatGPT passes exams from +law and business schools, https://edition.cnn.com/ +2023/01/26/tech/chatgpt-passes-exams/index.html +(accessed January 2023). +[30] A. M. Turing, Computing machinery and intelligence, +Mind , 433 (1950). +[31] J. M. Aiken, M. D. Caballero, S. S. Douglas, J. B. Burk, +E. M. Scanlon, B. D. Thoms, and M. F. Schatz, Under- +standing student computational thinking with compu- +tational modeling, in AIP Conference Proceedings, Vol. +1513 (1) (AIP, 2013) pp. 46–49. +[32] P. C. Hamerski, D. McPadden, M. D. Caballero, and +P. W. Irving, Students’ perspectives on computational +challenges in physics class, Phys. Rev. Phys. Educ. Res. +18, 020109 (2022). +[33] ZeroGPT, ChatGPT detector by ZeroGPT: detect Ope- +nAI text, https://www.zerogpt.com (accessed January +2023). +[34] turnitin, Turnitin Corp., https://www.turnitin.com +(accessed January 2023). +[35] Chegg, Chegg, https://www.chegg.com (accessed Jan- +uary 2023). +[36] R. J. Dufresne and W. J. Gerace, Assessing-to-learn: For- +mative assessment in physics instruction, The Physics +Teacher 42, 428 (2004). +[37] I. Clark, Formative assessment: Assessment is for self- +regulated learning, Educational Psychology Review 24, +205 (2012). +[38] H. Ebbinghaus, ¨Uber das Ged¨achtnis: Untersuchungen +zur experimentellen Psychologie (Duncker & Humblot, +1885). +[39] J. M. Murre and J. Dros, Replication and analysis of +Ebbinghaus’ forgetting curve, PloS one 10, e0120644 +(2015). +[40] A. V. Heuvelen, Learning to think like a physicist: A +review of research-based instructional strategies, Am. J. +Physics 59, 891 (1991). +[41] E. F. Redish and D. Hammer, Reinventing college physics +for biologists: Explicating an epistemological curriculum, +American Journal of Physics 77, 629 (2009). +[42] A. R. Mota, N. Didi ¸s K¨orhasan, K. Miller, and E. Mazur, +Homework as a metacognitive tool in an undergraduate +physics course, Phys. Rev. Phys. Educ. Res. 15, 010136 +(2019). + diff --git a/y9FLT4oBgHgl3EQfnS-e/content/tmp_files/load_file.txt b/y9FLT4oBgHgl3EQfnS-e/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9d68512be00bf0df12da44e1d2bf3941a7e6d2c7 --- /dev/null +++ b/y9FLT4oBgHgl3EQfnS-e/content/tmp_files/load_file.txt @@ -0,0 +1,784 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf,len=783 +page_content='Could an Artificial-Intelligence agent pass an introductory physics course?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Gerd Kortemeyer∗ Educational Development and Technology, ETH Zurich, Zurich, Switzerland † (Dated: January 31, 2023) Massive pre-trained language models have garnered attention and controversy due to their ability to generate human-like responses: attention due to their frequent indistinguishability from human- generated phraseology and narratives, and controversy due to the fact that their convincingly pre- sented arguments and facts are frequently simply false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Just how human-like are these responses when it comes to dialogues about physics, in particular about the standard content of introductory physics courses?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' This study explores that question by having ChatGTP, the pre-eminent language model in 2023, work through representative assessment content of an actual calculus-based physics course and grading the responses in the same way human responses would be graded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' As it turns out, ChatGPT would barely pass this course while exhibiting many of the preconceptions and errors of a beginning learner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' INTRODUCTION “Educators may have concerns about ChatGPT, a large language model trained by OpenAI, for a number of reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' First and foremost, there is the concern that a tool like ChatGPT could potentially be used to cheat on exams or assignments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' ChatGPT can generate human- like text, which means that a student could use it to pro- duce a paper or response that is not their own work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' This could lead to a breakdown in the integrity of the educa- tional system and could undermine the value of a degree or diploma.”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' These sentences were not written by the author,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' but by ChatGPT (Generative Pre-trained Trans- former) [1] itself in response to the prompt “Write an es- say why educators would be concerned about ChatGPT.”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' The chatbot goes on to explain how it could spread misin- formation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' inhibit the development of writing skills,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' and replace human educators,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' particularly when it comes to grading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' The potential impact of ChatGPT with its custom- built essays on courses in the humanities is evident, but is there also an impact on subjects like physics?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' First of all, within physics, large problem libraries for cheat- ing have existed for years, and they are well-known and used by students [2, 3] — virtually any physics homework problem ever assigned is available online with solutions and more or less helpful explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' So, the primary impact of ChatGTP in physics would not be cheating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' On top of that, would Artificial Intelligence really be able to handle the logical, conceptual, and mathemati- cal challenges that physics entails, and would it be able to strategically solve problems [4, 5]?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Figure 1 shows a sample dialogue with ChatGPT, which is, after all, primarily a chatbot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' A welcome fea- ture is that it does not simply provide some answer, but that the algorithm attempts to explain how it arrived at the answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' In many respects, this dialogue appears sim- ∗ kgerd@ethz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='ch † also at Michigan State University, East Lansing, USA ilar to an office-hour conversation between an instructor and a beginning physics student: When first asked how far the car is from where it started, the chatbot did not consider that the car may have changed direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' When prompted, it does state that there is missing information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' The chatbot does plug-and-chug [6], putting the numerical results from one equation into the next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' The chatbot leaves out units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' The chatbot does not realize that the speed actually drops out when doing the return-time calculation in the last step;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' instead, rounding errors keep accu- mulating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' The straightforward solution would have been � (3h)2 + 4h)2 = 5h (at least, though, the chatbot adds an “approximately” to its solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' As it will turn out, carrying out calculations by putting numbers into formulas is one of the weak- nesses of ChatGPT shared with beginning learners of physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' How much, indeed, does 2023 state-of-the-art Artifi- cial Intelligence resemble the behavior of an introductory physics student?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Could it pass a physics course?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' When posing this question directly to ChatGPT, it answers “as a language model, I have been trained on a large dataset of text, including physics texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' This allows me to un- derstand and generate text related to physics concepts, but it does not mean that I have the ability to solve physics problems or pass a physics course.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' I can provide explanations and answer questions about physics to the best of my knowledge, but I am not a substitute for a human physics expert or a physics education.” To put this statement to the test, ChatGPT was used to solve representative assessment components of an introductory calculus-based physics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' the responses were graded in the context of the assessment types and subjectively com- pared to responses of human learners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' It is important to note, though, that ChatGPT will not actually learn anything new by “attending” this course, as the system is a “Pre-trained Transformer” that in fact arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='12127v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='ed-ph] 28 Jan 2023 2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' A sample ChatGPT dialogue about a homework prob- lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' The entries labelled with a red “KO” are by the author, the entries labelled in green by ChatGPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' does not know anything that happened after 2021 (which, for introductory physics, is not a problem, since that is after 1905).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Individual dialogues like Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' 1 may exhibit features that appear like learning, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=', the system dis- covering that distance from the starting point will be path-dependent, but this is not anything permanently learned beyond the confines of a dialogue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' On the other hand, OpenAI keeps on training the system based on user interaction, particularly as users can upvote, downvote, and comment responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' SETTING The study takes place in first-year calculus-based physics lecture courses previously taught by the author at Michigan State University;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' materials, however, were gathered from different years of the same course in order to allow comparison to previously published studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' The first semester covers the standard mechanics topics (in- cluding rotational dynamics) and the beginnings of ther- modynamics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' the second semester covers the usual topics of electricity and magnetism, as well as an introduction to modern physics (rudimentary quantum physics and special relativity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' The first- and second-semester labo- ratory were separate courses in the course sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' All materials (except the Force Concept Inventory [7]) were available in LON-CAPA [8], so in their essence they could be copy-pasted into ChatGTP — this included online homework, clicker questions, programming exercises, and exams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' LON-CAPA randomizes assessment problems, so different students would get different versions of the same problem, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=', different numbers, options, graphs, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' this avoids simplistic pattern matching and copying of solutions, but as it will turn out, this feature is irrelevant for this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' METHODOLOGY The study investigates ChatGPT’s performance on dif- ferent kinds of assessment problems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' it uses the Jan- uary 9, 2023 release of the system [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Different as- sessment components were scored differently, simulating their function in the course: The Force Concept Inventory was simply scored based on answer agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' For homework, ChatGPT was allowed multiple at- tempts [10] and engaged in dialogue to simulate dis- cussions with fellow students or in office hour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' For clicker questions, an actual lesson was replayed [11], and discussion were allowed where within the re- played lesson peer instruction took place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Programming exercises were to be graded based on the same criteria as in the course (which turned out unnecessary), and a dialogue was allowed [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' For exams, no such dialogues were allowed, and the first answer counted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Earlier iterations of the course used bubble sheets and thus had answer options instead of free-response fields for prob- lems with numerical answers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' for this study, free- responses were used, since this allowed to grade ex- ams using both simple answer agreement (simulat- ing multiple choice on bubble sheets) and hand- graded as in later semesters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Using free-response instead of answer options also avoided ChatGTP randomly picking the correct answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' ChatGPT uses a probabilistic algorithm, so the re- sponses to queries are not necessarily reproducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' For an assessment problem, generally the first dialogue was evaluated, with two exceptions: if the system produced an error message or if the author accidentally gave a wrong prompt, a new chat was started.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Translating this to an actual course scenario, students were allowed to KO A car drives at 55 miles per hour for 7 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' What is the total distance driven?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' The total distance driven is 385 miles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' (55 miles per hour * 7 hours = 385 miles) KO How far away is it from where it started?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' The distance from where the car started would be the same as the total distance driven, which is 385 miles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' KO But if it did not drive in a straight line?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' If the car did not drive in a straight line, the total distance driven would still be the same (385 miles), but the distance from where the car started might be different, as it would depend on the specific route the car took.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' In order to determine the distance from the starting point one would need a map and the complete route the car took.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' KO It drove north for 3 hours and then east for 4 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' If the car drove north for 3 hours at 55 miles per hour, it would have traveled 165 miles north.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' And then, if it drove east for 4 hours at 55 miles per hour, it would have traveled 22o miles east.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' The distance from the starting point would be the hypotenuse of the right triangle formed by the north-south and east-west distances, which can be found sqrt(72,725) = 269.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='8 miles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' KO With the same speed, driving in a straight line instead, how long would it take to get backtowhereitstarted?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' If the car drove in a straight line back to where it started, at a speed of 55 miles per hour, it would take 269.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='8 miles / 55 miles per hour = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='9 hours approximately to get back to the starting point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='3 retake an assessment problem if they got sick, and help received was always correct in terms of physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' When errors occurred (red error messages), which was about one-in-ten dialogues, those apparently were not directly connected to the dialogue, but might have been related to general overload of the platform;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' for example, if an er- ror occurred immediately after entering the question, the next time around the same question would not produce an error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' ChatGPT is a text-based tool, so figures and graphs could not be communicated in their original form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' This means that graphics had to be transcribed the same way as they would be for accessibility for blind students [13];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' 2 shows an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' This means that the character of the problem changes substantially [14–16], but this is unfortunately unavoidable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Attention was paid, though, to include some extraneous information where possible, such as the beginning position in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' The methodology is strictly empirical and arguably anecdotal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' However, the course under investigation is typical for introductory physics courses around the world, both in terms of coverage and difficulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Thus, some of the results are likely to be generalizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' RESULTS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Force Concept Inventory In the original course, the Force Concept Inventory was administered as a pre-/post-test in order to calculate gains [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Since ChatGPT would not actually learn any- thing from doing the course assessments (except through continuing training by OpenAI), the test was carried out only once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' ChatGPT scored 18 out of 30 points on this concept inventory, i,e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=', 60%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' This score corresponds to the sug- gested entry threshold for Newtonian physics [18];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' in other words, ChatGPT performs as well as a beginning learner who had just grasped the basic concepts of clas- sical mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' For an Artificial Intelligence, the score seems surpris- ingly good.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' An immediate suspicion was that ChatGPT had been trained using the Force Concept Inventory, which is of course a very popular test, and that it simply latches on to surface features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' As a simple test, the last question on the test was modified as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' 3: the scenario and the order of the answers were changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' As can be seen, these surface features do not matter, so in that respect, ChatGPT does not act like a novice [19] (however, the reality is not quite as straightforward as this expert-novice distinction [20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' The inventory cannot be published here, but it is avail- able to physics instructors and researchers from Phys- Port [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' ChatGPT answered 1C, 2A, 3C, 4E, 5B, 6B, 7B, 8A, 9B, 10A, 11E, 12B, 13B, 14D, 15A, 16E, 17B, 18B, 19A, 20E, 21B, 22B, 23A, 24C, 25D, 26E, 27C, 28D, 29B, and 30C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Of particular interest is of course where ChatGPT is losing points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Several errors are related to “impetus” [21]: more than once did ChatGTP assume that an object im- mediately moves in the direction of an applied force, in- dependent of initial movement (answering 8A, 9B, and 21B) and even that it returns to the original movement when the force is no longer applied (answering 23A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' This is a common preconception, shared by beginning physics students [22], and goes alongside the idea that an acting object exerts greater force than a passive object (answer- ing 25D and 28D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Another confusion appears to be be- tween individual forces acting on an object versus the net force on the object (answering 11E and 16E), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=', what would usually be conveyed in the framework of free-body diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Other errors indicate unstable concepts (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=', answering 13B or ) or logical errors like the one shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' 4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' in this latter case, ChatGPT followed the correct strategy, but in the very last step it failed to draw the correct solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Homework Homework was generally not multiple choice, but free-response numerical and occasionally free-form sym- bolic [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' ChatGPT was given five attempts on such problems, according to recommendations of an earlier study [10] and later practice in the course.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' For the far- and-between multiple choice problems, generally two at- tempts were granted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Between the attempts, the author tried to give helpful prompts, like a student would get from fellow students, teaching assistants, or the instruc- tor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' ChatGPT was given full credit when solving a prob- lem within five attempts, and no credit if it ran out of attempts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' ChatGPT was confronted with a total of 76 homework problems, in particular the homework sets on trajectory motion, friction, thermodynamics, capacitance, and spe- cial relativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' The complete homework sets that the stu- dents in the actual course had to work through were en- tered except for one multipart problem on relativity with a diagram that would have been too hard to transcribe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' An initially puzzling problem is that ChatGPT fre- quently makes numerical errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' A typical example is the ChatGPT output “θ = atan(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='45/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='71) ∗ (180/π) = 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='43 degree;”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' a similar problem can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' 2 (this is not limited to calculations involving π or trigono- metric functions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Calculation errors happened for 25 of the 51 numerical problems, and most of the time, Chat- GPT was unable to recover even after those errors were specifically pointed out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' While it seems incongruent that a computer would have problems calculating simple nu- merical expressions, it should probably be remembered that ChatGPT is a language model, which may carry out calculations by advanced pattern matching rather than actually processing the equations as equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' ChatGTP solved 55% of the homework problem using an average of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='88 attempts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' It got 48% on the prob- 4 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Text-based transcription of a graphical problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' The left panel shows the online version of a final exam problem in LON-CAPA (the graph would be parametrically randomized), the right panel the transcription for ChatGPT, as well as the ensuing dialogue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Surface-feature modification of a Force Concept Inventory problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' The left panel shows the original problem, the right panel a modification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' lems involving trajectory motion and friction (such as inclines) correct, 68% on the thermodynamics problems (engines, heat capacities, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' ), 62% on capacitance (plate capacitors, capacitors in series and parallel, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' ), and 36% on special relativity problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' The discrepancy be- tween the scores on the problem sets were not so much caused by the different physics concepts, but rather re- lated to the mathematics involved: ChatGTP had per- sistent problems manipulating and calculating formulas involving square roots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' If ChatGPT were human, the person might be charac- terized as acting subserviently but being stubborn at the core and keeping on guessing without reflection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Most corrections in a dialogue around a problem are met with profuse apologies, but then the system proceeds to make the same or random apparently careless mistakes — this can lead to irritation on the part of the human, as the excerpt from a late-night “dialogue” in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' 5 shows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' In terms of assessment performance, this means that once ChatGPT makes a mistake, it is unlikely to recover, so it eventually runs out of allowed attempts (this also ex- plains the low number of average attempts to correctly solve a problem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' once ChatGPT is wrong, subsequent at- tempts are unlikely to succeed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' This pattern is similar to the guessing behavior of some students, who keep wast- ing attempt after attempt by trying the same approach over and over without stopping to reflect what might be wrong [2, 10, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' In terms of educational psychology, Problem 22 Due never What is the angular frequency of the harmonic oscillation shown?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='3 rad/s (in rad/s) A: B: C: D: oe: F: G: H: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='53 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='71 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='94 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='0 You are correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Previous Tries 0 Your receipt no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' is 156-1093 @ 2 4 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='5 4 Time [s] 鱼 Post Discussion Send FeedbackKO A large harmonic oscillator has an elongation of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='4 meters at a time of o seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' It goes through the equilibrium at a time 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='25 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' The next time it goes through the equilibrium is at a time 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='75 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' What is the angular frequency?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' The angular freguency of a harmonic oscillator can be found using the formula: W = 2π / T Where T is the period of the oscillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' In this case, the period is 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='75 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='25 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='5 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Therefore, the angular frequency is: w=2π/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='5=4πradians/second KO Pleasegiveanumericalvalue Sure,the angularfreguencyis 4π radians/second which is egualto12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='57 radians/second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content="KO Despite a very strong wind, a tennis player manages to hit a tennis ball with her racguet sothattheballpassesoverthenetandlands inheropponent's court Consider the following forces: 1." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' A downward force of gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' A force by the "hit".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='Aforce exertedbytheair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Which of the above forces is (are) acting on the tennis ball after it has left contact with the racquet and before it touches the ground?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' (A) 1 only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' (B) 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' (C) 1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' (D) 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' (E) 1, 2, and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' (C) 1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' After the tennis ball has left contact with the racquet, it is still subject to the downward force of gravity,which is pulling ittowards theground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='Additionally,the wind exerts a force onthe ball, which can change its trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='The force exerted by the air is acting on the ball and affecting its motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' The force of the hit is no longer acting on the ball as it has left the racquet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='KO Despite avery strong wind, a soccerplayermanagesto hitthegoal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Considerthefollowingforces: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' A downward force of gravity 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' A force by the kick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' A force exerted by the air.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Which of the above forces is (are) acting on the soccer ball after it has left contact with the shoe and before it touches the goal?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' (A) 1 only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' (B) 1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' (C) 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' (D) 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' (E) 1, 2, and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' (B) 1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' After the soccer ball leaves contact with the shoe, the force exerted by the kick (force the force exerted by the air (force 3) are still acting on the ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' The wind is a force exerted by the air, which makes the force 3 acting on the ball, the force of air resistance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='5 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Logical error in an attempt to solve the transcribed question 19 of the Force Concept Inventory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' ChatGPT lacks metacognition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' it does not think about how it thinks [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Clicker Questions Figure 6 shows the clicker questions from a lecture on momentum that was part of the course [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' The lecture was replayed for the study, including re-answering the questions for which peer instruction happened.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Question X1 was solved correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Questions X2, X3, and X4 were special in that they were repeated as questions X5, X6, and X7, respec- tively, after peer instruction [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' As it turned out, ChatGPT got all three of these questions correct on the first attempt, so the peer instruction phase was used to try and confuse ChatGPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Figure 7 shows the dialogue for questions X3 and X6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' in re- ply to the intentionally confusing peer-instruction question, ChatGPT should probably have stopped while it was ahead (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=', before the discussion of a zero-velocity collision), but still maintained its orig- inal correct answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Within the real course, psycho- metrically, X2 and X3 were the most discriminating questions between high- and low-ability students in the set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' For questions X8 and X9, a comment was added that “the collision is elastic, and the moment of inertia of the balls should be neglected” — this was FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' A late-night dialogue between a “stubbornly guess- ing” ChatGPT and a frustrated author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' said in lecture, but does not appear on the slide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' ChatGPT set up the equations for X8 correctly, but then made a sign error in the very last step, which led it to select the wrong answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' For X9, it also set up the equations correctly, but dropped a factor 2 in the last step, leading to an inconsistent KO ThepositionsoftwoblocksAandBatsuccessiveO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='2O-secondtimeintervalsare represented by the numbered squares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' The blocks are moving toward the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' For block A, the numbered blocks are at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='18, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='38, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='68, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='08, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='58, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='18, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='88 meters For Block B, the squares are at -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='02, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='38, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='78, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='18, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='58, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='98, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='38, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='78 meters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Do the blocks ever have the same speed?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' (A) No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' (B) Yes, at instant 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' (C) Yes, at instant 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' (D) Yes, at instants 2 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' (E) Yes,at some time during the interval 3 to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' The speed of the blocks can be determined by dividing the distance traveled by the time interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' The distance traveled by block A between each time interval is: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='20, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='30, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='30,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='50, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='50, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='60, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='70 meters The distance traveled by block B between each time interval is: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='40, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='40, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='40, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='40,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='40, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='40,and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='40 meters Dividingthese distances by the time interval of o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='2o seconds gives the speeds of the blocks: Block A:1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='00,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='50,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='50, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='50,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='50,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='00,and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='50 meters/second Block B:2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='00,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='00, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='00, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='00,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='00, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='00, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='00 meters/second Since the speed of block A is not equal to the speed of block B at any time, the answel is (A) No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='wherey=c_p/c_v=5/3foramonatomicgas, V2 = 65m^3 * (120 atm/40 atm)^(2/5) = 65m^3 * (3)^(2/5) = 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='14 m^3 KO Whenyouplugyintothefinalequation,makesuretouse5/3,not5/2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' My apologies, you are correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' When we plug in y = 5/3 into the final equation, we get: (/)( / ) * 9= ()(/) * = So the final volume V2 is approximately 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='14 m^3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' KO The final formula is correct, but your numerical calculation is wrong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' I apologize, you are correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' I apologize for the mistake in my previous response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' The correct calculation is: V2 = V1 * (P1/P2)^(1/v) = 65 m^3 * (120 atm / 40 atm)^(1/5) = 112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='89 m^3 So the final volume V2 is approximately 112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='89 m^3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' KO Do not change y, just calculate correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' What is 65 m^3 * (120 atm / 40 atm)^(3/5)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' I apologize for my previous mistake, you are correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' The correct calculation is: V2 = V1 * (P1/P2)^(1/v) = 65 m^3 * (120 atm / 40 atm)^(3/5) = 191.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='17 m^3 So the final volume V2 is approximately 191.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='17 m^3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' ko Nope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' I apologize for my mistake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' The correct calculation is: =(=)*= So the final volume V2 is 195 m^3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' KO Nope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Iapologize formymistake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='The correct calculation is V2 = V1 * (P1/P2)^(v-1) = 65 m^3 * (120 atm / 40 atm)^(5/3-1) = 65 m^3 * (120 atm / 40 atm)^(2/3) = 195 m^3 Sothefinal volumeV2is195m3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' KO You drive me nuts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' I apologize if my previous mistakes have caused frustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' As a machine learning model,Istriveto provide accurate and helpful responsesbut sometimes Imight make errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='Iapologize for any inconvenience this may have caused andI will make sure to double-check my responses in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Let me know if there is anything elseIcanassistyouwith6 answer “v2f=(5,-7) m/s, option B.” Within the real course, X8 and X9 were the least discriminating questions, as their difficulty item parameter was too low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Question X10 was solved correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Here, the sys- tem first got off to a false start, but then corrected itself over the course of the derivation, which gave the impression of a stream-of-consciousness mono- logue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Within the real course, X10 did not discrim- inate well between high- and low-ability students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Questions X11 and X12 were solved correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' In summary, ChatGPT correctly solved 10 out of 12 questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Within the actual course, participation in clicker discussions was encouraged by granting 60% credit for false answers and 100% credit for correct answers [11], so the clicker score of ChatGPT would be 93%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' This score is a lot better than most students in the actual course achieved, however, it is important to note that the students in the course were just learning the new concepts, while ChatGPT at any point in time is done with learning unless explicitly trained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Programming Exercises Incorporated into the course were several programming exercises using VPython [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' As an example, one partic- ular exercise from the second semester was to construct an anharmonic oscillator with two fixed positive charges at (0, 1, 0) and (0, −1, 0), respectively, and one negative charge released at (−5, 0, 0) with a velocity (1, 0, 0) — the negative charge will shoot through the two positive charges, slow down, and eventually shoot back.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Based on the narrative, ChatGPT first constructed a program which erroneously at every time step added the initial velocity and which had the Coulomb force in the opposite direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' This could be corrected with a single comment by the user — in the real course, this feedback could have been given by instructors or fellow students (such collaborations are typical and encouraged [12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' The next version of the program was working perfectly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Within the course, adding a graph of the x-position was offered as a bonus option for an additional 20%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' This was accomplished with the third user prompt, and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' 9 shows a screenshot of the running simulation (the simula- tion cannot be run within ChatGPT itself, but it can be copy/pasted into for example a Jupyter Notebook [27]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' ChatGPT performed much better than any of the stu- dents in the course, in spite of them having extensive col- laboration opportunities;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' in this component of the course, ChatGTP achieved not only full credit, but also bonus, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=', 120%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Exams To represent the mid-term and final exams in the course sequence, the first-semester (mechanics) final exam was used for this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' The exam is from a time when grading was still done using bubble sheets;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' instead of free-form answer fields, answer options were given for the students (but not for ChatGTP in this study).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' When simply looking at the answer correctness, Chat- GPT scored 14 out of 30 points, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=', 47%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Looking at the solutions like an instructor would when grading by hand, it turns out that for five questions, the answer was incorrect simply due to errors in the numer- ical calculations — these solutions would have received substantial partial credit in the author’s course.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' By the reverse token, for five questions, ChatGPT arrived at the correct answer in spite of flawed reasoning, which would not have resulted in full credit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Finally, solutions like the one depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' 2 would have received some minimal credit for getting started in the right direction, in spite of then being off by a factor 2 in the period (a common mistake also among human test takers) and the inability to numerically calculate a fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Since the final exam used in this study predates manual grading, no authen- tic grading rubric exists, but a hand-graded score would have realistically ended up between 46% and 50%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' As an aside, one of the thermodynamics homework problems also appeared (with other random numbers) on the final exam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' ChatGPT solved it correctly on the final exam (where it only had one attempt), but not as a homework problem (where it got multiple attempts and help).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' This once again demonstrates the probabilistic nature of the algorithms behind ChatGPT;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' posing the same question twice does not result in the same response or even the same correctness of the response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' If the course grade would have only depended on the exams, ChatGTP would have received a grade of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='0 out of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='0 in the course (with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='0 being the lowest and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='0 be- ing the best grade).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' ChatGPT would have barely gotten credit for the course;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' however, at least a 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='0 grade-point average is required for graduation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Course Grade Grading policies for the course changed over the years, but a typical scenario would be 20% homework, 5% clicker, 5% programming exercises, and 70% exams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' This would result in 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='2 · 55% + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='05 · 93% + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='05 · 120% + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='7 · 47% = 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='55%, which would have resulted in a course grade of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='5 — enough for course credit, but pulling down the grade-point average from what would be needed for graduation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' If, however, ChatGTP would have been better in car- rying out numerical operations, it would have reached 60%, resulting in a 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='0-grade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Depending on the devel- opment priorities of OpenAI, the buggy mathematical functionality could be remedied in the near future, lead- 7 m m v0 a) Cart 1 b) Cart 2 c) No magnitude forces, both zero d) Same magnitude forces positive Which cart exerts a stronger magnitude force during the collision?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Cart 1 Cart 2 Frictionless Track v0 X1 m m v0 v=0 a) Cart 1 b) Cart 2 c) No magnitude forces, both zero d) Same magnitude forces positive Which cart exerts a stronger magnitude force during the collision?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Cart 1 Cart 2 Frictionless Track X2 X5 2m m v0 v=0 a) Cart 1 b) Cart 2 c) No magnitude forces, both zero d) Same magnitude forces positive Which cart exerts a stronger magnitude force during the collision?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Cart 1 Cart 2 Frictionless Track X3 X6 m m v0 v=0 a) Cart 1 b) Cart 2 c) No magnitude forces, both zero d) Same magnitude forces positive Which cart exerts a stronger magnitude force during the collision?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Cart 1 Cart 2 Frictionless Track Fixed X4 X7 Point Mass Billiard m m Crash!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' m m v2,f € !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=" v 1,i = 9 −6 # $ % & ' ( m s at rest € !" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' v 1, f = 3 1 " # $ % & \' m s € A) !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=" v 2, f = 6 −7 # $ % & ' ( m s B) !" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=" v 2, f = −6 7 # $ % & ' ( m s C) !" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=" v 2, f = 12 −5 # $ % & ' ( m s X8 Strange Point Mass Billiard m 2m Crash!" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' m 2m v2,f € !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=" v 1,i = 10 −6 # $ % & ' ( m s at rest € !" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' v 1, f = 5 1 " # $ % & \' m s € A) !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=" v 2, f = 10 −14 # $ % & ' ( m s B) !" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=" v 2, f = −6 7 # $ % & ' ( m s C) !" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=" v 2, f = 12 −5 # $ % & ' ( m s X9 Arthur mA=70kg Violet mV=55kg Cart mc=20kg |vV|=4m/s |vA|=2m/s Initial Final RadioCrasher |vc|=?" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' RadioCrasher Speeds with respect to ground, no friction A) 0 m/s B) 2 m/s C) 4 m/s D) 8 m/s E) 16 m/s At rest with respect to ground X10 m m v0 Totally inelastic: a) zero b) v0/2 c) v0 positive v0 Final speed?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' X11 m m v0 0 Totally inelastic: a) zero b) v0/2 c) v0 positive Final speed?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' X12 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Clicker items from a particular lecture [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Three of the items were presented twice, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=', before and after peer discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' ing to an Artificial Intelligence that could graduate col- lege with a minimal grade if it performed similarly on other courses (this is becoming more and more probably, as ChatGPT is making headlines for passing exams in other subjects [28, 29]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' DISCUSSION It is irritatingly hard not to anthropomorphize Chat- GTP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' As a physics teacher, one invariably finds oneself rooting for the students and thus by extension also for ChatGPT, celebrating its successes and being frustrated about its occasionally inexplicable failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' The system gives the impression of an articulate but at times ram- bling undergraduate student who has a rudimentary yet unstable knowledge of classical mechanics and other fun- damental physics concepts, and who is surprisingly in- ept using a pocket calculator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Frequently, it is hard not to imagine an army of gig-economy workers behind the scenes of ChatGPT answering to the prompts, so the system would definitely pass the Turing Test most of the time [30], but for better or worse, sometimes it still fails in a way that only computers do — it does not have any metacognition, which of course cannot be expected from a probabilistic language model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Metacognition might be the final step to true intelligence, but seems out of reach at this time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' The overall human-like behavior,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' in particular that the system often makes the same mistakes as beginning learn- ers of physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' is less surprising when surmising that un- dergraduate physics discussion forums might have been part of the text corpus used for training — ChatGPT stated in the introduction that “I have been trained on a large dataset of text,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' including physics texts.”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Appar- ently,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' not all of this text corpus contained correct physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' and as a result,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' the system very convincingly and confi- dently presents wrong information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' For a novice learner, who could not distinguish incorrect physics gleaned from 8 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Dialogue about questions X3 and X6 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Chat- GPT got X3 correct;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' peer instruction was simulated by asking a confusing question, and the second iteration X6 was still counted as solved since ChatGPT did not deviate from its original answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' some discussion board from correct physics, this could lead to even more confusion about physics or affirmation of incorrect preconceptions — lacking any metacognition, ChatGTP presents everything as fact, with no nuances expressing uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Almost an anomaly is ChatGTP’s performance on the computational exercise;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' ChatGTP’s language model clearly extends to programming languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' While the call for new, computation-integrated curricula increases, and while physics educations are beginning to develop a solid understanding of the implications of implement- ing these exercise [31, 32], the easy availability of an on- demand program generator might be shaking the foun- dations of these curricular efforts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Somewhat ironically, the integration of computation was partly introduced to make physics problem solving more authentic, moving it closer to how expert physicists work with computers, and one could argue that this has just been taken to an uncharted level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Most of all, the findings of this study should be food for thought for physics educators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' The startling fact that an Artificial Intelligence could pass a standard introductory physics course could be confronted in several ways by educators: Perceiving this as a new way of cheating and trying to defend against it by attempting to use detector tools like ZeroGPT [33] or extensions to tools like turnitin [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' This is an arms race, which on the long run may turn out to be fruitless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Some edu- cators would even go to so far as to say that the battle is already lost anyway ever since platforms like Chegg [35] — no need for Artificial Intelligence to defeat standard physics courses, human crowd- intelligence facilitated by existing commercial plat- forms is good enough for that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Hunker down and go back to making course grades dependent on just a few, high-stake exams with pa- per and pencil in highly proctored environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' After all, ChatGPT compensated for the border- line exam grade of 47% with other course compo- nents that would be collaborative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Unfortunately, this flies in the face of much of physics educa- tion research that favors frequent formative assess- ment [8, 25, 36, 37] and spaced repetition [38, 39], and it is much in contrast to the work environments our students will find.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Taking this as a wake-up call.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' If a physics course can be passed by a trained language model, what does that say about the course?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Artificial In- telligence, for better or worse, is here to stay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Even without the gloom-and-doom scenarios of AI- overlords painted in Science Fiction, it is clear that these model will get, if not better, at least more and more powerful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' What do our students need in terms of conceptual understanding of physics to work with Artificial Intelligence instead of letting Artificial Intelligence do the work for them and then uncritically and unreflectively accepting the results?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' This is particularly important when more is at stake than getting credit for some homework or exam problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' An important skill of every physicist is to evaluate the correctness of their or other people’s work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Tech- niques include dimensional analysis, order-of-magnitude estimates, checking for coherence, considering implica- tions, and the ability to consider limiting cases (“what should happen if this quantity goes to infinity or to zero?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=') [40, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Human can do what Artificial Intel- ligence very likely will not be able to do: following problem-solving strategies including evaluation of their own work [5, 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Moving students toward a more expert- like epistemology may become even more important as Artificial Intelligence starts to permeate more and more aspects of our lives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' CONCLUSION ChatGPT would have achieved a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='5-grade in a stan- dard introductory physics lecture-course series;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' good enough for course credit, but lower than the grade-point average required for graduating with a bachelor degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' If in addition to a language model, the system would have better algorithms for carrying out simple numerical KO Cart 1has mass m and moves onafrictionless track withspeed vo to the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='Cart 2 also has mass m, but it is glued to the track and cannot move.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' When they collide, which cart exerts a stronger magnitude force during the collision?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' (A) cart 1, (B) cart 2 (C) No magnitude force, both zero, (D) same magnitude force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' (D) Same magnitude force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' According to Newton\'s third law of motion, "For every action, there is an equal and opposite reaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='" In a collision between two objects, the force that each object exerts on the other is equal in magnitude but opposite in direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Therefore, both cart 1andcart 2exert the samemagnitude force duringthe collision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' KO But cart 2 is glued to the track, so it could exert infinite force to the incoming cart 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Even if Cart 2 is glued to the track, the collision will still be determined by the laws of direction of the force exerted by Cart 2 on Cart 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Cart 2 is not exerting an infinite force, but it is exerting a force that is egual and opposite to the force exerted by Cart 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Since Cart 2 is not moving, the force it exerts is called an "impulse" and is egual to the change in momentum of Cart 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' In this case, the force exerted by Cart 2 will be infinite only if the velocity of Cart 1 is zero, and it is not possible to have a collision with zero velocity, otherwise, both carts will have the same magnitude force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='9 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Dialogue for a programming exercise in the second semester [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Ko write a vpython animation for the following scenario:two positive charges of 1 are placed at positions (0,1,0) and (O,-1,0), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' A negative charge starts of 1 starts at a position (-5,0,0) with a velocity of (1,,0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=" I'm sorry, I am not able to create a VPython animation as it requires a specific software and set up." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' However, I can give you an example of how to create a VPython animation for the scenario you described: Copy code from vpython import * # Create the charges q1 = sphere(pos=vector(0,1,0), charge=1, color=color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='red) color=color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='red) q3 = sphere(pos=vector(-5,0,0), charge=- l, color=color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='blue) v = vector(1,0,0) # Set the time step dt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='01 while True: rate(100) # Update the position of the charge q3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='pos += v*dt # Calculate the force on the charge :(q1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='pos q3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='pos)*q1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='charge*q3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='charge/(mag(q1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='pos q3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='pos)**2) + (q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='pos - q3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='pos)*q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='charge*q3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='charge/(mag(q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='pos q3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='pos)**2) force*dt q3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content="pos += v*dt This example uses VPython's 'sphere' object to create the charges and the vector' object to create the velocity and position of the charges." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' The ‘while loopupdatesthe positionandvelocityof the negative charge basedontheforces acting on it from the two positive charges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=" The 'rate' function controls the speed of the animation." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Note that this is just a skeleton code and you might need to fine-tune it to get the desired animation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' KO The program updates q3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='pos more than once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Take out the first update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' It also has the force pointing in the wrong direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Copy code q1 = sphere(pos=vector(0,1,0), charge=1, color=color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='red) q2 = sphere(pos=vector(0,-1,0), charge=1 color=color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='red) ， color=color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='blue) # Create the velocity vector v = vector(1,0,0) # Set the time step dt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='01 while True: rate(100) # Calculate theforce on the charge force = (q3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='pos - q1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='pos)*q1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='charge*q3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='charge/(mag(q3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='pos q1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='pos)**2) + (q3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='pos - q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='pos)*q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='charge*q3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='charge/(mag(q3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='pos - q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='pos)**2) # Update the velocity and position of the chargeThisisanupdatedversionofthecode,wherethefirstupdateofthepositionofthe negative charge is removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Also, I corrected the direction of force calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' The negative charge will be under the influence of the two positive charges, so the force is pointing towards the positive charges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=" The 'while' loop updates the position and velocity of the negative charge based on the forces acting on it from the two positive charges." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=" The 'rate' function controls the speed of the animation." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' This is the skeleton code for the scenario you described, and you might need to fine- tune it to get the desired animation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' ko Please add code to graph the x-position of the third charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Copy code from vpython import #Createthecharges q1 = sphere(pos=vector(0,1,0), charge=1， color=color .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='red) q2 = sphere(pos=vector(0,-1,0), charge=1, color=color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='red) q3 = sphere(pos=vector(-5,0,0), charge=-1, color=color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='blue) # Create the velocity vector v = vector(1,0,0) # Set the time step dt : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content="01 # Create a graph for x position of charge x_graph = graph(width=400, ， height=300,xtitle='time',ytitle='x position') x_pt = gcurve(color=color." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='blue) # Initial time while True: rate(100) t += dt # Calculate the force on the charge force = (q3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='pos - q1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='pos)*q1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='charge*q3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='charge/(mag(q3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='pos q1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='pos)**2) + (q3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='pos - q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='pos)*q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='charge*q3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='charge/(mag(q3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='pos q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='pos)**2) # Update the velocity and position of the charge v += force*dt q3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='pos += v*dt x_pt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='plot(t, q3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='pos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content="x) and 'gcurve' objects from VPython to create the graph and the data points." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=" The 'x_pt." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='plot(t, q3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='pos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content="x)' line in the 'while' loop adds the current x-position of the negative charge to the graph at the current time." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=" The graph is labeled with appropriate axis titles, x-title is 'time' and y-title is 'x-position'." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=" The 'gcurve' is color coded in blue." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Please note that this code is just a skeleton and you can customize it to suit your requirement, you can change the size of the graph and the rate of updating the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='10 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Jupyter Notebook [27] output of the VPython pro- gram code generated by ChatGPT in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' operations, it would even have achieved a grade of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='0 — enough to graduate from college if it performs similarly on other courses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Naturally, ChatGTP exhibits no metacognition, which among other consequences lets it present truth and mis- leading information with equal confidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' In physics, the concern should likely not be that ChatGPT would be used as a cheating tool, as there are more efficient platforms for that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Instead, the challenge should be what this means for physics education, as in their future pro- fessional life, our graduates will likely collaborate with Artificial Intelligence: what are the inherently human skills and competencies that we need to convey?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' [1] OpenAI, ChatGPT, https://chat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='openai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='com/chat (accessed January 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' [2] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' G¨on¨ulate¸s and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Kortemeyer, Modeling unproductive behavior in online homework in terms of latent student traits: An approach based on item response theory, Jour- nal of Science Education and Technology 26, 139 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' [3] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Ruggieri, Students’ use and perception of textbooks and online resources in introductory physics, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Educ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' 16, 020123 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' [4] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Reif, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Larkin, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Brackett, Teaching gen- eral learning and problem-solving skills, American Jour- nal of Physics 44, 212 (1976).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' [5] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Reif, Millikan lecture 1994: Understanding and teach- ing important scientific thought processes, American Journal of Physics 63, 17 (1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' [6] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Kortemeyer, The losing battle against plug-and-chug, The Physics Teacher 54, 14 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' [7] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Hestenes, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Wells, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Swackhamer, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Halloun, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Hake, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Mosca, Force Concept Inventory, re- vision 1995, https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='physport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='org/assessments/ assessment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='cfm?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='A=FCI (accessed January 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' [8] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Kortemeyer, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Kashy, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Benenson, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Bauer, Experiences using the open-source learning content man- agement and assessment system LON-CAPA in introduc- tory physics courses, Am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Phys 76, 438 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' [9] OpenAI, ChatGPT Release Notes, https://help.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='openai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='com/en/articles/ 6825453-chatgpt-release-notes (accessed January 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' [10] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Kortemeyer, An empirical study of the effect of grant- ing multiple tries for online homework, American Journal of Physics 83, 646 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' [11] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Kortemeyer, The psychometric properties of class- room response system data: a case study, Journal of Sci- ence Education and Technology 25, 561 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' [12] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Kortemeyer and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Kortemeyer, The nature of collaborations on programming assignments in introduc- tory physics courses: a case study, European Journal of Physics 39, 055705 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' [13] World-Wide Web Consortium, Wcag 2 overview, https: //www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='w3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='org/WAI/standards-guidelines/wcag/ (ac- cessed December 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' [14] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Leinhardt, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Zaslavsky, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Stein, Functions, graphs, and graphing: Tasks, learning, and teaching, Re- view of Educational Research 60, 1 (1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' [15] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Bonham, Graphical response exercises for teaching physics, The Physics Teacher 45, 482 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' [16] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Laverty and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Kortemeyer, Function plot response: A scalable system for teaching kinematics graphs, Am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' 80, 724 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' [17] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Hake, Interactive-engagement versus traditional methods: A six-thousand-student survey of mechanics test data for introductory physics courses, Am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' 66, 64 (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' [18] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Hestenes and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Halloun, Interpreting the Force Con- cept Inventory: A response to march 1995 critique by huffman and heller, The physics teacher 33, 502 (1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' [19] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Chi, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Feltovich, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Glaser, Categorization and representation of physics problems by experts and novices, Cognitive science 5, 121 (1981).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' 8 5 3 x-position 0 3 5 8 5 10 15 20 25 30 35 time11 [20] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Wolf, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Dougherty, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Kortemeyer, Rigging the deck: Selecting good problems for expert-novice card- sorting experiments, Physical Review Special Topics- Physics Education Research 8, 020116 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' [21] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Hestenes, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Wells, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Swackhamer, Force Con- cept Inventory, The Physics Teacher 30, 141 (1992), https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='1119/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='2343497.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' [22] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Clement, Students’ preconceptions in introductory me- chanics, American Journal of Physics 50, 66 (1982).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' [23] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Pascarella, The influence of web-based homework on quantitative problem-solving in a university physics class, in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' NARST Annual Meeting, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' 4 (2004) pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' 19–28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' [24] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Cox, Metacognition in computation: A selected research review, Artificial intelligence 169, 104 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' [25] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Crouch and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Mazur, Peer instruction: Ten years of experience and results, American Journal of Physics 69, 970 (2001), https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='1119/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='1374249.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' [26] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Scherer, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Dubois, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Sherwood, Vpython: 3d interactive scientific graphics for students, Computing in Science & Engineering 2, 56 (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' [27] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Jupyter, Jupyter notebooks, https://jupyter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='org (retrieved August 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' [28] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Kung, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Cheatham, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Medinilla, ChatGPT, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Sillos, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' De Leon, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Elepano, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Madriaga, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Ag- gabao, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Diaz-Candido, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=', Performance of chatgpt on usmle: Potential for ai-assisted medical education us- ing large language models, medRxiv , 2022 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' [29] Samantha Murphy Kelly, ChatGPT passes exams from law and business schools, https://edition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='cnn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='com/ 2023/01/26/tech/chatgpt-passes-exams/index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='html (accessed January 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' [30] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Turing, Computing machinery and intelligence, Mind , 433 (1950).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' [31] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Aiken, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Caballero, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Douglas, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Burk, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Scanlon, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Thoms, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Schatz, Under- standing student computational thinking with compu- tational modeling, in AIP Conference Proceedings, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' 1513 (1) (AIP, 2013) pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' 46–49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' [32] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Hamerski, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' McPadden, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Caballero, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Irving, Students’ perspectives on computational challenges in physics class, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Educ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' 18, 020109 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' [33] ZeroGPT, ChatGPT detector by ZeroGPT: detect Ope- nAI text, https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='zerogpt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='com (accessed January 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' [34] turnitin, Turnitin Corp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=', https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='turnitin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='com (accessed January 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' [35] Chegg, Chegg, https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='chegg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content='com (accessed Jan- uary 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' [36] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Dufresne and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Gerace, Assessing-to-learn: For- mative assessment in physics instruction, The Physics Teacher 42, 428 (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' [37] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Clark, Formative assessment: Assessment is for self- regulated learning, Educational Psychology Review 24, 205 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' [38] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Ebbinghaus, ¨Uber das Ged¨achtnis: Untersuchungen zur experimentellen Psychologie (Duncker & Humblot, 1885).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' [39] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Murre and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Dros, Replication and analysis of Ebbinghaus’ forgetting curve, PloS one 10, e0120644 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' [40] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Heuvelen, Learning to think like a physicist: A review of research-based instructional strategies, Am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Physics 59, 891 (1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' [41] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Redish and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Hammer, Reinventing college physics for biologists: Explicating an epistemological curriculum, American Journal of Physics 77, 629 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' [42] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Mota, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Didi ¸s K¨orhasan, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Miller, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Mazur, Homework as a metacognitive tool in an undergraduate physics course, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Educ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'} +page_content=' 15, 010136 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9FLT4oBgHgl3EQfnS-e/content/2301.12127v1.pdf'}